African Journal Of Mathematical Physics Volume 8(2010)101-114
Casimir force in confined biomembranes K. El Hasnaoui, Y...

Author:
elhasnaoui khalid

African Journal Of Mathematical Physics Volume 8(2010)101-114

Casimir force in confined biomembranes K. El Hasnaoui, Y. Madmoune, H. Kaidi, M. Chahid, and M. Benhamou Laboratoire de Physique des Polym`eres et Ph´enom`enes Critiques Facult´e des Sciences Ben M’sik, P.O. 7955, Casablanca, Morocco [email protected]

abstract We reexamine the computation of the Casimir force between two parallel interacting plates delimitating a liquid with an immersed biomembrane. We denote by D their separation, which is assumed to be much smaller than the bulk roughness, in order to ensure the membrane confinement. This repulsive force originates from the thermal undulations of the membrane. To this end, we first introduce a field theory, where the field is noting else but the height-function. The field model depends on two parameters, namely the membrane bending rigidity constant, κ, and some elastic constant, µ ∼ D−4 . We first compute the static Casimir force (per unit area), Π, and find that the latter decays with separation D as : Π ∼ D−3 , with a known amplitude scaling as κ−1 . Therefore, the force has significant values only for those biomembranes of small enough κ. Second, we consider a biomembrane (at temperature T ) that is initially in a flat state away from thermal equilibrium, and we are interested in how the dynamic force, Π (t), grows in time. To do calculations, use is made of a non-dissipative Langevin equation (with noise) that is solved by the time height-field. We first show that the membrane roughness, L⊥ (t), increases with time as : L⊥ (t) ∼ t1/4 (t < τ ), with the final time τ ∼ D4 (required time over which the final equilibrium state is reached). Also, we find that the force increases in time according to : Π (t) ∼ t1/2 (t < τ ). The discussion is extended to the real situation where the biomembrane is subject to hydrodynamic interactions caused by the surrounding liquid. In this case, we show that : L⊥ (t) ∼ t1/3 (t < τh ) and Πh (t) ∼ t2/3 (t < τh ), with the new final time τh ∼ D3 . Consequently, the hydrodynamic interactions lead to substantial changes of the dynamic properties of the confined membrane, because both roughness and induced force grow more rapidly. Finally, the study may be extended, in a straightforward way, to bilayer surfactants confined to the same geometry. Key words: Biomembranes - Confinement - Casimir force - Dynamics.

I. INTRODUCTION

The cell membranes are of great importance to life, because they separate the cell from the surrounding environment and act as a selective barrier for the import and export of materials. More details concerning the structural organization and basic functions of biomembranes can be found in Refs. [1 − 7]. It is well-recognized by the scientiﬁc community that the cell membranes essentially present as a phospholipid bilayer combined with a variety of proteins and cholesterol (mosaic fluid model). In particular, the function of the cholesterol molecules is to ensure the bilayer ﬂuidity. A phospholipid is an amphiphile

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c a GNPHE publication 2010, [email protected] ⃝

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African Journal Of Mathematical Physics Volume 8(2010)101-114

molecule possessing a hydrophilic polar head attached to two hydrophobic (fatty acyl) chains. The phospholipids move freely on the membrane surface. On the other hand, the thickness of a bilayer membrane is of the order of 50 Angstroms. These two properties allow to consider it as a two-dimensional ﬂuid membrane. The ﬂuid membranes, self-assembled from surfactant solutions, may have a variety of shapes and topologies [8], which have been explained in terms of bending energy [9, 10]. In real situations, the biomembranes are not trapped in liquids of inﬁnite extent, but they rather conﬁned to geometrical boundaries, such as white and red globules or liposomes (as drugs transport agents [11 − 14]) in blood vessels. For simplicity, we consider the situation where the biomembrane is conﬁned in a liquid domain that is ﬁnite in one spatial direction. We denote by D its size in this direction. For a tube, D being the diameter, and for a liquid domain delimitated by two parallel plates, this size is simply the separation between walls. Naturally, the length D must be compared to the bulk roughness, L0⊥ , which is the typical size of humps caused by the thermal ﬂuctuations of the membrane. The latter depends on the nature of lipid molecules forming the bilayer. The biomembrane is conﬁned only when D is much smaller than the bulk roughness L0⊥ . This condition is similar to that usually encountered in conﬁned polymers context [15]. The membrane undulations give rise to repulsive eﬀective interactions between the conﬁning geometrical boundaries. The induced force we term Casimir force is naturally a function of the size D, and must decays as this scale is increased. In this paper, we are interested in how this force decays with distance. To simplify calculations, we assume that the membrane is conﬁned to two parallel plates that are a ﬁnite distance D < L0⊥ apart. The word ”Casimir” is inspired from the traditional Casimir eﬀect. Such an eﬀect, predicted, for the ﬁrst time, by Hendrick Casimir in 1948 [16], is one of the fundamental discoveries in the last century. According to Casimir, the vacuum quantum ﬂuctuations of a conﬁned electromagnetic ﬁeld generate an attractive force between two parallel uncharged conducting plates. The Casimir eﬀect has been conﬁrmed in more recent experiments by Lamoreaux [17] and by Mohideen and Roy [18]. Thereafter, Fisher and de Gennes [19], in a short note, remarked that the Casimir eﬀect also appears in the context of critical systems, such as ﬂuids, simple liquid mixtures, polymer blends, liquid 4 He, or liquid-crystals, conﬁned to restricted geometries or in the presence of immersed colloidal particles. For these systems, the critical ﬂuctuations of the order parameter play the role of the vacuum quantum ﬂuctuations, and then, they lead to long-ranged forces between the conﬁning walls or between immersed colloids [20, 21]. To compute the Casimir force between the conﬁning walls, we ﬁrst elaborate a more general ﬁeld theory that takes into account the primitive interactions experienced by the conﬁned membrane. As we shall see below, in conﬁnement regime, the ﬁeld model depends only on two parameters that are the membrane bending rigidity constant and a coupling constant containing all inﬁrmation concerning the interaction potential exerting by the walls. In addition, the last parameter is a known function of the separation D. With the help of the constructed free energy, we ﬁrst computed the static Casimir force (per unit area), Π. The exact calculations show that the latter decays with separation D according to a power 2 law, that is Π ∼ κ−1 (kB T ) D−3 , with a known amplitude. Here, kB T denotes the thermal energy, and κ the membrane bending rigidity constant. Of course, this force increases with temperature, and has signiﬁcant values only for those biomembranes of small enough κ. The second problem we examined is the computation of the dynamic Casimir force, Π (t). More precisely, we considered a biomembrane at temperature T that is initially in a ﬂat state away from the thermal equilibrium, and we were interested in how the expected force grows in time, before the ﬁnal state is reached. Using a scaling argument, we ﬁrst showed that the membrane roughness, L⊥ (t), grows with time as : L⊥ (t) ∼ t1/4 (t < τ ), with the ﬁnal time τ ∼ D4 . The latter can be interpreted as the required time over which the ﬁnal equilibrium state is reached. Second, using a non-dissipative Langevin equation, we found that the force increases in time according to the power law : Π (t) ∼ t1/2 (t < τ ). Third, the discussion is extended to the real situation where the biomembrane is subject to hydrodynamic interactions caused by the ﬂow of the surrounding liquid. In this case, we show that : L⊥ (t) ∼ t1/3 (t < τh ) and Π (t) ∼ t2/3 (t < τh ), with the new ﬁnal time τh ∼ D3 . Consequently, the hydrodynamic interactions give rise to drastic changes of the dynamic properties of the conﬁned membrane, since both roughness and induced force grow more rapidly. This paper is organized as follows. In Sec. II, we present the ﬁeld model allowing the determination of the Casimir force from a static and dynamic point of view. The Sec. III and Sec. IV are devoted to the computation of the static and dynamic induced forces, respectively. We draw our conclusions in the last section. Some technical details are presented in Appendix.

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African Journal Of Mathematical Physics Volume 8(2010)101-114 II. THEORETICAL FORMULATION

Consider a ﬂuctuating ﬂuid membrane that is conﬁned to two interacting parallel walls 1 and 2. We denote by D their ﬁnite separation. Naturally, the separation D must be compared to the bulk membrane roughness, L0⊥ , when the system is unconﬁned (free membrane). The membrane is conﬁned only when the condition L > L0⊥ , we expect ﬁnite size corrections. We assume that these walls are located at z = −D/2 and z = D/2, respectively. Here, z means the perpendicular distance. For simplicity, we suppose that the two surfaces are physically equivalent. We design by V (z) the interaction potential exerted by one wall on the ﬂuid membrane, in the absence of the other. Usually, V (z) is the sum of a repulsive and an attractive potentials. A typical example is provided by the following potential [22] V (z) = Vh (z) + VvdW (z) ,

(2.0)

Vh (z) = Ah e−z/λh

(2.0)

where

represents the repulsive hydration potential due to the water molecules inserted between hydrophilic lipid heads [22]. The amplitude Ah and the potential-range λh are of the order of : Ah ≃ 0.2 J/m2 and λh ≃ 0.2 − 0.3 nm. In fact, the amplitude Ah is Ah = Ph × λh , with the hydration pressure Ph ≃ 108 −109 Pa. There, VvdW (z) accounts for the van der Waals potential between one wall and biomembrane, which are a distance z apart. Its form is as follows [ ] H 1 2 1 VvdW (z) = − − , (2.0) 2 + 2 12π z 2 (z + δ) (z + 2δ) with the Hamaker constant H ≃ 10−22 − 10−21 J, and δ ≃ 4 nm denotes the membrane thickness. For large distance z, this implies VvdW (z) ∼

W δ2 . z4

(2.0)

Generally, in addition to the distance z, the interaction potential V (z) depends on certain length-scales, (ξ1 , ..., ξn ), which are the interactions ranges. The ﬂuid membrane then experiences the following total potential ( ) ( ) D D D D U (z) = V −z +V +z , − ≤z≤ . (2.0) 2 2 2 2 In the Monge representation, a point on the membrane can be described by the three-dimensional position vector r = (x, y, z = h (x, y)), where h (x, y) ∈ [−D/2, D/2] is the height-ﬁeld. The latter then ﬂuctuates around the mid-plane located at z = 0. The Statistical Mechanics for the description of such a (tensionless) ﬂuid membrane is based on the standard Canham-Helfrich Hamiltonian [9, 23] ∫ [κ ] 2 H [h] = dxdy (∆h) + W (h) , (2.0) 2 with the membrane bending rigidity constant κ. The latter is comparable to the thermal energy kB T , where T is the absolute temperature and kB is the Boltzmann’s constant. There, W (h) is the interaction potential per unit area, that is W (h) =

U (h) , L2

where the potential U (h) is deﬁned in Eq. (2), and L is the lateral linear size of the biomembrane. Let us discuss the pair-potential W (h). 103

(2.0)

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Firstly, Eq. (2) suggests that this total potential is an even function of the perpendicular distance h, that is W (−h) = W (h) .

(2.0)

In particular, we have W (−D/2) = W (D/2). Secondly, when they exist, the zeros h0 ’s of the potential function U (h) are such that ( ) ( ) D D V − h0 = −V + h0 . 2 2

(2.0)

This equality indicates that, if h0 is a zero of the potential function, then, −h0 is a zero too. The number of zeros is then an even number. In addition, the zero h0 ’s are diﬀerent from 0, in all cases. Indeed, the quantity V (D/2) does not vanish, since it represents the potential created by one wall at the middle of the ﬁlm. We emphasize that, when the potential processes no zero, it is either repulsive or attractive. When this same potential vanishes at some points, then, it is either repulsive of attractive between two consecutive zeros. Thirdly, we ﬁrst note that, from relation (2), we deduce that the ﬁrst derivative of the potential function, with respect to distance h, is an odd function, that is W ′ (−h) = −W ′ (h). Applying this relation to the midpoint h = 0 yields : W ′ (0) = 0. Therefore, the potential W exhibits an extremum at h = 0, whatever the form of the function V (h). We ﬁnd that this extremum is a maximum, if V ′′ (D/2) < 0, and a minimum, if V ′′ (D/2) > 0. The potential U presents an horizontal tangent at h = 0, if only if V ′′ (D/2) = 0. On the other hand, the general condition giving the extrema {hm } is dV dV = . (2.0) dh h= D −hm dh h= D +hm 2

2

Since the ﬁrst derivative W ′ (h) is an odd function of distance h, it must have an odd number of extremum points. The point h = hm is a maximum, if d2 V d2 V < − , (2.0) dh2 h= D −hm dh2 h= D +hm 2

2

and a minimum, if d2 V d2 V >− . dh2 h= D −hm dh2 h= D +hm 2

(2.0)

2

At point h = hm , we have an horizontal tangent, if d2 V d2 V =− . dh2 h= D −hm dh2 h= D +hm 2

(2.0)

2

The above deductions depends, of course, on the form of the interaction potential V (h). Fourthly, a simple dimensional analysis shows that the total interaction potential can be rewritten on the following scaling form ( ) 1 h ξ1 ξn W (h) = 2Φ , , ..., , (2.0) kB T D D D D where (ξ1 , ..., ξn ) are the ranges of various interactions experienced by the membrane, and Φ (x1 , ..., xn+1 ) is a (n + 1)-factor scaling-function. Finally, we note that the pair-potential W (h) cannot be singular at h = 0. It is rather an analytic function in the h variable. Therefore, at ﬁxed ratios ξi /D, an expansion of the scaling-function Φ, around the value h = 0, yields ( ) γ h2 W (h) = + O h4 . 4 kB T 2D 104

(2.0)

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African Journal Of Mathematical Physics Volume 8(2010)101-114

We restrict ourselves to the class of potentials that exhibit a minimum at the mid-plane h = 0. This assumption implies that the coeﬃcient γ is positive deﬁnite, i.e. γ > 0. Of course, such a coeﬃcient depends on the ratios of the scale-lengths ξi to the separation D. In conﬁnement regime where the distance h is small enough, we can approximate the total interaction potential by its quadratic part. In these conditions, the Canham-Helfrich Hamiltonian becomes ∫ [ ] 1 2 H0 [h] = dxdy κ (∆h) + µh2 , (2.0) 2 with the elastic constant µ=γ

kB T . D4

(2.0)

The prefactor γ will be computed below. The above expression for the elastic constant µ gives an idea on its dependance on the ﬁlm thickness D. In addition, we state that this coeﬃcient may be regarded as a Lagrange multiplier that ﬁxes the value of the membrane roughness. Thanks to the above Hamiltonian, we calculate the mean-expectation value of the physical quantities, like the height-correlation function (propagator or Green function), deﬁned by G (x − x′ , y − y ′ ) = ⟨h (x, y) h (x′ , y ′ )⟩ − ⟨h (x, y)⟩ ⟨h (x′ , y ′ )⟩ . The latter solves the linear diﬀerential equation ( 2 ) κ b∆ + µ b G (x − x′ , y − y ′ ) = δ (x − x′ ) δ (y − y ′ ) ,

(2.0)

(2.0)

where δ (x) denotes the one-dimensional Dirac function, and ∆ = ∂ 2 /∂x2 + ∂ 2 /∂y 2 represents the twodimensional Laplacian operator. We have used the notations : κ b = κ/kB T and µ b = µ/kB T , to mean the reduced membrane elastic constants. From the propagator, we deduce the expression of the membrane roughness ⟨ ⟩ 2 (2.0) L2⊥ = h2 − ⟨h⟩ = G (0, 0) . Such a quantity measures the ﬂuctuations of the height-function (ﬂuctuations amplitude) around the equilibrium plane located at z = 0. We show in Appendix that the membrane roughness is exactly given by L2⊥ =

D2 , 12

(2.0)

provided that one is in the conﬁnement-regime, i.e. D L∥ . From the standard relation L2⊥ =

kB T 2 L , 16κ ∥

(2.0)

we deduce 2 L∥ = √ 3

(

κ kB T

)1/2 D.

(2.0)

In contrary to L⊥ , the length-scale L∥ depends on the geometrical characteristics of the membrane (through κ). The next steps consist in the computation of the Casimir force at and out equilibrium.

III. STATIC CASIMIR FORCE

When viewed under the microscope, the membranes of vesicles present thermally excited shape ﬂuctuations. Generally, objects such as interfaces, membranes or polymers undergo such ﬂuctuations, in order to increase their conﬁgurational entropy. For bilayer biomembranes and surfactants, the consequence of these undulations is that, they give rise to an induced force called Casimir force. To compute the desired force, we start from the partition function constructed with the Hamiltonian deﬁned in Eq. (13). This partition function is the following functional integral { } ∫ H0 [h] Z = Dh exp − , (3.0) kB T where integration is performed over all height-ﬁeld conﬁgurations. The associated free energy is such that : F = −kB T ln Z, which is, of course, a function of the separation D. If we denote by Σ = L2 the common area of plates, the Casimir force (per unit area) is minus the ﬁrst derivative of the free energy (per unit area) with respect to the ﬁlm-thickness D, that is Π=−

1 ∂F . Σ ∂D

(3.0)

This force per unit area is called disjoining pressure. In fact, Π is the required pressure to maintain the two plates at some distance D apart. In term of the partition function, the disjoining pressure rewrites Π 1 ∂ ln Z 1 ∂µ ∂ ln Z = = . kB T Σ ∂D Σ ∂D ∂µ

(3.0)

Using deﬁnition (19) together with Eqs. (23) and (24) yields Π=−

1 ∂µ 2 L . 2 ∂D ⊥

(3.0)

Explicitly, we obtain the desired formula 2

Π=

3 (kB T ) . 8 κD3

(3.0)

From this relation, we extract the expression of the disjoining potential (per unit area) [25] ∫ Vd (D) = −

D

Π (D′ ) dD′ =

∞

106

2

3 (kB T ) . 16 κD2

(3.0)

K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

The above expression of the Casimir force (per unit area) calls the following remarks. Firstly, this force decays with distance more slowly in comparison to the Coulombian one that decreases rather as D−2 . Secondly, this same force depends on the nature of lipids forming the bilayer (through κ). In this sense, contrarily to the Casimir eﬀect in Quantum Field Theory [16] and in Critical Phenomena [20], the present force is not universal. Incidentally, if this force is multiplied by κ, then, it will become a universal quantity. Thirdly, at ﬁxed temperature and distance, the force amplitude has signiﬁcant values only for those bilayer membrane of small bending rigidity constant. Fourthly, as it should be, such a force increases with increasing temperature. Indeed, at high temperature, the membrane undulations are strong enough. Finally, the numerical prefactor 3/8 (Helfrich’s cH -amplitude [9]) is close to the value obtained using Monte Carlo simulation [26]. In Fig. 1, we superpose the variations of the reduced static Casimir force Π/kB T upon separation D, for two lipid systems, namely SOPC and DAPC [27], at temperature T = 18◦ C. The respective membrane bending rigidity constants are : κ = 0.96 × 10−19 J and κ = 0.49 × 10−19 J. These values correspond to the renormalized bending rigidity constants : κ b = 23.9 and κ b = 12.2. The used methods for the measurement of these rigidity constants were entropic tension and micropipet [27]. These curves reﬂect the discussion made above.

FIG. 1. Reduced static Casimir force, Π/kB T , versus separation D, for two lipid systems that are SOPC (solid line) and DAPC (dashed line), of respective membrane bending rigidity constants : κ = 0.96 × 10−19 J and κ = 0.49 × 10−19 J, at temperature T = 18◦ C. The reduced force and separation are expressed in arbitrary units.

IV. DYNAMIC CASIMIR FORCE

To study the dynamic phenomena, the main physical quantity to consider is the time height-ﬁeld, h (r, t), where r = (x, y) ∈ R2 denotes the position vector and t the time. The latter represents the time observation of the system before it reaches its ﬁnal equilibrium state. We recall that the time height function h (r, t) solves a non-dissipative Langevin equation (with noise) [28] δH0 [h] ∂h (r, t) = −Γ + ν (r, t) , ∂t δh (r, t) 107

(4.0)

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African Journal Of Mathematical Physics Volume 8(2010)101-114

where Γ > 0 is a kinetic coeﬃcient. The latter has the dimension : [Γ] = L40 T0−1 , where L0 is some length and T0 the time unit. Here, ν (r, t) is a Gaussian random force with mean zero and variance ⟨ν (r, t) ν (r′ , t′ )⟩ = 2Γδ2 (r − r′ ) δ (t − t′ ) ,

(4.0)

and H0 is the static Hamiltonian (divided by kB T ), deﬁned in Eq. (13). The bare time correlation function, whose Fourier transform is the dynamic structure factor, is deﬁned by the expectation mean-value over noise ν G (r − r′ , t − t′ ) = ⟨h (r, t) h (r′ , t′ )⟩ν − ⟨h (r, t)⟩ν ⟨h (r′ , t′ )⟩ν ,

t > t′ .

(4.0)

The dynamic equation (28) shows that the time height function h is a functional of noise ν, and we write : h = h [ν]. Instead of solving the Langevin equation for h [ν] and then averaging over the noise distribution P [ν], the correlation and response functions can be directly computed by means of a suitable ﬁeld-theory, of action [28 − 31] { } ∫ [ ] ∫ δH0 e e A h, e h = dt d2 r e h∂t h + Γe h − hΓh , (4.0) δh so that, for an arbitrary observable, O [ζ], one has ∫ ⟨O⟩ν =

∫

e DhDe hOe−A[h,h] , [dν] O [φ [ν]] P [ν] = ∫ e DhDe he−A[h,h]

(4.0)

where e h (r, t) is an auxiliary ﬁeld, coupled to an external ﬁeld h (r, t). The correlation and response functions can be computed replacing the static Hamiltonian H0 appearing in Eq. (13), by a new one : ∫ H0 [h, J] = H0 [h] − d2 rJh. Consequently, for a given observable O, we have ⟨ ⟩ δ ⟨O⟩J =Γ e h (r, t) O . (4.0) δJ (r, t) J=0 [ ] The notation ⟨ . ⟩J means the average taken with respect to the action A h, e h, J associated with the Hamiltonian H0 [h, J]. In view of the structure of equality (33), e h is called response field. Now, if O = h, we get the response of the order parameter ﬁeld to the external perturbation J ⟨ ⟩ δ ⟨h (r′ , t′ )⟩J ′ ′ R (r − r , t − t ) = =Γ e h (r, t) h (r′ , t′ ) . (4.0) δJ (r, t) J=0 J=0 The causality implies that the response function vanishes for t < t′ . In fact, this function can be related to the time-dependent (connected) correlation function using the ﬂuctuation-dissipation theorem, according to which ⟨ ⟩ Γ e h (r, t) h (r′ , t′ ) = −θ (t − t′ ) ∂t ⟨h (r, t) h (r′ , t′ )⟩c . (4.0) The above important formula shows that the time correlation function C (r − r′ , t − t′ ) = ⟨φ (r, t) φ (r′ , t′ )⟩c may be determined by the knowledge of the response function. In particular, we show that ∫ t ⟨ ⟩ ⟨ 2 ⟩ 2 h (r, t′ ) h (r, t′ ) . (4.0) dt′ e L⊥ (t) = h (r, t) c = −2Γ −∞

The limit t → −∞ gives the natural value L2⊥ (−∞) = 0, since, as assumed, the initial state corresponds to a completely ﬂat interface. Consider now a membrane at temperature T that is initially in a ﬂat state away from thermal equilibrium. At a later time t, the membrane possesses a certain roughness, L⊥ (t). Of course, the latter is timedependent, and we are interested in how it increases in time. 108

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We point out that the thermal ﬂuctuations give rise to some roughness that is characterized by the appearance of anisotropic humps. Therefore, a segment of linear size L eﬀectuates excursions of size [32] L⊥ = BLζ .

(4.0)

Such a relation deﬁnes the roughness exponent ζ. Notice that L is of the order of the in-plane correlation length, L∥ . From relation (20), we deduce the exponent ζ and the amplitude B. Their respective values 1/2 are : ζ = 1 and B ∼ (kB T /κ) . In order to determine the growth of roughness L⊥ in time, the key is to consider the excess free energy (per unit area) due to the conﬁnement, ∆F . Such an excess is related to the fact that the conﬁning membrane suﬀers a loss of entropy. Formula (27) tells us how ∆F must decay with separation. The result reads [32] 2/ζ

∆F ∼ kB T /L2max ∼ kB T (B/L⊥ )

,

(4.0)

where Lmax represents the wavelength above which all shape ﬂuctuations are not accessible by the conﬁned membrane. The repulsive ﬂuctuation-induced interaction leads to the disjoining pressure Π=−

∂∆F −(1+2/ζ) ∼ L⊥ . ∂L⊥

(4.0)

In addition, a care analysis of the Langevin equation (28) shows that ∂∆F ∂L⊥ −(1+2/ζ) ∼ −Γ = Γ × Π ∼ ΓL⊥ . ∂t ∂L⊥

(4.0)

We emphasize that this scaling form agrees with Monte Carlo predictions [32, 33]. Solving this ﬁrst-order diﬀerential equation yields [34] L⊥ (t) ∼ Γθ⊥ tθ⊥ ,

θ⊥ =

1 ζ = . 2 + 2ζ 4

(4.0)

This implies the following scaling form for the linear size L (t) ∼ Γθ∥ tθ∥ ,

θ∥ =

1 1 = . 2 + 2ζ 4

(4.0)

Let us comment about the obtained result (39). Firstly, as it should be, the roughness increases with time (the exponent θ⊥ is positive deﬁnite). In addition, the exponent θ⊥ is universal, independently on the membrane bending rigidity constant κ. Secondly, we note that, in Eq. (39), we have ignored some non-universal amplitude that scales as κ−1/4 . This means that the time roughness is signiﬁcant only for those biomembranes of small bending rigidity constant. Fourthly, this time roughness can be interpreted as the perpendicular size of holes and valleys at time t. Fifthly, the roughness increases until a ﬁne time, τ . The latter can be interpreted as the time over which the system reaches its ﬁnal equilibrium state. This characteristic time then scales as τ ∼ Γ−1 L⊥

1/θ⊥

,

(4.0)

where we have ignored some non-universal amplitude that scales as κ. Here, L⊥ ∼ D is the ﬁnal roughness. Explicitly, we have τ ∼ Γ−1 D4 .

(4.0)

As it should be, the ﬁnal time increases with increasing ﬁlm thickness D. On the other hand, we can rewrite the behavior (39) as L⊥ (t) = L⊥ (τ )

( )θ⊥ t . τ

109

(4.0)

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African Journal Of Mathematical Physics Volume 8(2010)101-114

This equality means that the roughness ratio, as a function of the reduced time, is universal. Now, to compute the dynamic Casimir force, we start from a formula analog to that deﬁned in Eq. (24), that is b b Π (t) 1 ∂ ln Z 1 ∂µ ∂ ln Z = = , kB T Σ ∂D Σ ∂D ∂µ

(4.0)

with the new partition function b= Z

∫

e

DhDe he−A[h,h] .

(4.0)

A simple algebra taking into account the basic relation (35a) gives Π (t) 1 ∂µ 2 =− L (t) , kB T 2 ∂D ⊥

(4.0)

which is very similar to the static relation deﬁned in Eq. (25), but with a time-dependent membrane roughness, L⊥ (t). Combining formulae (43) and (46) leads to the desired expression for the time Casimir force (per unit area) Π (t) = Π (τ )

( )θf t , τ

(4.0)

where Π (τ ) is the ﬁnal static Casimir force, relation (25). The force exponent, θf , is such that θf = 2θ⊥ =

ζ 1 = . 1+ζ 2

(4.0)

The induced force then grows with time as t1/2 until it reaches its ﬁnal value Π (τ ). At ﬁxed time and separation D, the force amplitude depends, of course, on κ, and decreases in this parameter according to κ−3/2 . Also, we note that the above equality means that the force ratio as a function of the reduced time is universal. In Fig. 2, we draw the reduced dynamic Casimir force, Π (t) /Π (τ ), upon the renormalized time t/τ .

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FIG. 2. Reduced dynamic Casimir force, Π (t) /Π (τ ), upon the renormalized time t/τ .

Finally, consider again a membrane which is initially ﬂat but is now coupled to overdamped surface waves. This real situation corresponds to a conﬁned membrane subject to hydrodynamic interactions. The roughness now grows as [35] b ⊥ (t) ∼ tθb⊥ , L

θb⊥ =

ζ 1 = . 1 + 2ζ 3

(4.0)

Therefore, the roughness increases with time more rapidly than that relative to biomembranes free from hydrodynamic interactions. In this case, the dynamic Casimir force is such that Πh (t) = Π (τh )

(

t τh

)θbf ,

(4.0)

where Π (τh ) is the ﬁnal static Casimir force, relation (25). The new force exponent is θbf = 2θb⊥ =

2 2ζ = . 1 + 2ζ 3

(4.0)

There, τh ∼ D3 accounts for the new time-scale over which the conﬁned membrane reaches its ﬁnal equilibrium state. Therefore, the dynamic Casimir force decays with time as t2/3 , that is more rapidly than that where the hydrodynamic interactions are ignored, which scales rather as t1/2 . As we said before, this drastic change can be attributed to the overdamped surface waves that develop larger and larger humps. We depict, in Fig. 3, the variation of the reduced dynamic force (with hydrodynamic interactions), Πh (t) /Π (τh ), upon the renormalized time t/τh . 111

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FIG. 3. Reduced dynamic Casimir force (with hydrodynamic interactions), Πh (t) /Π (τ ), upon the renormalized time t/τh .

V. CONCLUSIONS

In this work, we have reexamined the computation of the Casimir force between two parallel walls delimitating a ﬂuctuating ﬂuid membrane that is immersed in some liquid. This force is caused by the thermal ﬂuctuations of the membrane. We have studied the problem from both static and dynamic point of view. We were ﬁrst interested in the time variation of the roughening, L⊥ (t), starting with a membrane that is inially in a ﬂat state, at a certain temperature. Of course, this length grows with time, and we found that : L⊥ (t) ∼ tθ⊥ (θ⊥ = 1/4), provided that the hydrodynamic interactions are ignored. For real systems, however, these interactions are important, and we have shown that the roughness increases more rapidly b ⊥ (t) ∼ tθb⊥ (θ⊥ = 1/3). The dynamic process is then stopped at a ﬁnal τ (or τh ) that represents as : L the required time over which the biomembrane reaches its ﬁnal equilibrium state. The ﬁnal time behaves as : τ ∼ D4 (or τh ∼ D3 ), with D the ﬁlm thickness. Now, assume that the system is explored at scales of the order of the wavelength q −1 , where q = (4π/λ) sin (θ/2) is the wave vector modulus, with λ the wavelength of the incident radiation and θ the scattering-angle. In these) conditions, the relaxation rate, τ (q), scales with q as : τ −1 (q) ∼ q 1/θ⊥ = q 4 ( b

or τh−1 (q) ∼ q 1/θ⊥ = q 3 . Physically speaking, the relaxation rate characterizes the local growth of the height ﬂuctuations. Afterwards, the question was addressed to the computation of the Casimir force, Π. At equilibrium, using an appropriate ﬁeld theory, we found that this force decays with separation D as : Π ∼ D−3 , with a known amplitude scaling as κ−1 , where κ is the membrane bending rigidity constant. Such a force is then very small in comparison with the Coulombian one. In addition, this force disappears when the 112

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temperature of the medium is suﬃciently lowered. The dynamic Casimir force, Π (t), was computed using a non-dissipative Langevin equation (with noise), solved by the time height-ﬁeld. We have shown that : Π (t) ∼ tθf (θf = 2θ⊥ = 1/2). When the hydrodynamic interactions eﬀects are ( ) important, we found that the dynamic force increases more rapidly as : θbf b b Πh (t) ∼ t θf = 2θ⊥ = 2/3 . Notice that we have ignored some details such as the role of inclusions (proteins, cholesterol, glycolipids, other macromolecules) and chemical mismatch on the force expression. It is well-established that these details simply lead to an additive renormalization of the bending rigidity constant. Indeed, we write κeﬀective = κ + δκ, where κ is the bending rigidity constant of the membrane free from inclusions, and δκ is the contribution of the incorporated entities. Generally, the shift δκ is a function of the inclusion concentration and compositions of species of diﬀerent chemical nature (various phospholipids forming the bilayer). Hence, to take into account the presence of inclusions and chemical mismatch, it would be suﬃcient to replace κ by κeﬀective , in the above established relations. As last word, we emphasize that the results derived in this paper may be extended to bilayer surfactants, although the two systems are not of the same structure and composition. One of the diﬀerences is the magnitude order of the bending rigidity constant.

APPENDIX

To show formula (17), we start from the partition function that we rewrite on the following form ∫ Z=

{

H [h] Dh exp − kB T

}

D/2 ∫

dzΦ (z) .

=

(5.0)

−D/2

Also, it is easy to see that the membrane mean-roughness is given by D/2 ∫

L2⊥ =

dzz 2 Φ (z)

−D/2 D/2 ∫

.

(5.0)

dzΦ (z)

−D/2

The restricted partition function is ∫ Φ (z) =

{ } H [h] Dhδ [z − h (x0 , y0 )] exp − . kB T

(5.0)

Here, H [h] is the original Hamiltonian deﬁned in Eq. (3). Of course, this deﬁnition is independent on the chosen point (x0 , y0 ), because of the translation symmetry along the parallel directions to plates. Notice that the above function is not singular, whatever the value of the perpendicular distance. Since we are interested in the conﬁnement-regime, ( ) that is when the separation D is much smaller than the membrane mean-roughness L0⊥ z ∼ h 0 . 2π L3

(2.0)

(2.0)

The above equality shows that the elastic constant k scales with separation L as : k ∼ L−3 . We note that the potential depth |U0 | has as eﬀect to renormalize the density amplitude [21]. III. TIME EVOLUTION OF THE PARTICLE DENSITY

Consider, now, an assembly of colloidal particles moving around a ﬂuctuating ﬂuid membrane. Under a sudden change of a suitable parameter, such as temperature, pressure or membrane environment, the 95

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system is out equilibrium. We assume that the system is subject to Brownian dynamics by changing the colloid-membrane interaction strength w, but the temperature and ﬁlm thickness remain ﬁxed. This change may be caused by aﬀecting the membrane environment. The problem can be studied through the local particle density, n (z, t). The latter represents the number of colloids per unit volume, at distance z and at time t. More precisely, we are interested in how this density evolves in time before the colloidal suspension reaches its ﬁnal equilibrium state. For simplicity, we will neglect mutual interactions between nanoparticles. This hypothesis makes sense at least for small particle densities. Hence, the only interaction experienced by the beads is an external potential caused by the membrane undulations. Within the harmonic approximation, this potential is deﬁned in Eqs. (9) and (10). To determine the time evolution of the local particle density, use will be made of the Smoluchowski equation [27, 28], which is a linear partial diﬀerential equation, of ﬁrst order and second order, with respect to time and perpendicular distance, respectively. Before writing and solving this equation, we shall need some backgrounds. We ﬁrst recall the expression of the equilibrium particle density } { U (z) neq (z) = A exp − , (3.0) kB T where A is a normalization constant and U (z) is the one-body potential due to the membrane undulations. If this potential is approximated by its harmonic form, the above deﬁnition becomes { } W (z) neq (z) = n0 exp − , (3.0) kB T where n0 is now the value of the particle density at the mid-plane z = 0. When the colloidal dispersion is out of equilibrium, in addition to distance, the density depends on time. This means that the nanoparticles execute Brownian dynamic but in the presence of the harmonic potential W (z). In order to compute this local density, we ﬁrst recall that the Brownian diﬀusion is correctly described by the Fick’s law. The latter stipulates that the ﬂux of matter, j (z, t), is directly proportional to the spatial gradient of density, that is j = −D

∂n 1 ∂W − , ∂z ζ ∂z

(3.0)

with D=

kB T ζ

(3.0)

the diﬀusion constant and ζ the friction coeﬃcient, of which the inverse 1/ζ is the mobility. If we design by a the particle radius and by ηs the solvent viscosity, the friction coeﬃcient ζ can be calculated from hydrodynamics [36] : ζ = 6πηs a. Equality (14) states that the diﬀusion constant characterizing the thermal motion is related to the quantity ζ, which expresses the response to an external ﬁeld. Such an equality is a consequence of the well-known dissipation-ﬂuctuation theorem [27, 28]. On the other hand, relation (13) must be combined with the local conservation law of matter ∂n ∂j + =0. ∂t ∂z Combining Eqs. (13) and (15) yields the Smoluchowski equation ( ) ∂n 1 ∂ ∂n ∂W = kB T +n ∂t ζ ∂z ∂z ∂z

(3.0)

(3.0)

solved by the local particle density n (z, t). At equilibrium, that is ∂n/∂t = 0, the above equation reduces to : kB T ∂n/∂z + n∂W/∂z = 0, whose solution is neq (z) = n0 exp exp {−W (z) /kB T }, which is nothing else but the density deﬁned in Eq. (12). Replacing the harmonic potential W (z) by its explicit form (9) gives ∂ 2 n k ∂n k ∂n =D 2 + z + n. ∂t ∂z ζ ∂z ζ 96

(3.0)

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This new Smoluchowski equation must be supplemented by two boundary conditions, which are n (z, t = ∞) = nf (z) .

n (z, t = 0) = ni (z) ,

(3.0)

If the temperature T and separation L are ﬁxed, the inial and ﬁnal equilibrium particle densities, ni (z) and nf (z), are completely determined by the initial and ﬁnal surface coupling constants wi and wf , respectively. Therefore, the dynamic is caused by a change of the membrane environment. Taking advantage of those mathematical techniques used in Ref. [21], we show that the solution to the Smoluchowski equation (17) is given by { ( t/τ )2 } [ ( )]−1/2 ∫ ∞ ze f − y −2t/τf ( ) [ni (y) − nf (y)] , n (z, t) = nf (z) + 2πDτf 1 − e dy exp − 2Dτf e2t/τf − 1 −∞ (3.0) where the initial and ﬁnal equilibrium particle densities are given by { } { } ki z 2 z2 i i ni (z) = n0 exp − = n0 exp − , (3.0) 2kB T 2Dτi

nf (z) =

nf0

} } { { kf z 2 z2 f = n0 exp − , exp − 2kB T 2Dτf

(3.0)

with the time-scales τi and τf τi =

ζ 1 = ki 12

√

2π D−1 3 L , 3 wi

τf =

ζ 1 = kf 12

√

2π D−1 3 L , 3 wf

(3.0)

where wi and wf > wi are the initial and ﬁnal surface coupling constants. This means that the colloidmembrane interaction is suddenly increased from wi to wf . The above relations suggest that the times τi and τf depend on the colloid-membrane interaction and ﬁlm thickness. In particular, the time-scale τf may be interpreted as the time beyond which the colloidal system reaches its ﬁnal equilibrium state. Then, a weak colloid-membrane interaction necessitates a great time before the colloidal system tends to its ﬁnal state. In fact, τf has another physical meaning, and can be regarded as the required time over which the particles are trapped in new holes and valleys. Now, after integration over the y variable, in Eq. (19), we obtain a closer form for the time particle density { } [ ( )]−1/2 1 z2 i −2t/τf ( ) n (z, t) = n0 1 + η e −1 exp − , (3.0) 1 + η e−2t/τf − 1 2Dτi with the reduced time-shift η=

τi − τf wf − wi = >0. τi wf

(3.0)

The quantity η then represents the relative shift of the colloid-membrane interaction strengths wi and wf . The density amplitude, in Eq. (23), was obtained using the matter conservation law ∫

∫

+∞

−∞

ni (z) dz =

+∞ −∞

nf (z) dz ≡ Γ .

(3.0)

Here, Γ represents the adsorbance, which is deﬁned as the total number of colloids (per unit area) located near membrane. Combining Eqs. (20), (21) and (25) yields the relationship √ √ 2πDτf nf0 = 2πDτi ni0 = Γ . (3.0) Let us comment the density expression (23). First, we note that, it is easy to see that the solution (23) satisﬁes the two boundary conditions (18). 97

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The initial and ﬁnal equilibrium states are deﬁned in Eqs. (20) and (21). Second, when it is reduced by√ ni0 , the time particle density depends on three dimensionless factors, namely the renormalized distance z/ 2Dτi , the time-ratio t/τf and the time-shift η = (τi − τf ) /τi . Therefore, all microscopic details (colloid-membrane interaction) are entirely contained in τi and τf . Finally, we emphasize that the time particle density curve exhibits a maximum at z = 0, and it is symmetric around this point, whatever be the values of t/τf and η. We√depict, in Fig. 2, the reduced time particle density n (z, t) /ni0 versus the renormalized distance z/ 2Dτi , choosing three values of the time-ratio t/τf : 0, 0.5, and ∞. The former corresponds to the initial state and the second to the ﬁnal one. These curves are drawn with parameter η = 0.5. This value means that the ﬁnal surface coupling constant wf is two times more important than the initial one wi , that is wf = 2wi .

√ FIG. 2. Reduced local particle density, n (z, t) /ni0 , versus the renormalized distance z/ 2Dτi , with three values of the time-ratio t/τf : 0 (dashed line), 0.5 (solid line), ∞ (line in dots). These curves are drawn choosing the value η = 0.5 (wf = 2wi ).

IV. CONCLUSIONS

This work is dedicated to the Brownian dynamics study of small colloidal particles in contact with an attractive penetrable ﬂuid membrane. The host liquid was assumed to be delimitated by two parallel reﬂecting walls, which are a ﬁnite distance apart. The membrane surrounded by beads is conﬁned only if the ﬁlm thickness is much smaller than the bulk membrane mean-roughness. Physics was discussed in terms of three relevant parameters, which are the absolute temperature, T , the separation between walls, L, and colloid-membrane interaction strength, w. In our study, we have ﬁxed the temperature and ﬁlm thickness to some values, and varied the surface coupling constant. This can be experimentally achieved modifying the membrane environment. For the present study, we have started from three hypothesizes : (1) the particles are point-like, (2) they are of very low-density (in order to forget their mutual interactions), and (3) strongly interact with the membrane. To achieve the investigation of the Brownian dynamics, use was made of a theoretical formalism based on the Smoluchowski equation. The latter is solved by the time particle density we were interested in. We have exactly computed this physical quantity, around the mid-plane of the ﬁlm, where the essential of phenomenon occurs. Within this distance-domain, the mean-force external potential was approximated 98

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by an harmonic one. This means that we were in the presence of Brownian particles moving in an harmonic potential, that is, √in addition to the usual diﬀusion, these experience small oscillations with a frequency ν scaling as ν ∼ w/L3 . Hence, the harmonic approximation used for the mean-force potential is largely justiﬁed for those ﬂuid interfaces of small enough colloid-membrane interaction strength. As we have shown, the time particle density depends on a time-scale, τ , scaling as τ ∼ L3 /w. We have interpreted this scale-time as the required time over which the nanoparticles reach their ﬁnal equilibrium state. Also, τ can be regarded as the time-interval where the particles are trapped in holes and valleys of size slightly smaller than the ﬁlm thickness L. 0 We note that, when the separation L is much greater than the bulk roughness ξ⊥ , in addition to the above evoked parameters, the physical phenomenon depends on the speciﬁc membrane characteristic (via the bending rigidity constant κ). In this case, the Brownian dynamics can be caused by a change of the parameter κ. This situation has less physical interest, since, in this case, ﬁnite size eﬀects contribute to the leading behavior only by exponentially small corrections. As last word, we emphasize that the results derived in this paper may be extended to bilayer surfactants, although the two systems are not of the same structure and composition.

ACKNOWLEDGMENTS

We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions, during the ”First International Workshop On Soft-Condensed Matter Physics and Biological Systems”, 14-17 November 2006, Marrakech, Morocco. One of us (M.B.) would like to thank the Professor C. Misbah for fruitful correspondences, and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kinds of hospitalities during his regular visits.

REFERENCES

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M.S. Bretscher and S. Munro, Science 261, 12801281 (1993). J. Dai and M.P. Sheetz, Meth. Cell Biol. 55, 157171 (1998). 3 M. Edidin, Curr. Opin. Struc. Biol. 7, 528532 (1997). 4 C.R. Hackenbrock, Trends Biochem. Sci. 6, 151154 (1981). 5 C. Tanford, The Hydrophobic Eﬀect, 2d ed., Wiley, 1980. This book includes, in addition, a good discussion about the interactions of proteins and membranes. 6 D.E. Vance and J. Vance, eds., Biochemistry of Lipids, Lipoproteins, and Membranes, Elsevier, 1996. 7 F. Zhang, G.M. Lee, and K. Jacobson, BioEssays 15, 579588 (1993). 8 S. Safran, Statistical Thermodynamics of Surfaces, Interfaces and Membranes, Addison-Wesley, Reading, MA, 1994. 9 W. Helfrich, Z. Naturforsch. 28c, 693 (1973). 10 U. Seifert, Advances in Physics 46, 13 (1997). 11 H. Ringsdorf and B. Schmidt, How to Bridge the Gap Between Membrane, Biology and Polymers Science, P.M. Bungay et al., eds, Synthetic Membranes : Science, Engineering and Applications, p. 701, D. Reiidel Pulishing Compagny, 1986. 12 D.D. Lasic, American Scientist 80, 250 (1992). 13 V.P. Torchilin, Eﬀect of Polymers Attached to the Lipid Head Groups on Properties of Liposomes, D.D. Lasic and Y. Barenholz, eds, Handbook of Nonmedical Applications of Liposomes, Volume 1, p. 263, RCC Press, Boca Raton, 1996. 14 R. Joannic, L. Auvray, and D.D. Lasic, Phys. Rev. Lett. 78, 3402 (1997). 15 P.-G. de Gennes, Scaling Concept in Polymer Physics, Cornell University Press, 1979. 16 C. Fradin, A. Abu-Arish, R. Granek, and M. Elbaum, Biophys. J. 84, 2005 (2003). 17 D.F. Evans and H. Wennerstr¨ om, The Colloidal Domain, Wiley, New-York, 1999. 18 T. Bickel, M. Benhamou, and H. Kaidi, Phys. Rev. E 70, 051404 (2004). 2

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H. Kaidi, T. Bickel, and M. Benhamou, Europhys. Lett. 69, 15 (2005) A. Bendouch, H. Kaidi, T. Bickel, and M. Benhamou, J. Stat. Phys.: Theory and Experiment P01016, 1 (2006). 21 A. Bendouch, M. Benhamou, and H. Kaidi, Elec. J. Theor. Phys. 5, 215 (2008). 22 A. Einstein, Ann. Physik. 17, 549 (1905) ; 19, 371 (1906); see, also, Investigation on the Theory of the Brownian Movement, E.P. Dutton and Copy Inc, New York, 1926. 23 N. Wax, Noise and Stochastic Processes, Dover Publishing Co., New York, 1954. 24 M. Lax, Rev. Mod. Phys. 32, 25 (1960) ; 38, 541 (1966). 25 R. Kubo, Rep. Prog. Phys. 29, 255 (1966). 26 C.W. Gardiner, Handbook of Stochastic Methods, 3d ed., Springer, 2004. 27 H. Risken and T. Frank, The Focker-Planck Equation : Methods of Solutions and Applications, 2d ed., Springer, 1989. 28 M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press Oxford, 1986. 29 P.B. Canham, J. Theoret. Biol. 26, 61 (1970). 30 K. El Hasnaoui, Y. Madmoune, H. Kaidi, M. Chahid, and M. Benhamou, Induced forces in conﬁned biomembranes, submitted for publication. 31 K. Binder, in : Phase Transitions and Critical Phenomena, Vol. 8, edited by C. Domb and J.L. Lebowitz, Academic Press, London, 1983. 32 H.W. Diehl, in : Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, Vol. 10, Academic Press, London, 1986. 33 S. Dietrich, in : Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, Vol. 12, Academic Press, London, 1988. 34 C. Itzykson and J.M. Drouﬀe, Statistical Field Theory : 1 and 2, Cambridge University Press, 1989. 35 J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. 36 G.K. Batchelor, An Introduction to Fluid Dynamics, Chap. 4, Cambridge University Press, 1970. 20

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Statistical mechanics of bilayer membranes in troubled aqueous media M. Benhamou ∗ , K. Elhasnaoui, H. Kaidi, M. Chahid Laboratoire de Physique des Polymères et Phénomènes Critiques, Faculté des Sciences Ben M’sik, P.O. 7955, Casablanca, Morocco

article

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Article history: Received 30 March 2010 Available online 14 April 2010 Keywords: Bilayer membranes Vesicles Impurities Equilibrium Statistical mechanics

abstract We consider a bilayer membrane surrounded by small impurities, assumed to be attractive or repulsive. The purpose is a quantitative study of the effects of these impurities on the statistical properties of the supported membrane. Using the replica trick combined with a variational method, we compute the membrane mean-roughness and the height correlation function for almost-flat membranes, as functions of the primitive elastic constants of the membrane and some parameter that is proportional to the volume fraction of impurities and their interaction strength. As results, the attractive impurities increase the shape fluctuations due to the membrane undulations, while repulsive ones suppress these fluctuations. Second, we compute the equilibrium diameter of (spherical) vesicles surrounded by small random particles starting from the curvature equation. Third, the study is extended to a lamellar phase composed of two parallel fluid membranes, which are separated by a finite distance. This lamellar phase undergoes an unbinding transition. We demonstrate that the attractive impurities increase the unbinding critical temperature, while repulsive ones decrease this temperature. Finally, we say that the presence of small impurities in an aqueous medium may be a mechanism to suppress or to produce an unbinding transition, even the temperature and polarizability of the aqueous medium are fixed, in lamellar phases formed by parallel lipid bilayers. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Usually, the aqueous media supporting biological membranes are assumed to be homogeneous. Actually, any real system inevitably contains impurities. Under well-controlled conditions, the particles can be removed from the surrounding medium. But, if these entities are present, it is also interesting to study their effect on the statistical properties of the biomembranes, such as fluctuations’ spectrum and dynamical behavior. In general, random inhomogeneities tend to disorder the system. It is important to make a distinction between annealed and quenched disorders. The former is used when impurities and host constituents (phospholipids) are in equilibrium [1]. This means that their respective mobilities are comparable. If it is not the case, that is host constituents and impurities are out of equilibrium, the disorder is rather quenched [1]. When the Statistical Mechanics is used, the latter consists to trace over all membrane undulations, before performing the summation over the impurities’ disorder. Although the quenched disorder is harder to analyze, it remains more realistic than its annealed counterpart. Indeed, the thermal and the noise averaging have very different roles. In this paper, the physical system we consider is a fluid membrane (flat or closed) trapped in a troubled aqueous medium. The latter is impregnated by a weak amount of impurities that may be attractive or repulsive regarding the membrane. The aim is to show how these entities can modify the statistical properties of the fluid membrane. These properties will be studied through the fluctuations’ amplitude. To model the system, we suppose that the impurities act as a random external potential with a Gaussian distribution (uncorrelated disorder). In addition, we suppose that the disorder is quenched. To

∗

Corresponding author. E-mail address: [email protected] (M. Benhamou).

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do calculations, the replica theory [2,3] that is based on a mathematical analytical continuation, usually encountered in Quantum Mechanics [4] and Critical Phenomena will be made use of [5,6]. Our finding are as follows. First, using the above evoked theory, we compute the mean-roughness of an almost-flat membrane, as a function of the primitive parameters of the pure membrane (free from impurities) and a certain parameter depending on the volume fraction of impurities and their interaction strength. The main conclusion is that, attractive impurities increase the shape fluctuations due to the thermal undulations, while repulsive ones tend to suppress these fluctuations and then lead to a strong membrane confinement. Second, we analyze the impurities effects on the equilibrium shape of closed vesicles solving the curvature equation (for spherical vesicles). We show that the vesicle is more stable in the presence of repulsive impurities, in comparison with attractive ones. Thereafter, the study is extended to lamellar phases formed by two parallel fluid membranes. We demonstrate that the presence of random impurities drastically affects the physical properties of the lamellar phase, in particular, the unbinding transition driving the system from a bind state to a state where the two membranes are completely separated. This paper is organized as follows. In Section 2, we describe the fundamentals of the used model. Section 3 deals with the computation of the fluctuations amplitude of a single almost-flat fluid membrane, surrounded by attractive or repulsive impurities. We compute, in Section 4, the equilibrium diameter of a closed vesicle in the presence of impurities. Extension of study to lamellar phases is the aim of Section 5. Finally, some concluding remarks are drawn in the last section. 2. Effective field theory Consider a fluctuating fluid membrane embedded in a three-dimensional liquid surrounded by very small impurities. For the sake of simplicity, we suppose that the impurities are point like. Within the framework of the Monge representation, a point on the membrane can be described by the position-vector (r , z = h (r )), where r = (x, y) ∈ R2 is the transverse vector and h (r ) is the height function. The Statistical Mechanics of fluid membranes free from impurities is based on the Canham–Helfrich Hamiltonian [7]

Z

d2 r

H0 [h] =

hκ 2

(∆h)2 +

µ 2

i

h2 ,

(1)

with κ the membrane bending modulus. The confinement energy (per unit area) µh2 /2 (µ > 0) is responsible for the localization of the membrane in some region of the Euclidean space, where it fluctuates around an equilibrium plane located at h = 0. Therefore, the height h takes either positive and negative values. For simplicity, the membrane is assumed to be tensionless. In fact, this assumption does not change conclusions made below. To model the impurities effects on the statistical properties of the system, we suppose that these tend to reinforce the membrane confinement, if they are repulsive, or to render this membrane more free, if the particles are rather attractive. These tendencies can be explained assuming that the confinement parameter is local in space and making the substitution

µ → µ + V (r ) ,

(2)

in the above Hamiltonian, where the variation V (r ) can be regarded as a random external potential. To simplify, the corresponding probability distribution is supposed to be Gaussian (uncorrelated disorder), that is V (r ) = 0,

V (r ) V (r 0 ) = −vδ2 r − r 0 .

(3)

Here, −v is a positive constant proportional to both concentration of impurities and strength of their interaction potential, and δ2 (r ) denotes the two-dimensional Dirac distribution. Therefore, the new effective Canham–Helfrich Hamiltonian reads

Z H [h] =

d2 r

κ 2

(∆h)2 +

1 2

(µ + V (r )) h2 ,

(4)

for attractive impurities. For repulsive ones, V (r ) must be replaced by iV (r ), with i2 = −1. Since the disorder distribution is Gaussian, all its odd moments vanish, but the even ones do not. This implies that all physical quantities, calculated with the pure imaginary potential iV , are entire numbers. Since the impurities and membranes are not in equilibrium, the disorder is rather quenched, that is we have to average not the partition function, Z , but its logarithm, ln Z . The latter defines the free energy. For a given (quenched) configuration of impurities, the partition function is

Z Z =

D he−A[h] ,

(5)

with the action A [h] =

H [h] kB T

Z =

2

d r

b κ 2

(∆h) + 2

1 2

2 b b µ + V (r ) h .

(6)

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We have used the notations

b κ=

κ kB T

,

µ

b µ=

kB T

V b V = .

,

(7)

kB T

In term of the reduced impurities potential b V , the second disorder law in Eq. (3) becomes: b V (r ) b V (r 0 ) = −b vδ r − r 0 , with

b v = v/ (kB T )2 . A simple dimensional analysis shows that: [b κ ] = L0 , [b µ] = L−4 and [b v ] = L−6 , where L is some length that may be the membrane thickness. The main quantity to consider is the average of the logarithm of the partition function, ln Z , over disorder. To compute such a quantity, we use the replica trick [2]. This method consists to formally write Zn − 1

ln Z = lim

n

n→0

.

(8)

After performing the average over disorder, we get

Z Zn

=

D h1 . . . D hn e−A[h1 ,...,hn ] ,

(9)

with the n-replicated effective action A [h1 , . . . , hn ] =

Z

" d2 r

n b κX

2 α=1

(∆hα )2 +

n b µX

2 α=1

h2α +

n b v X

8

α=1

! h2α

n X

β=1

!# h2β

,

(10)

where Greek indices denote replicas. The last term introduces an additional coupling constant b v < 0, which is directly responsible for the effective interaction between replicas due to the presence of impurities. The above action describes attractive impurities. For repulsive ones, the coupling b v must be replaced by −b v . The additive quartic term in the above action means that the presence of impurities is accompanied by an increase of entropy, when these are attractive, and by an entropy loss, if they are rather repulsive. The following paragraph is devoted to a quantitative determination of the fluctuations spectrum in the presence of impurities. 3. Single almost-flat membrane We start by considering the attractive impurities problem. Of course, the functional integral (9) cannot be exactly computed. One way is the use of a variational method. To this end, we consider a bilayer membrane in the presence of the following bare action Aη [h1 , . . . , hn ] =

Z

" d2 r

n b κX

2 α=1

(∆hα )2 +

n b ηX

2 α=1

# h2α

,

(11)

with the new confinement coupling b η > 0. With this action, the partition function, Zη , is exact. We have Zη = (Z0 )n ,

(12)

with

Z Z0 =

Z b κ b η 2 2 2 D h exp − d r . (∆h) + h 2

(13)

2

The latter accounts for the usual partition function of a bilayer membrane free from impurities. Introduce the mean-value of a functional X [h1 , . . . , hn ], calculated with the bare action Aη [h1 , . . . , hn ],

hX i0 =

1

Z

D h1 . . . D hn X [h1 , . . . , hn ] exp −Aη [h1 , . . . , hn ] .

Zη

(14)

In term of this mean-value, the averaged partition function may write

Z n = Zη exp − A − Aη

0

.

(15)

Using the standard inequality

X e

0

≥ e hX i0 ,

(16)

we get

Z n ≥ Zη exp − A − Aη

0

.

(17)

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This implies that

1 − Zn

− ln Z = lim

≤ lim

n

n→0

1 − Zη exp − A − Aη

.

0

n

n→0

(18)

On the other hand, we have

A − Aη

"

Z

d2 r

=

0

n

2 b µ −b ηX

2

α=1

hα

+

0

n X n

b vX

8 α=1 β=1

# 2

h2α hβ

0

.

(19)

It is easy to see that n X

2

α=1

hα

n X n X

= nσ 2 ,

0

α=1 β=1

h2α h2β

0

= n (n + 2) σ 2

2

,

(19a)

with the squared membrane roughness 1

σ 2 (η) =

1

p . 8 b κb η

(20)

With these considerations,

− ln Z ≤ − ln Z0 + Σ

b µ −b η 2

σ2 +

b v 4

σ2

2

,

(21)

where Σ is the area of the reference plane. Notice that the right-hand side of this inequality is b η-dependent function. Therefore, a good approximate value of −ln Z is

− ln Z

MF

= min − ln Z0 + Σ η

b µ −b η 2

σ + 2

b v 4

σ

2 2

.

(22)

(The subscript MF is for mean-field theory). The variational parameter b η is such that

∂ b µ −b η 2 b v 2 2 − ln Z0 + Σ σ + σ = 0. ∂b η 2 4

(23)

Combining the relationship

σ2 = −

2 ∂ ln Z0 Σ ∂b η

(24)

and Eq. (20) yields the minimum value of b η, which satisfies the following implicit relation

b η=b µ +b vσ 2 .

(25)

The squared membrane roughness, σ 2 , is then given by the parametric equations

σ 2 = 1 p1 , 8 b κb η b η=b µ +b vσ 2 .

(26)

Eliminating b η between these equations gives the implicit relation

σ2 =

1

1

q . 8 b κ b µ +b vσ 2

(27)

p

Introduce the usual squared roughness of a membrane free from impurities, σ02 = 1/8 b κb µ. Then, we have

σ 2 = σ02 p

1 1 +b vσ 2 /b µ

.

(28)

The latter may be rewritten, in terms of dimensionless variables σ 2 /σ02 , b κ , and w = −64b κb v σ02

σ2 1 = q , 2 σ0 1 − wσ 2 /σ02

(attractive impurities) .

3

> 0, as (29)

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Table 1 Some values of the coupling w and their corresponding ratios σ 2 /σ02 , for attractive impurities.

w

σ 2 /σ02

0.1 0.15 0.2 0.25 0.3

1.0575 1.0937 1.1378 1.1939 1.2715 √ 3

w∗

Notice that the coupling constant w > 0 is directly proportional to the volume fraction of impurities and their interaction strength. Let us comment about the obtained result. Firstly, this makes sense only when σ 2 /σ02 < 1/w . This condition is fulfilled for very weak disorders (w is small enough). Secondly, since w is positive definite, we have: σ 2 > σ02 . A comparison between this inequality and that just evoked above implies that 0 < w < 1. Thirdly, if we set x = σ 2 /σ02 , the above relation can be rewritten as

w=

x2 − 1 x3

.

(30)

Therefore, w is a direct function of x (Fig. 1). This formula may be experimentally used to estimate the impurities coupling w, knowing the experimental value of the ratio σ 2 /σ02 . If w is fixed to some value, the x-variable then solves the following third-degree algebraic equation

w x3 − x2 + 1 = 0 .

(31)

The existence of√its roots then depends on the value of the renormalized coupling w . We show that there exists a typical value w ∗ = 2/3 3 of the coupling w , such that: (1) For w > w ∗ (strong disorder), we have only one root that is negative ; √ (2) For w = w ∗ (medium disorder), we have one negative root and a positive unique root, which is x = σ 2 /σ02 = 3. Then, the maximal value of the membrane roughness is σ ∗ = 31/4 σ0 . This constraint means that the lipid bilayer cannot support undulations of perpendicular size (hunt size) greater than σ ∗ . This is realized if only if the impurities volume fraction φ is below some value φ ∗ , directly proportional to the threshold w ∗ ; (3) For w < w ∗ (weak disorder), we have one negative root andtwo positive ones. Only one positive root is acceptable (the smallest one). At this particular root, the free energy −kB T ln Z MF is minimal. We show that the absolute minimum satisfies the inequality

− 4wx6 + 3x4 − 1 > 0.

(32)

In this case, the ratio σ /σ can be obtained by numerically solving Eq. (31). We report in Table 1 some values of the impurities coupling w and the corresponding dimensionless squared membrane roughness σ 2 /σ02 . It is easy to see that, the ratio σ 2 /σ02 increases with increasing coupling w , provided that w is in the interval w < w ∗ . 2

2 0

Now, for repulsive impurities, the fundamental relationship (29) is replaced by

σ2 1 = q , 2 σ0 1 − wσ 2 /σ02

(repulsive impurities) ,

(33)

or equivalently

− wx3 − x2 + 1 = 0,

(34)

with the stability condition 4w x6 + 3x4 − 1 > 0.

(35)

The equality (34) can be transformed into the following direct function (Fig. 2)

w=

x2 − 1 x3

,

(36)

with x < 1 and w < 0. As it should be, the membrane roughness is reduced by the presence of repulsive impurities, that is σ 2 < σ02 , and in addition, σ 2 decreases with increasing impurities coupling −w . As a matter of fact, the effect of these particles is to reinforce

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Fig. 1. Impurities’ coupling w versus the ratio σ 2 /σ02 , for attractive impurities.

Fig. 2. Impurities’ coupling w versus the ratio σ 2 /σ02 , for repulsive impurities.

Table 2 Some values of the coupling w and their corresponding ratios σ 2 /σ02 , for attractive impurities.

w

σ 2 /σ02

−0.1 −0.15 −0.2 −0.25 −0.3 −0.4

0.9554 0.9364 0.9190 0.9032 0.8885 0.8622

the confinement of the considered fluid membrane. Therefore, a bilayer membrane collapses, when it is trapped in a troubled aqueous media with repulsive impurities. In this case, it is easy to see that the algebraic equation has only one positive root. In Table 2, we give some values of the coupling w and the corresponding ratio σ 2 /σ02 .

= h (r ) h r 0 − hh (r )i h r . The latter measures the fluctuations of the height function h around its mean-value hhi. Here, h.i denotes Another interesting physical quantity is the (connected) height correlation function: G r − r 0

0

the thermal expectation mean-value, which must not be confused with average over disorder.

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To compute the correlation functions, we start from the generating functional

Z Z [J] =

d r J (r ) h (r ) ,

Z

2

D h exp −A [h] +

(37)

with the action A [h] = H [h] /kB T , where H [h] is the effective Canham–Helfrich Hamiltonian defined in Eq. (4). Here, J is an auxiliary source coupled to the field h. Functional derivatives of Z [J] with respect to source J give all height correlation functions, in particular, the propagator G r −r

0

δ 2 ln Z [J] = δ J (r ) δ J (r 0 )

.

(38)

J =0

This definition indicates that the main object to compute is the averaged connected generating functional ln Z [J]. To compute the functional ln Z [J], as before, we use the replica method based on the limit ln Z [J] = lim

Z n [J] − 1 n

n→0

,

(39)

with

(

Z

D h1 . . . D hn exp −A [h1 , . . . , hn ] +

Z n [J] =

n Z X

) d2 r J (r ) hi (r ) ,

(40)

i=1

where A [h1 , . . . , hn ] is the action, relation (10). Using the standard cumulant method [5,6] based on the approximate formula

hexp {X }i0 = exp hX i0 + (1/2!) X 2 0 − hX i20 + · · · ,

(41)

we find that

Z n [J] ' Zη exp − A − Aη

Z Z 0 n 2 2 0 0 + · · · exp d r d r J r G r − r J r , ( ) 0 0

(42)

2

with the usual bare propagator G0 r − r

0

Z =

0 d2 q eiq.(r −r )

κ q4 + b η (2π )2 b

.

(43)

In Eq. (42), we have ignored high-order terms in J that do not contribute to the propagator. After performing the limit n = 0, we find that the averaged connected generating functional reads ln Z [J] ' − ln Z0 + Σ

b µ −b η 2

σ2 +

b v 4

σ2

2

+

1 2

Z

d2 r

Z

d2 r 0 J (r ) G0 r − r 0 J r 0 .

(44)

By simple functional derivation, we obtain the expected propagator G r −r

0

= G0 r − r

0

Z =

0 d2 q eiq.(r −r )

κ q4 + b η (2π )2 b

.

(45)

Then, the expected propagator identifies with the bare one. Here, the variational parameter b η satisfies the implicit equation (25). We then recover the relationship: σ 2 = G (0), where σ 2 is the squared membrane roughness computed above. The next step consists to extend the study to a closed vesicle surrounded by attractive or repulsive impurities. 4. Single vesicle We start by recalling some basic backgrounds dealt with the equilibrium shape of (spherical) vesicles, which may studied using Differential Geometry techniques. The vesicle is essentially formed by two adjacent monolayers (inner and outer) that are formed by amphiphile lipid molecules. These permanently diffuse with the molecules of the surrounded aqueous medium. Such a diffusion then provokes thermal fluctuations (undulations) of the membrane. This means that the latter experiences fluctuations around an equilibrium plane. Consider a biomembrane of arbitrary topology. A point of this membrane (surface) can be described by two local coordinates (u1 , u2 ). At each point of the surface, there exists two particular curvatures (minimal and maximal), called

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principal curvatures, denoted C1 = 1/R1 and C2 = 1/R2 . The quantities R1 and R2 are the principal curvature radii. With the help of the principal curvatures, one constructs two invariants that are the mean-curvature 1

(C1 + C2 ) , 2 and the Gauss curvature C =

(46)

K = C1 C2 .

(47)

The principal curvatures C1 and C2 are nothing else but the eigenvalues of the curvature tensor [8]. To comprehend the geometrical and physical properties of the biomembranes, one needs a good model. The widely accepted one is the fluid mosaic model proposed by Singer and Nicolson in 1972 [9]. This model consists to regard the cell membrane as a lipid bilayer, where the lipid molecules can move freely in the membrane surface like a fluid, while the proteins and other amphiphile molecules (cholesterol, sugar molecules, . . . ) are simply embedded in the lipid bilayer. We note that the elasticity of cell membranes crucially depends on the bilayers in this model. The elastic properties of bilayer biomembranes were first studied, in 1973, by Helfrich [7]. The author recognized that the lipid bilayer could be regarded as smectic-A liquid crystals at room temperature, and proposed the following curvature free energy (without impurities) F =

κ

Z

(2C )2 dA + κG

2

Z

Z K dA +

γ dA + p

Z

dV .

(48)

where dA denotes the area element, and V is the volume enclosed within the lipid bilayer. In the above definition, κ accounts for the bending rigidity constant, κG for the Gaussian curvature, γ for the surface tension, and p for the pressure difference between the outer and inner sides of the vesicle. The spontaneous curvature is ignored. The first-order variation gives the shape equation of lipid vesicles [10] p − 2γ C + 4κ C C 2 − K + κ∇ 2 (2C ) = 0,

(49)

with the surface Laplace–Bertolami operator

∂ ∇ = √ g ∂ ui

1

2

∂ gg ∂ uj

√

ij

,

(50)

where g ij is the metric tensor on the surface and g = det g ij . For open or tension-line vesicles, local differential equation (49) must be supplemented by additional boundary √ conditions we do not write [11]. The above equation have known three analytic solutions corresponding to sphere [10], 2-torus [12–15] and biconcave disk [16]. For spherical vesicles, the solution to the above curvature equation is exact, and we find that the equilibrium radius is R0 =

2γ p

.

(51)

Now, assume that the vesicle is trapped in a troubled aqueous medium. Usually, to take into account the presence of impurities, a low-order coupling between their volume fraction, φ , and the mean-curvature C is added to the above free energy F , that is

Z F → F ± kB T

φ C dA.

(52)

The positive sign is for attractive impurities and the negative sign for repulsive ones. The new free energy is then F =

κ

Z

(2C ± 2C0 ) dA + κG 2

2

Z

Z K dA +

b γ dA + p

Z dV

(53)

with the notation C0 = k B T

φ , 4κ

b γ = γ − 2κ C02 .

(54)

Then, the impurities generate an extra spontaneous curvature. This means that these give arise to an asymmetry of the vesicle when its membrane is crossed. In addition, these impurities additively renormalize the interfacial tension coefficient. Minimizing this new curvature free energy yields p − 2b γ C + κ (2C ± C0 ) 2C 2 ∓ C0 C − 2K + κ∇ 2 (2C ) = 0.

(55) 2

For a spherical vesicle, the mean curvature is a constant, and we have K = C . In this case, the mean-curvature C is a root of a polynomial of degree 2. This means that we have an exact solution we do not write. In particular, for very small volume fractions (φ 1), we find that the vesicle equilibrium radius writes R R0

=1±

pkB T 8γ

2

φ + O φ2 ,

(56)

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or

R − R0 pkB T R ' 8γ 2 φ. 0

(57)

The signs (+) and (−) describe attractive and repulsive impurities, respectively. Here, R0 = 2γ /p denotes the equilibrium radius for a spherical vesicle free from impurities. The above results call the following remarks. Firstly, as it should be, the equilibrium radius of the vesicle increases with increasing volume fraction of impurities, when these are attractive. This means that the vesicle is swollen, but it radius remains close to the unperturbed one R0 , as long as the volume fraction of impurities is very small. For repulsive impurities, however, the vesicle is collapsed. Secondly, formula (57) makes sense only when the volume fraction φ is below some threshold φ ∗ = 8γ 2 /kB Tp. Since the latter must be small√in comparison with unity, the vesicle bears interfacial tensions of coefficients that do not exceed some typical value γ ∗ = kB Tp/8. Finally, at fixed volume fraction φ < φ ∗ , this formula may be used to estimate the experimental value of the ratio p/γ 2 , by a simple measurement of the equilibrium radii R (with impurities) and R0 (without impurities). The following paragraph will be devoted to the study of lamellar phases made of two parallel lipid bilayers. 5. Lamellar phases The natural question is the extension of the study to the case when we have more than one membrane immersed in a aqueous media, which is impregnated by small random impurities. In our analysis, we start from a lamellar phase composed of two parallel (neutral) fluid membranes free from impurities. The effects of these entities on physics will be discussed below. The cohesion between these bilayer membranes is ensured by long-ranged attractive van der Waals forces [17], which are balanced, at short membrane separation, by strong repulsion coming from hydration forces [18] and by steric shape fluctuations ones resulting from the membrane undulations [7]. For two parallel bilayer membranes separated by a finite distance l, the total interaction energy per unit area is V (l) = VH (l) + VW (l) + VS (l) .

(58)

The first part VH (l) = AH e−l/λH

(59)

represents the hydration potential (per unit area) that acts at small separations of the order of 1 nm. The corresponding amplitude AH and potential-range λH are about AH ' 0.2 J/m2 and λH ' 0.3 nm. The second part VW (l) = −

W 12π

1 l2

−

2

(l + δ)2

+

1

(l + 2δ)2

(60)

is the attractive van der Waals potential (per unit area) that originates from polarizabilities of lipid molecules and water molecules. Here, W accounts for the Hamaker constant that is in the range W ' 10−22 − 10−21 J, and δ for the bilayer thickness. The latter is of the order of δ ' 4 mm. The last part is the steric shape fluctuations potential (per unit area) [7] Vs (l) = cH

(kB T )2 , κ l2

(61)

with kB the Boltzmann’s constant, T the absolute temperature, and κ the common bending rigidity constant of the two membranes. But in the case of two bilayers of different bending rigidity constants κ1 and κ2 , we have κ = κ1 κ2 / (κ1 + κ2 ). There, the coefficient cH is a known numerical coefficient [7]. We note that the lamellar phase remains stable at the minimum of the potential, provided that the potential depth is comparable to the thermal energy kB T . This depends, in particular, on the value of amplitude W of the direct van der Waals energy. The Hamaker constant W may be varied changing the polarizability of the aqueous medium. In a pioneered theoretical paper, Lipowsky and Leibler [19] have shown that there exists a certain threshold Wc beyond which the van der Waals attractive interactions are sufficient to bind the membranes together, while below this characteristic amplitude, the membrane undulations dominate the attractive forces, and then, the membranes separate completely. According to the authors, Wc is in the interval Wc ' (6.3 − 0.61) × 10−21 J when the bending rigidity constant is in the range κ ' (1 − 20) × 10−19 J. We note that the typical value Wc corresponds to some temperature, Tc , called unbinding critical temperature [19,20]. In particular, it was found [19] that, when the critical amplitude is approached from above, the mean-separation between the two membranes, hli0 , diverges according to

hli0 = ξ⊥0 ∼ (Tc − T )−ψ ,

T → Tc− ,

(62)

with ψ a critical exponent whose value is [19]: ψ ' 1.00 ± 0.03. The latter was computed using field-theoretical Renormalization-Group.

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Now, assume that the lamellar phase is trapped in a troubled aqueous medium. As first implication, the impurities drastically modify the unbinding transition phenomenon, in particular, the critical temperature. For simplicity, suppose that the two adjacent fluid membranes are physically identical. Then, we have to consider two distinct physical situations: The impurities attract or push the two membranes. As we have shown above, for attractive impurities, the membrane roughness ξ⊥ is important, and then, the steric shape fluctuation energy dominates. Therefore, we expect that the unbinding transition occurs at a low critical temperature Tc∗ we will determine below. For repulsive impurities, however, we have an opposite tendency, that is the unbinding transition takes place at a high temperature greater than Tc (absence of impurities). This can be explained by the fact that the membrane roughness is less important, and then, the direct van der Waals energy dominates. In any case, the mean-separation hli scales as

hli = ξ⊥ ∼ Tc∗ − T

−ψ

,

T → Tc− .

(63) ∗

The new unbinding critical temperature Tc can be estimated as follows. Using formulae (29) and (33), we show that, at first order in the impurities coupling w ,

ξ⊥ w ' 1 + + O w2 , 0 4 ξ⊥ ξ⊥ w ' 1 − + O w2 , 0 4 ξ⊥

(attractive impurities) ,

(64)

(repulsive impurities) .

(65)

Combing Eqs. (62) to (65), we find that the difference between the two critical temperatures Tc and Tc∗ (without and with impurities) is as follows Tc − Tc∗ ∼ w + O w 2 ,

Tc − Tc∗ ∼ −w + O w

(attractive impurities) ,

2

,

(repulsive impurities) .

(66) (67)

The prefactors in the above behaviors remain unknown. The temperature shift is then proportional to the volume fraction of impurities and their interaction strength through the coupling w . Then, we are in a situation similar to finite size scaling. For charged membranes forming the lamellar phase, it was found [19] that the mean-separation between two adjacent bilayers scales as [19]

hli0 = ξ⊥0 ∼ (χ − χc )−ψ ,

(68)

in the vicinity of χc , with χ the ionic concentration of the aqueous medium and χc its critical value. For instance, for DPPC in CaCl2 solutions, χc is in the interval [21]: χc ' 84 − 10 mM. In the presence of impurities, we state that the mean-separation behaves as

hli = ξ⊥ ∼ χ − χc∗

−ψ

,

(69)

with the same exponent ψ . In this case, we have

χc∗ − χc ∼ w + O w2 , (attractive impurities) , χc∗ − χc ∼ −w + O w 2 , (repulsive impurities) .

(70) (71)

We note that the same comments may by done in this case. 6. Conclusions In this paper, we presented a large scope about the effects of impurities on the statistical properties of fluid membranes. In fact, these drastically affect the living systems behavior. As an example, we can quote a very recent experimental study [22], where the authors have undertaken a series of comparative experiments, in order to explore the effect of impurities in the form of proteins and lipids on the crystallization of membrane proteins in vapor diffusion. For the present study, the impurities were assumed to be attractive or repulsive. Using the replica trick combined with a variational method, we have computed the membrane mean-roughness, as a function of the parameters associated with the pure membrane and some parameter that is proportional to the volume fraction of impurities and their interaction strength. The main conclusion is that, attractive impurities increase the shape fluctuations due to the membrane undulations, but repulsive ones tend to suppress these fluctuations. Also, we have computed the equilibrium diameter of (spherical) vesicles surrounded by small random particles solving the curvature equation. Thereafter, we extended discussion to lamellar phase formed by two parallel fluid membranes that are a finite distance apart. This lamellar phase may undergo an unbinding transition. We have shown that, attractive impurities increase the unbinding critical temperature, while repulsive ones tend to decrease this temperature.

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We point out that, the incorporation of a small amount of impurities in an aqueous medium may be a mechanism to suppress or to produce an unbinding transition (even the temperature and polarizability of the aqueous medium are fixed) within lamellar phases composed of fluid membranes. Finally, the present study may be extended to bilayer membranes of arbitrary topology. The essential conclusion is that, attractive impurities tend to swell the membrane, while in the presence of repulsive ones, this membrane is collapsed. Also, the used method can be applied to multilayers constituted by several parallel lamellar phases. Acknowledgements We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions, during the ‘‘First International Workshop On Soft-Condensed Matter Physics and Biological Systems’’, 14–17 November 2006, Marrakech, Morocco. One of us (M.B.) would like to thank Professor C. Misbah for fruitful correspondences, and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kind hospitality during his regular visits. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, 1996. V.J. Emery, Phys. Rev. B 11 (1975) 239. S.F. Edvards, P.W. Anderson, J. Phys. (France) 5 (1975) 965. L.D. Landau, E.M. Lifshitz, Quantum Mechanics, 3rd edition, Pergamin Press, 1991. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. C. Itzykson, J.-M. Drouffe, Statistical Field Theory: 1 and 2, Cambridge University Press, 1989. W. Helfrich, Z. Natureforsch 28c (1973) 693. K. Wolfgang, Differential Geometry: Curves – Surfaces – Manifolds, American Mathematical Society, 2005. S.J. Singer, G.L. Nicolson, Science 175 (1972) 720. O.-Y. Zhong-Can, W. Helfrich, Phys. Rev. Lett. 59 (1987) 2486. Z.C. Tu, Z.C. Ou-Yang, Phys. Rev. E 68 (2003) 061915. O.-Y. Zhong-Can, Phys. Rev. A 41 (1990) 4517. Z. Lin, R.M. Hill, H.T. Davis, L.E. Scriven, Y. Talmon, Langmuir 10 (1994) 1008. M. Mutz, D. Bensimon, Phys. Rev. A 43 (1991) 4525. A.S. Rudolph, B.R. Ratna, B. Kahn, Nature 352 (1991) 52. H. Naito, M. Okuda, O.Y. Zhong-Can, Phys. Rev. E 48 (1993) 2304. J.N. Israelachvili, Intermolecular and Surface Forces, 2nd edition, Academic Press, London, 1991. R.P. Rand, V.A. Parsegian, Biochim. Biophys. Acta 988 (1989) 351. R. Lipowsky, S. Leibler, Phys. Rev. Lett. 56 (1986) 2541. R. Lipowsky, E. Sackmann (Eds.), An Extensive List of References on the Subject can be Found in: Structure and Dynamics of Membranes: Generic and Specific Interactions, vol. 1B, Elsevier, 1995. [21] L.J. Lis, W.T. Lis, V.A. Parsegian, R.P. Rand, Biochemistry 20 (1981) 1771. [22] A. Christopher, et al., Acta Cryst. D 65 (2009) 1062–1073.

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An extended study of the phase separation between phospholipids and grafted polymers on a bilayer biomembrane

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Phys. Scr. 83 (2011) 065801 (6pp)

doi:10.1088/0031-8949/83/06/065801

An extended study of the phase separation between phospholipids and grafted polymers on a bilayer biomembrane M Benhamou, I Joudar, H Kaidi, K Elhasnaoui, H Ridouane and H Qamar Laboratoire de Physique des Polymères et Phénomènes Critiques, Faculté des Sciences, Ben M’sik, PO Box 7955, Casablanca, Morocco E-mail: [email protected]

Received 4 April 2010 Accepted for publication 9 May 2011 Published 26 May 2011 Online at stacks.iop.org/PhysScr/83/065801 Abstract We re-examine here the phase separation between phospholipids and adsorbed polymer chains on a fluid membrane with a change in some suitable parameter (temperature). Our purpose is to quantify the significant effects of the solvent quality and of the polydispersity of adsorbed loops formed by grafted polymer chains on the segregation phenomenon. To this end, we elaborate on a theoretical model that allows us to derive the expression for the mixing free energy. From this, we extract the phase diagram shape in the composition–temperature plane. Our main conclusion is that the polymer chain condensation is very sensitive to the solvent quality and to the polydispersity of loops of adsorbed chains. PACS numbers: 87.16.Dg, 47.57.Ng, 64.60.F

bounded by sphingolipids such as cholera and tetanus toxins. Sphingolipids and cholesterol favor the aggregation of proteins in microdomains called rafts. In fact, these play the role of a platform for the attachment of proteins while the membranes are moved around the cell and also during signal transduction. Proteins (long macromolecules) are another principal component of cell membranes. Transmembrane proteins are amphipathic and are formed by hydrophobic and hydrophilic regions having the same orientation as other lipid molecules. These proteins are also called integral proteins. Their function is to transport substances, such as ions and macromolecules, across the membrane. There exist other types of proteins that may be attached to the cytoplasm surface by fatty acyl chains or to the external cell surface by oligosaccharides. These are termed peripheral membrane proteins. They have many functions; in particular, they protect the membrane surface, regulate cell signaling and participate in many other important cellular events. In addition, some peripheral membrane proteins (those having basic residues) tend to bind electrostatically to negatively charged membranes, such as the inner leaflet of the plasma membrane.

1. Introduction Biological membranes are of great importance to life, because they separate the cell from the outside environment and separate the compartments inside the cell in order to protect important processes and specific events. Nowadays, it is largely recognized that biological membranes are present as a lipid bilayer composed of two adjacent leaflets [1, 2], which are formed by amphiphile molecules possessing hydrophilic polar-heads pointing outward and hydrophobic fatty acyl chains that form the core. The majority of lipid molecules are phospholipids. These have a polar-head group and two non-polar hydrocarbon tails, whose length is of the order of 5 nm. Also, the cell membranes incorporate another type of lipid, cholesterol [1, 2]. The cholesterol molecules have several functions in the membrane. For example, they give rigidity or stability to the cell membrane and prevent crystallization of hydrocarbons. The biomembranes also contain glycolipids (sugars), which are lipid molecules that microaggregate in the membrane, and may be protective and act as insulators. Certain kinds of molecules are 0031-8949/11/065801+06$33.00

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© 2011 The Royal Swedish Academy of Sciences

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We note that the majority of macromolecules forming the bilayer are simply anchored on the membrane and form a soft branched polymer brush [3, 4]. The study of grafted polymers on soft interfaces was motivated by the fact that they have potential applications in biological materials, such as liposomes [5–8]. These soft materials were discovered by A Bangham. Currently, they are major tools in biology, biochemistry and medicine (as drugs transport agents). Liposomes are artificial vesicles of spherical shape that can be produced from natural nontoxic phospholipids and cholesterol. But the lipid bilayers have a short lifetime, because of the weak stability of the vesicles and their extermination by white blood cells. To have stable vesicles in time, one useful method consists of protecting them with a coat of flexible polymer chains (coat size of the order of 50 nm [1]), which prevents the adhesion of marker proteins [6, 7]. In fact, these polymer chains stabilize the liposomes, due to the excluded volume forces between monomers [8]. Liposomes can also be synthesized from A–B diblock-copolymers immersed in a selective solvent that prefers to be contacted by the polymer A. The hydrophobic B-polymer chains then aggregate and form a thin bilayer, while the hydrophilic polymer chains A float in the solvent. These copolymer-based liposomes have properties that are slightly different from those of the lipid ones [9] (high resistance, high rigidity and weak permeability to water). Depending on the choice of copolymers, these liposomes are resistant to detergents [10]. The grafting of polymers onto lipid membranes was considered in a very recent paper [11]. More precisely, the purpose was to investigate of the phase separation between phospholipids and anchored polymers. As assumptions, the aqueous medium was assumed to be a good solvent, and in addition, the polymer chains were anchored to the interface only by one extremity, a big amphiphile molecule. The latter is chemically different from the phospholipid molecules. In this paper, however, we assume that the polymer chains may be adsorbed on the membrane by many monomers that are directly attached to some polar-heads of the host lipid molecules (mobile anchors), and then organize in polydisperse loops. The surrounding liquid may be a good solvent or a theta solvent. As we shall see below, the loops’ polydispersity and solvent quality drastically affect the phase behavior of the system. Under a change in a suitable parameter, such as temperature, the phospholipids and adsorbed polymer chains phase separate into macroscopic domains alternately rich in the two components. For the determination of the phase diagram shape, we elaborate on a theoretical model that takes into account both the loops’ polydispersity and the solvent quality. This paper is organized as follows. In section 2, we derive the expression for the mixing free energy of the phospholipid–anchor mixture. To investigate the phase diagram shape is the aim of section 3. Finally, some concluding remarks are given in section 4.

bound to the polar-head of a lipid molecule (for many proteins, the lipid anchors are glycosylphosphatidylinositol units) [12, 13]; (ii) by hydrophobic side-groups of the polymers which are integrated into the bilayer [14, 15]; (iii) by membrane spanning hydrophobic domains of the polymer (membrane-bound proteins, for example); and (iv) by a strong adsorption that drives the polymer from a desorbed state to an adsorbed one [16]. For a single polymer chain, the adsorption (on solid substrates) can be theoretically studied using a scaling argument [17], for instance. In addition to the polymerization degree of the polymer chain, N , and the excluded volume parameter, v, the study was based on an additional parameter, δ, which is the energy (per kB T unit) required to adsorb one monomer on the surface. For a strong adsorption, δ −1 defines the adsorbed-layer thickness. An adsorption transition takes place at some typical value δ ∗ of δ scaling as [17]: δ ∗ ∼ RF−1 ∼ a N −ν F (νF = 3/5), where RF is the Flory radius of the polymer chain. For dilute and semi-dilute polymer solutions, the adsorption phenomenon depends, in addition to the parameter δ, on the polymer concentration. In this paper, we consider situation (iv), where each monomer has the same probability of being adsorbed on the membrane surface [18]1 . More precisely, a given monomer is susceptible to becoming linked to a polar-head of a phospholipid molecule (anchor). As a result, the polymer chains form polydisperse loops (with eventually one or two tails floating in the aqueous medium). In fact, this assumption conforms to what was considered in [19]. The case where no loops are present (adsorption only by one extremity) was considered in [11]. Contrary to the adsorption on the solid surface, the anchored polymer chains are mobile on the host membrane and may undergo the aggregation transition that we are interested in. To be more general, the fluid membrane is assumed to be in contact with a good solvent or a theta solvent. The purpose is to write a general expression for the mixing free energy. The latter will allow the determination of the phase diagram related to the aggregation of anchored polymer chains. First, we start with the free energy (per unit area) of the polymer layer (for solid substrates), which is given by [20] 0 Z N F0 1 S (n) 2 β 2 0 ' 2 [b S (n)] + [−b S (n)] ln − dn. kB T b 1 S1 (1) For the configuration study, Guiselin [21] considered that each loop can be viewed as two linear strands (two half-loop). Here, S(n) is the number of strands having more than n monomers per unit area, and b represents the monomer size. The number N denotes the length of the longest strand. Hereafter, we shall use the notation S1 = 8/a, which denotes the total number of grafted chains per unit area, with 8 being the volume fraction of anchors and a their area. If the mixture is assumed to be incompressible, then 1 − 8 is the volume fraction of the phospholipid molecules. Let us come back to the free energy expression (1); note that the first term of the right-hand side represents the

2. Mixing free energy

1

The adsorption of an adequate polymer on a fluid membrane made of two phospholipids of different chemical nature may be a mechanism of phase separation between these two unlike components as was pointed out.

Polymers can adhere to biomembranes in several ways: (i) by lipid anchors; that is, the polymer is covalently 2

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contribution of the (two- or three-body) repulsive interactions between monomers belonging to the polymer layer. The second term is simply the entropy contribution describing all possible rearrangements of the grafted chains in the polymer layer. There, the exponent β depends on the solvent quality [20]. When the grafting is accomplished in a dilute solution with a good solvent or a theta solvent, the values of β are β = 11/6 (blob model) or β = 2, respectively. For polydisperse polymer layers, the distribution S(n) is known in the literature [20]. This can be obtained by minimizing the above free energy with respect to this distribution. Without presenting details, we simply sketch the general result that [20] 8 S(n) ∼ 1/(β−1) . (2) an For usual solvents, we have S(n) ∼

8 an 6/5

(good solvents),

8 (theta solvents). an We shall rewrite the distribution S(n) as S(n) ∼

S(n) =

8 f (n), a

Explicitly, we have ζ (N ) = 1 (polymer brushes), ζ (N ) ∼ N β/(1−β)

(3)

(4)

(5)

F = 8 ln 8 + (1 − 8) ln(1 − 8) + χ 8(1 − 8) kB T + u8β + η(N ) 8.

(6)

β N 1/(1−β) ln N . β −1

(7)

where the positive coefficients C and D are such that (in dimension 2) [23] Z ∞ π D = −π rU (r ) dr, C = σ 2 (covolume). (15) 2 σ

In the asymptotic limit N → ∞, this coefficient goes to η(N ) → a −1 [β + ln(β − 1)],

(8)

provided that β > 1 (note that β = 11/6, for good solvents, and β = 2, for theta ones). This asymptotic limit is always positive definite and inversely proportional to the polar-head area a. In fact, the positive sign of the coefficient η(N ) agrees with the entropy loss due to the polymer chains grafting. As we shall see below, this linear contribution does not change the phase diagram in the composition-critical parameter plane. Here, the coupling constant u is as follows: b2 a

β−1

1 N

Z

u= with ζ (N ) =

N

N ζ (N ),

[ f (n)]β dn.

(13)

We note that, for polymer chains anchored by one extremity, a big amphiphile molecule (the polymer brush case), the first contribution of entropy, 8 ln 8, should be divided by some factor q, which represents the ratio of the anchor area to the area of polar-heads of the host phospholipids. In the present case, we have q = 1. In equality (11), χ accounts for the Flory interaction parameter 1 D χ = χ0 − 2 C − (14) , A kB T

where the linear term in 8 describes the contribution of entropy to the free energy, where the coefficient η(N ) is as follows (see the appendix): −1 [β + ln(β − 1)] 1 − N 1/(1−β) η(N ) = a +

(12)

Note that ζ (N ) < 1, since, in all cases, β > 1. Therefore, the polydispersity of loops decreases the repulsive interaction energy. Now, we have all the ingredients for the determination of the expression for the mixing free energy. To this end, we imagine that the interface is present as a two-dimensional (2D) Flory–Huggins lattice [17, 22], where each site is occupied by an anchor or by a phospholipid molecule. Hence, we can regard the volume fraction of anchors 8 as the probability that a given site is occupied by an anchor. Therefore, 1 − 8 is the occupation probability of phospholipids. Before the determination of the desired mixing free energy (per site), we note that the latter is the sum of three contributions, which are the mixing entropy (per site), the volume free energy (per site) and the interaction energy (per site) coming from the membrane undulations. Actually, the induced attractive forces due to the membrane undulations are responsible for the condensation of anchors. These forces balance the repulsive ones between monomers along the connected polymer chains. We then write

with f (n) ∼ 1/n 1/(β−1) , for polydisperse polymer layers, and f (n) = 1 (for all n), for polymer brushes. Therefore, F0 can be approximated by F0 = a −1 u8β + η(N )8, kB T

(polydisperse systems).

(11)

Here, U (r ) is the pair potential induced by the membrane undulations ([11]; [24] and references therein) r < σ, ∞, σ 4 U (r ) = (16) −AH , r > σ, r where σ is the hard disc diameter, which is proportional to the square root of the anchor area a. There, the potential amplitude AH plays the role of the Hamaker constant. It was found that the latter decays with the bending rigidity constant according to ([11]; [24] and reference therein)

(9)

(10)

AH ∼ κ −2 .

1

3

(17)

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But this amplitude is also sensitive to temperature. We remark that the above mixing free energy is not symmetric under the change 8 → 1 − 8. Some straightforward algebra gives the following expression for the attraction parameter D: D = AH

π κ −2 ∼ . 2σ 2 σ2

interactions widen the compatibility domain, and then, the separation transition appears at low temperature. Now, to see the influence of the solvent quality, we rewrite the interaction parameter u as u = u 0 ζ (N ) ∼ u 0 N β/(1−β) < u 0 , where u 0 is the interaction parameter relative to a monodisperse system. Thus, the polydispersity of loops has a tendency to reduce the compatibility domain in comparison with the monodisperse case. The critical volume fraction, 8c , can be obtained by minimizing the critical parameter χ (8) with respect to the 8-variable. We then obtain

(18)

Therefore, those membranes with a small bending modulus induce significant attraction between anchors. In formula (12), χ0 > 0 is the Flory interaction parameter describing the chemical segregation between amphiphile molecules that are phospholipids and anchors. This segregation parameter is usually written as χ0 = α0 +

1 1 = 0. − 2 + β(β − 1) (β − 2) u8β−3 c 2 8 (1 − 8c ) c For good solvents (β = 11/6), we have

γ0 , T

(19) 55 1 1 − 2− u8−7/6 = 0. c 2 8 216 (1 − 8c ) c

where the coefficients α0 and γ0 depend on the chemical nature of the various species. Also, the total interaction parameter χ can be written as χ =α+

γ , T

C , A2

γ = γ0 +

(20)

D . kB A2

(24)

Therefore, the critical volume fraction is the abscissa of the intersection point of the curve of the equation (2x − 1)/ x 5/6 (1 − x)2 and the horizontal straight line of the equation y = (55/216)u. Note that this critical volume fraction is unique, and in addition, it must be greater than the value 1/2 (for mathematical compatibility). The coordinates of the critical point are (8c , χc ), where 8c solves the above equation and χc = χ (8c ). The latter can be determined by combining equations (19) and (21). For theta solvents (β = 2), the coordinates of the critical point are exact,

with the new coefficients α = α0 −

(23)

(21)

These coefficients then depend on the chemical nature of the unlike components and on the membrane’s characteristics through its bending modulus κ. If we admit that the coupling constant u has a slight dependence on temperature, we will draw the phase diagram in the plane of variables (8, χ). Indeed, all of the temperature dependence is contained in the Flory interaction parameter χ.

8c = 12 ,

χc = 2 + u,

(25)

where the interaction parameter u scales as u=

3. Phase diagram

b2 −1 N . a

(26)

The above relation clearly shows that the polymer chains’ condensation rapidly takes place only when the characteristic mass N is high enough. The same tendency is also seen in the case of good solvents. It is straightforward to show that the critical fraction and the critical parameter are shifted to lower values in the case of polydisperse systems, whatever be the quality of the surrounding solvent. In figure 1, we present the spinodal curve for monodisperse (with no loops) and polydisperse (with loops) systems, with a fixed parameter N . We have chosen the good solvents situation. For theta solvents, the same tendency is seen. We present in figure 2 the spinodal curve for a polydisperse system (with loops), at various values of the parameter N . As expected the critical parameter is shifted to higher values when we augmented the typical polymerization degree. Finally, we compare, in figure 3, the spinodal curves for a polydisperse system (with loops) for the case of theta solvents and those for the case of good solvents, at fixed parameters b and N . All the curves in the figure reflect our discussions above.

With the help of the above mixing free energy, we can determine the shape of the phase diagram in the (8, χ )-plane that is associated with the aggregation process that drives the anchors from a dispersed phase (gas) to a dense one (liquid). We focus only on the spinodal curve along which the thermal compressibility diverges. The spinodal curve equation can be obtained by equating to zero the second derivative of the mixing free energy with respect to the anchor volume fraction 8; that is, ∂ 2 F/∂82 = 0. Then, we obtain the following expression for the critical Flory interaction parameter: 1 1 χ(8) = + β(β − 1) u8β−2 . (22) 2 8(1 − 8) Above this critical interaction parameter appear two phases: one is homogeneous and the other is separated. Of course, the linear term in 8 appearing in equality (11) does not contribute to the critical parameter expression. We remark that, in usual solvents, this critical interaction parameter is increased due to the presence of (twoor three-body) repulsive interactions between monomers belonging to the polymer layer. This means that these 4

Phys. Scr. 83 (2011) 065801

χ

M Benhamou et al

χ

80

20

16

60

12 40

8 20

4

0 0,0

0,2

0,4

0,6

0,8

0 0,0

1,0

Φ

12

8

4

0,4

0,6

0,8

0,8

1,0

polymer chains. The latter were taken into account through the well-known form of the chains’ length distribution. In doing so, we first computed the expression for the mixing free energy by adopting the Flory–Huggins lattice image usually encountered in polymer physics [17, 22]. Such an expression shows that there is competition between four contributions: entropy, chemical mismatch between unlike species, interaction energy induced by the membrane undulations and the interaction energy between monomers belonging to the grafted layer. Such a competition governs the phases succession. We emphasize that the present work and a previous work [11] differ from another previous work that was concerned with the same problem, but in which the substrate was assumed to be a rigid surface [3]. Therefore, the membrane undulations were neglected. As we have seen, these undulations increase the segregation parameter χ by an additive term, χm , scaling as κ −2 . This means that the phase separation is accentuated due to the presence of thermal fluctuations. Compared with the previous work [11], which was concerned with soft brushes with monodisperse end-grafted polymers, the present work is more general, since it takes into account both the polydispersity of loops forming the adsorbed polymer chains and the solvent quality. Thus, the present study was achieved in a unified way. As we have seen, these details drastically affect the phase diagram architecture. Now, let us discuss further the influence of solvent quality on the critical phase behavior. We recall that the solvent quality appears in the free energy (11) through the repulsion parameter u, defined in equation (7). We find, in the N → ∞ limit, that u g ∼ N 1/5 u θ , where the subscripts g and θ stand for good and theta solvents, respectively. This implies that the good solvent plays the role of a stabilizer regarding the phase separation. Finally, this work must be considered as a natural extension of a study reported previously [11], which was concerned with monodisperse end-grafted polymer chains

16

0,2

0,6

Figure 3. Superposition of spinodal curves for a polydisperse system for the case of theta solvents (solid line) on those for the case of good solvents (dashed line). For these curves, we chose N = 100 and b2 = 0.5a.

20

0 0,0

0,4

Φ

Figure 1. Spinodal curves (in a good solvent) for monodisperse (dashed line) and polydisperse (solid line) systems, when N = 100, with the parameter b2 = 0.5a.

χ

0,2

1,0

Φ Figure 2. Spinodal curves for a polydisperse system, when N = 50 (solid line), 100 (dashed line) and 150 (dotted line), with the parameter b2 = 0.5a . We assumed that the surrounding liquid is a good solvent. For theta solvents, the tendency is the same.

4. Discussion and conclusions This paper is devoted to the thermodynamic study of the aggregation of the polymer chains adsorbed on a soft surface. Such an aggregation is caused by competition between the chemical segregation between phospholipids and grafted polymer chains, their volumic interactions and the membrane undulations. More precisely, we addressed the question of how these polymer chains can be driven from a dispersed phase (gas) to a dense one (liquid), under a change in a suitable parameter, e.g. the absolute temperature. To be more general, we achieved the study in a unified way; that is, we have considered more realistic physical situations: the solvent quality (good or theta solvents) and the polydispersity of the adsorbed loops formed by the grafted 5

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trapped in a good solvent. Therefore, the present work presents a wide perspective on the phenomenon of segregation between the host phospholipids and grafted polymer chains on bilayer membranes.

References [1] Lipowsky R and Sasckmann S (ed) 1995 Structure and Dynamics of Membranes: From Cells to Vesicles vol 1A (New York: Elsevier) [2] Benhamou M 2008 Lipids Insights 1 2 [3] Nicolas A and Fourcade B 2003 Eur. Phys. J. E 10 355 [4] Aubouy M et al 2000 Phys. Rev. Lett. 84 4858 Aubouy M et al 1996 Macromolecules 29 7261 [5] Ringsdorf H and Schmidt B 1986 How to bridge the gap between membrane, biology and polymers science Synthetic Membranes: Science, Engineering and Applications ed P M Bungay et al (Dordrecht: D Reidel) p 701 [6] Lasic D D 1992 Am. Sci. 80 250 [7] Torchilin V P 1996 Effect of polymers attached to the lipid head groups on properties of liposomes Handbook of Nonmedical Applications of Liposomes vol 1, ed D D Lasic and Y Barenholz (Boca Raton, FL: RCC Press) p 263 [8] Joannic R, Auvray L and Lasic D D 1997 Phys. Rev. Lett. 78 3402 [9] Discher B M et al 1999 Science 284 1143 [10] Elferink M G L et al 1992 Biochim. Biophys. Acta 1106 23 [11] Benhamou M, Joudar J and Kaidi H 2007 Eur. Phys. J. E 24 343 [12] Blume G and Cev G 1990 Biochim. Biophys. Acta 1029 91 [13] Lasic D D, Martin F J, Gabizon A, Huang S K and Papahadjopoulos D 1991 Biochim. Biophys. Acta 1070 187 [14] Decher G et al 1989 Angew. Makromol. Chem. 166 71 [15] Simon J, Kuhner M, Ringsdorf H and Sackmann E 1995 Chem. Phys. Lipids 76 241 [16] Garel1 T, Kardar M and Orland H 1995 Europhys. Lett. 29 303 Chatellier X and Andelman D 1995 Europhys. Lett. 32 567 Lipowsky R 1995 Europhys. Lett. 30 197 Xie A F and Granick S 2002 Nature Mater. 1 129 [17] de Gennes P-G 1979 Scaling Concept in Polymer Physics (Ithaca, NY: Cornell University Press) [18] Binder W H, Barragan V and Menger F M 2003 Angew. Chem. Int. Edn Engl. 42 5802 [19] Manghi M and Aubouy M 2001 Adv. Coll. Interf. Sci. 94 21 [20] Manghi M and Aubouy M 2003 Phys. Rev. E 68 041802 [21] Guiselin O 1992 Euro. Phys. Lett. 17 225 Guiselin O 1992 PhD Thesis Paris VI University [22] Flory P J 1953 Principles of Polymer Chemistry (Ithaca, NY: Cornell University Press) [23] Balian R 2006 From Microphysics to Macrophysics—Methods and Applications of Statistical Physics (Berlin: Springer) [24] Marchenko V I and Misbah C 2002 Eur. Phys. J. E 8 477

Acknowledgments We are indebted to Professors T Bickel, J-F Joanny and C Marques for helpful discussions during the First International Workshop on Soft-Condensed Matter Physics and Biological Systems (14–17 November 2006, Marrakech, Morocco). MB thanks Professor C Misbah for fruitful correspondence and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kind hospitality during his visit. We are grateful to the referee for critical remarks and useful suggestions that helped us to improve the scientific content of this paper.

Appendix The aim is to determine the coefficient η(N ) appearing in formula (6). We start with the entropy contribution to the free energy 0 Z N 2 0 Fentropic 1 S (n) ' 2 −b S (n) ln − dn, (A.1) kB T b 1 S1 where the loop-size distribution is defined in equation (2). Then, the above expression can be rewritten as Fentropic = η(N ) 8, kB T

(A.2)

with N

1 n β/(1−β) dn. n β/(1−β) ln β −1 1 (A.3) Straightforward algebra gives η(N ) = a −1 [β + ln(β − 1)] 1 − N 1/(1−β) η(N ) = a −1

1 β −1

+

Z

β N 1/(1−β) ln N . β −1

(A.4)

This concludes the determination of the coefficient η(N ).

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African Journal Of Mathematical Physics Volume 10(2011)55-64

Statistics of a single D-manifold restricted to two parallel biomembranes or a tubular vesicle M. Benhamou∗ , K. Elhasnaoui, H. Kaidi, M. Chahid Laboratoire de Physique des Polym`eres et Ph´enom`enes Critiques Facult´e des Sciences Ben M’sik, P.O. Box 7955, Casablanca, Morocco ∗ [email protected]

abstract The purpose is an extensive conformational study of a single polymer immersed in an aqueous medium (good solvent) delimitated by bilayer membranes. To be more general, we assume that the polymer is of arbitrary topology we term D-polymeric fractal or Dmanifold, where D is the spectral dimension (for instance, D = 1, for linear polymers, and 4/3, for branched ones). The main quantity to consider is the parallel extension of the conﬁned polymer. To make explicit calculations, we suppose that the polymer is restricted to a tubular vesicle or two parallel biomembranes. We ﬁrst show that, for the ﬁrst geometry, the polymer is conﬁned only when the tubular vesicle is in equilibrium state. For the second geometry, the conﬁnement is possible if only if the two parallel membranes are in their binding state, that is below the unbinding or adhesion temperature. In any case, the parallel gyration radius of the conﬁned polymer is computed using an extended Flory-de Gennes theory. As result, this radius strongly depends on the polymer topology (through the spectral dimension D) and on the membranes sizes, which are the equilibrium diameter (function of bending modulus, pressure diﬀerence between inner and outer sides of the membrane, and interfacial tension coeﬃcient), for the ﬁrst geometry, and the mean-separation (function of temperature and interaction strength between the adjacent membranes), for the second one. Finally, we give the expression of the conﬁnement free energy, as a function of the polymer size, and discuss the eﬀects of external pressure or lateral tension on the radius expression for two conﬁning parallel membranes. Key words: D-Polymeric fractals, Biomembranes, Vesicles, Conﬁnement.

I. INTRODUCTION

The polymer conﬁnement ﬁnds many applications in various domains, such as biological functions, ﬁltration, gel permeation chromatography, heterogeneous catalysis, and oil recuperation. The physics of polymer conﬁnement is a rich and exciting problem. Recently, much attention has been paid to the structure and dynamics of polymer chains conﬁned to two surfaces, or inside cylindrical pores [1 − 7]. Thereafter, the study has been extended to more complex polymers, called D-polymeric fractals or D-manifolds [8] that are restricted to the same geometries. Here, D is the spectral dimension that measures the degree of the connectivity of monomers inside the polymer [9]. For instance, this intrinsic dimension is 1 for linear polymers and 4/3 for branched ones. The D-manifolds may be polymerized (or

0

c a GNPHE publication 2011, [email protected] ⃝

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African Journal Of Mathematical Physics Volume 10(2011)55-64

crumpled) vesicles [10]. The polymer conﬁnement between two surfaces and in cylinders with sinusoidal undulations was also investigated [11, 12]. The conﬁning geometries may be soft-bodies, as bilayer biomembranes and surfactants, and spherical and tubular vesicles. A particular question has been addressed to polymer conﬁnement in surfactant bilayers of a lyotropic lamellar phase [13], where the authors reported on small-angle X-ray scattering and freefracture electron microscopy studies of a nonionic surfactant/water/polyelectrolyte system in the lamellar phase region. The fundamental remark is that, the polymer molecules cause both local deformation and softening of the bilayer. In the same context, it was experimentally demonstrated [14] that the polymer conﬁnement may induce a nematic transition of microemulsion droplets. More precisely, the authors showed that upon conﬁnement, spherical droplets deform to prolate ellipsoid droplets. The origin of such a structural transition may be attributed to a loss of the conformational entropy of polymer chains due to the conﬁnement. In other experiments [15 − 17], a new phase has been observed adding an neutral hydrosoluble polymer (PVP) in the lyotropic lamellar phase (CPCI/hexanol/water). Also, one has studied the eﬀect of a neutral water-soluble polymer on the lamellar phase of a zwitterionic surfactant system [18]. The polymer conﬁnement is also relevant for living systems and governs many biological processes. As example, we can quote membrane nanotubes that play a major role in intercellular traﬃc, in particular for lipid and proteins exchange between various compartments in eukaryotic cells [19 − 22]. The traﬃc of macromolecules and vesicles in nanotubes is ensured by molecular motors [23]. Also, these are responsible for the extraction of nanotubes [24]. The formation of tubular membrane tethers or spicules [25 − 33] results from the action of localized forces that are perpendicular to the membrane. These forces may originate from the polymerization of ﬁbers [34], as actin [35], tubulin [36], or sickle hemoglobin [37]. Inspired by biological processes, as macromolecules and vesicles transport, we aim at a conformational study of conﬁned polymers in aqueous media delimitated by biomembranes. More precisely, the purpose is to see how these biomembranes can modify the conformational properties of the constrained polymer. To this end, we choose two geometries : a tubular vesicle (Geometry I), and two parallel biomembranes (Geometry II). To be more general, we consider polymers or arbitrary connectivity called D-polymeric fractals or D-manifolds [38]. Linear and branched polymers, and polymerized vesicles constitute typical examples. We note that this conformational study in necessary for the description of dynamic properties of polymers restricted to these geometries. As we shall below, the polymer conﬁnement is entirely controlled by the conﬁning biomembranes state. Indeed, for Geometry I, the polymer may be conﬁned only when the conﬁning tubular vesicle is at equilibrium. For Geometry II, the conﬁnement is possible if only if the two parallel membranes are in the binding state. This paper is organized as follows. In Sec. II, we brieﬂy recall the conformational study of unconﬁned D-polymeric fractals in good solvent. Sec. III deals with the conformational study of polymers conﬁned inside a tubular vesicle. In Sec. IV, we extend the study to D-polymeric fractals conﬁned to two parallel biomembranes. Some concluding remarks are drawn in the last section.

II. UNCONFINED POLYMERIC FRACTAL

Consider a single polymeric fractal of arbitrary topology (linear polymers, branched polymers, polymer networks, ...). We assume that the considered polymer is trapped in a good solvent. We denote by RF ∼ aM 1/dF

(1)

its gyration (or Flory) radius, where dF is the Hausdorﬀ fractal dimension, M is the molecular-weight (total mass) of the considered polymer, and a denotes the monomer size. The mass M is related to the linear dimension N by : M = N D , where D is the spectral dimension [9]. The latter is deﬁned as the Hausdorﬀ dimension corresponding to the maximal extension of the fractal. Naturally, the Hausdorﬀ dimension depends on the Euclidean dimensionality d, the spectral dimension D and the solvent quality. When the polymer is ideal (without excluded volume forces), its Hausdorﬀ dimension, d0F , is a known simpler function of D [9] d0F =

2D . 2−D 56

(2)

M. Benhamou et al.

African Journal Of Mathematical Physics Volume 10(2011)55-64

For linear polymers (D = 1), d0F = 2 [2], for ideal branched ones (D = 4/3), d0F = 4 [9], and for crumpled membranes (D = 2), d0F = ∞. Because of the positivity of the Hausdorﬀ dimension, the above expression makes sense only for D < 2. Indeed, this condition is fulﬁlled for any complex polymer with spectral dimension in the interval 1 ≤ D < 2 [9]. A polymeric fractal in good solvent is swollen, because of the presence of the excluded volume forces. The polymer size increases with increasing total mass M according to the power law (1). The ﬁrst implication of the polymer swelling is that, the actual Hausdorﬀ dimension dF is quite diﬀerent from the Gaussian one, deﬁned in Eq. (2). However, there exists a special value of the Euclidean dimensionality called upper critical dimension duc [39, 40], beyond which the polymeric fractal becomes ideal. This upper dimension is naturally a D-dependent function, which can be determined [39] using a criterion of Ginzburg type, usually encountered in critical phenomena [41, 42]. According to Ref. [39], duc is given by duc =

4D . 2−D

(3)

For instance, the upper critical dimension is 4 for linear polymers [2], and 8 for branched ones [39]. We emphasize that, in general, the Hausdorﬀ fractal dimension dF cannot be exactly computed. Many techniques have been used to determine its approximate value, in particular, the Flory-de Gennes (FD) theory [2]. Using a generalized FD approach, it was found that the fractal dimension is given by [39] dF = D

d+2 , D+2

(4)

below the critical dimension, and it equals the Gaussian fractal dimension d0F described above. For dimension 3, we have dF (3) =

5D . D+2

(5)

For instance, for linear polymers, dF (3) = 5/3, and dF (3) = 2, for branched ones (animals). In the following paragraph, we shall focus our attention on the conformational study of a single polymeric fractal conﬁned to a long tubular vesicle.

III. CONFINED POLYMERIC FRACTAL IN GEOMETRY I A. Useful backgrounds

Before studying the conformation of a single polymeric fractal, we recall some basic backgrounds concerning the equilibrium shape of tubular vesicles. This can be done using Diﬀerential Geometry machineries. The tubular vesicle is essentially formed by two adjacent leaﬂets (inner and outer) that are composed of amphiphile lipid molecules. These permanently diﬀuse with the molecules of the surrounded aqueous medium. Such a diﬀusion then provokes thermal ﬂuctuations (undulations) of the membrane. This means that the latter experiences ﬂuctuations around an equilibrium plane we are interested in. Consider a biomembrane of arbitrary topology. A point of this membrane can be described by two local coordinates (u1 , u2 ). From surfaces theory point of view, at each point, there exists two particular curvatures (minimal and maximal), called principal curvatures, denoted C1 = 1/R1 and C2 = 1/R2 . The quantities R1 and R2 are the principal curvature radii. With the help of the principal curvatures, one constructs two invariants that are the mean-curvature C=

1 (C1 + C2 ) , 2

(6)

and the Gauss curvature K = C1 C2 .

57

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African Journal Of Mathematical Physics Volume 10(2011)55-64

We recall that C1 and C2 are nothing else but the eigenvalues of the curvature tensor [43]. To comprehend the geometrical and physical properties of the biomembranes, one needs a good model. The widely accepted one is the ﬂuid mosaic model proposed by Singer and Nicholson in 1972 [44]. This model consists to regard the cell membrane as a lipid bilayer, where the lipid molecules can move freely in the membrane surface like a ﬂuid, while the proteins and other amphiphile molecules (cholesterol, sugar molecules, ...) are simply embedded in the lipid bilayer. We note that the elasticity of cell membranes crucially depends on the bilayers in this model. The elastic properties of bilayer biomembranes were ﬁrst studied, in 1973, by Helfrich [45]. The author recognized that the lipid bilayer could be regarded as smectic-A liquid crystals at room temperature, and proposed the following curvature free energy ∫ ∫ ∫ ∫ κ 2 (2C + 2C0 ) dA + κG KdA + γdA + p dV . (8) F = 2 where dA denotes the area element, and V is the volume enclosed within the lipid bilayer. In the above deﬁnition, κ accounts for the bending rigidity constant, C0 for the spontaneous curvature, κG for the Gaussian curvature, γ for the surface tension, and p for the pressure diﬀerence between the outer and inner sides of the vesicles. The ﬁrst order variation gives the shape equation of lipid vesicles [46] ( ) p − 2γC + κ (2C + C0 ) 2C 2 − C0 C − 2K + κ∇2 (2C) = 0 , (9) with the surface Laplace-Bertlami operator ) ∂ gg , (10) ∂uj ( ) where g ij is the metric tensor on the surface and g = det g ij . For open or tension-line vesicles, local diﬀerential equation (9) must be supplemented by additional boundary conditions we√do not write [47]. The above equation have known three analytic solutions corresponding to sphere [46], 2-torus [48 − 51] and biconcave disk [52]. For cylindrical (or tubular) vesicles, one of the principal curvature is zero, and we have 1 ∂ ∇ =√ g ∂ui 2

C=−

( √

ij

1 , R

K=0,

(11)

where R is the radius of the cylinder. If we ignore the boundary conditions (assumption valid for very long tubes), the uniform solution to equation (9) is ( H=2

4κ p

)1/3 ,

(12)

where H is the equilibrium diameter. We have neglected the surface tension and spontaneous curvature contributions, in order to have a simpliﬁed expression for the equilibrium diameter. The above relation makes sense as long as the pressure diﬀerence is smaller than a critical value pc that scales as [53] : pc ∼ κ/R3 . The latter is in the range 1 to 2 Pa. The meaning of the critical pressure is that, beyond pc , the vesicle is unstable. This implies that the equilibrium diameter must be greater than 1/3 the critical one Hc = 2 (4κ/pc ) . In what follows, we shall use the idea that consists to regard the tubular vesicle as a rigid cylinder of eﬀective diameter H that depends on the characteristics of the bilayer through the parameters κ and p.

B. Parallel extension to the cylinder-axis

Consider now a polymeric fractal of arbitrary topology conﬁned inside a tubular vesicle of equilibrium diameter H. The host solvent is assumed to be a good solvent. The polymer is conﬁned if only if its three-dimensional Flory radius RF 3 is much larger than the diameter H, that is H ?

E : F GH E :9 < 9D

.

(8)

The notation 〈… 〉 means the thermal average, which is performed with the defined in Eq. (2). The general expression for Canham-Helfrich Hamiltonian this propagator we do not recall is known in literature [11-13]. The particular value of the propagator , 0 = JK defines the mean-roughness of the membrane, JK . The latter and the in-plane correlation length (beyond which the two-point correlation function exponentially fails), J∥ , are related by : JK = J∥ /16"̂ . To solve integral equation (7a), we set = is arbitrary. It is straightforward to show that ∑

Q

N

,

N

=(

N,

− ′ =,

where the subscript 1 ≤ P ≤ − ′ ,

N

(9)

with the matrix elements Q

N

+7 ,

−

N

.

(10)

− ′ ,

(11)

.

(12)

The inversion of Eq. (7a) gives − ′ = ∑N

,

RS

N

,

N

with the matrix coefficients R

N

= QN = (

N

+ 7N ,

−

N

Matrix R is then the transpose of matrix Q. Combining now relationships (7a) and (11) yields the expression for the height-correlation function , , ′ =,

− ′ −∑

,N

, 5

−

T N,

N

− ′ ,

(13)

with coefficients T

N

=

7 + 7N RS

N

.

(14)

These define a squared matrix, T. In Appendix A, we show that this matrix is symmetric and positive definite. The main result (13) calls the following remarks. Firstly, as it must be, the height-correlation function is not invariant under translations in the parallel directions to the fluid membrane and substrate, but the bare correlation function , does. The translation symmetry breaking can be directly seen on integral equation (7). Secondly, since matrix T is symmetric, the propagator is invariant permutting the points and ′. Thirdly, all dependence of this propagator in elastic constants 's is entirely contained in matrix elements T N 's. Fourthly, the positivity property of the second term in the r.h.s of expression (13) indicates that the presence of tethers considerably reduces the thermal fluctuations. Sixthly, the expected propagator is completely determined by the knowledge of matrix T and bare propagator , . Finally, as it should be, at infinity, that is → ∞ and ′ → ∞, , , ′ goes to zero. 4. Fluctuation spectra and discussion The local membrane mean-roughness, JK , is simply the value of propagator , , ′ for = ′. From expression (13), we obtain the following exact result JK

= JK

−∑

,N

,

−

T N,

−

N

.

(15)

Matrix elements T N ’s depend on all distances between tethers W − N W’s and their elastic constants ’s. Before discussing the above finding, we rewrite it on the following scaling form

6

JK

= JK X1 − ∑

Y

,N

Z NY T

−

−

N

,

(16)

with the dimensionless quantities

Y

−

Z T

N

= JK T

=,

−

N

,

(17)

/ JK

.

(18)

Z N ’s The latter is small than unity and Y 0 = 1. Notice that coefficients T depend only on dimensionless elastic constants 7 JK ’s and renormalized bare

propagators Y − N ’s. Now, let us discuss the obtained result. Firstly, to comprehend the general expression (16), we examine some cases : (i) One tether: Z is a scalar quantity, and we Let be the vector-position. In this case, matrix T have Z = 7 JK /[1 + 7 JK \ . T

(19)

Combing this equality with formula (13) yields JK

= JK ]1 −

a 9 7 _` ^

a 9 7 _` G^

Y

−

.

(20)

(ii) Two tethers: We denote by and the position-vectors of the two tethers. In this case, we Z are as follows find that the matrix elements of T Z T

=

a _`

b

9

[ 7 + 7 7 JK \ , Z T

Z =T

=−

Z T

a _`

b

F

with 7

=

7 7 Y

a _`

b

9

−

[ 7 + 7 7 JK \ , (21a) ,

(21b)

c = 1 + 7 JK

+ 7 JK

+ 7 7 JK

d

− 7 7 JK d Y

−

. (21c)

Then, the associated local roughness is JK

Z Y Z Y 1−T − −T − = JK e Z Y −2T − Y −

f

/

.

(22)

(iii) The tethers have the same length and their extremities coincide at one : point ≡ This case corresponds to a -watermelon tether, and we find that the associated local roughness is given by JK

= JK h1 − iY

−

,

(23)

with the constant i=∑

,N

Z T

N

.

(24)

Z. A Such a constant is nothing else but the sum of all coefficients of matrix T comparison between relations (20) and (23) suggests that a -watermelon tether Z , such that is equivalent to one tether of effective elastic constant j Z JK /[1 + j Z JK \ . i=j

(25)

i = 7 JK /[1 + 7 JK \ .

(26)

Z= 7. j

(27)

Z to that of one tether, 7 . We Such a equality relates the new elastic constant j show in appendix B that the quantity i is exactly given by

By comparison, we have

Secondly, in all cases, the positivity of the second term in the r.h.s of equality (16) implies that the local roughness, JK , remains above the bulk one, JK , at 8

any point of the membrane. Therefore, the presence of tethers has tendency to suppress the shape fluctuations of the supported fluid membrane. Thirdly, for any number of tethers, when the considered point is far from the positions of the tethers extremities ’s, that is | − | ≫ J∥ , the local roughness JK fails exponentially to the bulk one, JK . Finally, let us assume that the tethers are long-flexible polymer chains (in dilute solution) of common polymerization degree, m. In this case, the associated elastic constants are identical and scale as noS ~qS m S , where q represents the monomer size. For infinitely long-polymer chains, that is m → ∞, Z goes to JK /no r, where r is the matrix unity. In this limit, to first matrix T order in m S , the general result (16) becomes JK

= JK s1 −

JK /no

∑

Y

−

+ t mS u .

(28)

The above equality can be seen as a Taylor series in JK /no -variable. 5. Concluding remarks

We recall that the purpose of this paper is the investigation of correlations between points belonging to a supported membrane that is linked to a solid surface via long-flexible polymer chains. The presence of the connected macromolecules can be regarded as a new mechanism to bind the membrane for specific interests. This binding phenomenon depends naturally on the value of the molecular-weight of the connected polymer chains. High-molecular-weights have tendency to increase the shape-fluctuations amplitude. But, low-molecular-weights suppress these fluctuations. According to formula expressing the local fluctuations amplitude, the quality constitutes a pertinent parameter of discussion. In fact, the presence of a good solvent reduce Our predictions reveal that the fluctuations amplitude is local, and it is more important in regions of the membrane with low-grafting density. This means that such amplitude depends strongly on the distribution of the anchoring points. In our description, we assumed that the anchoring points on the inner monolayer of the supported membrane are immobile. In fact, these points free move since they are fixed on polar-heads of some phospholipids. Such a problem will be discussed elsewhere. 9

For simplicity, we have ignored primitive interactions between the supported membrane and the wall, as van der Waals, hydration and shape-fluctuations interactions. Under certain physical circumstances, theses interactions can be neglected (rigid tethers…). Finally, questions such as dynamics, extension of study to more than one membrane and other developments are under consideration.

10

Appendix A The aim is to demonstrate that matrix T, defined in Eq. (14), is symmetric and v and ,̅ , of positive definite. To this end, we introduce two matrices, j respective elements 7 ( N and , − N , with indices 1 ≤ x ≤ and 1 ≤ P ≤ , and ( N = 1, if x = P, and ( N = 0, otherwise. It is easy to see that the matrix B expresses as follows v, R = r + ,̅ j

(A.1)

where r is the matrix unity. On the other hand, the matrix T can be rewritten as v r + ,̅ j v v RS = j T=j

S

.

(A.2)

v S and ,̅ are Such a form clearly suggests that Ty = T, since both matrices j symmetric. Then, matrix T is symmetric. To show that T is positive definite, we consider its inverse matrix TS = v S . Explicitly, we have Rj v S + ,̅ . TS = j

(A.3)

The associated matrix elements are TS

N

= 7S (

N

+,

−

.

N

(A.4)

To prove that matrix T is positive definite, it will be sufficient to show that z≡∑

TS

,N

N N

>0,

(A.5)

for any set of integer variables ,…, , and z = 0, if only if ,…, = 0, … ,0 . Start from the Fourier integral representation of the bare propagator ,

−

N

=

8 9 :; =

; .|A ; } BA ; ~• >?

E : F GH E :9 < 9 D

We show that 11

.

(A.6)

z=∑

• ; .A ; } •………………………………………………………… >? ; .A ;} „? ∑• 8 9 :; |∑}‚ƒ €} = }‚ƒ €~ =

+

E : F GH E :9 D

< 9

.

(A.7)

Here, ̅ denotes the complex conjugate of , and ̅ > 0 is its squared module. Since the elastic constants "̂ , $2 of the fluid membrane are positive, z > 0. The latter vanishes, if only if = 0 x = 1, … , . Matrix TS is then positive definite. Therefore, T is a positive definite matrix. This ends the proof of the desired mathematical properties of matrix T. Appendix B To show formula (25), use is made of relation (A.4) of Appendix A. We start with identity ∑

T

,N

N

TS

N†

=(

†

.

(B.1)

According to Eq. (A.4), we have ∑

,N

T N [ 7 S (N† + JK \ = (

A double summation over indices x and ‡ gives ∑

,N

Z T

N

†

.

= i = 7 JK /[1 + 7 JK \ .

We have Eq. (17). This ends the proof of formula (25).

12

(B.2)

(B.3)

References [1] More details can be found in : M. Tanaka and E. Sackmann, Nature 437, 656 (2005). [2] E. Sackmann, Science 271, 43 (1996). [3] S. Terrettaz, M. Mayer and H. Vogel, Langmuir 19, 55675569 (2003). [4] P.C. Gufler, D. Pum, U.B. Sleytr and B. Schuster, Biochimica et Biophysica Acta 1661, 154 (2004). [5] V. Atanasov, P.P. Atanasova, I.K. Vockenroth, N. Knorr and I. Köper, Bioconjugate Chemistry 17, 631 (2006). [6] L.J.C. Jeuken, R.J. Bushby and S.D. Evans, Electrochemistry Communications 9, 610 (2007). [7] P. Théato, R. Zentel and S. Schwarz, Macromolecular Bioscience 2, 387 (2002). [8] R.-J. Merath and U. Seifert, Eur. Phys. J. E 23, 103 (2007), and references therein. [9] P.B. Canham, J. Theoret. Biol. 26, 61 (1970). [10] W. Helfrich, Z. Natureforsch 28c, 693 (1973). [11] T. Bickel, M. Benhamou, and H. Kaidi, Phys. Rev. E 70, 051404 (2004). [12] H. Kaidi, T. Bickel, and M. Benhamou, Europhys. Lett. 69, 15 (2005). [13] A. Bendouch, H. Kaidi, T. Bickel, and M. Benhamou, J. Stat. Phys.: Theory and Experiment P01016, 1 (2006).

13

Statistical Theory and Method Abstracts – Zentralblatt Database c 2012 FIZ Karlsruhe

ZMATH 1197.76166 Madmoune, Y.; El Hasnaoui, K.; Bendouch, A.; Kaidi, H.; Chahid, M.; Benhamou, M. Brownian dynamics of nanoparticles in contact with a confined biomembrane. Afr. J. Math. Phys. 8, No. 1, 91-100, Article No. 1010, electronic only (2010). Summary: The system we consider is a fluid membrane confined to two parallel reflecting walls that are separated by a finite distance, L, assumed to be small in comparison to the bulk roughness. The attractive membrane is surrounded by small colloidal particles (nanoparticles). The purpose is the study of Brownian dynamics of these particles, under a change of a suitable parameter, such as temperature, T , or colloidmembrane interaction strength, w. The Brownian dynamics is investigated through the knowledge of the time particle density, which solves the Smoluchowski equation. Solving this equation around the mid-plane, where the essential of phenomenon occurs, we obtain the exact form of the local particle density, as a function of the perpendicular distance and time. In the derived expression, appears some time-scale, τ , which scales as τ ∼ L3 /w. This scale-time can be interpreted as the required time over which the colloidal suspension reaches their final equilibrium state. Also, τ can be regarded as the time-interval over which the particles are trapped in holes and valleys. Classification: 76Z05 82D99 92C30 60K35 Keywords: biomembranes; nanoparticles; confinement; Brownian dynamics http://www.fsr.ac.ma/GNPHE/ajmpVolume8N1-2010/ajmp1010.pdf

–1–

November 13, 2012

INTERNATIONAL JOURNAL of ACADEMIC RESEARCH

Vol. 5. No. 3. May, 2013

T. El Hafi, M. Benhamou, K. Elhasnaoui, H. Kaidi. Fluctuation spectra of supported membranes via long-flexible polymers. International Journal of Academic Research Part A; 2013; 5(3), 5-10. DOI: 10.7813/2075-4124.2013/5-3/A.1

FLUCTUATION SPECTRA OF SUPPORTED MEMBRANES VIA LONG-FLEXIBLE POLYMERS 1

1,2

1

T. El Hafi , M. Benhamou , K. Elhasnaoui , H. Kaidi

1,3

1

2

LPPPC, Sciences Faculty Ben M’sik, P.O. Box 7955, Casablanca, ENSAM, Moulay Ismail University, P.O. Box 25290, Al Mansour, Meknes, 3 CRMEF, P.O. Box 255, Meknes (MOROCCO) [email protected] DOI: 10.7813/2075-4124.2013/5-3/A.1

ABSTRACT We consider a fluid membrane that is supported to a solid substrate via long-flexible polymers (as proteins or lipopolymers). In this paper, we aim at a quantitative investigations of the associated fluctuation spectra. The study is achieved through the knowledge of the height correlation function between points on the membrane. First, we compute exactly such a function for all pairs of points of the membrane, versus the parameters of the problem, which are the elastic constants of the membrane (bending modulus and interfacial tension) and the molecularweight of the connected polymer chains. Second, from this correlation function, we extract the exact expression of the fluctuations amplitude and discuss its dependence on molecular-weight. Third, we write exact scaling laws for both height correlation function and fluctuations amplitude. Finally, we emphafsize that the presence of the connected polymers may be a new mechanism to bind membranes in the vicinity of substrates for specific interests. Key words: Supported membranes; Tethers; Polymers; Correlation function; Fluctuation spectra 1. INTRODUCTION Lipid-bilayer membranes supported on solid substrates are widely used as cell-surface models connecting biological and artificial materials. Usually, the supported biomembranes can be placed either directly on solids or on ultrathin polymer supports that play the role of the extracellular matrix [1]. The main goal is to control, organize and study the properties and function of membranes and membrane-associated proteins. Supported lipid-membranes have other practical interests [2]. Indeed, they permit the biofunctionalization of inorganic solids and polymeric materials, and provide a natural environment for the immobilization of proteins (such as hormone receptors and antibodies). They allow the preparation of ultrathin, high-electric-resistance layers on conductors and the incorporation or receptors into these insulating layers for the design of biosensors. In addition, they are used in Electrochemistry for the detection of protein activity by measuring the resistance and capacitance changes [3-6]. Currently, supported membranes can be assemled as follows : (i) the inner monolayer of the lipid-bilayer fixed to the substrate by covalent chemical bonds or by ion bridges, (ii) freely supported lipid-protein bilayers separated from the substrate by ultrathin water layers (about ten nanometers), and (iii) bilayer membranes linked to the substrate by grafted or adsobed polymers, as lipopolymers [7]. The fixation of these lipopolymers on flat surfaces was studied using Surface Plasmon Spectroscopy. The film thickness of the adsorbed lipopolymer is about 12 to 20Å. Statistical Mechanics of supported membranes has been the subject of many theoretical investigations [8]. The majority of these works considered tethers are springs. In this paper, however, the tethers are long-flexible polymer chains. We shall assume that the two extremities of each polymer chain on the inner monolayer of the supported bilayer membrane and wall remain parallel each to other. The purpose is the study of the fluctuation spectra of supported membrane pairs. This is done through the computation of the height-height correlation function. We have computed this function exactly, from which we extracted an important quantity that is the fluctuations amplitude, even when the polymer chains are polydisperse. Such an amplitude is naturally space-dependent function. The remaining of presentation proceeds as follows. Section 2 deals with the description of the used extended Hamiltonian. In Section 3, we present the exact computation of the correlation function of the supported membrane. Section 4 is dedicated to the fluctuation spectra and discussion. Some concluding remarks are drawn in the last section. Finally, some technical details are relegated in Appendices A and B.

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Vol. 5. No. 3. May, 2013

2. EXTENDED CANHAM-HELFRICH HAMILTONIAN Start with a fluctuating fluid membrane that is embedded in a three-dimensional homogeneous aqueous medium, delimitated by a plan rigid substrate (semi-infinite geometry). We assume that the membrane is linked to the solid surface by polymer chains. We denote by ⃗ the position-vector of the second extremity of chain on the supported interface. For simplicity, we suppose that the two extremities of a given chain remain along the same axis that is perpendicular to the fluid membrane and substrate. To be more general, the positions of anchoring extremities are assumed to be distributed at random. Within the framework of the Monge representation, a point on the membrane can be described by ⃗, = ℎ( ⃗) , where ⃗ = ( , ) is the two-dimensional transverse vector and ℎ( ⃗) is the height-function (perpendicular distance to the surface). The Statistical Mechanics of the supported fluid membrane is based on an extended Hamiltonian [8] [ℎ] =

[ℎ] + ∑

ℎ (⃗ ),

(1)

with the bare Canham-Helfrich Hamiltonian [9,10] [ℎ] =

⃗ [ (∆ℎ) + (∇ℎ) ],

∫

(2)

where is the membrane bending modulus, is the interfacial tension coefficient. The second part in the r.h.s of Hamiltonian expression (1) accounts for the elastic contribution due to the presence of the connected hydrocarbon chains (springs), which is not local. There, denotes the elastic constant of polymer chain . The generalized Hamiltonian (1) describes the so-called model D in literature [8]. It will be convenient to rewrite the total Hamiltonian (1) as follows [ℎ] =

∫

⃗ [ (∆ℎ) + (∇ℎ) + ℎ ],

(3)

with the local function ( ⃗) = ∑

( ⃗ − ⃗ ).

(4)

Here, ( ⃗) denotes the two-dimensional Dirac-function. Then, the term ℎ /2 plays the role of a local confinement potential that maintains the position of the fluid membrane around some typical distance, ℎ . 3. EXACT HEIGHT-CORRELATION FUNCTION With the help of the above total Hamiltonian, we compute the expectation mean-value of various functionals of the height-field ℎ, in particular the two-point correlation function ( ⃗, ⃗′) = 〈ℎ( ⃗)ℎ( ⃗′)〉 − 〈ℎ( ⃗)〉〈ℎ( ⃗′)〉.

(5)

Such a function measures the height-fluctuations. As we shall see below, this correlation function allows the determination of the local membrane-roughness. Notice that the presence of tethers breaks the translation symmetry in the parallel directions to the supported membrane. Now, the aim is the computation of the expected correlation function. To this end, we start from the following second-order differential equation ̂ ∆ − Δ + ( ⃗)

( ⃗, ⃗′) =

( ⃗ − ⃗′),

(6)

)+( / ). We have introduced the rescaling with the two-dimensional Laplacian operator Δ = ( / parameters : ̂ = / , = / , = / , where is the absolute temperature and is the Boltzmann’ constant. There, = / . Differential equation (6) is directly obtained from Hamiltonian (1) by simple functional derivatives. To compute the height-correlation function, the first step consists to transform differential equation (6) into the following integral equation solved by the propagator , ( ⃗, ⃗′) =

( ⃗ − ⃗′) − ∫

( ⃗, ⃗′) =

( ⃗ − ⃗′) − ∑

⃗′′

( ⃗ − ⃗′′) ( ⃗′′) ( ⃗′′ − ⃗′),

(7)

( ⃗ − ⃗ ) ( ⃗ − ⃗′),

(7a)

or equivalently,

where

( ⃗ − ⃗′) is the (bare) propagator of a non-supported fluid membrane (

6 | PART A. NATURAL AND APPLIED SCIENCES

= 0),

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Vol. 5. No. 3. May, 2013

( ⃗ − ⃗′) = 〈ℎ( ⃗)ℎ( ⃗′)〉 − 〈ℎ( ⃗)〉 〈ℎ( ⃗′)〉 = ∫ (

⃗.( ⃗ ⃗′)

⃗ )

.

(8)

The notation 〈… 〉 means the thermal average, which is performed with the Canham-Helfrich Hamiltonian defined in Eq. (2). The general expression for this propagator we do not recall is known in literature [11-13]. The particular value of the propagator (0) = ( ) defines the mean-roughness of the membrane, . The latter and the in-plane correlation length (beyond which the two-point correlation function exponentially fails), ∥ , are related by : (

) = ∥ /16 ̂ . To solve integral equation (7a), we set ⃗ = ⃗ , where the subscript 1 ≤ to show that ∑

( ⃗ − ⃗′) =

≤

is arbitrary. It is straightforward

⃗ − ⃗′ ,

(9)

with the matrix elements =

⃗ −⃗ .

+

(10)

The inversion of Eq. (7a) gives [

( ⃗ − ⃗′) = ∑

]

⃗ − ⃗′ ,

(11)

with the matrix coefficients =

=

⃗ −⃗ .

+

(12)

Matrix is then the transpose of matrix . Combining now relationships (7a) and (11) yields the expression for the height-correlation function ( ⃗, ⃗′) =

( ⃗ − ⃗′) − ∑

,

(⃗ − ⃗ )

⃗ − ⃗′ ,

(13)

with coefficients =

+

[

]

.

(14)

These define a squared matrix, . In Appendix A, we show that this matrix is symmetric and positive definite. The main result (13) calls the following remarks. Firstly, as it must be, the height-correlation function is not invariant under translations in the parallel directions to the fluid membrane and substrate, but the bare correlation function does. The translation symmetry breaking can be directly seen on integral equation (7). Secondly, since matrix is symmetric, the propagator is invariant permutting the points ⃗ and ⃗′. Thirdly, all dependence of this propagator in elastic constants 's is entirely contained in matrix elements 's. Fourthly, the positivity property of the second term in the r.h.s of expression (13) indicates that the presence of tethers considerably reduces the thermal fluctuations. Sixthly, the expected propagator is completely determined by the knowledge of matrix and bare propagator . Finally, as it should be, at infinity, that is ⃗ → ∞ and ⃗′ → ∞, ( ⃗, ⃗′) goes to zero. 4. FLUCTUATION SPECTRA AND DISCUSSION The local membrane mean-roughness, ( ⃗), is simply the value of propagator expression (13), we obtain the following exact result ( ⃗) = (

) −∑

,

(⃗ − ⃗ )

⃗− ⃗ .

( ⃗, ⃗′) for ⃗ = ⃗′. From

(15)

Matrix elements ’s depend on all distances between tethers ⃗ − ⃗ ’s and their elastic constants Before discussing the above finding, we rewrite it on the following scaling form ( ⃗) =

1−∑

,

(⃗ − ⃗ )

⃗− ⃗ ,

’s.

(16)

with the dimensionless quantities

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INTERNATIONAL JOURNAL of ACADEMIC RESEARCH =(

)

Vol. 5. No. 3. May, 2013

,

(⃗ − ⃗ ) =

(17)

( ⃗ − ⃗ )/( ) .

(18)

The latter is small than unity and (0) = 1. Notice that coefficients ’s depend only on dimensionless ⃗ − ⃗ ’s. elastic constants ( ) ’s and renormalized bare propagators Now, let us discuss the obtained result. Firstly, to comprehend the general expression (16), we examine some cases : (i) One tether: Let ⃗ be the vector-position. In this case, matrix is a scalar quantity, and we have = ( ) / 1+ (

) .

(19)

Combing this equality with formula (13) yields ( ⃗) =

of

(

1−

) (

(⃗ − ⃗ ) .

)

(20)

(ii) Two tethers: We denote by ⃗ and ⃗ the position-vectors of the two tethers. In this case, we find that the matrix elements are as follows =

+

( ) , =

=

( ) ,

+

(21a)

( ⃗ − ⃗ ),

=−

(21b)

with =1+

( ) +

( ) +

( ) −

( )

( ⃗ − ⃗ ).

(21c)

Then, the associated local roughness is ( ⃗) =

(⃗ − ⃗ ) − (⃗ − ⃗ )

1− −2

(⃗ − ⃗ ) (⃗ − ⃗ )

/

.

(22)

(iii) The tethers have the same length and their extremities coincide at one point ⃗ ( ⃗ ≡ ⃗ ) : This case corresponds to a -watermelon tether, and we find that the associated local roughness is given by ( ⃗) =

(⃗ − ⃗ ),

1−

(23)

with the constant =∑

.

,

(24)

Such a constant is nothing else but the sum of all coefficients of matrix . A comparison between relations (20) and (23) suggests that a -watermelon tether is equivalent to one tether of effective elastic constant , such that =

( ) / 1+ ( ) .

Such a equality relates the new elastic constant quantity is exactly given by =

(

(25)

to that of one tether, . We show in appendix B that the

) / 1+

(

) .

(26)

By comparison, we have =

.

(27)

Secondly, in all cases, the positivity of the second term in the r.h.s of equality (16) implies that the local roughness, ( ⃗), remains above the bulk one, , at any point ⃗ of the membrane. Therefore, the presence of tethers has tendency to suppress the shape fluctuations of the supported fluid membrane.

8 | PART A. NATURAL AND APPLIED SCIENCES

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Vol. 5. No. 3. May, 2013

Thirdly, for any number of tethers, when the considered point ⃗ is far from the positions of the tethers extremities ⃗ ’s, that is | ⃗ − ⃗ | ≫ ∥ , the local roughness ( ⃗) fails exponentially to the bulk one, . Finally, let us assume that the tethers are long-flexible polymer chains (in dilute solution) of common polymerization degree, . In this case, the associated elastic constants are identical and scale as ~ , where represents the monomer size. For infinitely long-polymer chains, that is → ∞, matrix goes to ( / ) , where is the matrix unity. In this limit, to first order in , the general result (16) becomes ( ⃗) =

1− (

/

) ∑

(⃗ − ⃗ ) + (

The above equality can be seen as a Taylor series in

/

).

(28)

-variable.

5. CONCLUDING REMARKS We recall that the purpose of this paper is the investigation of correlations between points belonging to a supported membrane that is linked to a solid surface via long-flexible polymer chains. The presence of the connected macromolecules can be regarded as a new mechanism to bind the membrane for specific interests. This binding phenomenon depends naturally on the value of the molecular-weight of the connected polymer chains. High-molecular-weights have tendency to increase the shape-fluctuations amplitude. But, low-molecular-weights suppress these fluctuations. According to formula expressing the local fluctuations amplitude, the quality constitutes a pertinent parameter of discussion. In fact, the presence of a good solvent reduce Our predictions reveal that the fluctuations amplitude is local, and it is more important in regions of the membrane with low-grafting density. This means that such amplitude depends strongly on the distribution of the anchoring points. In our description, we assumed that the anchoring points on the inner monolayer of the supported membrane are immobile. In fact, these points free move since they are fixed on polar-heads of some phospholipids. Such a problem will be discussed elsewhere. For simplicity, we have ignored primitive interactions between the supported membrane and the wall, as van der Waals, hydration and shape-fluctuations interactions. Under certain physical circumstances, theses interactions can be neglected (rigid tethers…). Finally, questions such as dynamics, extension of study to more than one membrane and other developments are under consideration. REFERENCES 1. More details can be found in : M. Tanaka and E. Sackmann, Nature 437, 656 (2005). 2. E. Sackmann, Science 271, 43 (1996). http://dx.doi.org/10.1126/science.271.5245.43. 3. S. Terrettaz, M. Mayer and H. Vogel, Langmuir 19, 55675569 (2003). http://dx.doi.org/10.1021/ la034197v. 4. P.C. Gufler, D. Pum, U.B. Sleytr and B. Schuster, Biochimica et Biophysica Acta 1661, 154 (2004). http://dx.doi.org/10.1016/j.bbamem.2003.12.009. 5. V. Atanasov, P.P. Atanasova, I.K. Vockenroth, N. Knorr and I. Köper, Bioconjugate Chemistry 17, 631 (2006). http://dx.doi.org/10.1021/bc050328n. 6. L.J.C. Jeuken, R.J. Bushby and S.D. Evans, Electrochemistry Communications 9, 610 (2007). http://dx.doi.org/10.1016/j.elecom.2006.10.045. 7. P. Théato, R. Zentel and S. Schwarz, Macromolecular Bioscience 2, 387 (2002). http://dx.doi.org/ 10.1002/1616-5195(200211)2:83.0.CO;2-5. 8. R.-J. Merath and U. Seifert, Eur. Phys. J. E 23, 103 (2007), and references therein. http://dx.doi.org/ 10.1140/epje/i2006-10084-2. 9. P.B. Canham, J. Theoret. Biol. 26, 61 (1970). http://dx.doi.org/10.1016/S0022-5193(70)80032-7. 10. W. Helfrich, Z. Natureforsch 28c, 693 (1973). 11. T. Bickel, M. Benhamou, and H. Kaidi, Phys. Rev. E 70, 051404 (2004). http://dx.doi.org/10.1103/ PhysRevE.70.051404. 12. H. Kaidi, T. Bickel, and M. Benhamou, Europhys. Lett. 69, 15 (2005). http://dx.doi.org/10.1209/ epl/i2004-10305-4. 13. A. Bendouch, H. Kaidi, T. Bickel, and M. Benhamou, J. Stat. Phys.: Theory and Experiment P01016, 1 (2006).

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APPENDIX A The aim is to demonstrate that matrix , defined in Eq. (14), is symmetric and positive definite. To this end, ⃗ − ⃗ , with indices 1 ≤ ≤ and we introduce two matrices, and ̅ , of respective elements and 1 ≤ ≤ , and = 1, if = , and = 0, otherwise. It is easy to see that the matrix B expresses as follows = + ̅

,

(A.1)

where is the matrix unity. On the other hand, the matrix = ( + ̅

=

can be rewritten as

) .

(A.2) and ̅ are symmetric. Then, matrix

Such a form clearly suggests that = , since both matrices is symmetric. To show that is positive definite, we consider its inverse matrix

=

. Explicitly, we have

+ ̅ .

=

(A.3)

The associated matrix elements are [ To prove that matrix

]

=

⃗ −⃗ .

+

(A.4)

is positive definite, it will be sufficient to show that ≡∑

[

,

for any set of integer variables ( , … , integral representation of the bare propagator ⃗ −⃗

]

> 0,

), and

= 0, if only if ( , … ,

⃗. ⃗

⃗ )

= ∫(

(A.5) ) = (0, … ,0). Start from the Fourier

⃗

.

(A.6)

We show that =∑

⃗ )

+ ∫(

⃗. ⃗

∑

⃗. ⃗

∑

.

(A.7)

Here, ̅ denotes the complex conjugate of , and ̅ > 0 is its squared module. Since the elastic constants ( ̂ , ) of the fluid membrane are positive, > 0. The latter vanishes, if only if = 0 ( = 1, … , ). Matrix is then positive definite. Therefore, is a positive definite matrix. This ends the proof of the desired mathematical properties of matrix .

APPENDIX B To show formula (25), use is made of relation (A.4) of Appendix A. We start with identity ∑

,

∑

,

[

]

=

.

+(

)

(B.1)

According to Eq. (A.4), we have

A double summation over indices ∑

,

and =

=

.

(B.2)

gives =

(

) / 1+

We have Eq. (17). This ends the proof of formula (25).

10 | PART A. NATURAL AND APPLIED SCIENCES

( ) .

(B.3)

J Membrane Biol (2013) 246:383–389 DOI 10.1007/s00232-013-9544-9

The Quantum Casimir Effect May Be a Universal Force Organizing the Bilayer Structure of the Cell Membrane Piotr H. Pawlowski • Piotr Zielenkiewicz

Received: 1 November 2012 / Accepted: 29 March 2013 / Published online: 24 April 2013 Ó The Author(s) 2013. This article is published with open access at Springerlink.com

Abstract A mathematic–physical model of the interaction between cell membrane bilayer leaflets is proposed based on the Casimir effect in dielectrics. This model explains why the layers of a lipid membrane gently slide one past another rather than penetrate each other. The presented model reveals the dependence of variations in the free energy of the system on the membrane thickness. This function is characterized by the two close minima corresponding to the different levels of interdigitation of the lipids from neighbor layers. The energy barrier of the compressing transition between the predicted minima is estimated to be 5.7 kT/lipid, and the return energy is estimated to be 3.1 kT/lipid. The proposed model enables estimation of the value of the membrane elastic thickness modulus of compressibility, which is 1.7 9 109 N/m2, and the value of the interlayer friction coefficient, which is 1.9 9 108 Ns/m3. Keywords effect

Cell membrane Lipid bilayer Casimir

Introduction Casimir-Polder (Casimir 1948; Casimir and Polder 1948) forces are universal physical forces arising from a quantized field. They act even between two uncharged metallic P. H. Pawlowski (&) P. Zielenkiewicz Institute of Biochemistry and Biophysics, Polish Academy of Sciences, PAS, Pawinskiego 5a, 02-106 Warszawa, Poland e-mail: [email protected] P. Zielenkiewicz Plant Molecular Biology Laboratory, Warsaw University, Warsaw, Poland

plates in a vacuum, placed a few micrometers apart, without any external electromagnetic field. The idea that Casimir forces may play an important role in different biomembrane systems is quite new. Based on the fundamental work of Lifshitz (1956) dealing with retarded van der Waals forces between macroscopic bodies, first it was applied to the formation of cellular ‘‘rouleaux’’ (Bradonjic´ et al. 2009) and confined biomembranes (El Hasnaoui et al. 2010). In a natural manner, it supplements the theory of nonretarded van der Waals interactions in a lipid–water system (Parsegian and Ninham 1970), sometimes offering an interesting counterproposal. Thus, forces related to the zero-point energy of quantum fluctuations may play an important role in biology, and the analysis of these forces offers a new view into biological phenomena at the cellular level. Herein, we propose a simple second-quantized explanation for the fact that cell membrane bilayer leaflets slip (Otter and Shkulipa 2007) past one another, rather than penetrate each other. This membrane feature is of great importance as it determines the anisotropy of the membrane’s rheological properties. The relative freedom of movement of molecules along membrane leaflets and the relative restriction of displacement in the transverse direction account for the great lateral fluidity and the small perpendicular compressibility of a membrane (Evans and Hochmuth 1978). These dual mechanical properties, both fluid-like and solid-like, have an impact on the structure and function of the proteins embedded in the lipid bilayer matrix (Andersen and Koeppe 2007). This impact finally determines the status of the cell membrane as an active barrier, a natural organizer and an important participant in all processes of life. The proposed model of the interaction of the cell membrane bilayer leaflets considers the membrane interior to be a three-layer dielectric sandwich. The free energy of

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P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

the system is related to the electromagnetic field excitations in the ground state. The free energy depends on the varying thickness of the central layer, where lipid chains penetrate the opposite layers and where the density of lipid chains, and the dielectric permittivity, varies in space. The theoretical values of the two energy minima, the membrane elastic thickness modulus of compressibility and the interlayer friction coefficient, were set according to this model.

The Model The cell membrane interior was considered to be a threelayer dielectric sandwich, which consists of parallel slabs as in Fig. 1. The two lateral peripheral regions of this system are fully occupied by all the lipids of the local leaflet. The remaining central region contains the hydrocarbon tails that penetrate from the neighboring layers. The dielectric constant, ec, of this central layer differs from the dielectric constant, ep, of the other parts of the membrane. We assumed that the length of the lipids in each leaflet may vary within a narrow range, L ± dL/2, where L represents the average lipid length and dL represents the width of the length distribution. The distribution of the lipid lengths was assumed to be uniform. Thus, the perpendicular cross section of a membrane bilayer matrix resembles ‘‘the two overlapping combs with some broken teeth.’’

ep

L

ec

ep

δL

Fig. 1 Three-layer dielectric sandwich model of a cell membrane. Only the longest and shortest lipids are shown. L average lipid length, dL width of the lipid length range, ec dielectric constant of the central region, ep dielectric constant of the peripheral region

123

The thickness of the central region, dc, falls to a minimum value equal to dL in the configuration in which the total membrane thickness, d, equals 2L (Fig. 2a). This configuration we called ‘‘configuration MTCR’’ (minimal thickness of the central region). To clarify further considerations, configuration MTCR was treated as the reference configuration. In this configuration, dc may increase both with an increase (Fig. 2b) and with a decrease (Fig. 2c) in the total membrane thickness. Thus, the possible variation in the thickness of the central region in configuration MTCR is unidirectional. For convenience, we introduced the variable x to describe the difference between the actual total membrane thickness and the thickness of the membrane in configuration MTCR; that is, x = d-2L. Then to formalize the description of the central region, one may write dc = dL ? |x|, where |x| denotes the absolute value. The dielectric constant, ec, was assumed to vary in space within the central region due to variations in the lipid density. Variations along the direction perpendicular to the membrane plane were postulated. At first approximation, ec was simply characterized by the spatial average heci. In general, when the total membrane thickness differs from the thickness of the reference configuration by x, the average heci changes as described by the following formula: 8 jxj=dL > x\0 < 1 þ ep 1 1 þ 1þjxj=dL hec i ¼ ep ð1Þ x¼0 > : 1 þ e 1 1 x[0 p 1þx=dL For details, see Appendix. The plot of Eq. 1 indicates (Fig. 3) that the average heci increases with the compression of the membrane thickness (x \ 0) and decreases with membrane thickness extension (x [ 0). For the membrane in the configuration MTCR (x = 0), heci strictly equals ep. Quantum electrodynamic considerations reveal that two dielectric plates separated by a dielectric medium may be attracted as a result of the decrease in the zero-point energy of the quantum electromagnetic field excitations (Srivastava et al. 1985). In the case of the cell membrane, by approximating the dielectric constant of the central region with the average value heci, the free energy, F, of the field per unit area may be described by the following equation (Bradonjic´ et al. 2009): p2 ep hec i 2 hc pﬃﬃﬃﬃﬃﬃﬃﬃ F¼ 3 720 ep þ hec i hec iðdLÞ ð1 þ jxj=dLÞ3

ð2Þ

where h is the reduced Planck constant and c is the speed of light in a vacuum; the dependence of heci on x is described by Eq. 1.

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

385

d = 2L

d > 2L

d < 2L

MTCR

dc = δ L

a

b

dc > δ L

Fig. 2 Unidirectional variations in the thickness of the central region of the membrane. Only the longest and shortest lipids are shown. a Configuration MTCR. Total membrane thickness d = 2L. Central region thickness dc = dL. b Bilayer leaflets farther apart than in configuration MTCR. Total membrane thickness d [ 2L. Central

dc > δ L

c

region thickness dc [ dL. c Bilayer leaflets closer together than in configuration MTCR. Total membrane thickness d \ 2L. Central region thickness dc [ dL. L average lipid length, dL width of the lipid length range, MTCR minimal thickness of the central region

According to the model, in both cases, the thickness of the ˚) central region of the membrane (dc = 9.0 and dc = 7.9 A is larger than the assumed thickness in the configuration ˚ ). MTCR (dc = 5 A

εc 2εp - 1

εp

Discussion 1

-10

0

10

x/ δ L

Fig. 3 The space average of the dielectric constant in the central region, heci, as a function of the ratio x/dL. x the difference between the total membrane thickness and the membrane thickness in configuration MTCR, dL width of the lipid length range, ep dielectric constant in peripheral regions of the membrane

We assumed that the central region of the membrane may be treated as a kind of leaflet interface. One interesting question is how the interface is stabilized to avoid leaflet penetration and sticking. The proposed model may offer a simple explanation based on the Casimir effect in dielectrics. At reasonable values of the applied parameters (see ‘‘Results’’ section), the model predicts two free energy minima. For the membrane organized in the lower energy ˚ ), its leaflets are separated by 4 A ˚ minimum (x = 4 A

Results

0,03 0,01 2

F [J/m ]

Equations 1 and 2 show that, for ideally flat leaflets (dL = 0), Casimir forces couple membrane layers at zero distance (x = 0), with the free energy tending to negative ˚ , ep = 2), infinity. More reasonable calculations (dL = 5 A taking into account the heterogeneity of the lipid lengths (Blaurock 1982) and specifying the dielectric constant of the peripheral regions of the membrane (Huang and Levitt 1977) were also performed. These calculations indicate (Fig. 4) that there are two finite minima of free energy, both for the membrane in configurations close to the MTCR. The first (lower) minimum (F = -4.68 9 10-2 J/ ˚ ), and m2) is for leaflets being slightly separated (x = 4.0 A -2 2 the second one (F = -2.54 9 10 J/m ) is for leaflets ˚ ). being moderately pushed together (x = -2.9 A

0,05

-0,01 -0,03 -0,05 -0,07 -50

-30

-10

10

30

50

70

90

x [A] Fig. 4 The free energy, F, of the field per unit area as a function of a change, x, in the membrane thickness relative to the thickness of the MTCR membrane

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P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

greater than in configuration MTCR. At the assumed het˚ ), this result means erogeneity of the lipid lengths (dL = 5 A that 20 % of lipids (1-x/dL) penetrate into the neighboring ˚ (dL-x). leaflet (Fig. 5a) but by no more than 1 A According to the results presented in Fig. 4, transition ˚ ) requires into the second (higher) minimum (x = –2.9 A approximately 5.7 kT/lipid (calculated for T = 300 K and ˚ 2). In this minimum energy conarea per lipid a = 50 A figuration, the membrane leaflets are in closer contact. All ˚ lipids penetrate the opposite layer but no deeper than 7.9 A (Fig. 5b). The return to the other minimum requires 3.1 kT/ lipid. The revealed characteristic energies are several times higher than those estimated assuming the average tension c (0.003 mN/m) of living cells (Blanchard and Rauch 2012). The result ca kT indicates that the Casimir effect may be an important contributor to the membrane dynamics, along with the hydrophobic effect. To study the mechanical properties of the presented model membrane, the elastic thickness modulus of

a

100%

n1

-4.5

n2

-0.5 0.5

4.5

ξ [A]

3.95

ξ [A]

b

100%

n1

-3.95

n2

-1.05

1.05

Fig. 5 The area densities n1 and n2 of lipids belonging to a given leaflet as a function of a distance, n, from the membrane midplane. The range occupied by the longest and shortest lipids is shown above. ˚ . b For the a For the first minimum (lower) of free energy, x = 4 A ˚ second minimum (higher) of free energy, x = -2.9 A

123

compressibility, k, was numerically estimated using a differential second derivative of the free energy at the first minimum (k = d F00 ). Assuming a bilayer thickness of d = 5 nm (Kuchel and Ralston 1998), k = 1.7 9 109 N/ m2; i.e., this estimated value of k is of the same order as that measured using volume dilatometry of lipid bilayers (Srinivasan et al. 1974). This result indicates that the central region of the cell membrane resembles an ‘‘incompressible’’ core that does not allow lipid interleaflet penetration. The results of the presented model also enable estimation of the value of the interlayer friction coefficient, f (friction force per unit area and unit velocity). Approximating the end of the lipid penetrating the neighbor leaflet with a hemisphere of a radius r that experiences the action of the Stokes force from the liquid passing with the velocity 2v, one may obtain f = 6pgr/a, where g is the membrane shear viscosity. Taking the typical value of viscosity g = 0.1 Ns/m2 estimated from the diffusion coefficient of membrane-spanning proteins in phospholipid bilayers (Waugh 1982) and assuming that the membrane is ˚ , f = 1.9 9 108 Ns/ in the first minimum with r = 0.5 A 3 m , which is within the range of reported experimental values (Shkulipa et al. 2005; Otter and Shkulipa 2007). For the second minimum, one may expect an eightfold higher value because of a deeper penetration. According to the model, a membrane bilayer in a basic state (lower minimum) should possess relatively small interleaflet friction. The probability that, due to thermal fluctuations, some regions of the membrane reach second minimum is relatively small. At first glance, one may worry about some physical and mathematical problems with the proposed approach. It is obvious that some points require additional discussion, especially phenomena at a physical level neglected in our model. First of all, is it justified to assume that the lipids of opposite leaflets interpenetrate each other at all? Steric interactions seem to be the dominant suppressor of interdigitation. However, a spontaneous or induced interdigitated phase of bilayers consisting of double-tail lipids was confirmed in computer simulation and differential scanning calorimetry experiments (Kranenburg 2004; Kranenburg et al. 2004; Mavromoustakos et al. 2011). Moreover, a simple estimation below shows that the possible steric effect is not energetically dominant, as one may expect. Let us assume that during interdigitation the part p of lipids is compressed and their length decreases by an assumed certain value, k. Let us also assume that each of two lipid acyl chains contains the number b of C–C bonds characterized by a certain bond stiffness, g. Then, elastic deformation energy per lipid molecule may be calculated as pgk2/b. For typical conditions, p = 1/10 (meaning that 50 % of penetrating lipids in the lower minimum are

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

387

˚´ and b = 15; this compressed), g = 100 N/m, k = 0.5 A energy equals 0.8 kBT. These are only 7 % of the predicted value of the energy barrier in a lower minimum and, as such, may be neglected at first approximation. The next question is, how much does the derived shape of the free energy, F (Fig. 4), depend on the specific way that the interpenetration occurs? It was assumed that the distribution of lipid lengths was uniform and, in this way, interdigitation varied linearly with a distance from the membrane midplane (Figs. 5, 6). What will change if we assume the more spectacular variation? In extreme cases, when a single-point distribution (dL = 0) is assumed, one infinite minimum of energy at midplane will be obtained and the energy will increase with distance, like –1/|x|3. Thus, narrowing the distribution lowers minima and approaches them together. It should be stressed that the parameters heci, ep and dL in Eq. 2 may also effectively describe a more diverse system. Applied formula for the free energy per unit area, Eq. 2, assuming ep ? ? and heci ? ?, gives the famous Casimir result for the energy of attraction between ideal mirrors. This energy is a result of the change in the zero-point energy of an empty quantum vacuum. From the other hand, it is well known that real cell membranes are under the permanent influence of electrostatic interactions. Assuming a natural transmembrane electric potential V = 100 MV, a membrane thickness dm = 10-8, an effective membrane dielectric constant em = 2 and a vacuum permittivity e0 = 10-11 [F/m], it is easy to estimate the area density of the energy of an electric field, 0.5e0em (V2/dm). It is equal to 10-5 J/m2 . This result is three orders less than the energy of the considered Casimir effect.

np

n1

n2

ξ -L-x /2

−δ L/2-x /2

−δ L/2+x/2

0

δ L/2-x /2

δ L/2 +x/2

L +x/2

Fig. 6 The area densities n1 and n2 of lipids belonging to a given leaflet as a function of the distance, n, from the membrane midplane. The range occupied by the longest and shortest lipids is shown above. np total number of lipids belonging to a given leaflet per unit surface area, L average lipid length, dL width of the lipid length range, x = d - 2L, where d is the total membrane thickness

It is necessary to underline that the discussed formula is correct only within certain assumed constraints. One of them, zero temperature approximation (Landau and Lifshitz 1960) may be justified for considered conditions (kBT hc/dc). The second assumption, i.e., the same dielectric constant at all frequencies, is formally valid for ideal dielectrics or in the case of large separations. However, for values of constant permittivity not so far from unity (vacuum value), possible error for small distances is disregarded. Moreover, the bulk term in energy is disregarded, which may be dominant for large separations. As the ratio of energies of attraction of water slabs and lipid slabs (both estimated using Eq. 2) is as small as 1:10, for the sake of simplicity, the influence of water outside the membrane was not considered. Anisotropy in the dielectric constant is also beyond the scope of this article, and the model for the dielectric constant in the region of interdigitation is very simple, based on a linear superposition of dielectric constants in terms of the effective densities of tails. We realize that real membranes, especially cell membranes, are obviously complex, heterogeneous, nonideal dielectrics with complicated frequency responses and real conductivity. They are certainly not perfectly plain and smooth plates. Despite the above simplifications, the general predictions of our model, i.e., the magnitude of the free energy and the existence of two energy minima, seem to be still reasonable and wait for more detailed further investigation and confirmation. A lateral Casimir effect for corrugated planes or nonretarded local pairwise van der Waals forces might provide a better description of the physics there and provide a more accurate description. For example, the last one can replace an inverse-cube law, describing variation in energy with distance, by an inverse-square law. Using this method, a simple estimation of energy of so-called hydrophobic bonding, at a Hamaker function ´˚ equal to 7 9 10-21 J and a distance of 50 A (Parsegian and Ninham 1970), gives the value of the density of free energy close to 10-5 J/m2. This quantity is three orders less than the energy of lipid–lipid interactions and two orders less than the energy of water–water interactions estimated in our model. We think that future numerical brute-force simulations might make an important contribution toward a better understanding of the mentioned discrepancy. In light of these findings, it is evident that the Casimir effect may play an important role in many biological phenomena and may be a universal force that organizes the tensegrity structure of biological systems. Some scientists might expect spectacular ‘‘levitation’’ forces between the leaflets, but it appears that there is instead a ‘‘quantum trap’’ preventing membrane leaflet interdigitation and collapse as well as maintaining a significant gap between leaflets, which leaves molecules a plane with freedom of

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P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

movement. Even without taking into account interlayer lipid collisions, hydrophobic interactions and the stabilizing role of proteins, Casimir forces may prevent lipid chain mishmash or molecular escape. This hypothesis appears to be fruitful and is worth experimental verification. Acknowledgements We thank Prof. Bogdan Lesyng and Dr. Marek Kalinowski from the University of Warsaw for inspiration and valuable discussion. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

8 > : np

ð6Þ Independently of the sign of x, the central region consists of three layers: 8 < dL=2 j xj=2 n\ dL=2 þ j xj=2 dL=2 þ j xj=2 n dL=2 j xj=2 ð7Þ : dL=2 j xj=2\n dL=2 þ j xj=2 Calculating the sum in Eq. 4, with the help of Eqs. 5, 6 in respective layers, depending on the sign of x, one can obtain: 8 n > np þ 2p 1 þ nx=2 > dL=2 > < x n ¼ np 1 dL > > > : n þ np 1 nþx=2 p dL=2 2

Appendix The Average Value of the Dielectric Constant in the Central Region of a Lipid Bilayer

L x=2 n\ dL=2 þ x=2 dL=2 þ x=2 n dL=2 þ x=2 dL=2 þ x=2\n L þ x=2

dL=2 jxj=2 n\ dL=2 þ j xj=2 dL=2 þ jxj=2 n dL=2 jxj=2

x\0

dL=2 jxj=2\n dL=2 þ j xj=2

ð8Þ Let variable n represent the distance from the membrane midplane; then, the central region may be defined as the membrane layer within the range dL=2 j xj= 2 n dL=2 þ j xj=2. Here, dL represents the width of the distribution of lipid lengths, and x = d - 2L, where d is the total membrane thickness and L is the average lipid length. In general, x may be positive (thickness expansion), negative (thickness compression) or zero (configuration MTCR) and falls within the range x [ dL - 2L. Assuming that the dielectric constant ec in the central region varies locally with n, ec depends on the number of lipids, n, that are passing through the unit area of plane n = constant. This dependence may be described as n ec ¼ 1 þ ðep 1Þ ð3Þ np where ep is the dielectric constant of the peripheral region of the membrane fully occupied only by the lipids belonging to a given leaflet and np is the total number of lipids belonging to a given leaflet per unit surface area. The area density n of lipids at distance n can be described by the equation n ¼ n1 þ n2

ð4Þ

where n1 and n2 are area densities of lipids belonging to a given leaflet. For a uniform distribution of lipid lengths, the constituent densities n1 and n2 (Fig. 6) can be described as follows: 8 > < np L x=2 n\ dL=2 x=2 np nþx=2 n1 ¼ 2 1 dL=2 dL=2 x=2 n dL=2 x=2 > : 0 dL=2 x=2\n L þ x=2 ð5Þ

123

n ¼ np

dL=2 n dL=2

8 np nþx=2 > 1 > dL=2 2 > < x n ¼ np 1 dL > > > : np 1 þ nx=2 2 dL=2

x¼0

ð9Þ

dL=2 x=2 n\ dL=2 þ x=2 dL=2 þ x=2 n dL=2 x=2

x[0

dL=2 x=2\n dL=2 þ x=2

ð10Þ The space average of n in the range dL=2 j xj=2 n dL=2 þ j xj=2 is calculated as dL=2þ R jxj=2

hni ¼

ndn

dL=2j xj=2

dL þ j xj

and according to Eqs. 8, 9 and 10 equals 8 j xj=dL > x\0 < np 1 þ 1þjxj=dL hni ¼ np x¼0 > :n 1 x[0 p 1þx=dL

ð11Þ

ð12Þ

Then, the space average of the dielectric constant ec according to Eqs. 3 and 12 can be described as 8 jxj=dL > x\0 < 1 þ ep 1 1 þ 1þjxj=dL hec i ¼ ep ð13Þ x¼0 > : 1 þ e 1 1 x[0 p 1þx=dL

References Andersen OS, Koeppe RE (2007) Bilayer thickness and membrane protein function: an energetic perspective. Annu Rev Biophys Biomol Struct 36:107–130

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect Blanchard A, Rauch C (2012) Membrane lipid asymmetry and permeability to drugs: a matter of size. In: Deavaux PF, Herrmann A (eds) Transmembrane dynamics of lipids. Wiley, Hoboken, pp 251–274 Blaurock AE (1982) Evidence of bilayer structure and of membrane interactions from X-ray diffraction analysis. Biochem Biophys Acta 650:167–207 Bradonjic´ K, Swain JD, Widom A, Srivastava YN (2009) The Casimir effect in biology: the role of molecular quantum electrodynamics in linear aggregations of red blood cells. J Phys Conf Ser 161:12035 Casimir HBG (1948) On the attraction between two perfectly conducting plates. Proc R Netherlands Acad Arts Sci 51:793–795 Casimir HBG, Polder D (1948) The influence of retardation on the London-van der Waals forces. Phys Rev 73:360–372 El Hasnaoui K, Madmoune Y, Kaidi H, Chahid M, Benhamou M (2010) Casimir force in confined biomembranes. Afr J Math Phys 8:101–114 Evans EA, Hochmuth RM (1978) Mechanical properties of membranes. Curr Topics Membr Transport 10:1–65 Huang W, Levitt DG (1977) Theoretical calculation of the dielectric constant of a bilayer membrane. Biophys J 17:111–128 Kranenburg M (2004) Phase transitions of lipid bilayers. A mesoscopic approach. Thesis, University of Amsterdam

389 Kranenburg M, Vlaar M, Smit B (2004) Simulating induced interdigitation in membranes. Biophys J 87:1596–1605 Kuchel PW, Ralston GB (1998) Schaum’s outline of theory and problems of biochemistry. McGraw-Hill, New York Landau LD, Lifshitz EM (1960) Electrodynamics of continuous media. Pergamon Press, Oxford Lifshitz EM (1956) The theory of molecular attractive forces between solids. Sov. Phys 2:73–83 Mavromoustakos T, Chatzigeorgiu P, Koukoulista C, Durdagi S (2011) Partial interdigitation of lipid bilayers. Int J Quantum Chem 111:1172–1183 Otter WK, Shkulipa SA (2007) Intermonolayer friction and surface shear viscosity of lipid. Biophys J 93:423–433 Parsegian VA, Ninham BW (1970) Temperature-dependent van der Waals forces. Biophys J 10:664–674 Shkulipa SA, Otter WK, Briels WJ (2005) Surface viscosity, diffusion, and intermonolayer friction: simulating sheared amphiphilic bilayers. Biophys J 89:823–829 Srinivasan KR, Kay RL, Nagle JF (1974) The pressure dependence of the lipid bilayer phase transition. Biochemistry 13:3494–3496 Srivastava Y, Widom A, Friedman MH (1985) Microchips as precision quantum electrodynamic probes. Phys Rev Lett 55:2246–2248 Waugh RE (1982) Surface viscosity measurements from large bilayer vesicle tether formation experiments. Biophys J 38:29–37

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Colloidal aggregation in critical crosslinked polymer blends M. Benhamou∗ , F. Elhajjaji, K. Elhasnaoui, and A. Derouiche Polymer Physics and Critical Phenomena Laboratory Sciences Faculty Ben M’sik, P.O. Box 7955, Casablanca, Morocco

We consider a low-density assembly of small colloidal particles that are immersed in a critical crosslinked polymer blend made of two polymers of different chemical nature. We assume that, near the spinodal temperature at which the mixture exhibits a microphase separation, the particles preferentially adsorb one of the two polymers. The consequence is that, the beads aggregate in the non-preferred phase. The aim is an extensive study of the thermodynamical properties of the colloidal aggregation, under a change of a suitable parameter such as temperature. This phase transition drives the colloids from a dispersed phase (gas) to a dense one (liquid). To this end, we elaborate a new field theory that allows the determination of the effective free energy of the mobile colloids, as a functional of their density. From this effective free energy, we draw the phase diagram in particle volume fractiontemperature plane, and investigate all critical properties of the colloidal flocculation, from a static and dynamic point of view. Finally, the discussion is extended to impregnated crosslinked polymer blends in solution.

I. INTRODUCTION

The aggregation phenomenon within the traditional colloidal solutions are due to the van der Waals attractive forces, which generally originate from the fact that the particles possess a dipolar moment [1]. The same phenomenon may produces within the so-called critical systems with immersed colloidal particles, such as a ﬂuid near the liquid-gas critical

∗ Author

for correspondence; E-mail : [email protected]

1

point, a mixture of simple liquids (or polymers) near the consolute point, liquid 4 He near the λ-transition or liquid-crystals. For these systems, the critical ﬂuctuations of the order parameter generate long-range attractive forces between colloids, termed critical Casimir forces [2]. The word Casimir is attributed to the well-known Casimir eﬀect [3], according to which two perfectly conducting parallel metals attract each other, due to the vacuum quantum ﬂuctuations. The latter play the role of the critical ﬂuctuations for physical systems exhibiting a critical point. Physics of the colloidal aggregation in critical two-component liquid mixtures is a very exciting and rich problem, and has received a great deal of attention from a theoretical and experimental point of view. Theoretically, the critical Casimir eﬀect has been studied by several methods, such as conformal invariance in dimension 2 [4 − 8], ﬁeld-theoretical Renormalization-Group (RG) in dimension 3 [2, 9 − 15], and Monte Carlo simulation [16, 17]. Recently, the study was successfully extended to polymer blends [18]. Experimentally, the critical Casimir force was measured in a series of experiments [19 − 27]. Among these, we can quote certain experiments [19 − 23] concerning the behavior of silica beads of small diameter, immersed in a binary liquid mixture made of lutidine and water. In fact, the silica beads have tendency to adsorb rather lutidine, when one is in the vicinity of the consolute point of the mixture free from particles. This is the critical adsorption [28 − 41]. The experimental observation shows that, near criticality, the colloids undergo a reversible aggregation in the water-rich side. It was then natural to consider more complex systems such as crosslinked polymer blends (CPBs) conﬁned to two parallel walls [42] or with immersed colloids [43]. CPBs are made of two chemically incompatible polymers that are densely crosslinked in their one-phase region, by γ-irradiation [44], for instance. When the temperature is lowered, below some characteristic temperature depending on the reticulation dose, the CPB phase separates into mesoscopic phases alternatively rich in unlike polymers. This is the microphase separation (MPS). We note that the ﬁrst theory of MPS in CPBs was introduced by de Gennes [45], followed by several extended works [46]. The incorporation of a small amount of particles in a CPB may have some relevance for industry. Indeed, these particles can reinforce the mechanical properties of the considered

2

CPB. But, in most practical realizations, the immersed particles are instable and ﬂocculate under some special physical conditions. It is then useful to study their stability conditions. In fact, the presence of colloids inﬂuences the primitive segregation interactions between unlike connected polymer chains and their chemical potentials, and the internal rigidity of the polymer network. In this paper, we consider a low-density assembly of colloidal particles, immersed in a critical CPB made of two polymers A and B, of diﬀerent chemical nature. It is assumed that, near the spinodal temperature at which the CPB undergoes a MPS, the colloids preferentially adsorb one polymer, saying A. Thus, we are in the presence of a critical adsorption. The consequence is that, the particles aggregate in the non-preferred B-rich phase. For instance, the system may be a hydrogenated polyoleﬁn-deuterated polyoleﬁn mixture incorporating silicon particles, considered in some recent experiment by Wendlandt and coworkers [47]. The induced force responsible for the colloidal aggregation was computed in Ref. [43], at the spinodal point. The main result is that, the induced force decays as r−3 exp (−r/ξ ∗ ), where r is the interparticle distance and ξ ∗ ∼ an1/2 is the microdomains size (mesh size) [45]. Here, a denotes the monomer size and n is the mean number of monomers per strand (section of chains between consecutive crosslinks). The purpose is a study of the thermodynamical properties of the colloidal aggregation, from a static and dynamic point of view. This colloidal aggregation is a phase transition driving the colloids from a dispersed phase (gas) to a dense one (liquid). To this end, we ﬁrst elaborate a ﬁeld theory governing the ﬂocculation phenomenon. Combining this ﬁeld theory with the usual cumulant method [48, 49], we derive the expression of the eﬀective free energy of the moving colloids. The latter is naturally a functional of the volume fraction of particles. We assume that the colloidal system is disordered. The problem is then the speciﬁcation of the disorder nature. To be more realist, we choose the most diﬃcult one, that is a quenched disorder [50]. This means that one must average the logarithm of the partition function over the disorder. The opposite case is an annealed disorder, where the partition function itself is averaged over the external disorder variables. This disorder is relatively simple, and assumes that the colloids and polymers are at equilibrium. In fact, such an assumption is not reliable, since, in principle, the diﬀusion time of colloids is diﬀerent

3

from that of polymers. To do explicit calculations, we suppose that the disorder is rather quenched and follows a Gaussian distribution. After averaging over disorder, we ﬁnd that the eﬀective free energy of colloids is similar to that relatively to the traditional Flory-Huggins theory usually encountered in Polymer Physics [51, 52]. The obtained eﬀective free energy of the dispersed colloids allows the determination of the phase diagram shape in the particle volume fraction-temperature plane and all critical properties of the aggregation transition. Finally, the discussion is extended to troubled CPBs in solution. The remaining of presentation proceeds as follows. We derive, in Sec. II, the expression of the eﬀective free energy of the immersed colloids. The phase diagram is investigated in Sec. II. We discuss, in Sec. III, the static and dynamic scattering properties. Some concluding remarks are drawn in the last section.

II. EFFECTIVE FREE ENERGY

Consider M spherical colloidal particles immersed in a CPB. For simplicity, these particles are assumed to be small and of the same radius. Thus, we are concerned with a monodisperse system. The fundamental assumption is that, near the bulk spinodal temperature, the particles preferentially attract the polymer A. This means that one is in the presence of a critical adsorption, where the colloids are surrounded by the preferred polymer A. The consequence is that, the particles located in the non-preferred B-rich microdomains aggregate. In fact, this aggregation is due to a long-ranged attractive force experienced by the colloids, which was computed in Ref. [43]. To investigate the aggregation phenomenon, we introduce an order parameter or composition ﬂuctuation, ϕ, which is simply the diﬀerence of compositions, ΦA and ΦB , of the two polymers, that is ϕ = ΦA − ΦB . The order parameter ϕ (r) is then a scalar ﬁeld depending on the position vector, r, of the representative point of the mixture. To investigate the physical properties of the troubled CPB, we must precise the form of the Hamiltonian, which is a generalization of that introduced by de Gennes for studying MPS in pure CPBs [45]. In the absence of colloids, the de Gennes’ (bare) Hamiltonian writes [45]

4

Hb [ϕ] = a−d kB T

d

d r

t 2 a2 C ϕ + (∇ϕ)2 + P 2 2 2 2

.

(1)

The latter results from a competition between the usual phase separation and the elastic properties of the polymer network. In the above deﬁnition, the integration is performed over the d-dimensional inﬁnite euclidean space Rd . In the above deﬁnition, T stands for the absolute temperature, kB for the Boltzmann’s constant, and a for the monomer size. The notation t=

1 (χc − χ) < 0 2

(1a)

denotes the distance from the critical temperature, where χ is the standard Flory interaction parameter, which is inversely proportional to the absolute temperature T , and χc = 2/N is its critical value if the system is uncrosslinked. Here, N denotes the common polymerization degree of the polymer chains before they are crosslinked. Therefore, we are concerned with a monodisperse system. The ﬁrst two terms in relation (1) constitute the expansion of the traditional Flory-Huggins free energy [51, 52] around the critical composition. The gradient term accounts for the interfacial energy between A-rich and B-rich phases, and the third one describes the gel elasticity. The elastic contribution in this relation was ﬁrst introduced by de Gennes by analogy with the polarization of a dielectric medium. The internal rigidity constant, C, was related to the average number n of monomers per strand by [45] C ∼ a−2 n−2 .

(1b)

The quantity 1/n was interpreted as the reticulation dose [44]. In equality (1), the vector P denotes the average displacement between the centres of masses of A and B strands; it is the analog of the polarization in the dielectric problem. The monomers A and B play the role of the positive and negative charges of the dielectric medium, respectively. The charge ﬂuctuations correspond to the ﬂuctuations of composition. The quantities P and ϕ are not independent each other, but related by the Maxwell’s relation: divP = −ϕ (r). Such a relation enables us to transform the gel elasticity contribution as follows C C d 2 d rP → − dd rϕ∆−1 r ϕ , 2 V 2 V where ∆−1 r is simply the inverse Laplacian operator, deﬁned by 5

(1c)

∆−1 r ϕ (r)

=−

dd r′ K (r, r′ ) ϕ (r′ ) ,

(1d)

with the Green function K (r, r′ ), such that −∆r K (r, r′ ) = δd (r − r′ ) .

(1e)

It was found [45] that the system phase separates at the spinodal temperature, that is for 1/2 ts = −2 Ca2 ∼D ,

(1f)

where D = n−1 is the reticulation dose. This relation then deﬁnes a critical line in the (C, t)plane along which the CPB undergoes a MPS. For t > ts , the CPB is in the disordered phase, and for t < ts , it is rather in the ordered one. We recall that the microdomains size or mesh −1/4

size is as follows [45]: ξ ∗ = a (Ca2 )

∼ an1/2 .

For a CPB with immersed point-like colloids, the proposed Hamiltonian is M

M

H [ϕ] Hb [ϕ] ϕ2 (ri ) + C0 P 2 (ri ) . = + t0 kB T kB T i=1 i=1

(2)

Here, M denotes the total number of particles dispersed in the medium. The two last terms in the right-hand side describe the contribution of colloids to the Hamiltonian, where the positive quantities t0 and C0 measure the perturbation of the temperature parameter t and rigidity constant C due to the presence of the moving particles. For shapely particles, t0 and C0 express the interactions between the mixture and their surface. A simple dimensional analysis shows that: [t0 ] = L0 and [C0 ] = L−2 , where L is some length. It will be convenient to rewrite the above deformed Hamiltonian on the following form H [ϕ] Hb [ϕ] = + dd rρ (r) O (r) , (3) kB T kB T with the local particle density ρ (r) =

N

δd (r − ri )

(3a)

i=1

and local ﬁeld operator O (r) = t0 ϕ2 (r) + C0 P 2 (r) . 6

(3b)

Now, to determine the free energy of a CPB, F, where the particles positions, (r1 , ..., rM ), are ﬁxed in space, we shall need the expression of the partition function, Z. The latter is deﬁned by the following functional integral d 1 Z (r1 , ..., rM ) = Dϕe−Ab [ϕ]+ d rρ(r)O(r) , M!

(4)

with the bare action Ab [ϕ] = Hb [ϕ] /kB T . The above functional integral is performed over all possible conﬁgurations of ϕ-ﬁeld. The quantity M ! accounts for the usual symmetry factor. It will be useful to introduce the bulk expectation mean-value −1

Xb = Zb × DϕX [ϕ] e−Ab [ϕ] ,

(5)

with the partition function of the free CPB Zb = Dϕe−Ab [ϕ] .

(5a)

With these considerations, the partition rewrites as Zb d exp Z (r1 , ..., rM ) = d rρ (r) O (r) . M! b

(6)

We note that the bulk mean-value in the above expression can be computed using the cumulant method usually encountered in Statistical Field Theory [48, 49], based on the approximative formula

X 2 2 e b = eX b +(1/2!)[ X b −X b ]+... .

(7)

Then, we have Zb Z (r1 , ..., rM ) = exp M!

1 d rρ (r) O (r)b + 2 d

d

d r

d r ρ (r) ρ (r ) O (r) O (r )b,c + ... , d ′

′

′

(8) where

O (r) O (r′ )b,c = O (r) O (r′ )b − O (r)b O (r′ )b

(9)

accounts for the connected two-point correlation function constructed with the ﬁeld operator O (r). In relation (8), only one and two-body interactions are taken into account. High order terms describing the three-body interactions and more are then ignored. 7

To compute the desired free energy, we start from the standard formula F = −kB T ln Z and ﬁnd F [ρ] Fb Fint [ρ] = + − S/kB , kB T kB T kB T

(10)

where Fb = −kB T ln Zb represents the free energy of the host mixture, and Fint is the contribution of the eﬀective interactions that can be written as Fint [ρ] 1 d d = − d rρ (r) O (r)b,c − d r dd r′ ρ (r) ρ (r′ ) O (r) O (r′ )b,c + ... . kB T 2

(11)

Then, this formula is a combination of a one-body potential, O (r)b , and a two-body one,

O (r) O (r′ )b,c . We have

O (r)b,c = t0 ϕ2 (r) b + C0 P 2 (r) b ,

(12)

O (r) O (r′ )b,c = t20 ϕ2 (r) ϕ2 (r′ ) b,c + 2t0 C0 ϕ2 (r) P 2 (r′ ) b,c + C02 P 2 (r) P 2 (r′ ) b,c . (13) In equality (10), S accounts for the entropy of particles, and it is essentially the logarithm of the symmetry factor M !, to which the Stirling’s formula will be applied. Now, the aim is the computation of the eﬀective free energy of the trapped colloids. To do calculations, we ﬁrst assume that their positions in space are disordered at random, with a quenched disorder. The latter is more realistic than the annealed one, since the monomers and colloids are not in equilibrium. We denote by ρ0 = M/Ω the number density of particles, where Ω is the total volume occupied by the troubled CPB. According to the known central limit theorem, the random density ﬂuctuations around the mean-value ρ0 can be assumed to be governed by a Gaussian (or bimodal) distribution, that is 1 2 P [ρ (r)] = P0 exp − 2 [ρ (r) − ρ0 ] . 2ρ0

(14)

Here, the diﬀerence ρ (r) − ρ0 is the density ﬂuctuation and P0 is a normalization constant. Then, the ﬁrst and second moments of this distribution are as follows ρ (r) = ρ0 ,

(15)

ρ (r) ρ (r′ ) = ρ20 + ρ0 δd (r − r′ ) .

(16)

8

Therefore, we are concerned with a non-correlated disorder. For a quenched disorder, we have to average not the partition function Z but its logarithm ln Z = −F [ρ] /kB T , where F [ρ] is the free energy of expression (11). Then, we have Fb F [ρ] Fint [ρ] = + − S/kB , kB T kB T kB T

(17)

with 1 d d rρ (r) O (r)b,c − d r dd rρ (r) ρ (r′ ) O (r) O (r′ )b,c 2

ρ0 Ω ρ20 2 d d d = −ρ0 d r O (r)b,c − d r O (r) b,c − d r dd r′ O (r) O (r′ )b,c . (18) 2 2

Fint [ρ] ≃− kB T

d

To determine the architecture of the phase diagram of the colloidal system, we need the eﬀective free energy (per site), ∆F , that can be obtained from vF [ρ]/Ω by ignoring the constant Fb and terms proportional to density ρ0 . Here, v denotes the volume of colloids. In order to precise the form of ∆F , we ﬁrst introduce the dimensionless particle density (volume fraction of colloids) Ψ = ρ0 × v. After symmetrization (Ψ → 1 − Ψ), the entropy (per site) writes [51, 52] vS = −Ψ ln Ψ − (1 − Ψ) ln (1 − Ψ) . kB Ω

(19)

With these considerations, the eﬀective free energy (per site) is given by ∆F = Ψ ln Ψ + (1 − Ψ) ln (1 − Ψ) + uΨ (1 − Ψ) , kB T with the colloid interaction parameter 1 u= dd r O (0) O (r)b,c > 0 2v

(20)

(21)

that essentially presents as the spatial integral of the two-point correlation function of the ﬁeld operator O. The derived eﬀective free energy calls the following remarks. Firstly, its expression clearly shows its analogy with that relatively to the standard FloryHuggins theory (FH) [51, 52] describing the free polymer blends; u then plays the role of the Flory interaction parameter. Therefore, the present theory of free energy (20) is a lattice model, where a given site is occupied by one particle, with a probability Ψ. Thus, 1 − Ψ is 9

the probability to have an empty site. In fact, if Ψ is the composition of the dense phase (liquid phase), that of the dispersed one is 1 − Ψ. Secondly, formula (20) indicates that the critical ﬂuctuations of composition give arise to an eﬀective interaction energy between colloids that is responsible for their aggregation. Thirdly, we note that the dimensionless parameter u is temperature-dependent and always positive definite. This means that, above some critical value, u∗ , deﬁned later, one assists to a coexistence between dilute (gas) and dense (liquid) phases. Finally, we emphasize that the interaction parameter u is a linear combination of three bulk correlation functions (integrated over space coordinates), that is u=

1 2 t0 X00 + 2t0 C0 X01 + C02 X11 , 2v

with the coeﬃcients

X00 = dd r ϕ2 (0) ϕ2 (r) b,c , X11 =

X01 = X10 =

dd r ϕ2 (0) P 2 (r) b,c ,

dd r P 2 (0) P 2 (r) b,c .

Replacing P 2 by −ϕ∆−1 r ϕ and using the Wick theorem [48, 49] yields X00 = 2 dd r [ ϕ (0) ϕ (r)b ]2 , X01 = X10 = 2

X11 = 2

dd r ϕ (0) ϕ (r)b ϕ (0) ∆−1 r ϕ (r) b ,

−1 dd r ϕ (0) ϕ (r)b ∆−1 r ϕ (0) ∆r ϕ (r) b .

(22)

(22a)

(22b)

(22c)

(22d)

(22e)

Explicitly, we have X00 (T, C) = 2sd a

2d

∞

q d−1 dq , (a2 q 2 + t + C/q 2 )2

0

2d

X01 (T, C) = X10 (T, C) = 2sd a

0

2d

X11 (T, C) = 2sd a

∞ 0

∞

q d−3 dq , (a2 q 2 + t + C/q 2 )2

q d−5 dq . (a2 q 2 + t + C/q 2 )2

10

(22f)

(22g)

(22h)

Here, sd = 2π d/2 Γ (d/2) denotes the area of the unit sphere embedded in the d-dimensional euclidean space Rd , where Γ (z) is the Euler gamma function [53]. In mean-ﬁeld theory, the critical behavior emerges in the d → 4 limit, and we ﬁnd, near the spinodal point (t → t+ s ), the following expressions for the coeﬃcients Xij ’s X00 = 2π 3

a7 ξ ∗−3

X01 = X10 = π 3

X11 = π 3

a7 ξ ∗−1 (t − ts )

a7 ξ ∗ (t − ts )

3/2

t → t+ s ,

,

(t − ts )3/2

3/2

,

,

t → t+ s ,

t → t+ s .

Therefore, the colloid interaction parameter scales as

ts − t −3/2 u = u0 , t → t+ s , ts

(23a)

(23b)

(23c)

(24)

with the positive amplitude u0 = We have used the notation

a4 π3 2 2 2t + 2t C + C . 0 0 0 0 25/2 v 0 = C0 ξ ∗ 2 . C

(24a)

(24b)

The central result (24) call two remarks.

Firstly, as it should be, the colloid interaction parameter becomes more and more important as the spinodal temperature (at which the CPB undergoes a MPS) is reached (from above), and diverges at this same temperature. Secondly, all dependance on the perturbation parameters (t0 , C0 ), due to the presence of ﬁllers, is entirely contained in the positive amplitude u0 deﬁned in Eq. (24a). This dependence is quadratic in these parameters. In addition, this amplitude increases with increasing variables t0 and C0 . In Fig. 1, we draw the reduced colloid interaction parameter, u/u0 , upon the reduced (positive) temperature shift ∆t = (ts − t) /ts . Remark that this curve is universal, independently on the physical and geometrical properties of colloids and their interactions strength with the host CPB. 11

The constructed eﬀective free energy allows the determination of the phase diagram shape in the composition-temperature plane. This is precisely the aim of the following paragraph.

III. PHASE DIAGRAM

Now, the equilibrium volume fraction of particles solves the equation ∂∆F/∂Ψ = 0. Explicitly, we have

Ψ 1 ln . u= 2Ψ − 1 1−Ψ

(25)

This equation deﬁnes the coexistence curve in the (Ψ, u)-plane, along which the dispersed and dense phases coexist. On the other hand, the spinodal curve, along which the isotherm compressibility diverges, can be obtained solving ∂ 2 ∆F/∂Ψ2 = 0. Then, the spinodal curve equation is u=

1 . 2Ψ (1 − Ψ)

(26)

Finally, the location of the critical point, K, can be determined solving the two simultaneous equations ∂ 2 ∆F/∂Ψ2 = 0 and ∂ 3 ∆F/∂Ψ3 = 0. We ﬁnd that its coordinates (Ψ∗ , u∗ ) are such as Ψ∗ =

1 , 2

u∗ = 2 .

(27)

The critical value u∗ = 2 corresponds to a well-deﬁned critical temperature, T ∗ , which satisﬁes the implicit equation 1 2 t0 X00 (T ∗ , C) + 2t0 C0 X01 (T ∗ , C) + C02 X11 (T ∗ , C) = 2 . 2v

(28)

Combining the dominant parts of the coeﬃcients X00 , X01 and X11 we computed above with equality (28) yields the critical temperature, T ∗ , at which the colloids ﬂocculate u 2/3 0 ∗ t = ts 1 − . 2

(29)

We have ts < t∗ < 0, provided that u0 < 2. Notice that t∗ = (χc − χ∗ ) /2, with χc = 2/N and χ∗ ∼ T ∗ −1 . Here, the amplitude u0 is deﬁned in equality (24). The above relationship 12

then deﬁnes a critical surface in the (t0 , C0 , t∗ )-space on which the colloidal system presents a phase transition. Therefore, at T = T ∗ and Ψ∗ = 0.5, the colloidal system exhibits a second order phase transition from a dispersed phase to a dense one, at the critical temperature T ∗ , which is very close to the spinodal temperature Ts , since the perturbation parameters t0 and C0 are assumed to be small enough. Come back to Eq. (25) deﬁning the coexistence curve and try to rewrite it in the vicinity of the critical point. Close to this point, we obtain u = u∗ +

8 (Ψ − Ψ∗ )2 , 3

Ψ → Ψ∗ ,

(30)

or equivalently, 1 Ψ=Ψ ± 2 ∗

3 (u − u∗ )1/2 , 2

u → u∗ .

(30a)

Notice that u − u∗ ∼ T ∗ − T . The phase diagram of the colloidal system is drawn in Fig. 2, where the solid and dashed lines represent the binodal and spinodal, respectively. These lines meet at the critical point K = (Ψ∗ , u∗ ). The region between the two curves is a metastable domain. We note that the phase diagram architecture we investigated above has been determined using the FH approach that is a mean-ﬁeld theory (MFT), which underestimates the ﬂuctuations of the composition that are strong near criticality. To go beyond this theory, and in order to get a correct critical phase behavior, use will be made, in the next section, of the scaling approach that is a direct consequence of the RG theory [48, 49].

IV. SCATTERING PROPERTIES A. Static properties

Now, consider another main physical quantity that is the structure factor, S (q), which informs us on the critical properties of the colloidal system. This can be measured in lightscattering and neutron-scattering experiments. The structure factor is nothing else but the Fourier transform of the two-point correlation function G (r) = Ψ (0) Ψ (r)− Ψ (0) Ψ (r), which measures the ﬂuctuations of the particle composition Ψ. We write 13

S (q) =

dd q (2π)d

eiq.r G (r) ,

(31)

where q denotes the wave-vector, whose amplitude is given by |q| = (4π/λ) sin (θ/2), where λ is the wavelength of the incident radiation and θ is the scattering-angle. In particular, the zero-scattering-angle limit (q → 0) of the structure factor deﬁnes the thermal compressibility of the colloidal system, κT , that is S (0) = kB T κT .

(32)

This compressibility diverges as the critical temperature T ∗ is approached, and we have κT ∼ |T − T ∗ |−γt ,

(32a)

where γt is a critical exponent whose value depends only the space dimensionality d and not on the chemical details of the system. Other critical exponents can be deﬁned. For example, the order parameter ψ = Ψ − Ψ∗ behaves, around T ∗ , as 0 , T ≥ T∗ , ψ∼ (T ∗ − T )βt , T ≤ T ∗ .

(33)

with a second critical exponent βt , in the limit ∆µ = µ − µc → 0, where µ is the chemical potential and µc is its critical value [48, 49]. In term of the colloid interaction parameter u and composition Ψ, we have u − u∗ ∼ (Ψ − Ψ∗ )1/βt

(33a)

that is the equation of the coexistence curve close to the critical point. At T = T ∗ , the order parameter behaves as ψ ∼ |∆µ|1/δt , with a critical exponent δt . On the other hand, the size of domains rich in colloids or the thermal correlation length measuring the correlations extent, ξt , diverges at criticality according to ξt ∼ σ |T − T ∗ |−νt ,

(34)

with the critical exponent νt and an atomic-scale σ that may be the particle diameter. Also, the speciﬁc heat diverges at the critical point with some critical exponent denoted 14

αt . Finally, we deﬁne the last critical exponent, ηt , which characterizes the small-distance behavior of the correlation function, that is G (r) ∼ r2−d−ηt ,

r 0, where b0 is the covolume and a0 = −2π r0 r2 U (r), with the

primitive pair-potential U (r). Here, r0 is the unique zero of U (r), that is U (r0 ) = 0. In this case, the total colloid interaction parameter is then u + up . The presence of the primitive interactions naturally induces a change of the phase behavior. In particular, the location of the critical ﬂocculation temperature is shifted to its higher values. In other word, the critical ﬂuctuations of the host CPB have tendency to accentuate the colloidal aggregation. All conclusions concerning the scattering properties remain the same. Finally, this work must be regarded as a natural extension of a published work dealt

with the colloidal aggregation in uncrosslinked polymer blends with immersed colloids [59].

ACKNOWLEDGMENTS

We are much indebted to Professors Daoud for stimulating correspondences. One of us (M.B) would like to thank the Laboratoire L´eon Brillouin (Saclay) for their kinds of hospitality during his regular visits.

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18

[6] J.L. Cardy, Nucl. Phys. B 275, 200 (1986). [7] T.W. Burkhardt and T. Xue, Phys. Rev. Lett. 66, 895 (1991); Nucl. Phys. B 345, 653 (1991). [8] T.W. Burkhardt and E. Eisenriegler, Nucl. Phys. B 424 [FS], 487 (1994). [9] M. Krech and S. Dietrich, Phys. Rev. A 46, 1886 (1992). [10] M. Krech, Phys. Rev. E 56, 1642 (1997). [11] K. Symanzik, Nucl. Phys. B 190 [FS], 1 (1981). [12] M. Krech and S. Dietrich, Phys. Rev. Lett. 66, 345 (1991); 67, 1055 (1991); Phys. Rev. A 46, 1922 (1992). [13] E. Eisenriegler and M. Stapper, Phys. Rev. B 50, 10009 (1994). [14] T.W. Burkhardt and E. Eisenriegler, Phys. Rev. Lett. 74, 3189 (1995). [15] E. Eisenriegler and U. Ritschel, Phys. Rev. B 51, 13717 (1995). [16] K.K. Mon, Phys. Rev. Lett. 54, 2671 (1985) ; M. Krech and D.P. Landau, Phys. Rev. E 53, 4414 (1996). [17] R.R. Netz, Phys. Rev. Lett. 76, 3646 (1996). [18] M. Benhamou, Casimir Effect in Polymer Physics, Recent Res. Devel. Physics 8, 93 (2009). [19] D. Beysens and D. Est`eve, Phys. Rev. Lett. 54, 2123 (1985). [20] V. Gurfein, D. Beysens, and F. Perrot, Phys. Rev. A 40, 2543 (1989). [21] M.L. Broide, Y. Garrabos, and D. Beysens, Phys. Rev. E 47, 3768 (1993). [22] T. Narayanan, A. Kumar, E.S.R. Gopal, D. Beysens, P. Guenoun, and G. Zalczer, Phys. Rev. E 48, 1989 (1993). [23] D. Beysens, J.-M. Petit, T. Narayanan, A. Kumar, and M.L. Broide, J. Phys. Chem. 98, 382 (1994). [24] P.D. Gallagher, M.L. Kurnaz, and J.V. Maher, Phys. Rev. A 46, 7750 (1992).

19

[25] M.L. Kurnaz and J.V. Maher, Phys. Rev. E 51, 5916 (1995). [26] H. Gr¨ ull and D. Woermann, J. Phys. Chem. 101, 814 (1997). [27] A list of recent experimental works can be found in: G.A. Williams, Physica C 404, 415 (2004). [28] M.E. Fisher and P.-G. de Gennes, C. R. Acad. Sci. (Paris) S´erie B 287, 207 (1978) ; also, P.-G. de Gennes, C. R. Acad. Sci. (Paris) II 292, 701 (1981). [29] H.W. Diehl, in: Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, Vol. 10, Academic Press, London, 1986. [30] S. Dietrich, in: Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, Vol. 12, Academic Press, London, 1988. [31] D. Beysens and S. Leibler, J. Phys. Lett. (Paris) 43, L-133 (1982). [32] C. Franck and S.E. Schnatterly, Phys. Rev. Lett. 48, 763 (1982). [33] M. Schlossman, X.-L. Wu, and C. Franck, Phys. Rev. B 31, 1478 (1985). [34] J.A. Dixon, M. Schlossman, X.-L. Wu, and C. Franck, Phys. Rev. B 31, 1509 (1985). [35] An extensive list of recent experiments using some new techniques can be found in : A. Hanke, M. Krech, F. Schlessener, and S. Dietrich, Phys. Rev. E 60, 5163 (1999). [36] S. Bl¨ umel and G.H. Findenegg, Phys. Rev. Lett. 54, 447 (1985). [37] E. Br´ezin and S. Leibler, Phys. Rev. B 27, 594 (1983); S. Leibler, PhD Thesis, Paris University XI, 1984. [38] J. Rudnik and D. Jasnow, Phys. Rev. Lett. 48, 1595 (1982). [39] S. Leibler and L. Peliti, J. Phys. C 15, L-403 (1982). [40] L. Peliti and S. Leibler, J. Phys. C 16, L-2635 (1983). [41] Critical adsorption on curved objects, such as single spherical and rod-like colloidal particles, has been investigated in: A. Hanke and S. Dietrich, Phys. Rev. E 59, 5081 (1999); A. Hanke,

20

Phys. Rev. Lett. 48, 2180 (2000). [42] M. Benhamou, A. Agouzouk, H. Kaidi, M. Boughou, S. El Fassi, and A. Derouiche, Electronic Journal of Theoretical Physics 5, 141, (2008). [43] S. El Fassi, M. Benhamou, M. Boughou, H. Kaidi, M. El Yaznasni, and H. Ridouane, Acta Physica Polinica 118, 333 (2010). [44] R.M. Briber and B.J. Bauer, Macromolecules 21, 3296 (1988). [45] P.-G. de Gennes, J. Phys. Lett. (Paris) 40, 69 (1979). [46] An exhaustive list of works dealt with MPS in CPBs can be found in: M. Benhamou, A. Derouiche, A. Bettachy, and F. Elhajjaji, Eur. Phys. J. E 34, 3 (2011). [47] M. Wendlandt, T. Kerle, M. Heuberger, and J. Klein, J. Polymer Science: Part B 38, 831 (2000). [48] C. Itzykson and J.-M. Drouffe, Statistical Field Theory: 1 and 2, Cambridge University Press, 1989. [49] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. [50] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge, University Press, 1996. [51] P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, 1953. [52] P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, New York, 1979. [53] I.S. Gradstein and I.M. Ryzhik, Tables of Integrals, Series and Products, Academic, New York, 1980. [54] J.-F. Joanny, L. Leibler, and R. Ball, J. Chem. Phys. 81, 4640 (1984). [55] D. Broseta, L. Leibler, and J.-F. Joanny, Macromolecules 20, 1935 (1987). [56] J. des Cloizeaux and G. Jannink, Polymers in Solution: Their Modelling and Structure, Oxford

21

University Press, 1999. [57] L. Sch¨afer, U. Lehr, and C. Kappeler, J. Phys. I (Paris) 1, 211 (1991); A.I. Mudrov and K.B. Varnashev, J. Phys. A: Math. Gen. 34, L-347 (2001). [58] R. Balian, From Microphysics to Macrophysics - Methods and Applications of Statistical Physics, Springer Verlag, 2006. [59] M. Benhamou, H. Ridouane, A. Derouiche, and M. Rahmoune, J. Chem. Phys. 122, 244913 (2005).

22

FIGURE CAPTIONS

Fig. 1 : Variation of the reduced colloid interaction parameter, u/u0 , upon the temperature shift ∆t = (ts − t) /ts .

Fig. 2 : Phase diagram shape, where the solid line is the binodal and the dashed one is the spinodal. The two curves meet at the ”liquid-gas” critical point, K, of coordinates (Ψc = 0.5, u∗ = 2).

23

u/u0 20

15

10

5

0 0,0

0,2

0,4

0,6

0,8

1,0

∆t Figure 1

u 7 6 5 .

4 3

K ..

2 0,00

0,25

0,50

0,75

Figure 2

1,00

Ψ

Casimir force in confined biomembranes K. El Hasnaoui, Y. Madmoune, H. Kaidi, M. Chahid, and M. Benhamou Laboratoire de Physique des Polym`eres et Ph´enom`enes Critiques Facult´e des Sciences Ben M’sik, P.O. 7955, Casablanca, Morocco [email protected]

abstract We reexamine the computation of the Casimir force between two parallel interacting plates delimitating a liquid with an immersed biomembrane. We denote by D their separation, which is assumed to be much smaller than the bulk roughness, in order to ensure the membrane confinement. This repulsive force originates from the thermal undulations of the membrane. To this end, we first introduce a field theory, where the field is noting else but the height-function. The field model depends on two parameters, namely the membrane bending rigidity constant, κ, and some elastic constant, µ ∼ D−4 . We first compute the static Casimir force (per unit area), Π, and find that the latter decays with separation D as : Π ∼ D−3 , with a known amplitude scaling as κ−1 . Therefore, the force has significant values only for those biomembranes of small enough κ. Second, we consider a biomembrane (at temperature T ) that is initially in a flat state away from thermal equilibrium, and we are interested in how the dynamic force, Π (t), grows in time. To do calculations, use is made of a non-dissipative Langevin equation (with noise) that is solved by the time height-field. We first show that the membrane roughness, L⊥ (t), increases with time as : L⊥ (t) ∼ t1/4 (t < τ ), with the final time τ ∼ D4 (required time over which the final equilibrium state is reached). Also, we find that the force increases in time according to : Π (t) ∼ t1/2 (t < τ ). The discussion is extended to the real situation where the biomembrane is subject to hydrodynamic interactions caused by the surrounding liquid. In this case, we show that : L⊥ (t) ∼ t1/3 (t < τh ) and Πh (t) ∼ t2/3 (t < τh ), with the new final time τh ∼ D3 . Consequently, the hydrodynamic interactions lead to substantial changes of the dynamic properties of the confined membrane, because both roughness and induced force grow more rapidly. Finally, the study may be extended, in a straightforward way, to bilayer surfactants confined to the same geometry. Key words: Biomembranes - Confinement - Casimir force - Dynamics.

I. INTRODUCTION

The cell membranes are of great importance to life, because they separate the cell from the surrounding environment and act as a selective barrier for the import and export of materials. More details concerning the structural organization and basic functions of biomembranes can be found in Refs. [1 − 7]. It is well-recognized by the scientiﬁc community that the cell membranes essentially present as a phospholipid bilayer combined with a variety of proteins and cholesterol (mosaic fluid model). In particular, the function of the cholesterol molecules is to ensure the bilayer ﬂuidity. A phospholipid is an amphiphile

0

c a GNPHE publication 2010, [email protected] ⃝

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K. El Hasnaoui et al.

African Journal Of Mathematical Physics Volume 8(2010)101-114

molecule possessing a hydrophilic polar head attached to two hydrophobic (fatty acyl) chains. The phospholipids move freely on the membrane surface. On the other hand, the thickness of a bilayer membrane is of the order of 50 Angstroms. These two properties allow to consider it as a two-dimensional ﬂuid membrane. The ﬂuid membranes, self-assembled from surfactant solutions, may have a variety of shapes and topologies [8], which have been explained in terms of bending energy [9, 10]. In real situations, the biomembranes are not trapped in liquids of inﬁnite extent, but they rather conﬁned to geometrical boundaries, such as white and red globules or liposomes (as drugs transport agents [11 − 14]) in blood vessels. For simplicity, we consider the situation where the biomembrane is conﬁned in a liquid domain that is ﬁnite in one spatial direction. We denote by D its size in this direction. For a tube, D being the diameter, and for a liquid domain delimitated by two parallel plates, this size is simply the separation between walls. Naturally, the length D must be compared to the bulk roughness, L0⊥ , which is the typical size of humps caused by the thermal ﬂuctuations of the membrane. The latter depends on the nature of lipid molecules forming the bilayer. The biomembrane is conﬁned only when D is much smaller than the bulk roughness L0⊥ . This condition is similar to that usually encountered in conﬁned polymers context [15]. The membrane undulations give rise to repulsive eﬀective interactions between the conﬁning geometrical boundaries. The induced force we term Casimir force is naturally a function of the size D, and must decays as this scale is increased. In this paper, we are interested in how this force decays with distance. To simplify calculations, we assume that the membrane is conﬁned to two parallel plates that are a ﬁnite distance D < L0⊥ apart. The word ”Casimir” is inspired from the traditional Casimir eﬀect. Such an eﬀect, predicted, for the ﬁrst time, by Hendrick Casimir in 1948 [16], is one of the fundamental discoveries in the last century. According to Casimir, the vacuum quantum ﬂuctuations of a conﬁned electromagnetic ﬁeld generate an attractive force between two parallel uncharged conducting plates. The Casimir eﬀect has been conﬁrmed in more recent experiments by Lamoreaux [17] and by Mohideen and Roy [18]. Thereafter, Fisher and de Gennes [19], in a short note, remarked that the Casimir eﬀect also appears in the context of critical systems, such as ﬂuids, simple liquid mixtures, polymer blends, liquid 4 He, or liquid-crystals, conﬁned to restricted geometries or in the presence of immersed colloidal particles. For these systems, the critical ﬂuctuations of the order parameter play the role of the vacuum quantum ﬂuctuations, and then, they lead to long-ranged forces between the conﬁning walls or between immersed colloids [20, 21]. To compute the Casimir force between the conﬁning walls, we ﬁrst elaborate a more general ﬁeld theory that takes into account the primitive interactions experienced by the conﬁned membrane. As we shall see below, in conﬁnement regime, the ﬁeld model depends only on two parameters that are the membrane bending rigidity constant and a coupling constant containing all inﬁrmation concerning the interaction potential exerting by the walls. In addition, the last parameter is a known function of the separation D. With the help of the constructed free energy, we ﬁrst computed the static Casimir force (per unit area), Π. The exact calculations show that the latter decays with separation D according to a power 2 law, that is Π ∼ κ−1 (kB T ) D−3 , with a known amplitude. Here, kB T denotes the thermal energy, and κ the membrane bending rigidity constant. Of course, this force increases with temperature, and has signiﬁcant values only for those biomembranes of small enough κ. The second problem we examined is the computation of the dynamic Casimir force, Π (t). More precisely, we considered a biomembrane at temperature T that is initially in a ﬂat state away from the thermal equilibrium, and we were interested in how the expected force grows in time, before the ﬁnal state is reached. Using a scaling argument, we ﬁrst showed that the membrane roughness, L⊥ (t), grows with time as : L⊥ (t) ∼ t1/4 (t < τ ), with the ﬁnal time τ ∼ D4 . The latter can be interpreted as the required time over which the ﬁnal equilibrium state is reached. Second, using a non-dissipative Langevin equation, we found that the force increases in time according to the power law : Π (t) ∼ t1/2 (t < τ ). Third, the discussion is extended to the real situation where the biomembrane is subject to hydrodynamic interactions caused by the ﬂow of the surrounding liquid. In this case, we show that : L⊥ (t) ∼ t1/3 (t < τh ) and Π (t) ∼ t2/3 (t < τh ), with the new ﬁnal time τh ∼ D3 . Consequently, the hydrodynamic interactions give rise to drastic changes of the dynamic properties of the conﬁned membrane, since both roughness and induced force grow more rapidly. This paper is organized as follows. In Sec. II, we present the ﬁeld model allowing the determination of the Casimir force from a static and dynamic point of view. The Sec. III and Sec. IV are devoted to the computation of the static and dynamic induced forces, respectively. We draw our conclusions in the last section. Some technical details are presented in Appendix.

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African Journal Of Mathematical Physics Volume 8(2010)101-114 II. THEORETICAL FORMULATION

Consider a ﬂuctuating ﬂuid membrane that is conﬁned to two interacting parallel walls 1 and 2. We denote by D their ﬁnite separation. Naturally, the separation D must be compared to the bulk membrane roughness, L0⊥ , when the system is unconﬁned (free membrane). The membrane is conﬁned only when the condition L > L0⊥ , we expect ﬁnite size corrections. We assume that these walls are located at z = −D/2 and z = D/2, respectively. Here, z means the perpendicular distance. For simplicity, we suppose that the two surfaces are physically equivalent. We design by V (z) the interaction potential exerted by one wall on the ﬂuid membrane, in the absence of the other. Usually, V (z) is the sum of a repulsive and an attractive potentials. A typical example is provided by the following potential [22] V (z) = Vh (z) + VvdW (z) ,

(2.0)

Vh (z) = Ah e−z/λh

(2.0)

where

represents the repulsive hydration potential due to the water molecules inserted between hydrophilic lipid heads [22]. The amplitude Ah and the potential-range λh are of the order of : Ah ≃ 0.2 J/m2 and λh ≃ 0.2 − 0.3 nm. In fact, the amplitude Ah is Ah = Ph × λh , with the hydration pressure Ph ≃ 108 −109 Pa. There, VvdW (z) accounts for the van der Waals potential between one wall and biomembrane, which are a distance z apart. Its form is as follows [ ] H 1 2 1 VvdW (z) = − − , (2.0) 2 + 2 12π z 2 (z + δ) (z + 2δ) with the Hamaker constant H ≃ 10−22 − 10−21 J, and δ ≃ 4 nm denotes the membrane thickness. For large distance z, this implies VvdW (z) ∼

W δ2 . z4

(2.0)

Generally, in addition to the distance z, the interaction potential V (z) depends on certain length-scales, (ξ1 , ..., ξn ), which are the interactions ranges. The ﬂuid membrane then experiences the following total potential ( ) ( ) D D D D U (z) = V −z +V +z , − ≤z≤ . (2.0) 2 2 2 2 In the Monge representation, a point on the membrane can be described by the three-dimensional position vector r = (x, y, z = h (x, y)), where h (x, y) ∈ [−D/2, D/2] is the height-ﬁeld. The latter then ﬂuctuates around the mid-plane located at z = 0. The Statistical Mechanics for the description of such a (tensionless) ﬂuid membrane is based on the standard Canham-Helfrich Hamiltonian [9, 23] ∫ [κ ] 2 H [h] = dxdy (∆h) + W (h) , (2.0) 2 with the membrane bending rigidity constant κ. The latter is comparable to the thermal energy kB T , where T is the absolute temperature and kB is the Boltzmann’s constant. There, W (h) is the interaction potential per unit area, that is W (h) =

U (h) , L2

where the potential U (h) is deﬁned in Eq. (2), and L is the lateral linear size of the biomembrane. Let us discuss the pair-potential W (h). 103

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Firstly, Eq. (2) suggests that this total potential is an even function of the perpendicular distance h, that is W (−h) = W (h) .

(2.0)

In particular, we have W (−D/2) = W (D/2). Secondly, when they exist, the zeros h0 ’s of the potential function U (h) are such that ( ) ( ) D D V − h0 = −V + h0 . 2 2

(2.0)

This equality indicates that, if h0 is a zero of the potential function, then, −h0 is a zero too. The number of zeros is then an even number. In addition, the zero h0 ’s are diﬀerent from 0, in all cases. Indeed, the quantity V (D/2) does not vanish, since it represents the potential created by one wall at the middle of the ﬁlm. We emphasize that, when the potential processes no zero, it is either repulsive or attractive. When this same potential vanishes at some points, then, it is either repulsive of attractive between two consecutive zeros. Thirdly, we ﬁrst note that, from relation (2), we deduce that the ﬁrst derivative of the potential function, with respect to distance h, is an odd function, that is W ′ (−h) = −W ′ (h). Applying this relation to the midpoint h = 0 yields : W ′ (0) = 0. Therefore, the potential W exhibits an extremum at h = 0, whatever the form of the function V (h). We ﬁnd that this extremum is a maximum, if V ′′ (D/2) < 0, and a minimum, if V ′′ (D/2) > 0. The potential U presents an horizontal tangent at h = 0, if only if V ′′ (D/2) = 0. On the other hand, the general condition giving the extrema {hm } is dV dV = . (2.0) dh h= D −hm dh h= D +hm 2

2

Since the ﬁrst derivative W ′ (h) is an odd function of distance h, it must have an odd number of extremum points. The point h = hm is a maximum, if d2 V d2 V < − , (2.0) dh2 h= D −hm dh2 h= D +hm 2

2

and a minimum, if d2 V d2 V >− . dh2 h= D −hm dh2 h= D +hm 2

(2.0)

2

At point h = hm , we have an horizontal tangent, if d2 V d2 V =− . dh2 h= D −hm dh2 h= D +hm 2

(2.0)

2

The above deductions depends, of course, on the form of the interaction potential V (h). Fourthly, a simple dimensional analysis shows that the total interaction potential can be rewritten on the following scaling form ( ) 1 h ξ1 ξn W (h) = 2Φ , , ..., , (2.0) kB T D D D D where (ξ1 , ..., ξn ) are the ranges of various interactions experienced by the membrane, and Φ (x1 , ..., xn+1 ) is a (n + 1)-factor scaling-function. Finally, we note that the pair-potential W (h) cannot be singular at h = 0. It is rather an analytic function in the h variable. Therefore, at ﬁxed ratios ξi /D, an expansion of the scaling-function Φ, around the value h = 0, yields ( ) γ h2 W (h) = + O h4 . 4 kB T 2D 104

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We restrict ourselves to the class of potentials that exhibit a minimum at the mid-plane h = 0. This assumption implies that the coeﬃcient γ is positive deﬁnite, i.e. γ > 0. Of course, such a coeﬃcient depends on the ratios of the scale-lengths ξi to the separation D. In conﬁnement regime where the distance h is small enough, we can approximate the total interaction potential by its quadratic part. In these conditions, the Canham-Helfrich Hamiltonian becomes ∫ [ ] 1 2 H0 [h] = dxdy κ (∆h) + µh2 , (2.0) 2 with the elastic constant µ=γ

kB T . D4

(2.0)

The prefactor γ will be computed below. The above expression for the elastic constant µ gives an idea on its dependance on the ﬁlm thickness D. In addition, we state that this coeﬃcient may be regarded as a Lagrange multiplier that ﬁxes the value of the membrane roughness. Thanks to the above Hamiltonian, we calculate the mean-expectation value of the physical quantities, like the height-correlation function (propagator or Green function), deﬁned by G (x − x′ , y − y ′ ) = ⟨h (x, y) h (x′ , y ′ )⟩ − ⟨h (x, y)⟩ ⟨h (x′ , y ′ )⟩ . The latter solves the linear diﬀerential equation ( 2 ) κ b∆ + µ b G (x − x′ , y − y ′ ) = δ (x − x′ ) δ (y − y ′ ) ,

(2.0)

(2.0)

where δ (x) denotes the one-dimensional Dirac function, and ∆ = ∂ 2 /∂x2 + ∂ 2 /∂y 2 represents the twodimensional Laplacian operator. We have used the notations : κ b = κ/kB T and µ b = µ/kB T , to mean the reduced membrane elastic constants. From the propagator, we deduce the expression of the membrane roughness ⟨ ⟩ 2 (2.0) L2⊥ = h2 − ⟨h⟩ = G (0, 0) . Such a quantity measures the ﬂuctuations of the height-function (ﬂuctuations amplitude) around the equilibrium plane located at z = 0. We show in Appendix that the membrane roughness is exactly given by L2⊥ =

D2 , 12

(2.0)

provided that one is in the conﬁnement-regime, i.e. D L∥ . From the standard relation L2⊥ =

kB T 2 L , 16κ ∥

(2.0)

we deduce 2 L∥ = √ 3

(

κ kB T

)1/2 D.

(2.0)

In contrary to L⊥ , the length-scale L∥ depends on the geometrical characteristics of the membrane (through κ). The next steps consist in the computation of the Casimir force at and out equilibrium.

III. STATIC CASIMIR FORCE

When viewed under the microscope, the membranes of vesicles present thermally excited shape ﬂuctuations. Generally, objects such as interfaces, membranes or polymers undergo such ﬂuctuations, in order to increase their conﬁgurational entropy. For bilayer biomembranes and surfactants, the consequence of these undulations is that, they give rise to an induced force called Casimir force. To compute the desired force, we start from the partition function constructed with the Hamiltonian deﬁned in Eq. (13). This partition function is the following functional integral { } ∫ H0 [h] Z = Dh exp − , (3.0) kB T where integration is performed over all height-ﬁeld conﬁgurations. The associated free energy is such that : F = −kB T ln Z, which is, of course, a function of the separation D. If we denote by Σ = L2 the common area of plates, the Casimir force (per unit area) is minus the ﬁrst derivative of the free energy (per unit area) with respect to the ﬁlm-thickness D, that is Π=−

1 ∂F . Σ ∂D

(3.0)

This force per unit area is called disjoining pressure. In fact, Π is the required pressure to maintain the two plates at some distance D apart. In term of the partition function, the disjoining pressure rewrites Π 1 ∂ ln Z 1 ∂µ ∂ ln Z = = . kB T Σ ∂D Σ ∂D ∂µ

(3.0)

Using deﬁnition (19) together with Eqs. (23) and (24) yields Π=−

1 ∂µ 2 L . 2 ∂D ⊥

(3.0)

Explicitly, we obtain the desired formula 2

Π=

3 (kB T ) . 8 κD3

(3.0)

From this relation, we extract the expression of the disjoining potential (per unit area) [25] ∫ Vd (D) = −

D

Π (D′ ) dD′ =

∞

106

2

3 (kB T ) . 16 κD2

(3.0)

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The above expression of the Casimir force (per unit area) calls the following remarks. Firstly, this force decays with distance more slowly in comparison to the Coulombian one that decreases rather as D−2 . Secondly, this same force depends on the nature of lipids forming the bilayer (through κ). In this sense, contrarily to the Casimir eﬀect in Quantum Field Theory [16] and in Critical Phenomena [20], the present force is not universal. Incidentally, if this force is multiplied by κ, then, it will become a universal quantity. Thirdly, at ﬁxed temperature and distance, the force amplitude has signiﬁcant values only for those bilayer membrane of small bending rigidity constant. Fourthly, as it should be, such a force increases with increasing temperature. Indeed, at high temperature, the membrane undulations are strong enough. Finally, the numerical prefactor 3/8 (Helfrich’s cH -amplitude [9]) is close to the value obtained using Monte Carlo simulation [26]. In Fig. 1, we superpose the variations of the reduced static Casimir force Π/kB T upon separation D, for two lipid systems, namely SOPC and DAPC [27], at temperature T = 18◦ C. The respective membrane bending rigidity constants are : κ = 0.96 × 10−19 J and κ = 0.49 × 10−19 J. These values correspond to the renormalized bending rigidity constants : κ b = 23.9 and κ b = 12.2. The used methods for the measurement of these rigidity constants were entropic tension and micropipet [27]. These curves reﬂect the discussion made above.

FIG. 1. Reduced static Casimir force, Π/kB T , versus separation D, for two lipid systems that are SOPC (solid line) and DAPC (dashed line), of respective membrane bending rigidity constants : κ = 0.96 × 10−19 J and κ = 0.49 × 10−19 J, at temperature T = 18◦ C. The reduced force and separation are expressed in arbitrary units.

IV. DYNAMIC CASIMIR FORCE

To study the dynamic phenomena, the main physical quantity to consider is the time height-ﬁeld, h (r, t), where r = (x, y) ∈ R2 denotes the position vector and t the time. The latter represents the time observation of the system before it reaches its ﬁnal equilibrium state. We recall that the time height function h (r, t) solves a non-dissipative Langevin equation (with noise) [28] δH0 [h] ∂h (r, t) = −Γ + ν (r, t) , ∂t δh (r, t) 107

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where Γ > 0 is a kinetic coeﬃcient. The latter has the dimension : [Γ] = L40 T0−1 , where L0 is some length and T0 the time unit. Here, ν (r, t) is a Gaussian random force with mean zero and variance ⟨ν (r, t) ν (r′ , t′ )⟩ = 2Γδ2 (r − r′ ) δ (t − t′ ) ,

(4.0)

and H0 is the static Hamiltonian (divided by kB T ), deﬁned in Eq. (13). The bare time correlation function, whose Fourier transform is the dynamic structure factor, is deﬁned by the expectation mean-value over noise ν G (r − r′ , t − t′ ) = ⟨h (r, t) h (r′ , t′ )⟩ν − ⟨h (r, t)⟩ν ⟨h (r′ , t′ )⟩ν ,

t > t′ .

(4.0)

The dynamic equation (28) shows that the time height function h is a functional of noise ν, and we write : h = h [ν]. Instead of solving the Langevin equation for h [ν] and then averaging over the noise distribution P [ν], the correlation and response functions can be directly computed by means of a suitable ﬁeld-theory, of action [28 − 31] { } ∫ [ ] ∫ δH0 e e A h, e h = dt d2 r e h∂t h + Γe h − hΓh , (4.0) δh so that, for an arbitrary observable, O [ζ], one has ∫ ⟨O⟩ν =

∫

e DhDe hOe−A[h,h] , [dν] O [φ [ν]] P [ν] = ∫ e DhDe he−A[h,h]

(4.0)

where e h (r, t) is an auxiliary ﬁeld, coupled to an external ﬁeld h (r, t). The correlation and response functions can be computed replacing the static Hamiltonian H0 appearing in Eq. (13), by a new one : ∫ H0 [h, J] = H0 [h] − d2 rJh. Consequently, for a given observable O, we have ⟨ ⟩ δ ⟨O⟩J =Γ e h (r, t) O . (4.0) δJ (r, t) J=0 [ ] The notation ⟨ . ⟩J means the average taken with respect to the action A h, e h, J associated with the Hamiltonian H0 [h, J]. In view of the structure of equality (33), e h is called response field. Now, if O = h, we get the response of the order parameter ﬁeld to the external perturbation J ⟨ ⟩ δ ⟨h (r′ , t′ )⟩J ′ ′ R (r − r , t − t ) = =Γ e h (r, t) h (r′ , t′ ) . (4.0) δJ (r, t) J=0 J=0 The causality implies that the response function vanishes for t < t′ . In fact, this function can be related to the time-dependent (connected) correlation function using the ﬂuctuation-dissipation theorem, according to which ⟨ ⟩ Γ e h (r, t) h (r′ , t′ ) = −θ (t − t′ ) ∂t ⟨h (r, t) h (r′ , t′ )⟩c . (4.0) The above important formula shows that the time correlation function C (r − r′ , t − t′ ) = ⟨φ (r, t) φ (r′ , t′ )⟩c may be determined by the knowledge of the response function. In particular, we show that ∫ t ⟨ ⟩ ⟨ 2 ⟩ 2 h (r, t′ ) h (r, t′ ) . (4.0) dt′ e L⊥ (t) = h (r, t) c = −2Γ −∞

The limit t → −∞ gives the natural value L2⊥ (−∞) = 0, since, as assumed, the initial state corresponds to a completely ﬂat interface. Consider now a membrane at temperature T that is initially in a ﬂat state away from thermal equilibrium. At a later time t, the membrane possesses a certain roughness, L⊥ (t). Of course, the latter is timedependent, and we are interested in how it increases in time. 108

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We point out that the thermal ﬂuctuations give rise to some roughness that is characterized by the appearance of anisotropic humps. Therefore, a segment of linear size L eﬀectuates excursions of size [32] L⊥ = BLζ .

(4.0)

Such a relation deﬁnes the roughness exponent ζ. Notice that L is of the order of the in-plane correlation length, L∥ . From relation (20), we deduce the exponent ζ and the amplitude B. Their respective values 1/2 are : ζ = 1 and B ∼ (kB T /κ) . In order to determine the growth of roughness L⊥ in time, the key is to consider the excess free energy (per unit area) due to the conﬁnement, ∆F . Such an excess is related to the fact that the conﬁning membrane suﬀers a loss of entropy. Formula (27) tells us how ∆F must decay with separation. The result reads [32] 2/ζ

∆F ∼ kB T /L2max ∼ kB T (B/L⊥ )

,

(4.0)

where Lmax represents the wavelength above which all shape ﬂuctuations are not accessible by the conﬁned membrane. The repulsive ﬂuctuation-induced interaction leads to the disjoining pressure Π=−

∂∆F −(1+2/ζ) ∼ L⊥ . ∂L⊥

(4.0)

In addition, a care analysis of the Langevin equation (28) shows that ∂∆F ∂L⊥ −(1+2/ζ) ∼ −Γ = Γ × Π ∼ ΓL⊥ . ∂t ∂L⊥

(4.0)

We emphasize that this scaling form agrees with Monte Carlo predictions [32, 33]. Solving this ﬁrst-order diﬀerential equation yields [34] L⊥ (t) ∼ Γθ⊥ tθ⊥ ,

θ⊥ =

1 ζ = . 2 + 2ζ 4

(4.0)

This implies the following scaling form for the linear size L (t) ∼ Γθ∥ tθ∥ ,

θ∥ =

1 1 = . 2 + 2ζ 4

(4.0)

Let us comment about the obtained result (39). Firstly, as it should be, the roughness increases with time (the exponent θ⊥ is positive deﬁnite). In addition, the exponent θ⊥ is universal, independently on the membrane bending rigidity constant κ. Secondly, we note that, in Eq. (39), we have ignored some non-universal amplitude that scales as κ−1/4 . This means that the time roughness is signiﬁcant only for those biomembranes of small bending rigidity constant. Fourthly, this time roughness can be interpreted as the perpendicular size of holes and valleys at time t. Fifthly, the roughness increases until a ﬁne time, τ . The latter can be interpreted as the time over which the system reaches its ﬁnal equilibrium state. This characteristic time then scales as τ ∼ Γ−1 L⊥

1/θ⊥

,

(4.0)

where we have ignored some non-universal amplitude that scales as κ. Here, L⊥ ∼ D is the ﬁnal roughness. Explicitly, we have τ ∼ Γ−1 D4 .

(4.0)

As it should be, the ﬁnal time increases with increasing ﬁlm thickness D. On the other hand, we can rewrite the behavior (39) as L⊥ (t) = L⊥ (τ )

( )θ⊥ t . τ

109

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African Journal Of Mathematical Physics Volume 8(2010)101-114

This equality means that the roughness ratio, as a function of the reduced time, is universal. Now, to compute the dynamic Casimir force, we start from a formula analog to that deﬁned in Eq. (24), that is b b Π (t) 1 ∂ ln Z 1 ∂µ ∂ ln Z = = , kB T Σ ∂D Σ ∂D ∂µ

(4.0)

with the new partition function b= Z

∫

e

DhDe he−A[h,h] .

(4.0)

A simple algebra taking into account the basic relation (35a) gives Π (t) 1 ∂µ 2 =− L (t) , kB T 2 ∂D ⊥

(4.0)

which is very similar to the static relation deﬁned in Eq. (25), but with a time-dependent membrane roughness, L⊥ (t). Combining formulae (43) and (46) leads to the desired expression for the time Casimir force (per unit area) Π (t) = Π (τ )

( )θf t , τ

(4.0)

where Π (τ ) is the ﬁnal static Casimir force, relation (25). The force exponent, θf , is such that θf = 2θ⊥ =

ζ 1 = . 1+ζ 2

(4.0)

The induced force then grows with time as t1/2 until it reaches its ﬁnal value Π (τ ). At ﬁxed time and separation D, the force amplitude depends, of course, on κ, and decreases in this parameter according to κ−3/2 . Also, we note that the above equality means that the force ratio as a function of the reduced time is universal. In Fig. 2, we draw the reduced dynamic Casimir force, Π (t) /Π (τ ), upon the renormalized time t/τ .

110

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African Journal Of Mathematical Physics Volume 8(2010)101-114

FIG. 2. Reduced dynamic Casimir force, Π (t) /Π (τ ), upon the renormalized time t/τ .

Finally, consider again a membrane which is initially ﬂat but is now coupled to overdamped surface waves. This real situation corresponds to a conﬁned membrane subject to hydrodynamic interactions. The roughness now grows as [35] b ⊥ (t) ∼ tθb⊥ , L

θb⊥ =

ζ 1 = . 1 + 2ζ 3

(4.0)

Therefore, the roughness increases with time more rapidly than that relative to biomembranes free from hydrodynamic interactions. In this case, the dynamic Casimir force is such that Πh (t) = Π (τh )

(

t τh

)θbf ,

(4.0)

where Π (τh ) is the ﬁnal static Casimir force, relation (25). The new force exponent is θbf = 2θb⊥ =

2 2ζ = . 1 + 2ζ 3

(4.0)

There, τh ∼ D3 accounts for the new time-scale over which the conﬁned membrane reaches its ﬁnal equilibrium state. Therefore, the dynamic Casimir force decays with time as t2/3 , that is more rapidly than that where the hydrodynamic interactions are ignored, which scales rather as t1/2 . As we said before, this drastic change can be attributed to the overdamped surface waves that develop larger and larger humps. We depict, in Fig. 3, the variation of the reduced dynamic force (with hydrodynamic interactions), Πh (t) /Π (τh ), upon the renormalized time t/τh . 111

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FIG. 3. Reduced dynamic Casimir force (with hydrodynamic interactions), Πh (t) /Π (τ ), upon the renormalized time t/τh .

V. CONCLUSIONS

In this work, we have reexamined the computation of the Casimir force between two parallel walls delimitating a ﬂuctuating ﬂuid membrane that is immersed in some liquid. This force is caused by the thermal ﬂuctuations of the membrane. We have studied the problem from both static and dynamic point of view. We were ﬁrst interested in the time variation of the roughening, L⊥ (t), starting with a membrane that is inially in a ﬂat state, at a certain temperature. Of course, this length grows with time, and we found that : L⊥ (t) ∼ tθ⊥ (θ⊥ = 1/4), provided that the hydrodynamic interactions are ignored. For real systems, however, these interactions are important, and we have shown that the roughness increases more rapidly b ⊥ (t) ∼ tθb⊥ (θ⊥ = 1/3). The dynamic process is then stopped at a ﬁnal τ (or τh ) that represents as : L the required time over which the biomembrane reaches its ﬁnal equilibrium state. The ﬁnal time behaves as : τ ∼ D4 (or τh ∼ D3 ), with D the ﬁlm thickness. Now, assume that the system is explored at scales of the order of the wavelength q −1 , where q = (4π/λ) sin (θ/2) is the wave vector modulus, with λ the wavelength of the incident radiation and θ the scattering-angle. In these) conditions, the relaxation rate, τ (q), scales with q as : τ −1 (q) ∼ q 1/θ⊥ = q 4 ( b

or τh−1 (q) ∼ q 1/θ⊥ = q 3 . Physically speaking, the relaxation rate characterizes the local growth of the height ﬂuctuations. Afterwards, the question was addressed to the computation of the Casimir force, Π. At equilibrium, using an appropriate ﬁeld theory, we found that this force decays with separation D as : Π ∼ D−3 , with a known amplitude scaling as κ−1 , where κ is the membrane bending rigidity constant. Such a force is then very small in comparison with the Coulombian one. In addition, this force disappears when the 112

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temperature of the medium is suﬃciently lowered. The dynamic Casimir force, Π (t), was computed using a non-dissipative Langevin equation (with noise), solved by the time height-ﬁeld. We have shown that : Π (t) ∼ tθf (θf = 2θ⊥ = 1/2). When the hydrodynamic interactions eﬀects are ( ) important, we found that the dynamic force increases more rapidly as : θbf b b Πh (t) ∼ t θf = 2θ⊥ = 2/3 . Notice that we have ignored some details such as the role of inclusions (proteins, cholesterol, glycolipids, other macromolecules) and chemical mismatch on the force expression. It is well-established that these details simply lead to an additive renormalization of the bending rigidity constant. Indeed, we write κeﬀective = κ + δκ, where κ is the bending rigidity constant of the membrane free from inclusions, and δκ is the contribution of the incorporated entities. Generally, the shift δκ is a function of the inclusion concentration and compositions of species of diﬀerent chemical nature (various phospholipids forming the bilayer). Hence, to take into account the presence of inclusions and chemical mismatch, it would be suﬃcient to replace κ by κeﬀective , in the above established relations. As last word, we emphasize that the results derived in this paper may be extended to bilayer surfactants, although the two systems are not of the same structure and composition. One of the diﬀerences is the magnitude order of the bending rigidity constant.

APPENDIX

To show formula (17), we start from the partition function that we rewrite on the following form ∫ Z=

{

H [h] Dh exp − kB T

}

D/2 ∫

dzΦ (z) .

=

(5.0)

−D/2

Also, it is easy to see that the membrane mean-roughness is given by D/2 ∫

L2⊥ =

dzz 2 Φ (z)

−D/2 D/2 ∫

.

(5.0)

dzΦ (z)

−D/2

The restricted partition function is ∫ Φ (z) =

{ } H [h] Dhδ [z − h (x0 , y0 )] exp − . kB T

(5.0)

Here, H [h] is the original Hamiltonian deﬁned in Eq. (3). Of course, this deﬁnition is independent on the chosen point (x0 , y0 ), because of the translation symmetry along the parallel directions to plates. Notice that the above function is not singular, whatever the value of the perpendicular distance. Since we are interested in the conﬁnement-regime, ( ) that is when the separation D is much smaller than the membrane mean-roughness L0⊥ z ∼ h 0 . 2π L3

(2.0)

(2.0)

The above equality shows that the elastic constant k scales with separation L as : k ∼ L−3 . We note that the potential depth |U0 | has as eﬀect to renormalize the density amplitude [21]. III. TIME EVOLUTION OF THE PARTICLE DENSITY

Consider, now, an assembly of colloidal particles moving around a ﬂuctuating ﬂuid membrane. Under a sudden change of a suitable parameter, such as temperature, pressure or membrane environment, the 95

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system is out equilibrium. We assume that the system is subject to Brownian dynamics by changing the colloid-membrane interaction strength w, but the temperature and ﬁlm thickness remain ﬁxed. This change may be caused by aﬀecting the membrane environment. The problem can be studied through the local particle density, n (z, t). The latter represents the number of colloids per unit volume, at distance z and at time t. More precisely, we are interested in how this density evolves in time before the colloidal suspension reaches its ﬁnal equilibrium state. For simplicity, we will neglect mutual interactions between nanoparticles. This hypothesis makes sense at least for small particle densities. Hence, the only interaction experienced by the beads is an external potential caused by the membrane undulations. Within the harmonic approximation, this potential is deﬁned in Eqs. (9) and (10). To determine the time evolution of the local particle density, use will be made of the Smoluchowski equation [27, 28], which is a linear partial diﬀerential equation, of ﬁrst order and second order, with respect to time and perpendicular distance, respectively. Before writing and solving this equation, we shall need some backgrounds. We ﬁrst recall the expression of the equilibrium particle density } { U (z) neq (z) = A exp − , (3.0) kB T where A is a normalization constant and U (z) is the one-body potential due to the membrane undulations. If this potential is approximated by its harmonic form, the above deﬁnition becomes { } W (z) neq (z) = n0 exp − , (3.0) kB T where n0 is now the value of the particle density at the mid-plane z = 0. When the colloidal dispersion is out of equilibrium, in addition to distance, the density depends on time. This means that the nanoparticles execute Brownian dynamic but in the presence of the harmonic potential W (z). In order to compute this local density, we ﬁrst recall that the Brownian diﬀusion is correctly described by the Fick’s law. The latter stipulates that the ﬂux of matter, j (z, t), is directly proportional to the spatial gradient of density, that is j = −D

∂n 1 ∂W − , ∂z ζ ∂z

(3.0)

with D=

kB T ζ

(3.0)

the diﬀusion constant and ζ the friction coeﬃcient, of which the inverse 1/ζ is the mobility. If we design by a the particle radius and by ηs the solvent viscosity, the friction coeﬃcient ζ can be calculated from hydrodynamics [36] : ζ = 6πηs a. Equality (14) states that the diﬀusion constant characterizing the thermal motion is related to the quantity ζ, which expresses the response to an external ﬁeld. Such an equality is a consequence of the well-known dissipation-ﬂuctuation theorem [27, 28]. On the other hand, relation (13) must be combined with the local conservation law of matter ∂n ∂j + =0. ∂t ∂z Combining Eqs. (13) and (15) yields the Smoluchowski equation ( ) ∂n 1 ∂ ∂n ∂W = kB T +n ∂t ζ ∂z ∂z ∂z

(3.0)

(3.0)

solved by the local particle density n (z, t). At equilibrium, that is ∂n/∂t = 0, the above equation reduces to : kB T ∂n/∂z + n∂W/∂z = 0, whose solution is neq (z) = n0 exp exp {−W (z) /kB T }, which is nothing else but the density deﬁned in Eq. (12). Replacing the harmonic potential W (z) by its explicit form (9) gives ∂ 2 n k ∂n k ∂n =D 2 + z + n. ∂t ∂z ζ ∂z ζ 96

(3.0)

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African Journal Of Mathematical Physics Volume 8(2010)91-100

This new Smoluchowski equation must be supplemented by two boundary conditions, which are n (z, t = ∞) = nf (z) .

n (z, t = 0) = ni (z) ,

(3.0)

If the temperature T and separation L are ﬁxed, the inial and ﬁnal equilibrium particle densities, ni (z) and nf (z), are completely determined by the initial and ﬁnal surface coupling constants wi and wf , respectively. Therefore, the dynamic is caused by a change of the membrane environment. Taking advantage of those mathematical techniques used in Ref. [21], we show that the solution to the Smoluchowski equation (17) is given by { ( t/τ )2 } [ ( )]−1/2 ∫ ∞ ze f − y −2t/τf ( ) [ni (y) − nf (y)] , n (z, t) = nf (z) + 2πDτf 1 − e dy exp − 2Dτf e2t/τf − 1 −∞ (3.0) where the initial and ﬁnal equilibrium particle densities are given by { } { } ki z 2 z2 i i ni (z) = n0 exp − = n0 exp − , (3.0) 2kB T 2Dτi

nf (z) =

nf0

} } { { kf z 2 z2 f = n0 exp − , exp − 2kB T 2Dτf

(3.0)

with the time-scales τi and τf τi =

ζ 1 = ki 12

√

2π D−1 3 L , 3 wi

τf =

ζ 1 = kf 12

√

2π D−1 3 L , 3 wf

(3.0)

where wi and wf > wi are the initial and ﬁnal surface coupling constants. This means that the colloidmembrane interaction is suddenly increased from wi to wf . The above relations suggest that the times τi and τf depend on the colloid-membrane interaction and ﬁlm thickness. In particular, the time-scale τf may be interpreted as the time beyond which the colloidal system reaches its ﬁnal equilibrium state. Then, a weak colloid-membrane interaction necessitates a great time before the colloidal system tends to its ﬁnal state. In fact, τf has another physical meaning, and can be regarded as the required time over which the particles are trapped in new holes and valleys. Now, after integration over the y variable, in Eq. (19), we obtain a closer form for the time particle density { } [ ( )]−1/2 1 z2 i −2t/τf ( ) n (z, t) = n0 1 + η e −1 exp − , (3.0) 1 + η e−2t/τf − 1 2Dτi with the reduced time-shift η=

τi − τf wf − wi = >0. τi wf

(3.0)

The quantity η then represents the relative shift of the colloid-membrane interaction strengths wi and wf . The density amplitude, in Eq. (23), was obtained using the matter conservation law ∫

∫

+∞

−∞

ni (z) dz =

+∞ −∞

nf (z) dz ≡ Γ .

(3.0)

Here, Γ represents the adsorbance, which is deﬁned as the total number of colloids (per unit area) located near membrane. Combining Eqs. (20), (21) and (25) yields the relationship √ √ 2πDτf nf0 = 2πDτi ni0 = Γ . (3.0) Let us comment the density expression (23). First, we note that, it is easy to see that the solution (23) satisﬁes the two boundary conditions (18). 97

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The initial and ﬁnal equilibrium states are deﬁned in Eqs. (20) and (21). Second, when it is reduced by√ ni0 , the time particle density depends on three dimensionless factors, namely the renormalized distance z/ 2Dτi , the time-ratio t/τf and the time-shift η = (τi − τf ) /τi . Therefore, all microscopic details (colloid-membrane interaction) are entirely contained in τi and τf . Finally, we emphasize that the time particle density curve exhibits a maximum at z = 0, and it is symmetric around this point, whatever be the values of t/τf and η. We√depict, in Fig. 2, the reduced time particle density n (z, t) /ni0 versus the renormalized distance z/ 2Dτi , choosing three values of the time-ratio t/τf : 0, 0.5, and ∞. The former corresponds to the initial state and the second to the ﬁnal one. These curves are drawn with parameter η = 0.5. This value means that the ﬁnal surface coupling constant wf is two times more important than the initial one wi , that is wf = 2wi .

√ FIG. 2. Reduced local particle density, n (z, t) /ni0 , versus the renormalized distance z/ 2Dτi , with three values of the time-ratio t/τf : 0 (dashed line), 0.5 (solid line), ∞ (line in dots). These curves are drawn choosing the value η = 0.5 (wf = 2wi ).

IV. CONCLUSIONS

This work is dedicated to the Brownian dynamics study of small colloidal particles in contact with an attractive penetrable ﬂuid membrane. The host liquid was assumed to be delimitated by two parallel reﬂecting walls, which are a ﬁnite distance apart. The membrane surrounded by beads is conﬁned only if the ﬁlm thickness is much smaller than the bulk membrane mean-roughness. Physics was discussed in terms of three relevant parameters, which are the absolute temperature, T , the separation between walls, L, and colloid-membrane interaction strength, w. In our study, we have ﬁxed the temperature and ﬁlm thickness to some values, and varied the surface coupling constant. This can be experimentally achieved modifying the membrane environment. For the present study, we have started from three hypothesizes : (1) the particles are point-like, (2) they are of very low-density (in order to forget their mutual interactions), and (3) strongly interact with the membrane. To achieve the investigation of the Brownian dynamics, use was made of a theoretical formalism based on the Smoluchowski equation. The latter is solved by the time particle density we were interested in. We have exactly computed this physical quantity, around the mid-plane of the ﬁlm, where the essential of phenomenon occurs. Within this distance-domain, the mean-force external potential was approximated 98

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African Journal Of Mathematical Physics Volume 8(2010)91-100

by an harmonic one. This means that we were in the presence of Brownian particles moving in an harmonic potential, that is, √in addition to the usual diﬀusion, these experience small oscillations with a frequency ν scaling as ν ∼ w/L3 . Hence, the harmonic approximation used for the mean-force potential is largely justiﬁed for those ﬂuid interfaces of small enough colloid-membrane interaction strength. As we have shown, the time particle density depends on a time-scale, τ , scaling as τ ∼ L3 /w. We have interpreted this scale-time as the required time over which the nanoparticles reach their ﬁnal equilibrium state. Also, τ can be regarded as the time-interval where the particles are trapped in holes and valleys of size slightly smaller than the ﬁlm thickness L. 0 We note that, when the separation L is much greater than the bulk roughness ξ⊥ , in addition to the above evoked parameters, the physical phenomenon depends on the speciﬁc membrane characteristic (via the bending rigidity constant κ). In this case, the Brownian dynamics can be caused by a change of the parameter κ. This situation has less physical interest, since, in this case, ﬁnite size eﬀects contribute to the leading behavior only by exponentially small corrections. As last word, we emphasize that the results derived in this paper may be extended to bilayer surfactants, although the two systems are not of the same structure and composition.

ACKNOWLEDGMENTS

We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions, during the ”First International Workshop On Soft-Condensed Matter Physics and Biological Systems”, 14-17 November 2006, Marrakech, Morocco. One of us (M.B.) would like to thank the Professor C. Misbah for fruitful correspondences, and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kinds of hospitalities during his regular visits.

REFERENCES

1

M.S. Bretscher and S. Munro, Science 261, 12801281 (1993). J. Dai and M.P. Sheetz, Meth. Cell Biol. 55, 157171 (1998). 3 M. Edidin, Curr. Opin. Struc. Biol. 7, 528532 (1997). 4 C.R. Hackenbrock, Trends Biochem. Sci. 6, 151154 (1981). 5 C. Tanford, The Hydrophobic Eﬀect, 2d ed., Wiley, 1980. This book includes, in addition, a good discussion about the interactions of proteins and membranes. 6 D.E. Vance and J. Vance, eds., Biochemistry of Lipids, Lipoproteins, and Membranes, Elsevier, 1996. 7 F. Zhang, G.M. Lee, and K. Jacobson, BioEssays 15, 579588 (1993). 8 S. Safran, Statistical Thermodynamics of Surfaces, Interfaces and Membranes, Addison-Wesley, Reading, MA, 1994. 9 W. Helfrich, Z. Naturforsch. 28c, 693 (1973). 10 U. Seifert, Advances in Physics 46, 13 (1997). 11 H. Ringsdorf and B. Schmidt, How to Bridge the Gap Between Membrane, Biology and Polymers Science, P.M. Bungay et al., eds, Synthetic Membranes : Science, Engineering and Applications, p. 701, D. Reiidel Pulishing Compagny, 1986. 12 D.D. Lasic, American Scientist 80, 250 (1992). 13 V.P. Torchilin, Eﬀect of Polymers Attached to the Lipid Head Groups on Properties of Liposomes, D.D. Lasic and Y. Barenholz, eds, Handbook of Nonmedical Applications of Liposomes, Volume 1, p. 263, RCC Press, Boca Raton, 1996. 14 R. Joannic, L. Auvray, and D.D. Lasic, Phys. Rev. Lett. 78, 3402 (1997). 15 P.-G. de Gennes, Scaling Concept in Polymer Physics, Cornell University Press, 1979. 16 C. Fradin, A. Abu-Arish, R. Granek, and M. Elbaum, Biophys. J. 84, 2005 (2003). 17 D.F. Evans and H. Wennerstr¨ om, The Colloidal Domain, Wiley, New-York, 1999. 18 T. Bickel, M. Benhamou, and H. Kaidi, Phys. Rev. E 70, 051404 (2004). 2

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H. Kaidi, T. Bickel, and M. Benhamou, Europhys. Lett. 69, 15 (2005) A. Bendouch, H. Kaidi, T. Bickel, and M. Benhamou, J. Stat. Phys.: Theory and Experiment P01016, 1 (2006). 21 A. Bendouch, M. Benhamou, and H. Kaidi, Elec. J. Theor. Phys. 5, 215 (2008). 22 A. Einstein, Ann. Physik. 17, 549 (1905) ; 19, 371 (1906); see, also, Investigation on the Theory of the Brownian Movement, E.P. Dutton and Copy Inc, New York, 1926. 23 N. Wax, Noise and Stochastic Processes, Dover Publishing Co., New York, 1954. 24 M. Lax, Rev. Mod. Phys. 32, 25 (1960) ; 38, 541 (1966). 25 R. Kubo, Rep. Prog. Phys. 29, 255 (1966). 26 C.W. Gardiner, Handbook of Stochastic Methods, 3d ed., Springer, 2004. 27 H. Risken and T. Frank, The Focker-Planck Equation : Methods of Solutions and Applications, 2d ed., Springer, 1989. 28 M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press Oxford, 1986. 29 P.B. Canham, J. Theoret. Biol. 26, 61 (1970). 30 K. El Hasnaoui, Y. Madmoune, H. Kaidi, M. Chahid, and M. Benhamou, Induced forces in conﬁned biomembranes, submitted for publication. 31 K. Binder, in : Phase Transitions and Critical Phenomena, Vol. 8, edited by C. Domb and J.L. Lebowitz, Academic Press, London, 1983. 32 H.W. Diehl, in : Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, Vol. 10, Academic Press, London, 1986. 33 S. Dietrich, in : Phase Transitions and Critical Phenomena, edited by C. Domb and J.L. Lebowitz, Vol. 12, Academic Press, London, 1988. 34 C. Itzykson and J.M. Drouﬀe, Statistical Field Theory : 1 and 2, Cambridge University Press, 1989. 35 J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. 36 G.K. Batchelor, An Introduction to Fluid Dynamics, Chap. 4, Cambridge University Press, 1970. 20

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Author's personal copy Physica A 389 (2010) 3465–3475

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Statistical mechanics of bilayer membranes in troubled aqueous media M. Benhamou ∗ , K. Elhasnaoui, H. Kaidi, M. Chahid Laboratoire de Physique des Polymères et Phénomènes Critiques, Faculté des Sciences Ben M’sik, P.O. 7955, Casablanca, Morocco

article

info

Article history: Received 30 March 2010 Available online 14 April 2010 Keywords: Bilayer membranes Vesicles Impurities Equilibrium Statistical mechanics

abstract We consider a bilayer membrane surrounded by small impurities, assumed to be attractive or repulsive. The purpose is a quantitative study of the effects of these impurities on the statistical properties of the supported membrane. Using the replica trick combined with a variational method, we compute the membrane mean-roughness and the height correlation function for almost-flat membranes, as functions of the primitive elastic constants of the membrane and some parameter that is proportional to the volume fraction of impurities and their interaction strength. As results, the attractive impurities increase the shape fluctuations due to the membrane undulations, while repulsive ones suppress these fluctuations. Second, we compute the equilibrium diameter of (spherical) vesicles surrounded by small random particles starting from the curvature equation. Third, the study is extended to a lamellar phase composed of two parallel fluid membranes, which are separated by a finite distance. This lamellar phase undergoes an unbinding transition. We demonstrate that the attractive impurities increase the unbinding critical temperature, while repulsive ones decrease this temperature. Finally, we say that the presence of small impurities in an aqueous medium may be a mechanism to suppress or to produce an unbinding transition, even the temperature and polarizability of the aqueous medium are fixed, in lamellar phases formed by parallel lipid bilayers. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Usually, the aqueous media supporting biological membranes are assumed to be homogeneous. Actually, any real system inevitably contains impurities. Under well-controlled conditions, the particles can be removed from the surrounding medium. But, if these entities are present, it is also interesting to study their effect on the statistical properties of the biomembranes, such as fluctuations’ spectrum and dynamical behavior. In general, random inhomogeneities tend to disorder the system. It is important to make a distinction between annealed and quenched disorders. The former is used when impurities and host constituents (phospholipids) are in equilibrium [1]. This means that their respective mobilities are comparable. If it is not the case, that is host constituents and impurities are out of equilibrium, the disorder is rather quenched [1]. When the Statistical Mechanics is used, the latter consists to trace over all membrane undulations, before performing the summation over the impurities’ disorder. Although the quenched disorder is harder to analyze, it remains more realistic than its annealed counterpart. Indeed, the thermal and the noise averaging have very different roles. In this paper, the physical system we consider is a fluid membrane (flat or closed) trapped in a troubled aqueous medium. The latter is impregnated by a weak amount of impurities that may be attractive or repulsive regarding the membrane. The aim is to show how these entities can modify the statistical properties of the fluid membrane. These properties will be studied through the fluctuations’ amplitude. To model the system, we suppose that the impurities act as a random external potential with a Gaussian distribution (uncorrelated disorder). In addition, we suppose that the disorder is quenched. To

∗

Corresponding author. E-mail address: [email protected] (M. Benhamou).

0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.03.049

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do calculations, the replica theory [2,3] that is based on a mathematical analytical continuation, usually encountered in Quantum Mechanics [4] and Critical Phenomena will be made use of [5,6]. Our finding are as follows. First, using the above evoked theory, we compute the mean-roughness of an almost-flat membrane, as a function of the primitive parameters of the pure membrane (free from impurities) and a certain parameter depending on the volume fraction of impurities and their interaction strength. The main conclusion is that, attractive impurities increase the shape fluctuations due to the thermal undulations, while repulsive ones tend to suppress these fluctuations and then lead to a strong membrane confinement. Second, we analyze the impurities effects on the equilibrium shape of closed vesicles solving the curvature equation (for spherical vesicles). We show that the vesicle is more stable in the presence of repulsive impurities, in comparison with attractive ones. Thereafter, the study is extended to lamellar phases formed by two parallel fluid membranes. We demonstrate that the presence of random impurities drastically affects the physical properties of the lamellar phase, in particular, the unbinding transition driving the system from a bind state to a state where the two membranes are completely separated. This paper is organized as follows. In Section 2, we describe the fundamentals of the used model. Section 3 deals with the computation of the fluctuations amplitude of a single almost-flat fluid membrane, surrounded by attractive or repulsive impurities. We compute, in Section 4, the equilibrium diameter of a closed vesicle in the presence of impurities. Extension of study to lamellar phases is the aim of Section 5. Finally, some concluding remarks are drawn in the last section. 2. Effective field theory Consider a fluctuating fluid membrane embedded in a three-dimensional liquid surrounded by very small impurities. For the sake of simplicity, we suppose that the impurities are point like. Within the framework of the Monge representation, a point on the membrane can be described by the position-vector (r , z = h (r )), where r = (x, y) ∈ R2 is the transverse vector and h (r ) is the height function. The Statistical Mechanics of fluid membranes free from impurities is based on the Canham–Helfrich Hamiltonian [7]

Z

d2 r

H0 [h] =

hκ 2

(∆h)2 +

µ 2

i

h2 ,

(1)

with κ the membrane bending modulus. The confinement energy (per unit area) µh2 /2 (µ > 0) is responsible for the localization of the membrane in some region of the Euclidean space, where it fluctuates around an equilibrium plane located at h = 0. Therefore, the height h takes either positive and negative values. For simplicity, the membrane is assumed to be tensionless. In fact, this assumption does not change conclusions made below. To model the impurities effects on the statistical properties of the system, we suppose that these tend to reinforce the membrane confinement, if they are repulsive, or to render this membrane more free, if the particles are rather attractive. These tendencies can be explained assuming that the confinement parameter is local in space and making the substitution

µ → µ + V (r ) ,

(2)

in the above Hamiltonian, where the variation V (r ) can be regarded as a random external potential. To simplify, the corresponding probability distribution is supposed to be Gaussian (uncorrelated disorder), that is V (r ) = 0,

V (r ) V (r 0 ) = −vδ2 r − r 0 .

(3)

Here, −v is a positive constant proportional to both concentration of impurities and strength of their interaction potential, and δ2 (r ) denotes the two-dimensional Dirac distribution. Therefore, the new effective Canham–Helfrich Hamiltonian reads

Z H [h] =

d2 r

κ 2

(∆h)2 +

1 2

(µ + V (r )) h2 ,

(4)

for attractive impurities. For repulsive ones, V (r ) must be replaced by iV (r ), with i2 = −1. Since the disorder distribution is Gaussian, all its odd moments vanish, but the even ones do not. This implies that all physical quantities, calculated with the pure imaginary potential iV , are entire numbers. Since the impurities and membranes are not in equilibrium, the disorder is rather quenched, that is we have to average not the partition function, Z , but its logarithm, ln Z . The latter defines the free energy. For a given (quenched) configuration of impurities, the partition function is

Z Z =

D he−A[h] ,

(5)

with the action A [h] =

H [h] kB T

Z =

2

d r

b κ 2

(∆h) + 2

1 2

2 b b µ + V (r ) h .

(6)

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We have used the notations

b κ=

κ kB T

,

µ

b µ=

kB T

V b V = .

,

(7)

kB T

In term of the reduced impurities potential b V , the second disorder law in Eq. (3) becomes: b V (r ) b V (r 0 ) = −b vδ r − r 0 , with

b v = v/ (kB T )2 . A simple dimensional analysis shows that: [b κ ] = L0 , [b µ] = L−4 and [b v ] = L−6 , where L is some length that may be the membrane thickness. The main quantity to consider is the average of the logarithm of the partition function, ln Z , over disorder. To compute such a quantity, we use the replica trick [2]. This method consists to formally write Zn − 1

ln Z = lim

n

n→0

.

(8)

After performing the average over disorder, we get

Z Zn

=

D h1 . . . D hn e−A[h1 ,...,hn ] ,

(9)

with the n-replicated effective action A [h1 , . . . , hn ] =

Z

" d2 r

n b κX

2 α=1

(∆hα )2 +

n b µX

2 α=1

h2α +

n b v X

8

α=1

! h2α

n X

β=1

!# h2β

,

(10)

where Greek indices denote replicas. The last term introduces an additional coupling constant b v < 0, which is directly responsible for the effective interaction between replicas due to the presence of impurities. The above action describes attractive impurities. For repulsive ones, the coupling b v must be replaced by −b v . The additive quartic term in the above action means that the presence of impurities is accompanied by an increase of entropy, when these are attractive, and by an entropy loss, if they are rather repulsive. The following paragraph is devoted to a quantitative determination of the fluctuations spectrum in the presence of impurities. 3. Single almost-flat membrane We start by considering the attractive impurities problem. Of course, the functional integral (9) cannot be exactly computed. One way is the use of a variational method. To this end, we consider a bilayer membrane in the presence of the following bare action Aη [h1 , . . . , hn ] =

Z

" d2 r

n b κX

2 α=1

(∆hα )2 +

n b ηX

2 α=1

# h2α

,

(11)

with the new confinement coupling b η > 0. With this action, the partition function, Zη , is exact. We have Zη = (Z0 )n ,

(12)

with

Z Z0 =

Z b κ b η 2 2 2 D h exp − d r . (∆h) + h 2

(13)

2

The latter accounts for the usual partition function of a bilayer membrane free from impurities. Introduce the mean-value of a functional X [h1 , . . . , hn ], calculated with the bare action Aη [h1 , . . . , hn ],

hX i0 =

1

Z

D h1 . . . D hn X [h1 , . . . , hn ] exp −Aη [h1 , . . . , hn ] .

Zη

(14)

In term of this mean-value, the averaged partition function may write

Z n = Zη exp − A − Aη

0

.

(15)

Using the standard inequality

X e

0

≥ e hX i0 ,

(16)

we get

Z n ≥ Zη exp − A − Aη

0

.

(17)

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This implies that

1 − Zn

− ln Z = lim

≤ lim

n

n→0

1 − Zη exp − A − Aη

.

0

n

n→0

(18)

On the other hand, we have

A − Aη

"

Z

d2 r

=

0

n

2 b µ −b ηX

2

α=1

hα

+

0

n X n

b vX

8 α=1 β=1

# 2

h2α hβ

0

.

(19)

It is easy to see that n X

2

α=1

hα

n X n X

= nσ 2 ,

0

α=1 β=1

h2α h2β

0

= n (n + 2) σ 2

2

,

(19a)

with the squared membrane roughness 1

σ 2 (η) =

1

p . 8 b κb η

(20)

With these considerations,

− ln Z ≤ − ln Z0 + Σ

b µ −b η 2

σ2 +

b v 4

σ2

2

,

(21)

where Σ is the area of the reference plane. Notice that the right-hand side of this inequality is b η-dependent function. Therefore, a good approximate value of −ln Z is

− ln Z

MF

= min − ln Z0 + Σ η

b µ −b η 2

σ + 2

b v 4

σ

2 2

.

(22)

(The subscript MF is for mean-field theory). The variational parameter b η is such that

∂ b µ −b η 2 b v 2 2 − ln Z0 + Σ σ + σ = 0. ∂b η 2 4

(23)

Combining the relationship

σ2 = −

2 ∂ ln Z0 Σ ∂b η

(24)

and Eq. (20) yields the minimum value of b η, which satisfies the following implicit relation

b η=b µ +b vσ 2 .

(25)

The squared membrane roughness, σ 2 , is then given by the parametric equations

σ 2 = 1 p1 , 8 b κb η b η=b µ +b vσ 2 .

(26)

Eliminating b η between these equations gives the implicit relation

σ2 =

1

1

q . 8 b κ b µ +b vσ 2

(27)

p

Introduce the usual squared roughness of a membrane free from impurities, σ02 = 1/8 b κb µ. Then, we have

σ 2 = σ02 p

1 1 +b vσ 2 /b µ

.

(28)

The latter may be rewritten, in terms of dimensionless variables σ 2 /σ02 , b κ , and w = −64b κb v σ02

σ2 1 = q , 2 σ0 1 − wσ 2 /σ02

(attractive impurities) .

3

> 0, as (29)

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Table 1 Some values of the coupling w and their corresponding ratios σ 2 /σ02 , for attractive impurities.

w

σ 2 /σ02

0.1 0.15 0.2 0.25 0.3

1.0575 1.0937 1.1378 1.1939 1.2715 √ 3

w∗

Notice that the coupling constant w > 0 is directly proportional to the volume fraction of impurities and their interaction strength. Let us comment about the obtained result. Firstly, this makes sense only when σ 2 /σ02 < 1/w . This condition is fulfilled for very weak disorders (w is small enough). Secondly, since w is positive definite, we have: σ 2 > σ02 . A comparison between this inequality and that just evoked above implies that 0 < w < 1. Thirdly, if we set x = σ 2 /σ02 , the above relation can be rewritten as

w=

x2 − 1 x3

.

(30)

Therefore, w is a direct function of x (Fig. 1). This formula may be experimentally used to estimate the impurities coupling w, knowing the experimental value of the ratio σ 2 /σ02 . If w is fixed to some value, the x-variable then solves the following third-degree algebraic equation

w x3 − x2 + 1 = 0 .

(31)

The existence of√its roots then depends on the value of the renormalized coupling w . We show that there exists a typical value w ∗ = 2/3 3 of the coupling w , such that: (1) For w > w ∗ (strong disorder), we have only one root that is negative ; √ (2) For w = w ∗ (medium disorder), we have one negative root and a positive unique root, which is x = σ 2 /σ02 = 3. Then, the maximal value of the membrane roughness is σ ∗ = 31/4 σ0 . This constraint means that the lipid bilayer cannot support undulations of perpendicular size (hunt size) greater than σ ∗ . This is realized if only if the impurities volume fraction φ is below some value φ ∗ , directly proportional to the threshold w ∗ ; (3) For w < w ∗ (weak disorder), we have one negative root andtwo positive ones. Only one positive root is acceptable (the smallest one). At this particular root, the free energy −kB T ln Z MF is minimal. We show that the absolute minimum satisfies the inequality

− 4wx6 + 3x4 − 1 > 0.

(32)

In this case, the ratio σ /σ can be obtained by numerically solving Eq. (31). We report in Table 1 some values of the impurities coupling w and the corresponding dimensionless squared membrane roughness σ 2 /σ02 . It is easy to see that, the ratio σ 2 /σ02 increases with increasing coupling w , provided that w is in the interval w < w ∗ . 2

2 0

Now, for repulsive impurities, the fundamental relationship (29) is replaced by

σ2 1 = q , 2 σ0 1 − wσ 2 /σ02

(repulsive impurities) ,

(33)

or equivalently

− wx3 − x2 + 1 = 0,

(34)

with the stability condition 4w x6 + 3x4 − 1 > 0.

(35)

The equality (34) can be transformed into the following direct function (Fig. 2)

w=

x2 − 1 x3

,

(36)

with x < 1 and w < 0. As it should be, the membrane roughness is reduced by the presence of repulsive impurities, that is σ 2 < σ02 , and in addition, σ 2 decreases with increasing impurities coupling −w . As a matter of fact, the effect of these particles is to reinforce

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Fig. 1. Impurities’ coupling w versus the ratio σ 2 /σ02 , for attractive impurities.

Fig. 2. Impurities’ coupling w versus the ratio σ 2 /σ02 , for repulsive impurities.

Table 2 Some values of the coupling w and their corresponding ratios σ 2 /σ02 , for attractive impurities.

w

σ 2 /σ02

−0.1 −0.15 −0.2 −0.25 −0.3 −0.4

0.9554 0.9364 0.9190 0.9032 0.8885 0.8622

the confinement of the considered fluid membrane. Therefore, a bilayer membrane collapses, when it is trapped in a troubled aqueous media with repulsive impurities. In this case, it is easy to see that the algebraic equation has only one positive root. In Table 2, we give some values of the coupling w and the corresponding ratio σ 2 /σ02 .

= h (r ) h r 0 − hh (r )i h r . The latter measures the fluctuations of the height function h around its mean-value hhi. Here, h.i denotes Another interesting physical quantity is the (connected) height correlation function: G r − r 0

0

the thermal expectation mean-value, which must not be confused with average over disorder.

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To compute the correlation functions, we start from the generating functional

Z Z [J] =

d r J (r ) h (r ) ,

Z

2

D h exp −A [h] +

(37)

with the action A [h] = H [h] /kB T , where H [h] is the effective Canham–Helfrich Hamiltonian defined in Eq. (4). Here, J is an auxiliary source coupled to the field h. Functional derivatives of Z [J] with respect to source J give all height correlation functions, in particular, the propagator G r −r

0

δ 2 ln Z [J] = δ J (r ) δ J (r 0 )

.

(38)

J =0

This definition indicates that the main object to compute is the averaged connected generating functional ln Z [J]. To compute the functional ln Z [J], as before, we use the replica method based on the limit ln Z [J] = lim

Z n [J] − 1 n

n→0

,

(39)

with

(

Z

D h1 . . . D hn exp −A [h1 , . . . , hn ] +

Z n [J] =

n Z X

) d2 r J (r ) hi (r ) ,

(40)

i=1

where A [h1 , . . . , hn ] is the action, relation (10). Using the standard cumulant method [5,6] based on the approximate formula

hexp {X }i0 = exp hX i0 + (1/2!) X 2 0 − hX i20 + · · · ,

(41)

we find that

Z n [J] ' Zη exp − A − Aη

Z Z 0 n 2 2 0 0 + · · · exp d r d r J r G r − r J r , ( ) 0 0

(42)

2

with the usual bare propagator G0 r − r

0

Z =

0 d2 q eiq.(r −r )

κ q4 + b η (2π )2 b

.

(43)

In Eq. (42), we have ignored high-order terms in J that do not contribute to the propagator. After performing the limit n = 0, we find that the averaged connected generating functional reads ln Z [J] ' − ln Z0 + Σ

b µ −b η 2

σ2 +

b v 4

σ2

2

+

1 2

Z

d2 r

Z

d2 r 0 J (r ) G0 r − r 0 J r 0 .

(44)

By simple functional derivation, we obtain the expected propagator G r −r

0

= G0 r − r

0

Z =

0 d2 q eiq.(r −r )

κ q4 + b η (2π )2 b

.

(45)

Then, the expected propagator identifies with the bare one. Here, the variational parameter b η satisfies the implicit equation (25). We then recover the relationship: σ 2 = G (0), where σ 2 is the squared membrane roughness computed above. The next step consists to extend the study to a closed vesicle surrounded by attractive or repulsive impurities. 4. Single vesicle We start by recalling some basic backgrounds dealt with the equilibrium shape of (spherical) vesicles, which may studied using Differential Geometry techniques. The vesicle is essentially formed by two adjacent monolayers (inner and outer) that are formed by amphiphile lipid molecules. These permanently diffuse with the molecules of the surrounded aqueous medium. Such a diffusion then provokes thermal fluctuations (undulations) of the membrane. This means that the latter experiences fluctuations around an equilibrium plane. Consider a biomembrane of arbitrary topology. A point of this membrane (surface) can be described by two local coordinates (u1 , u2 ). At each point of the surface, there exists two particular curvatures (minimal and maximal), called

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principal curvatures, denoted C1 = 1/R1 and C2 = 1/R2 . The quantities R1 and R2 are the principal curvature radii. With the help of the principal curvatures, one constructs two invariants that are the mean-curvature 1

(C1 + C2 ) , 2 and the Gauss curvature C =

(46)

K = C1 C2 .

(47)

The principal curvatures C1 and C2 are nothing else but the eigenvalues of the curvature tensor [8]. To comprehend the geometrical and physical properties of the biomembranes, one needs a good model. The widely accepted one is the fluid mosaic model proposed by Singer and Nicolson in 1972 [9]. This model consists to regard the cell membrane as a lipid bilayer, where the lipid molecules can move freely in the membrane surface like a fluid, while the proteins and other amphiphile molecules (cholesterol, sugar molecules, . . . ) are simply embedded in the lipid bilayer. We note that the elasticity of cell membranes crucially depends on the bilayers in this model. The elastic properties of bilayer biomembranes were first studied, in 1973, by Helfrich [7]. The author recognized that the lipid bilayer could be regarded as smectic-A liquid crystals at room temperature, and proposed the following curvature free energy (without impurities) F =

κ

Z

(2C )2 dA + κG

2

Z

Z K dA +

γ dA + p

Z

dV .

(48)

where dA denotes the area element, and V is the volume enclosed within the lipid bilayer. In the above definition, κ accounts for the bending rigidity constant, κG for the Gaussian curvature, γ for the surface tension, and p for the pressure difference between the outer and inner sides of the vesicle. The spontaneous curvature is ignored. The first-order variation gives the shape equation of lipid vesicles [10] p − 2γ C + 4κ C C 2 − K + κ∇ 2 (2C ) = 0,

(49)

with the surface Laplace–Bertolami operator

∂ ∇ = √ g ∂ ui

1

2

∂ gg ∂ uj

√

ij

,

(50)

where g ij is the metric tensor on the surface and g = det g ij . For open or tension-line vesicles, local differential equation (49) must be supplemented by additional boundary √ conditions we do not write [11]. The above equation have known three analytic solutions corresponding to sphere [10], 2-torus [12–15] and biconcave disk [16]. For spherical vesicles, the solution to the above curvature equation is exact, and we find that the equilibrium radius is R0 =

2γ p

.

(51)

Now, assume that the vesicle is trapped in a troubled aqueous medium. Usually, to take into account the presence of impurities, a low-order coupling between their volume fraction, φ , and the mean-curvature C is added to the above free energy F , that is

Z F → F ± kB T

φ C dA.

(52)

The positive sign is for attractive impurities and the negative sign for repulsive ones. The new free energy is then F =

κ

Z

(2C ± 2C0 ) dA + κG 2

2

Z

Z K dA +

b γ dA + p

Z dV

(53)

with the notation C0 = k B T

φ , 4κ

b γ = γ − 2κ C02 .

(54)

Then, the impurities generate an extra spontaneous curvature. This means that these give arise to an asymmetry of the vesicle when its membrane is crossed. In addition, these impurities additively renormalize the interfacial tension coefficient. Minimizing this new curvature free energy yields p − 2b γ C + κ (2C ± C0 ) 2C 2 ∓ C0 C − 2K + κ∇ 2 (2C ) = 0.

(55) 2

For a spherical vesicle, the mean curvature is a constant, and we have K = C . In this case, the mean-curvature C is a root of a polynomial of degree 2. This means that we have an exact solution we do not write. In particular, for very small volume fractions (φ 1), we find that the vesicle equilibrium radius writes R R0

=1±

pkB T 8γ

2

φ + O φ2 ,

(56)

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or

R − R0 pkB T R ' 8γ 2 φ. 0

(57)

The signs (+) and (−) describe attractive and repulsive impurities, respectively. Here, R0 = 2γ /p denotes the equilibrium radius for a spherical vesicle free from impurities. The above results call the following remarks. Firstly, as it should be, the equilibrium radius of the vesicle increases with increasing volume fraction of impurities, when these are attractive. This means that the vesicle is swollen, but it radius remains close to the unperturbed one R0 , as long as the volume fraction of impurities is very small. For repulsive impurities, however, the vesicle is collapsed. Secondly, formula (57) makes sense only when the volume fraction φ is below some threshold φ ∗ = 8γ 2 /kB Tp. Since the latter must be small√in comparison with unity, the vesicle bears interfacial tensions of coefficients that do not exceed some typical value γ ∗ = kB Tp/8. Finally, at fixed volume fraction φ < φ ∗ , this formula may be used to estimate the experimental value of the ratio p/γ 2 , by a simple measurement of the equilibrium radii R (with impurities) and R0 (without impurities). The following paragraph will be devoted to the study of lamellar phases made of two parallel lipid bilayers. 5. Lamellar phases The natural question is the extension of the study to the case when we have more than one membrane immersed in a aqueous media, which is impregnated by small random impurities. In our analysis, we start from a lamellar phase composed of two parallel (neutral) fluid membranes free from impurities. The effects of these entities on physics will be discussed below. The cohesion between these bilayer membranes is ensured by long-ranged attractive van der Waals forces [17], which are balanced, at short membrane separation, by strong repulsion coming from hydration forces [18] and by steric shape fluctuations ones resulting from the membrane undulations [7]. For two parallel bilayer membranes separated by a finite distance l, the total interaction energy per unit area is V (l) = VH (l) + VW (l) + VS (l) .

(58)

The first part VH (l) = AH e−l/λH

(59)

represents the hydration potential (per unit area) that acts at small separations of the order of 1 nm. The corresponding amplitude AH and potential-range λH are about AH ' 0.2 J/m2 and λH ' 0.3 nm. The second part VW (l) = −

W 12π

1 l2

−

2

(l + δ)2

+

1

(l + 2δ)2

(60)

is the attractive van der Waals potential (per unit area) that originates from polarizabilities of lipid molecules and water molecules. Here, W accounts for the Hamaker constant that is in the range W ' 10−22 − 10−21 J, and δ for the bilayer thickness. The latter is of the order of δ ' 4 mm. The last part is the steric shape fluctuations potential (per unit area) [7] Vs (l) = cH

(kB T )2 , κ l2

(61)

with kB the Boltzmann’s constant, T the absolute temperature, and κ the common bending rigidity constant of the two membranes. But in the case of two bilayers of different bending rigidity constants κ1 and κ2 , we have κ = κ1 κ2 / (κ1 + κ2 ). There, the coefficient cH is a known numerical coefficient [7]. We note that the lamellar phase remains stable at the minimum of the potential, provided that the potential depth is comparable to the thermal energy kB T . This depends, in particular, on the value of amplitude W of the direct van der Waals energy. The Hamaker constant W may be varied changing the polarizability of the aqueous medium. In a pioneered theoretical paper, Lipowsky and Leibler [19] have shown that there exists a certain threshold Wc beyond which the van der Waals attractive interactions are sufficient to bind the membranes together, while below this characteristic amplitude, the membrane undulations dominate the attractive forces, and then, the membranes separate completely. According to the authors, Wc is in the interval Wc ' (6.3 − 0.61) × 10−21 J when the bending rigidity constant is in the range κ ' (1 − 20) × 10−19 J. We note that the typical value Wc corresponds to some temperature, Tc , called unbinding critical temperature [19,20]. In particular, it was found [19] that, when the critical amplitude is approached from above, the mean-separation between the two membranes, hli0 , diverges according to

hli0 = ξ⊥0 ∼ (Tc − T )−ψ ,

T → Tc− ,

(62)

with ψ a critical exponent whose value is [19]: ψ ' 1.00 ± 0.03. The latter was computed using field-theoretical Renormalization-Group.

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Now, assume that the lamellar phase is trapped in a troubled aqueous medium. As first implication, the impurities drastically modify the unbinding transition phenomenon, in particular, the critical temperature. For simplicity, suppose that the two adjacent fluid membranes are physically identical. Then, we have to consider two distinct physical situations: The impurities attract or push the two membranes. As we have shown above, for attractive impurities, the membrane roughness ξ⊥ is important, and then, the steric shape fluctuation energy dominates. Therefore, we expect that the unbinding transition occurs at a low critical temperature Tc∗ we will determine below. For repulsive impurities, however, we have an opposite tendency, that is the unbinding transition takes place at a high temperature greater than Tc (absence of impurities). This can be explained by the fact that the membrane roughness is less important, and then, the direct van der Waals energy dominates. In any case, the mean-separation hli scales as

hli = ξ⊥ ∼ Tc∗ − T

−ψ

,

T → Tc− .

(63) ∗

The new unbinding critical temperature Tc can be estimated as follows. Using formulae (29) and (33), we show that, at first order in the impurities coupling w ,

ξ⊥ w ' 1 + + O w2 , 0 4 ξ⊥ ξ⊥ w ' 1 − + O w2 , 0 4 ξ⊥

(attractive impurities) ,

(64)

(repulsive impurities) .

(65)

Combing Eqs. (62) to (65), we find that the difference between the two critical temperatures Tc and Tc∗ (without and with impurities) is as follows Tc − Tc∗ ∼ w + O w 2 ,

Tc − Tc∗ ∼ −w + O w

(attractive impurities) ,

2

,

(repulsive impurities) .

(66) (67)

The prefactors in the above behaviors remain unknown. The temperature shift is then proportional to the volume fraction of impurities and their interaction strength through the coupling w . Then, we are in a situation similar to finite size scaling. For charged membranes forming the lamellar phase, it was found [19] that the mean-separation between two adjacent bilayers scales as [19]

hli0 = ξ⊥0 ∼ (χ − χc )−ψ ,

(68)

in the vicinity of χc , with χ the ionic concentration of the aqueous medium and χc its critical value. For instance, for DPPC in CaCl2 solutions, χc is in the interval [21]: χc ' 84 − 10 mM. In the presence of impurities, we state that the mean-separation behaves as

hli = ξ⊥ ∼ χ − χc∗

−ψ

,

(69)

with the same exponent ψ . In this case, we have

χc∗ − χc ∼ w + O w2 , (attractive impurities) , χc∗ − χc ∼ −w + O w 2 , (repulsive impurities) .

(70) (71)

We note that the same comments may by done in this case. 6. Conclusions In this paper, we presented a large scope about the effects of impurities on the statistical properties of fluid membranes. In fact, these drastically affect the living systems behavior. As an example, we can quote a very recent experimental study [22], where the authors have undertaken a series of comparative experiments, in order to explore the effect of impurities in the form of proteins and lipids on the crystallization of membrane proteins in vapor diffusion. For the present study, the impurities were assumed to be attractive or repulsive. Using the replica trick combined with a variational method, we have computed the membrane mean-roughness, as a function of the parameters associated with the pure membrane and some parameter that is proportional to the volume fraction of impurities and their interaction strength. The main conclusion is that, attractive impurities increase the shape fluctuations due to the membrane undulations, but repulsive ones tend to suppress these fluctuations. Also, we have computed the equilibrium diameter of (spherical) vesicles surrounded by small random particles solving the curvature equation. Thereafter, we extended discussion to lamellar phase formed by two parallel fluid membranes that are a finite distance apart. This lamellar phase may undergo an unbinding transition. We have shown that, attractive impurities increase the unbinding critical temperature, while repulsive ones tend to decrease this temperature.

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We point out that, the incorporation of a small amount of impurities in an aqueous medium may be a mechanism to suppress or to produce an unbinding transition (even the temperature and polarizability of the aqueous medium are fixed) within lamellar phases composed of fluid membranes. Finally, the present study may be extended to bilayer membranes of arbitrary topology. The essential conclusion is that, attractive impurities tend to swell the membrane, while in the presence of repulsive ones, this membrane is collapsed. Also, the used method can be applied to multilayers constituted by several parallel lamellar phases. Acknowledgements We are much indebted to Professors T. Bickel, J.-F. Joanny and C. Marques for helpful discussions, during the ‘‘First International Workshop On Soft-Condensed Matter Physics and Biological Systems’’, 14–17 November 2006, Marrakech, Morocco. One of us (M.B.) would like to thank Professor C. Misbah for fruitful correspondences, and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kind hospitality during his regular visits. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, 1996. V.J. Emery, Phys. Rev. B 11 (1975) 239. S.F. Edvards, P.W. Anderson, J. Phys. (France) 5 (1975) 965. L.D. Landau, E.M. Lifshitz, Quantum Mechanics, 3rd edition, Pergamin Press, 1991. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989. C. Itzykson, J.-M. Drouffe, Statistical Field Theory: 1 and 2, Cambridge University Press, 1989. W. Helfrich, Z. Natureforsch 28c (1973) 693. K. Wolfgang, Differential Geometry: Curves – Surfaces – Manifolds, American Mathematical Society, 2005. S.J. Singer, G.L. Nicolson, Science 175 (1972) 720. O.-Y. Zhong-Can, W. Helfrich, Phys. Rev. Lett. 59 (1987) 2486. Z.C. Tu, Z.C. Ou-Yang, Phys. Rev. E 68 (2003) 061915. O.-Y. Zhong-Can, Phys. Rev. A 41 (1990) 4517. Z. Lin, R.M. Hill, H.T. Davis, L.E. Scriven, Y. Talmon, Langmuir 10 (1994) 1008. M. Mutz, D. Bensimon, Phys. Rev. A 43 (1991) 4525. A.S. Rudolph, B.R. Ratna, B. Kahn, Nature 352 (1991) 52. H. Naito, M. Okuda, O.Y. Zhong-Can, Phys. Rev. E 48 (1993) 2304. J.N. Israelachvili, Intermolecular and Surface Forces, 2nd edition, Academic Press, London, 1991. R.P. Rand, V.A. Parsegian, Biochim. Biophys. Acta 988 (1989) 351. R. Lipowsky, S. Leibler, Phys. Rev. Lett. 56 (1986) 2541. R. Lipowsky, E. Sackmann (Eds.), An Extensive List of References on the Subject can be Found in: Structure and Dynamics of Membranes: Generic and Specific Interactions, vol. 1B, Elsevier, 1995. [21] L.J. Lis, W.T. Lis, V.A. Parsegian, R.P. Rand, Biochemistry 20 (1981) 1771. [22] A. Christopher, et al., Acta Cryst. D 65 (2009) 1062–1073.

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An extended study of the phase separation between phospholipids and grafted polymers on a bilayer biomembrane

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 Phys. Scr. 83 065801 (http://iopscience.iop.org/1402-4896/83/6/065801) View the table of contents for this issue, or go to the journal homepage for more

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PHYSICA SCRIPTA

Phys. Scr. 83 (2011) 065801 (6pp)

doi:10.1088/0031-8949/83/06/065801

An extended study of the phase separation between phospholipids and grafted polymers on a bilayer biomembrane M Benhamou, I Joudar, H Kaidi, K Elhasnaoui, H Ridouane and H Qamar Laboratoire de Physique des Polymères et Phénomènes Critiques, Faculté des Sciences, Ben M’sik, PO Box 7955, Casablanca, Morocco E-mail: [email protected]

Received 4 April 2010 Accepted for publication 9 May 2011 Published 26 May 2011 Online at stacks.iop.org/PhysScr/83/065801 Abstract We re-examine here the phase separation between phospholipids and adsorbed polymer chains on a fluid membrane with a change in some suitable parameter (temperature). Our purpose is to quantify the significant effects of the solvent quality and of the polydispersity of adsorbed loops formed by grafted polymer chains on the segregation phenomenon. To this end, we elaborate on a theoretical model that allows us to derive the expression for the mixing free energy. From this, we extract the phase diagram shape in the composition–temperature plane. Our main conclusion is that the polymer chain condensation is very sensitive to the solvent quality and to the polydispersity of loops of adsorbed chains. PACS numbers: 87.16.Dg, 47.57.Ng, 64.60.F

bounded by sphingolipids such as cholera and tetanus toxins. Sphingolipids and cholesterol favor the aggregation of proteins in microdomains called rafts. In fact, these play the role of a platform for the attachment of proteins while the membranes are moved around the cell and also during signal transduction. Proteins (long macromolecules) are another principal component of cell membranes. Transmembrane proteins are amphipathic and are formed by hydrophobic and hydrophilic regions having the same orientation as other lipid molecules. These proteins are also called integral proteins. Their function is to transport substances, such as ions and macromolecules, across the membrane. There exist other types of proteins that may be attached to the cytoplasm surface by fatty acyl chains or to the external cell surface by oligosaccharides. These are termed peripheral membrane proteins. They have many functions; in particular, they protect the membrane surface, regulate cell signaling and participate in many other important cellular events. In addition, some peripheral membrane proteins (those having basic residues) tend to bind electrostatically to negatively charged membranes, such as the inner leaflet of the plasma membrane.

1. Introduction Biological membranes are of great importance to life, because they separate the cell from the outside environment and separate the compartments inside the cell in order to protect important processes and specific events. Nowadays, it is largely recognized that biological membranes are present as a lipid bilayer composed of two adjacent leaflets [1, 2], which are formed by amphiphile molecules possessing hydrophilic polar-heads pointing outward and hydrophobic fatty acyl chains that form the core. The majority of lipid molecules are phospholipids. These have a polar-head group and two non-polar hydrocarbon tails, whose length is of the order of 5 nm. Also, the cell membranes incorporate another type of lipid, cholesterol [1, 2]. The cholesterol molecules have several functions in the membrane. For example, they give rigidity or stability to the cell membrane and prevent crystallization of hydrocarbons. The biomembranes also contain glycolipids (sugars), which are lipid molecules that microaggregate in the membrane, and may be protective and act as insulators. Certain kinds of molecules are 0031-8949/11/065801+06$33.00

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© 2011 The Royal Swedish Academy of Sciences

Phys. Scr. 83 (2011) 065801

M Benhamou et al

We note that the majority of macromolecules forming the bilayer are simply anchored on the membrane and form a soft branched polymer brush [3, 4]. The study of grafted polymers on soft interfaces was motivated by the fact that they have potential applications in biological materials, such as liposomes [5–8]. These soft materials were discovered by A Bangham. Currently, they are major tools in biology, biochemistry and medicine (as drugs transport agents). Liposomes are artificial vesicles of spherical shape that can be produced from natural nontoxic phospholipids and cholesterol. But the lipid bilayers have a short lifetime, because of the weak stability of the vesicles and their extermination by white blood cells. To have stable vesicles in time, one useful method consists of protecting them with a coat of flexible polymer chains (coat size of the order of 50 nm [1]), which prevents the adhesion of marker proteins [6, 7]. In fact, these polymer chains stabilize the liposomes, due to the excluded volume forces between monomers [8]. Liposomes can also be synthesized from A–B diblock-copolymers immersed in a selective solvent that prefers to be contacted by the polymer A. The hydrophobic B-polymer chains then aggregate and form a thin bilayer, while the hydrophilic polymer chains A float in the solvent. These copolymer-based liposomes have properties that are slightly different from those of the lipid ones [9] (high resistance, high rigidity and weak permeability to water). Depending on the choice of copolymers, these liposomes are resistant to detergents [10]. The grafting of polymers onto lipid membranes was considered in a very recent paper [11]. More precisely, the purpose was to investigate of the phase separation between phospholipids and anchored polymers. As assumptions, the aqueous medium was assumed to be a good solvent, and in addition, the polymer chains were anchored to the interface only by one extremity, a big amphiphile molecule. The latter is chemically different from the phospholipid molecules. In this paper, however, we assume that the polymer chains may be adsorbed on the membrane by many monomers that are directly attached to some polar-heads of the host lipid molecules (mobile anchors), and then organize in polydisperse loops. The surrounding liquid may be a good solvent or a theta solvent. As we shall see below, the loops’ polydispersity and solvent quality drastically affect the phase behavior of the system. Under a change in a suitable parameter, such as temperature, the phospholipids and adsorbed polymer chains phase separate into macroscopic domains alternately rich in the two components. For the determination of the phase diagram shape, we elaborate on a theoretical model that takes into account both the loops’ polydispersity and the solvent quality. This paper is organized as follows. In section 2, we derive the expression for the mixing free energy of the phospholipid–anchor mixture. To investigate the phase diagram shape is the aim of section 3. Finally, some concluding remarks are given in section 4.

bound to the polar-head of a lipid molecule (for many proteins, the lipid anchors are glycosylphosphatidylinositol units) [12, 13]; (ii) by hydrophobic side-groups of the polymers which are integrated into the bilayer [14, 15]; (iii) by membrane spanning hydrophobic domains of the polymer (membrane-bound proteins, for example); and (iv) by a strong adsorption that drives the polymer from a desorbed state to an adsorbed one [16]. For a single polymer chain, the adsorption (on solid substrates) can be theoretically studied using a scaling argument [17], for instance. In addition to the polymerization degree of the polymer chain, N , and the excluded volume parameter, v, the study was based on an additional parameter, δ, which is the energy (per kB T unit) required to adsorb one monomer on the surface. For a strong adsorption, δ −1 defines the adsorbed-layer thickness. An adsorption transition takes place at some typical value δ ∗ of δ scaling as [17]: δ ∗ ∼ RF−1 ∼ a N −ν F (νF = 3/5), where RF is the Flory radius of the polymer chain. For dilute and semi-dilute polymer solutions, the adsorption phenomenon depends, in addition to the parameter δ, on the polymer concentration. In this paper, we consider situation (iv), where each monomer has the same probability of being adsorbed on the membrane surface [18]1 . More precisely, a given monomer is susceptible to becoming linked to a polar-head of a phospholipid molecule (anchor). As a result, the polymer chains form polydisperse loops (with eventually one or two tails floating in the aqueous medium). In fact, this assumption conforms to what was considered in [19]. The case where no loops are present (adsorption only by one extremity) was considered in [11]. Contrary to the adsorption on the solid surface, the anchored polymer chains are mobile on the host membrane and may undergo the aggregation transition that we are interested in. To be more general, the fluid membrane is assumed to be in contact with a good solvent or a theta solvent. The purpose is to write a general expression for the mixing free energy. The latter will allow the determination of the phase diagram related to the aggregation of anchored polymer chains. First, we start with the free energy (per unit area) of the polymer layer (for solid substrates), which is given by [20] 0 Z N F0 1 S (n) 2 β 2 0 ' 2 [b S (n)] + [−b S (n)] ln − dn. kB T b 1 S1 (1) For the configuration study, Guiselin [21] considered that each loop can be viewed as two linear strands (two half-loop). Here, S(n) is the number of strands having more than n monomers per unit area, and b represents the monomer size. The number N denotes the length of the longest strand. Hereafter, we shall use the notation S1 = 8/a, which denotes the total number of grafted chains per unit area, with 8 being the volume fraction of anchors and a their area. If the mixture is assumed to be incompressible, then 1 − 8 is the volume fraction of the phospholipid molecules. Let us come back to the free energy expression (1); note that the first term of the right-hand side represents the

2. Mixing free energy

1

The adsorption of an adequate polymer on a fluid membrane made of two phospholipids of different chemical nature may be a mechanism of phase separation between these two unlike components as was pointed out.

Polymers can adhere to biomembranes in several ways: (i) by lipid anchors; that is, the polymer is covalently 2

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M Benhamou et al

contribution of the (two- or three-body) repulsive interactions between monomers belonging to the polymer layer. The second term is simply the entropy contribution describing all possible rearrangements of the grafted chains in the polymer layer. There, the exponent β depends on the solvent quality [20]. When the grafting is accomplished in a dilute solution with a good solvent or a theta solvent, the values of β are β = 11/6 (blob model) or β = 2, respectively. For polydisperse polymer layers, the distribution S(n) is known in the literature [20]. This can be obtained by minimizing the above free energy with respect to this distribution. Without presenting details, we simply sketch the general result that [20] 8 S(n) ∼ 1/(β−1) . (2) an For usual solvents, we have S(n) ∼

8 an 6/5

(good solvents),

8 (theta solvents). an We shall rewrite the distribution S(n) as S(n) ∼

S(n) =

8 f (n), a

Explicitly, we have ζ (N ) = 1 (polymer brushes), ζ (N ) ∼ N β/(1−β)

(3)

(4)

(5)

F = 8 ln 8 + (1 − 8) ln(1 − 8) + χ 8(1 − 8) kB T + u8β + η(N ) 8.

(6)

β N 1/(1−β) ln N . β −1

(7)

where the positive coefficients C and D are such that (in dimension 2) [23] Z ∞ π D = −π rU (r ) dr, C = σ 2 (covolume). (15) 2 σ

In the asymptotic limit N → ∞, this coefficient goes to η(N ) → a −1 [β + ln(β − 1)],

(8)

provided that β > 1 (note that β = 11/6, for good solvents, and β = 2, for theta ones). This asymptotic limit is always positive definite and inversely proportional to the polar-head area a. In fact, the positive sign of the coefficient η(N ) agrees with the entropy loss due to the polymer chains grafting. As we shall see below, this linear contribution does not change the phase diagram in the composition-critical parameter plane. Here, the coupling constant u is as follows: b2 a

β−1

1 N

Z

u= with ζ (N ) =

N

N ζ (N ),

[ f (n)]β dn.

(13)

We note that, for polymer chains anchored by one extremity, a big amphiphile molecule (the polymer brush case), the first contribution of entropy, 8 ln 8, should be divided by some factor q, which represents the ratio of the anchor area to the area of polar-heads of the host phospholipids. In the present case, we have q = 1. In equality (11), χ accounts for the Flory interaction parameter 1 D χ = χ0 − 2 C − (14) , A kB T

where the linear term in 8 describes the contribution of entropy to the free energy, where the coefficient η(N ) is as follows (see the appendix): −1 [β + ln(β − 1)] 1 − N 1/(1−β) η(N ) = a +

(12)

Note that ζ (N ) < 1, since, in all cases, β > 1. Therefore, the polydispersity of loops decreases the repulsive interaction energy. Now, we have all the ingredients for the determination of the expression for the mixing free energy. To this end, we imagine that the interface is present as a two-dimensional (2D) Flory–Huggins lattice [17, 22], where each site is occupied by an anchor or by a phospholipid molecule. Hence, we can regard the volume fraction of anchors 8 as the probability that a given site is occupied by an anchor. Therefore, 1 − 8 is the occupation probability of phospholipids. Before the determination of the desired mixing free energy (per site), we note that the latter is the sum of three contributions, which are the mixing entropy (per site), the volume free energy (per site) and the interaction energy (per site) coming from the membrane undulations. Actually, the induced attractive forces due to the membrane undulations are responsible for the condensation of anchors. These forces balance the repulsive ones between monomers along the connected polymer chains. We then write

with f (n) ∼ 1/n 1/(β−1) , for polydisperse polymer layers, and f (n) = 1 (for all n), for polymer brushes. Therefore, F0 can be approximated by F0 = a −1 u8β + η(N )8, kB T

(polydisperse systems).

(11)

Here, U (r ) is the pair potential induced by the membrane undulations ([11]; [24] and references therein) r < σ, ∞, σ 4 U (r ) = (16) −AH , r > σ, r where σ is the hard disc diameter, which is proportional to the square root of the anchor area a. There, the potential amplitude AH plays the role of the Hamaker constant. It was found that the latter decays with the bending rigidity constant according to ([11]; [24] and reference therein)

(9)

(10)

AH ∼ κ −2 .

1

3

(17)

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But this amplitude is also sensitive to temperature. We remark that the above mixing free energy is not symmetric under the change 8 → 1 − 8. Some straightforward algebra gives the following expression for the attraction parameter D: D = AH

π κ −2 ∼ . 2σ 2 σ2

interactions widen the compatibility domain, and then, the separation transition appears at low temperature. Now, to see the influence of the solvent quality, we rewrite the interaction parameter u as u = u 0 ζ (N ) ∼ u 0 N β/(1−β) < u 0 , where u 0 is the interaction parameter relative to a monodisperse system. Thus, the polydispersity of loops has a tendency to reduce the compatibility domain in comparison with the monodisperse case. The critical volume fraction, 8c , can be obtained by minimizing the critical parameter χ (8) with respect to the 8-variable. We then obtain

(18)

Therefore, those membranes with a small bending modulus induce significant attraction between anchors. In formula (12), χ0 > 0 is the Flory interaction parameter describing the chemical segregation between amphiphile molecules that are phospholipids and anchors. This segregation parameter is usually written as χ0 = α0 +

1 1 = 0. − 2 + β(β − 1) (β − 2) u8β−3 c 2 8 (1 − 8c ) c For good solvents (β = 11/6), we have

γ0 , T

(19) 55 1 1 − 2− u8−7/6 = 0. c 2 8 216 (1 − 8c ) c

where the coefficients α0 and γ0 depend on the chemical nature of the various species. Also, the total interaction parameter χ can be written as χ =α+

γ , T

C , A2

γ = γ0 +

(20)

D . kB A2

(24)

Therefore, the critical volume fraction is the abscissa of the intersection point of the curve of the equation (2x − 1)/ x 5/6 (1 − x)2 and the horizontal straight line of the equation y = (55/216)u. Note that this critical volume fraction is unique, and in addition, it must be greater than the value 1/2 (for mathematical compatibility). The coordinates of the critical point are (8c , χc ), where 8c solves the above equation and χc = χ (8c ). The latter can be determined by combining equations (19) and (21). For theta solvents (β = 2), the coordinates of the critical point are exact,

with the new coefficients α = α0 −

(23)

(21)

These coefficients then depend on the chemical nature of the unlike components and on the membrane’s characteristics through its bending modulus κ. If we admit that the coupling constant u has a slight dependence on temperature, we will draw the phase diagram in the plane of variables (8, χ). Indeed, all of the temperature dependence is contained in the Flory interaction parameter χ.

8c = 12 ,

χc = 2 + u,

(25)

where the interaction parameter u scales as u=

3. Phase diagram

b2 −1 N . a

(26)

The above relation clearly shows that the polymer chains’ condensation rapidly takes place only when the characteristic mass N is high enough. The same tendency is also seen in the case of good solvents. It is straightforward to show that the critical fraction and the critical parameter are shifted to lower values in the case of polydisperse systems, whatever be the quality of the surrounding solvent. In figure 1, we present the spinodal curve for monodisperse (with no loops) and polydisperse (with loops) systems, with a fixed parameter N . We have chosen the good solvents situation. For theta solvents, the same tendency is seen. We present in figure 2 the spinodal curve for a polydisperse system (with loops), at various values of the parameter N . As expected the critical parameter is shifted to higher values when we augmented the typical polymerization degree. Finally, we compare, in figure 3, the spinodal curves for a polydisperse system (with loops) for the case of theta solvents and those for the case of good solvents, at fixed parameters b and N . All the curves in the figure reflect our discussions above.

With the help of the above mixing free energy, we can determine the shape of the phase diagram in the (8, χ )-plane that is associated with the aggregation process that drives the anchors from a dispersed phase (gas) to a dense one (liquid). We focus only on the spinodal curve along which the thermal compressibility diverges. The spinodal curve equation can be obtained by equating to zero the second derivative of the mixing free energy with respect to the anchor volume fraction 8; that is, ∂ 2 F/∂82 = 0. Then, we obtain the following expression for the critical Flory interaction parameter: 1 1 χ(8) = + β(β − 1) u8β−2 . (22) 2 8(1 − 8) Above this critical interaction parameter appear two phases: one is homogeneous and the other is separated. Of course, the linear term in 8 appearing in equality (11) does not contribute to the critical parameter expression. We remark that, in usual solvents, this critical interaction parameter is increased due to the presence of (twoor three-body) repulsive interactions between monomers belonging to the polymer layer. This means that these 4

Phys. Scr. 83 (2011) 065801

χ

M Benhamou et al

χ

80

20

16

60

12 40

8 20

4

0 0,0

0,2

0,4

0,6

0,8

0 0,0

1,0

Φ

12

8

4

0,4

0,6

0,8

0,8

1,0

polymer chains. The latter were taken into account through the well-known form of the chains’ length distribution. In doing so, we first computed the expression for the mixing free energy by adopting the Flory–Huggins lattice image usually encountered in polymer physics [17, 22]. Such an expression shows that there is competition between four contributions: entropy, chemical mismatch between unlike species, interaction energy induced by the membrane undulations and the interaction energy between monomers belonging to the grafted layer. Such a competition governs the phases succession. We emphasize that the present work and a previous work [11] differ from another previous work that was concerned with the same problem, but in which the substrate was assumed to be a rigid surface [3]. Therefore, the membrane undulations were neglected. As we have seen, these undulations increase the segregation parameter χ by an additive term, χm , scaling as κ −2 . This means that the phase separation is accentuated due to the presence of thermal fluctuations. Compared with the previous work [11], which was concerned with soft brushes with monodisperse end-grafted polymers, the present work is more general, since it takes into account both the polydispersity of loops forming the adsorbed polymer chains and the solvent quality. Thus, the present study was achieved in a unified way. As we have seen, these details drastically affect the phase diagram architecture. Now, let us discuss further the influence of solvent quality on the critical phase behavior. We recall that the solvent quality appears in the free energy (11) through the repulsion parameter u, defined in equation (7). We find, in the N → ∞ limit, that u g ∼ N 1/5 u θ , where the subscripts g and θ stand for good and theta solvents, respectively. This implies that the good solvent plays the role of a stabilizer regarding the phase separation. Finally, this work must be considered as a natural extension of a study reported previously [11], which was concerned with monodisperse end-grafted polymer chains

16

0,2

0,6

Figure 3. Superposition of spinodal curves for a polydisperse system for the case of theta solvents (solid line) on those for the case of good solvents (dashed line). For these curves, we chose N = 100 and b2 = 0.5a.

20

0 0,0

0,4

Φ

Figure 1. Spinodal curves (in a good solvent) for monodisperse (dashed line) and polydisperse (solid line) systems, when N = 100, with the parameter b2 = 0.5a.

χ

0,2

1,0

Φ Figure 2. Spinodal curves for a polydisperse system, when N = 50 (solid line), 100 (dashed line) and 150 (dotted line), with the parameter b2 = 0.5a . We assumed that the surrounding liquid is a good solvent. For theta solvents, the tendency is the same.

4. Discussion and conclusions This paper is devoted to the thermodynamic study of the aggregation of the polymer chains adsorbed on a soft surface. Such an aggregation is caused by competition between the chemical segregation between phospholipids and grafted polymer chains, their volumic interactions and the membrane undulations. More precisely, we addressed the question of how these polymer chains can be driven from a dispersed phase (gas) to a dense one (liquid), under a change in a suitable parameter, e.g. the absolute temperature. To be more general, we achieved the study in a unified way; that is, we have considered more realistic physical situations: the solvent quality (good or theta solvents) and the polydispersity of the adsorbed loops formed by the grafted 5

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M Benhamou et al

trapped in a good solvent. Therefore, the present work presents a wide perspective on the phenomenon of segregation between the host phospholipids and grafted polymer chains on bilayer membranes.

References [1] Lipowsky R and Sasckmann S (ed) 1995 Structure and Dynamics of Membranes: From Cells to Vesicles vol 1A (New York: Elsevier) [2] Benhamou M 2008 Lipids Insights 1 2 [3] Nicolas A and Fourcade B 2003 Eur. Phys. J. E 10 355 [4] Aubouy M et al 2000 Phys. Rev. Lett. 84 4858 Aubouy M et al 1996 Macromolecules 29 7261 [5] Ringsdorf H and Schmidt B 1986 How to bridge the gap between membrane, biology and polymers science Synthetic Membranes: Science, Engineering and Applications ed P M Bungay et al (Dordrecht: D Reidel) p 701 [6] Lasic D D 1992 Am. Sci. 80 250 [7] Torchilin V P 1996 Effect of polymers attached to the lipid head groups on properties of liposomes Handbook of Nonmedical Applications of Liposomes vol 1, ed D D Lasic and Y Barenholz (Boca Raton, FL: RCC Press) p 263 [8] Joannic R, Auvray L and Lasic D D 1997 Phys. Rev. Lett. 78 3402 [9] Discher B M et al 1999 Science 284 1143 [10] Elferink M G L et al 1992 Biochim. Biophys. Acta 1106 23 [11] Benhamou M, Joudar J and Kaidi H 2007 Eur. Phys. J. E 24 343 [12] Blume G and Cev G 1990 Biochim. Biophys. Acta 1029 91 [13] Lasic D D, Martin F J, Gabizon A, Huang S K and Papahadjopoulos D 1991 Biochim. Biophys. Acta 1070 187 [14] Decher G et al 1989 Angew. Makromol. Chem. 166 71 [15] Simon J, Kuhner M, Ringsdorf H and Sackmann E 1995 Chem. Phys. Lipids 76 241 [16] Garel1 T, Kardar M and Orland H 1995 Europhys. Lett. 29 303 Chatellier X and Andelman D 1995 Europhys. Lett. 32 567 Lipowsky R 1995 Europhys. Lett. 30 197 Xie A F and Granick S 2002 Nature Mater. 1 129 [17] de Gennes P-G 1979 Scaling Concept in Polymer Physics (Ithaca, NY: Cornell University Press) [18] Binder W H, Barragan V and Menger F M 2003 Angew. Chem. Int. Edn Engl. 42 5802 [19] Manghi M and Aubouy M 2001 Adv. Coll. Interf. Sci. 94 21 [20] Manghi M and Aubouy M 2003 Phys. Rev. E 68 041802 [21] Guiselin O 1992 Euro. Phys. Lett. 17 225 Guiselin O 1992 PhD Thesis Paris VI University [22] Flory P J 1953 Principles of Polymer Chemistry (Ithaca, NY: Cornell University Press) [23] Balian R 2006 From Microphysics to Macrophysics—Methods and Applications of Statistical Physics (Berlin: Springer) [24] Marchenko V I and Misbah C 2002 Eur. Phys. J. E 8 477

Acknowledgments We are indebted to Professors T Bickel, J-F Joanny and C Marques for helpful discussions during the First International Workshop on Soft-Condensed Matter Physics and Biological Systems (14–17 November 2006, Marrakech, Morocco). MB thanks Professor C Misbah for fruitful correspondence and the Laboratoire de Spectroscopie Physique (Joseph Fourier University of Grenoble) for their kind hospitality during his visit. We are grateful to the referee for critical remarks and useful suggestions that helped us to improve the scientific content of this paper.

Appendix The aim is to determine the coefficient η(N ) appearing in formula (6). We start with the entropy contribution to the free energy 0 Z N 2 0 Fentropic 1 S (n) ' 2 −b S (n) ln − dn, (A.1) kB T b 1 S1 where the loop-size distribution is defined in equation (2). Then, the above expression can be rewritten as Fentropic = η(N ) 8, kB T

(A.2)

with N

1 n β/(1−β) dn. n β/(1−β) ln β −1 1 (A.3) Straightforward algebra gives η(N ) = a −1 [β + ln(β − 1)] 1 − N 1/(1−β) η(N ) = a −1

1 β −1

+

Z

β N 1/(1−β) ln N . β −1

(A.4)

This concludes the determination of the coefficient η(N ).

6

African Journal Of Mathematical Physics Volume 10(2011)55-64

Statistics of a single D-manifold restricted to two parallel biomembranes or a tubular vesicle M. Benhamou∗ , K. Elhasnaoui, H. Kaidi, M. Chahid Laboratoire de Physique des Polym`eres et Ph´enom`enes Critiques Facult´e des Sciences Ben M’sik, P.O. Box 7955, Casablanca, Morocco ∗ [email protected]

abstract The purpose is an extensive conformational study of a single polymer immersed in an aqueous medium (good solvent) delimitated by bilayer membranes. To be more general, we assume that the polymer is of arbitrary topology we term D-polymeric fractal or Dmanifold, where D is the spectral dimension (for instance, D = 1, for linear polymers, and 4/3, for branched ones). The main quantity to consider is the parallel extension of the conﬁned polymer. To make explicit calculations, we suppose that the polymer is restricted to a tubular vesicle or two parallel biomembranes. We ﬁrst show that, for the ﬁrst geometry, the polymer is conﬁned only when the tubular vesicle is in equilibrium state. For the second geometry, the conﬁnement is possible if only if the two parallel membranes are in their binding state, that is below the unbinding or adhesion temperature. In any case, the parallel gyration radius of the conﬁned polymer is computed using an extended Flory-de Gennes theory. As result, this radius strongly depends on the polymer topology (through the spectral dimension D) and on the membranes sizes, which are the equilibrium diameter (function of bending modulus, pressure diﬀerence between inner and outer sides of the membrane, and interfacial tension coeﬃcient), for the ﬁrst geometry, and the mean-separation (function of temperature and interaction strength between the adjacent membranes), for the second one. Finally, we give the expression of the conﬁnement free energy, as a function of the polymer size, and discuss the eﬀects of external pressure or lateral tension on the radius expression for two conﬁning parallel membranes. Key words: D-Polymeric fractals, Biomembranes, Vesicles, Conﬁnement.

I. INTRODUCTION

The polymer conﬁnement ﬁnds many applications in various domains, such as biological functions, ﬁltration, gel permeation chromatography, heterogeneous catalysis, and oil recuperation. The physics of polymer conﬁnement is a rich and exciting problem. Recently, much attention has been paid to the structure and dynamics of polymer chains conﬁned to two surfaces, or inside cylindrical pores [1 − 7]. Thereafter, the study has been extended to more complex polymers, called D-polymeric fractals or D-manifolds [8] that are restricted to the same geometries. Here, D is the spectral dimension that measures the degree of the connectivity of monomers inside the polymer [9]. For instance, this intrinsic dimension is 1 for linear polymers and 4/3 for branched ones. The D-manifolds may be polymerized (or

0

c a GNPHE publication 2011, [email protected] ⃝

55

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African Journal Of Mathematical Physics Volume 10(2011)55-64

crumpled) vesicles [10]. The polymer conﬁnement between two surfaces and in cylinders with sinusoidal undulations was also investigated [11, 12]. The conﬁning geometries may be soft-bodies, as bilayer biomembranes and surfactants, and spherical and tubular vesicles. A particular question has been addressed to polymer conﬁnement in surfactant bilayers of a lyotropic lamellar phase [13], where the authors reported on small-angle X-ray scattering and freefracture electron microscopy studies of a nonionic surfactant/water/polyelectrolyte system in the lamellar phase region. The fundamental remark is that, the polymer molecules cause both local deformation and softening of the bilayer. In the same context, it was experimentally demonstrated [14] that the polymer conﬁnement may induce a nematic transition of microemulsion droplets. More precisely, the authors showed that upon conﬁnement, spherical droplets deform to prolate ellipsoid droplets. The origin of such a structural transition may be attributed to a loss of the conformational entropy of polymer chains due to the conﬁnement. In other experiments [15 − 17], a new phase has been observed adding an neutral hydrosoluble polymer (PVP) in the lyotropic lamellar phase (CPCI/hexanol/water). Also, one has studied the eﬀect of a neutral water-soluble polymer on the lamellar phase of a zwitterionic surfactant system [18]. The polymer conﬁnement is also relevant for living systems and governs many biological processes. As example, we can quote membrane nanotubes that play a major role in intercellular traﬃc, in particular for lipid and proteins exchange between various compartments in eukaryotic cells [19 − 22]. The traﬃc of macromolecules and vesicles in nanotubes is ensured by molecular motors [23]. Also, these are responsible for the extraction of nanotubes [24]. The formation of tubular membrane tethers or spicules [25 − 33] results from the action of localized forces that are perpendicular to the membrane. These forces may originate from the polymerization of ﬁbers [34], as actin [35], tubulin [36], or sickle hemoglobin [37]. Inspired by biological processes, as macromolecules and vesicles transport, we aim at a conformational study of conﬁned polymers in aqueous media delimitated by biomembranes. More precisely, the purpose is to see how these biomembranes can modify the conformational properties of the constrained polymer. To this end, we choose two geometries : a tubular vesicle (Geometry I), and two parallel biomembranes (Geometry II). To be more general, we consider polymers or arbitrary connectivity called D-polymeric fractals or D-manifolds [38]. Linear and branched polymers, and polymerized vesicles constitute typical examples. We note that this conformational study in necessary for the description of dynamic properties of polymers restricted to these geometries. As we shall below, the polymer conﬁnement is entirely controlled by the conﬁning biomembranes state. Indeed, for Geometry I, the polymer may be conﬁned only when the conﬁning tubular vesicle is at equilibrium. For Geometry II, the conﬁnement is possible if only if the two parallel membranes are in the binding state. This paper is organized as follows. In Sec. II, we brieﬂy recall the conformational study of unconﬁned D-polymeric fractals in good solvent. Sec. III deals with the conformational study of polymers conﬁned inside a tubular vesicle. In Sec. IV, we extend the study to D-polymeric fractals conﬁned to two parallel biomembranes. Some concluding remarks are drawn in the last section.

II. UNCONFINED POLYMERIC FRACTAL

Consider a single polymeric fractal of arbitrary topology (linear polymers, branched polymers, polymer networks, ...). We assume that the considered polymer is trapped in a good solvent. We denote by RF ∼ aM 1/dF

(1)

its gyration (or Flory) radius, where dF is the Hausdorﬀ fractal dimension, M is the molecular-weight (total mass) of the considered polymer, and a denotes the monomer size. The mass M is related to the linear dimension N by : M = N D , where D is the spectral dimension [9]. The latter is deﬁned as the Hausdorﬀ dimension corresponding to the maximal extension of the fractal. Naturally, the Hausdorﬀ dimension depends on the Euclidean dimensionality d, the spectral dimension D and the solvent quality. When the polymer is ideal (without excluded volume forces), its Hausdorﬀ dimension, d0F , is a known simpler function of D [9] d0F =

2D . 2−D 56

(2)

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African Journal Of Mathematical Physics Volume 10(2011)55-64

For linear polymers (D = 1), d0F = 2 [2], for ideal branched ones (D = 4/3), d0F = 4 [9], and for crumpled membranes (D = 2), d0F = ∞. Because of the positivity of the Hausdorﬀ dimension, the above expression makes sense only for D < 2. Indeed, this condition is fulﬁlled for any complex polymer with spectral dimension in the interval 1 ≤ D < 2 [9]. A polymeric fractal in good solvent is swollen, because of the presence of the excluded volume forces. The polymer size increases with increasing total mass M according to the power law (1). The ﬁrst implication of the polymer swelling is that, the actual Hausdorﬀ dimension dF is quite diﬀerent from the Gaussian one, deﬁned in Eq. (2). However, there exists a special value of the Euclidean dimensionality called upper critical dimension duc [39, 40], beyond which the polymeric fractal becomes ideal. This upper dimension is naturally a D-dependent function, which can be determined [39] using a criterion of Ginzburg type, usually encountered in critical phenomena [41, 42]. According to Ref. [39], duc is given by duc =

4D . 2−D

(3)

For instance, the upper critical dimension is 4 for linear polymers [2], and 8 for branched ones [39]. We emphasize that, in general, the Hausdorﬀ fractal dimension dF cannot be exactly computed. Many techniques have been used to determine its approximate value, in particular, the Flory-de Gennes (FD) theory [2]. Using a generalized FD approach, it was found that the fractal dimension is given by [39] dF = D

d+2 , D+2

(4)

below the critical dimension, and it equals the Gaussian fractal dimension d0F described above. For dimension 3, we have dF (3) =

5D . D+2

(5)

For instance, for linear polymers, dF (3) = 5/3, and dF (3) = 2, for branched ones (animals). In the following paragraph, we shall focus our attention on the conformational study of a single polymeric fractal conﬁned to a long tubular vesicle.

III. CONFINED POLYMERIC FRACTAL IN GEOMETRY I A. Useful backgrounds

Before studying the conformation of a single polymeric fractal, we recall some basic backgrounds concerning the equilibrium shape of tubular vesicles. This can be done using Diﬀerential Geometry machineries. The tubular vesicle is essentially formed by two adjacent leaﬂets (inner and outer) that are composed of amphiphile lipid molecules. These permanently diﬀuse with the molecules of the surrounded aqueous medium. Such a diﬀusion then provokes thermal ﬂuctuations (undulations) of the membrane. This means that the latter experiences ﬂuctuations around an equilibrium plane we are interested in. Consider a biomembrane of arbitrary topology. A point of this membrane can be described by two local coordinates (u1 , u2 ). From surfaces theory point of view, at each point, there exists two particular curvatures (minimal and maximal), called principal curvatures, denoted C1 = 1/R1 and C2 = 1/R2 . The quantities R1 and R2 are the principal curvature radii. With the help of the principal curvatures, one constructs two invariants that are the mean-curvature C=

1 (C1 + C2 ) , 2

(6)

and the Gauss curvature K = C1 C2 .

57

(7)

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African Journal Of Mathematical Physics Volume 10(2011)55-64

We recall that C1 and C2 are nothing else but the eigenvalues of the curvature tensor [43]. To comprehend the geometrical and physical properties of the biomembranes, one needs a good model. The widely accepted one is the ﬂuid mosaic model proposed by Singer and Nicholson in 1972 [44]. This model consists to regard the cell membrane as a lipid bilayer, where the lipid molecules can move freely in the membrane surface like a ﬂuid, while the proteins and other amphiphile molecules (cholesterol, sugar molecules, ...) are simply embedded in the lipid bilayer. We note that the elasticity of cell membranes crucially depends on the bilayers in this model. The elastic properties of bilayer biomembranes were ﬁrst studied, in 1973, by Helfrich [45]. The author recognized that the lipid bilayer could be regarded as smectic-A liquid crystals at room temperature, and proposed the following curvature free energy ∫ ∫ ∫ ∫ κ 2 (2C + 2C0 ) dA + κG KdA + γdA + p dV . (8) F = 2 where dA denotes the area element, and V is the volume enclosed within the lipid bilayer. In the above deﬁnition, κ accounts for the bending rigidity constant, C0 for the spontaneous curvature, κG for the Gaussian curvature, γ for the surface tension, and p for the pressure diﬀerence between the outer and inner sides of the vesicles. The ﬁrst order variation gives the shape equation of lipid vesicles [46] ( ) p − 2γC + κ (2C + C0 ) 2C 2 − C0 C − 2K + κ∇2 (2C) = 0 , (9) with the surface Laplace-Bertlami operator ) ∂ gg , (10) ∂uj ( ) where g ij is the metric tensor on the surface and g = det g ij . For open or tension-line vesicles, local diﬀerential equation (9) must be supplemented by additional boundary conditions we√do not write [47]. The above equation have known three analytic solutions corresponding to sphere [46], 2-torus [48 − 51] and biconcave disk [52]. For cylindrical (or tubular) vesicles, one of the principal curvature is zero, and we have 1 ∂ ∇ =√ g ∂ui 2

C=−

( √

ij

1 , R

K=0,

(11)

where R is the radius of the cylinder. If we ignore the boundary conditions (assumption valid for very long tubes), the uniform solution to equation (9) is ( H=2

4κ p

)1/3 ,

(12)

where H is the equilibrium diameter. We have neglected the surface tension and spontaneous curvature contributions, in order to have a simpliﬁed expression for the equilibrium diameter. The above relation makes sense as long as the pressure diﬀerence is smaller than a critical value pc that scales as [53] : pc ∼ κ/R3 . The latter is in the range 1 to 2 Pa. The meaning of the critical pressure is that, beyond pc , the vesicle is unstable. This implies that the equilibrium diameter must be greater than 1/3 the critical one Hc = 2 (4κ/pc ) . In what follows, we shall use the idea that consists to regard the tubular vesicle as a rigid cylinder of eﬀective diameter H that depends on the characteristics of the bilayer through the parameters κ and p.

B. Parallel extension to the cylinder-axis

Consider now a polymeric fractal of arbitrary topology conﬁned inside a tubular vesicle of equilibrium diameter H. The host solvent is assumed to be a good solvent. The polymer is conﬁned if only if its three-dimensional Flory radius RF 3 is much larger than the diameter H, that is H ?

E : F GH E :9 < 9D

.

(8)

The notation 〈… 〉 means the thermal average, which is performed with the defined in Eq. (2). The general expression for Canham-Helfrich Hamiltonian this propagator we do not recall is known in literature [11-13]. The particular value of the propagator , 0 = JK defines the mean-roughness of the membrane, JK . The latter and the in-plane correlation length (beyond which the two-point correlation function exponentially fails), J∥ , are related by : JK = J∥ /16"̂ . To solve integral equation (7a), we set = is arbitrary. It is straightforward to show that ∑

Q

N

,

N

=(

N,

− ′ =,

where the subscript 1 ≤ P ≤ − ′ ,

N

(9)

with the matrix elements Q

N

+7 ,

−

N

.

(10)

− ′ ,

(11)

.

(12)

The inversion of Eq. (7a) gives − ′ = ∑N

,

RS

N

,

N

with the matrix coefficients R

N

= QN = (

N

+ 7N ,

−

N

Matrix R is then the transpose of matrix Q. Combining now relationships (7a) and (11) yields the expression for the height-correlation function , , ′ =,

− ′ −∑

,N

, 5

−

T N,

N

− ′ ,

(13)

with coefficients T

N

=

7 + 7N RS

N

.

(14)

These define a squared matrix, T. In Appendix A, we show that this matrix is symmetric and positive definite. The main result (13) calls the following remarks. Firstly, as it must be, the height-correlation function is not invariant under translations in the parallel directions to the fluid membrane and substrate, but the bare correlation function , does. The translation symmetry breaking can be directly seen on integral equation (7). Secondly, since matrix T is symmetric, the propagator is invariant permutting the points and ′. Thirdly, all dependence of this propagator in elastic constants 's is entirely contained in matrix elements T N 's. Fourthly, the positivity property of the second term in the r.h.s of expression (13) indicates that the presence of tethers considerably reduces the thermal fluctuations. Sixthly, the expected propagator is completely determined by the knowledge of matrix T and bare propagator , . Finally, as it should be, at infinity, that is → ∞ and ′ → ∞, , , ′ goes to zero. 4. Fluctuation spectra and discussion The local membrane mean-roughness, JK , is simply the value of propagator , , ′ for = ′. From expression (13), we obtain the following exact result JK

= JK

−∑

,N

,

−

T N,

−

N

.

(15)

Matrix elements T N ’s depend on all distances between tethers W − N W’s and their elastic constants ’s. Before discussing the above finding, we rewrite it on the following scaling form

6

JK

= JK X1 − ∑

Y

,N

Z NY T

−

−

N

,

(16)

with the dimensionless quantities

Y

−

Z T

N

= JK T

=,

−

N

,

(17)

/ JK

.

(18)

Z N ’s The latter is small than unity and Y 0 = 1. Notice that coefficients T depend only on dimensionless elastic constants 7 JK ’s and renormalized bare

propagators Y − N ’s. Now, let us discuss the obtained result. Firstly, to comprehend the general expression (16), we examine some cases : (i) One tether: Z is a scalar quantity, and we Let be the vector-position. In this case, matrix T have Z = 7 JK /[1 + 7 JK \ . T

(19)

Combing this equality with formula (13) yields JK

= JK ]1 −

a 9 7 _` ^

a 9 7 _` G^

Y

−

.

(20)

(ii) Two tethers: We denote by and the position-vectors of the two tethers. In this case, we Z are as follows find that the matrix elements of T Z T

=

a _`

b

9

[ 7 + 7 7 JK \ , Z T

Z =T

=−

Z T

a _`

b

F

with 7

=

7 7 Y

a _`

b

9

−

[ 7 + 7 7 JK \ , (21a) ,

(21b)

c = 1 + 7 JK

+ 7 JK

+ 7 7 JK

d

− 7 7 JK d Y

−

. (21c)

Then, the associated local roughness is JK

Z Y Z Y 1−T − −T − = JK e Z Y −2T − Y −

f

/

.

(22)

(iii) The tethers have the same length and their extremities coincide at one : point ≡ This case corresponds to a -watermelon tether, and we find that the associated local roughness is given by JK

= JK h1 − iY

−

,

(23)

with the constant i=∑

,N

Z T

N

.

(24)

Z. A Such a constant is nothing else but the sum of all coefficients of matrix T comparison between relations (20) and (23) suggests that a -watermelon tether Z , such that is equivalent to one tether of effective elastic constant j Z JK /[1 + j Z JK \ . i=j

(25)

i = 7 JK /[1 + 7 JK \ .

(26)

Z= 7. j

(27)

Z to that of one tether, 7 . We Such a equality relates the new elastic constant j show in appendix B that the quantity i is exactly given by

By comparison, we have

Secondly, in all cases, the positivity of the second term in the r.h.s of equality (16) implies that the local roughness, JK , remains above the bulk one, JK , at 8

any point of the membrane. Therefore, the presence of tethers has tendency to suppress the shape fluctuations of the supported fluid membrane. Thirdly, for any number of tethers, when the considered point is far from the positions of the tethers extremities ’s, that is | − | ≫ J∥ , the local roughness JK fails exponentially to the bulk one, JK . Finally, let us assume that the tethers are long-flexible polymer chains (in dilute solution) of common polymerization degree, m. In this case, the associated elastic constants are identical and scale as noS ~qS m S , where q represents the monomer size. For infinitely long-polymer chains, that is m → ∞, Z goes to JK /no r, where r is the matrix unity. In this limit, to first matrix T order in m S , the general result (16) becomes JK

= JK s1 −

JK /no

∑

Y

−

+ t mS u .

(28)

The above equality can be seen as a Taylor series in JK /no -variable. 5. Concluding remarks

We recall that the purpose of this paper is the investigation of correlations between points belonging to a supported membrane that is linked to a solid surface via long-flexible polymer chains. The presence of the connected macromolecules can be regarded as a new mechanism to bind the membrane for specific interests. This binding phenomenon depends naturally on the value of the molecular-weight of the connected polymer chains. High-molecular-weights have tendency to increase the shape-fluctuations amplitude. But, low-molecular-weights suppress these fluctuations. According to formula expressing the local fluctuations amplitude, the quality constitutes a pertinent parameter of discussion. In fact, the presence of a good solvent reduce Our predictions reveal that the fluctuations amplitude is local, and it is more important in regions of the membrane with low-grafting density. This means that such amplitude depends strongly on the distribution of the anchoring points. In our description, we assumed that the anchoring points on the inner monolayer of the supported membrane are immobile. In fact, these points free move since they are fixed on polar-heads of some phospholipids. Such a problem will be discussed elsewhere. 9

For simplicity, we have ignored primitive interactions between the supported membrane and the wall, as van der Waals, hydration and shape-fluctuations interactions. Under certain physical circumstances, theses interactions can be neglected (rigid tethers…). Finally, questions such as dynamics, extension of study to more than one membrane and other developments are under consideration.

10

Appendix A The aim is to demonstrate that matrix T, defined in Eq. (14), is symmetric and v and ,̅ , of positive definite. To this end, we introduce two matrices, j respective elements 7 ( N and , − N , with indices 1 ≤ x ≤ and 1 ≤ P ≤ , and ( N = 1, if x = P, and ( N = 0, otherwise. It is easy to see that the matrix B expresses as follows v, R = r + ,̅ j

(A.1)

where r is the matrix unity. On the other hand, the matrix T can be rewritten as v r + ,̅ j v v RS = j T=j

S

.

(A.2)

v S and ,̅ are Such a form clearly suggests that Ty = T, since both matrices j symmetric. Then, matrix T is symmetric. To show that T is positive definite, we consider its inverse matrix TS = v S . Explicitly, we have Rj v S + ,̅ . TS = j

(A.3)

The associated matrix elements are TS

N

= 7S (

N

+,

−

.

N

(A.4)

To prove that matrix T is positive definite, it will be sufficient to show that z≡∑

TS

,N

N N

>0,

(A.5)

for any set of integer variables ,…, , and z = 0, if only if ,…, = 0, … ,0 . Start from the Fourier integral representation of the bare propagator ,

−

N

=

8 9 :; =

; .|A ; } BA ; ~• >?

E : F GH E :9 < 9 D

We show that 11

.

(A.6)

z=∑

• ; .A ; } •………………………………………………………… >? ; .A ;} „? ∑• 8 9 :; |∑}‚ƒ €} = }‚ƒ €~ =

+

E : F GH E :9 D

< 9

.

(A.7)

Here, ̅ denotes the complex conjugate of , and ̅ > 0 is its squared module. Since the elastic constants "̂ , $2 of the fluid membrane are positive, z > 0. The latter vanishes, if only if = 0 x = 1, … , . Matrix TS is then positive definite. Therefore, T is a positive definite matrix. This ends the proof of the desired mathematical properties of matrix T. Appendix B To show formula (25), use is made of relation (A.4) of Appendix A. We start with identity ∑

T

,N

N

TS

N†

=(

†

.

(B.1)

According to Eq. (A.4), we have ∑

,N

T N [ 7 S (N† + JK \ = (

A double summation over indices x and ‡ gives ∑

,N

Z T

N

†

.

= i = 7 JK /[1 + 7 JK \ .

We have Eq. (17). This ends the proof of formula (25).

12

(B.2)

(B.3)

References [1] More details can be found in : M. Tanaka and E. Sackmann, Nature 437, 656 (2005). [2] E. Sackmann, Science 271, 43 (1996). [3] S. Terrettaz, M. Mayer and H. Vogel, Langmuir 19, 55675569 (2003). [4] P.C. Gufler, D. Pum, U.B. Sleytr and B. Schuster, Biochimica et Biophysica Acta 1661, 154 (2004). [5] V. Atanasov, P.P. Atanasova, I.K. Vockenroth, N. Knorr and I. Köper, Bioconjugate Chemistry 17, 631 (2006). [6] L.J.C. Jeuken, R.J. Bushby and S.D. Evans, Electrochemistry Communications 9, 610 (2007). [7] P. Théato, R. Zentel and S. Schwarz, Macromolecular Bioscience 2, 387 (2002). [8] R.-J. Merath and U. Seifert, Eur. Phys. J. E 23, 103 (2007), and references therein. [9] P.B. Canham, J. Theoret. Biol. 26, 61 (1970). [10] W. Helfrich, Z. Natureforsch 28c, 693 (1973). [11] T. Bickel, M. Benhamou, and H. Kaidi, Phys. Rev. E 70, 051404 (2004). [12] H. Kaidi, T. Bickel, and M. Benhamou, Europhys. Lett. 69, 15 (2005). [13] A. Bendouch, H. Kaidi, T. Bickel, and M. Benhamou, J. Stat. Phys.: Theory and Experiment P01016, 1 (2006).

13

Statistical Theory and Method Abstracts – Zentralblatt Database c 2012 FIZ Karlsruhe

ZMATH 1197.76166 Madmoune, Y.; El Hasnaoui, K.; Bendouch, A.; Kaidi, H.; Chahid, M.; Benhamou, M. Brownian dynamics of nanoparticles in contact with a confined biomembrane. Afr. J. Math. Phys. 8, No. 1, 91-100, Article No. 1010, electronic only (2010). Summary: The system we consider is a fluid membrane confined to two parallel reflecting walls that are separated by a finite distance, L, assumed to be small in comparison to the bulk roughness. The attractive membrane is surrounded by small colloidal particles (nanoparticles). The purpose is the study of Brownian dynamics of these particles, under a change of a suitable parameter, such as temperature, T , or colloidmembrane interaction strength, w. The Brownian dynamics is investigated through the knowledge of the time particle density, which solves the Smoluchowski equation. Solving this equation around the mid-plane, where the essential of phenomenon occurs, we obtain the exact form of the local particle density, as a function of the perpendicular distance and time. In the derived expression, appears some time-scale, τ , which scales as τ ∼ L3 /w. This scale-time can be interpreted as the required time over which the colloidal suspension reaches their final equilibrium state. Also, τ can be regarded as the time-interval over which the particles are trapped in holes and valleys. Classification: 76Z05 82D99 92C30 60K35 Keywords: biomembranes; nanoparticles; confinement; Brownian dynamics http://www.fsr.ac.ma/GNPHE/ajmpVolume8N1-2010/ajmp1010.pdf

–1–

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T. El Hafi, M. Benhamou, K. Elhasnaoui, H. Kaidi. Fluctuation spectra of supported membranes via long-flexible polymers. International Journal of Academic Research Part A; 2013; 5(3), 5-10. DOI: 10.7813/2075-4124.2013/5-3/A.1

FLUCTUATION SPECTRA OF SUPPORTED MEMBRANES VIA LONG-FLEXIBLE POLYMERS 1

1,2

1

T. El Hafi , M. Benhamou , K. Elhasnaoui , H. Kaidi

1,3

1

2

LPPPC, Sciences Faculty Ben M’sik, P.O. Box 7955, Casablanca, ENSAM, Moulay Ismail University, P.O. Box 25290, Al Mansour, Meknes, 3 CRMEF, P.O. Box 255, Meknes (MOROCCO) [email protected] DOI: 10.7813/2075-4124.2013/5-3/A.1

ABSTRACT We consider a fluid membrane that is supported to a solid substrate via long-flexible polymers (as proteins or lipopolymers). In this paper, we aim at a quantitative investigations of the associated fluctuation spectra. The study is achieved through the knowledge of the height correlation function between points on the membrane. First, we compute exactly such a function for all pairs of points of the membrane, versus the parameters of the problem, which are the elastic constants of the membrane (bending modulus and interfacial tension) and the molecularweight of the connected polymer chains. Second, from this correlation function, we extract the exact expression of the fluctuations amplitude and discuss its dependence on molecular-weight. Third, we write exact scaling laws for both height correlation function and fluctuations amplitude. Finally, we emphafsize that the presence of the connected polymers may be a new mechanism to bind membranes in the vicinity of substrates for specific interests. Key words: Supported membranes; Tethers; Polymers; Correlation function; Fluctuation spectra 1. INTRODUCTION Lipid-bilayer membranes supported on solid substrates are widely used as cell-surface models connecting biological and artificial materials. Usually, the supported biomembranes can be placed either directly on solids or on ultrathin polymer supports that play the role of the extracellular matrix [1]. The main goal is to control, organize and study the properties and function of membranes and membrane-associated proteins. Supported lipid-membranes have other practical interests [2]. Indeed, they permit the biofunctionalization of inorganic solids and polymeric materials, and provide a natural environment for the immobilization of proteins (such as hormone receptors and antibodies). They allow the preparation of ultrathin, high-electric-resistance layers on conductors and the incorporation or receptors into these insulating layers for the design of biosensors. In addition, they are used in Electrochemistry for the detection of protein activity by measuring the resistance and capacitance changes [3-6]. Currently, supported membranes can be assemled as follows : (i) the inner monolayer of the lipid-bilayer fixed to the substrate by covalent chemical bonds or by ion bridges, (ii) freely supported lipid-protein bilayers separated from the substrate by ultrathin water layers (about ten nanometers), and (iii) bilayer membranes linked to the substrate by grafted or adsobed polymers, as lipopolymers [7]. The fixation of these lipopolymers on flat surfaces was studied using Surface Plasmon Spectroscopy. The film thickness of the adsorbed lipopolymer is about 12 to 20Å. Statistical Mechanics of supported membranes has been the subject of many theoretical investigations [8]. The majority of these works considered tethers are springs. In this paper, however, the tethers are long-flexible polymer chains. We shall assume that the two extremities of each polymer chain on the inner monolayer of the supported bilayer membrane and wall remain parallel each to other. The purpose is the study of the fluctuation spectra of supported membrane pairs. This is done through the computation of the height-height correlation function. We have computed this function exactly, from which we extracted an important quantity that is the fluctuations amplitude, even when the polymer chains are polydisperse. Such an amplitude is naturally space-dependent function. The remaining of presentation proceeds as follows. Section 2 deals with the description of the used extended Hamiltonian. In Section 3, we present the exact computation of the correlation function of the supported membrane. Section 4 is dedicated to the fluctuation spectra and discussion. Some concluding remarks are drawn in the last section. Finally, some technical details are relegated in Appendices A and B.

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2. EXTENDED CANHAM-HELFRICH HAMILTONIAN Start with a fluctuating fluid membrane that is embedded in a three-dimensional homogeneous aqueous medium, delimitated by a plan rigid substrate (semi-infinite geometry). We assume that the membrane is linked to the solid surface by polymer chains. We denote by ⃗ the position-vector of the second extremity of chain on the supported interface. For simplicity, we suppose that the two extremities of a given chain remain along the same axis that is perpendicular to the fluid membrane and substrate. To be more general, the positions of anchoring extremities are assumed to be distributed at random. Within the framework of the Monge representation, a point on the membrane can be described by ⃗, = ℎ( ⃗) , where ⃗ = ( , ) is the two-dimensional transverse vector and ℎ( ⃗) is the height-function (perpendicular distance to the surface). The Statistical Mechanics of the supported fluid membrane is based on an extended Hamiltonian [8] [ℎ] =

[ℎ] + ∑

ℎ (⃗ ),

(1)

with the bare Canham-Helfrich Hamiltonian [9,10] [ℎ] =

⃗ [ (∆ℎ) + (∇ℎ) ],

∫

(2)

where is the membrane bending modulus, is the interfacial tension coefficient. The second part in the r.h.s of Hamiltonian expression (1) accounts for the elastic contribution due to the presence of the connected hydrocarbon chains (springs), which is not local. There, denotes the elastic constant of polymer chain . The generalized Hamiltonian (1) describes the so-called model D in literature [8]. It will be convenient to rewrite the total Hamiltonian (1) as follows [ℎ] =

∫

⃗ [ (∆ℎ) + (∇ℎ) + ℎ ],

(3)

with the local function ( ⃗) = ∑

( ⃗ − ⃗ ).

(4)

Here, ( ⃗) denotes the two-dimensional Dirac-function. Then, the term ℎ /2 plays the role of a local confinement potential that maintains the position of the fluid membrane around some typical distance, ℎ . 3. EXACT HEIGHT-CORRELATION FUNCTION With the help of the above total Hamiltonian, we compute the expectation mean-value of various functionals of the height-field ℎ, in particular the two-point correlation function ( ⃗, ⃗′) = 〈ℎ( ⃗)ℎ( ⃗′)〉 − 〈ℎ( ⃗)〉〈ℎ( ⃗′)〉.

(5)

Such a function measures the height-fluctuations. As we shall see below, this correlation function allows the determination of the local membrane-roughness. Notice that the presence of tethers breaks the translation symmetry in the parallel directions to the supported membrane. Now, the aim is the computation of the expected correlation function. To this end, we start from the following second-order differential equation ̂ ∆ − Δ + ( ⃗)

( ⃗, ⃗′) =

( ⃗ − ⃗′),

(6)

)+( / ). We have introduced the rescaling with the two-dimensional Laplacian operator Δ = ( / parameters : ̂ = / , = / , = / , where is the absolute temperature and is the Boltzmann’ constant. There, = / . Differential equation (6) is directly obtained from Hamiltonian (1) by simple functional derivatives. To compute the height-correlation function, the first step consists to transform differential equation (6) into the following integral equation solved by the propagator , ( ⃗, ⃗′) =

( ⃗ − ⃗′) − ∫

( ⃗, ⃗′) =

( ⃗ − ⃗′) − ∑

⃗′′

( ⃗ − ⃗′′) ( ⃗′′) ( ⃗′′ − ⃗′),

(7)

( ⃗ − ⃗ ) ( ⃗ − ⃗′),

(7a)

or equivalently,

where

( ⃗ − ⃗′) is the (bare) propagator of a non-supported fluid membrane (

6 | PART A. NATURAL AND APPLIED SCIENCES

= 0),

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Vol. 5. No. 3. May, 2013

( ⃗ − ⃗′) = 〈ℎ( ⃗)ℎ( ⃗′)〉 − 〈ℎ( ⃗)〉 〈ℎ( ⃗′)〉 = ∫ (

⃗.( ⃗ ⃗′)

⃗ )

.

(8)

The notation 〈… 〉 means the thermal average, which is performed with the Canham-Helfrich Hamiltonian defined in Eq. (2). The general expression for this propagator we do not recall is known in literature [11-13]. The particular value of the propagator (0) = ( ) defines the mean-roughness of the membrane, . The latter and the in-plane correlation length (beyond which the two-point correlation function exponentially fails), ∥ , are related by : (

) = ∥ /16 ̂ . To solve integral equation (7a), we set ⃗ = ⃗ , where the subscript 1 ≤ to show that ∑

( ⃗ − ⃗′) =

≤

is arbitrary. It is straightforward

⃗ − ⃗′ ,

(9)

with the matrix elements =

⃗ −⃗ .

+

(10)

The inversion of Eq. (7a) gives [

( ⃗ − ⃗′) = ∑

]

⃗ − ⃗′ ,

(11)

with the matrix coefficients =

=

⃗ −⃗ .

+

(12)

Matrix is then the transpose of matrix . Combining now relationships (7a) and (11) yields the expression for the height-correlation function ( ⃗, ⃗′) =

( ⃗ − ⃗′) − ∑

,

(⃗ − ⃗ )

⃗ − ⃗′ ,

(13)

with coefficients =

+

[

]

.

(14)

These define a squared matrix, . In Appendix A, we show that this matrix is symmetric and positive definite. The main result (13) calls the following remarks. Firstly, as it must be, the height-correlation function is not invariant under translations in the parallel directions to the fluid membrane and substrate, but the bare correlation function does. The translation symmetry breaking can be directly seen on integral equation (7). Secondly, since matrix is symmetric, the propagator is invariant permutting the points ⃗ and ⃗′. Thirdly, all dependence of this propagator in elastic constants 's is entirely contained in matrix elements 's. Fourthly, the positivity property of the second term in the r.h.s of expression (13) indicates that the presence of tethers considerably reduces the thermal fluctuations. Sixthly, the expected propagator is completely determined by the knowledge of matrix and bare propagator . Finally, as it should be, at infinity, that is ⃗ → ∞ and ⃗′ → ∞, ( ⃗, ⃗′) goes to zero. 4. FLUCTUATION SPECTRA AND DISCUSSION The local membrane mean-roughness, ( ⃗), is simply the value of propagator expression (13), we obtain the following exact result ( ⃗) = (

) −∑

,

(⃗ − ⃗ )

⃗− ⃗ .

( ⃗, ⃗′) for ⃗ = ⃗′. From

(15)

Matrix elements ’s depend on all distances between tethers ⃗ − ⃗ ’s and their elastic constants Before discussing the above finding, we rewrite it on the following scaling form ( ⃗) =

1−∑

,

(⃗ − ⃗ )

⃗− ⃗ ,

’s.

(16)

with the dimensionless quantities

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)

Vol. 5. No. 3. May, 2013

,

(⃗ − ⃗ ) =

(17)

( ⃗ − ⃗ )/( ) .

(18)

The latter is small than unity and (0) = 1. Notice that coefficients ’s depend only on dimensionless ⃗ − ⃗ ’s. elastic constants ( ) ’s and renormalized bare propagators Now, let us discuss the obtained result. Firstly, to comprehend the general expression (16), we examine some cases : (i) One tether: Let ⃗ be the vector-position. In this case, matrix is a scalar quantity, and we have = ( ) / 1+ (

) .

(19)

Combing this equality with formula (13) yields ( ⃗) =

of

(

1−

) (

(⃗ − ⃗ ) .

)

(20)

(ii) Two tethers: We denote by ⃗ and ⃗ the position-vectors of the two tethers. In this case, we find that the matrix elements are as follows =

+

( ) , =

=

( ) ,

+

(21a)

( ⃗ − ⃗ ),

=−

(21b)

with =1+

( ) +

( ) +

( ) −

( )

( ⃗ − ⃗ ).

(21c)

Then, the associated local roughness is ( ⃗) =

(⃗ − ⃗ ) − (⃗ − ⃗ )

1− −2

(⃗ − ⃗ ) (⃗ − ⃗ )

/

.

(22)

(iii) The tethers have the same length and their extremities coincide at one point ⃗ ( ⃗ ≡ ⃗ ) : This case corresponds to a -watermelon tether, and we find that the associated local roughness is given by ( ⃗) =

(⃗ − ⃗ ),

1−

(23)

with the constant =∑

.

,

(24)

Such a constant is nothing else but the sum of all coefficients of matrix . A comparison between relations (20) and (23) suggests that a -watermelon tether is equivalent to one tether of effective elastic constant , such that =

( ) / 1+ ( ) .

Such a equality relates the new elastic constant quantity is exactly given by =

(

(25)

to that of one tether, . We show in appendix B that the

) / 1+

(

) .

(26)

By comparison, we have =

.

(27)

Secondly, in all cases, the positivity of the second term in the r.h.s of equality (16) implies that the local roughness, ( ⃗), remains above the bulk one, , at any point ⃗ of the membrane. Therefore, the presence of tethers has tendency to suppress the shape fluctuations of the supported fluid membrane.

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Vol. 5. No. 3. May, 2013

Thirdly, for any number of tethers, when the considered point ⃗ is far from the positions of the tethers extremities ⃗ ’s, that is | ⃗ − ⃗ | ≫ ∥ , the local roughness ( ⃗) fails exponentially to the bulk one, . Finally, let us assume that the tethers are long-flexible polymer chains (in dilute solution) of common polymerization degree, . In this case, the associated elastic constants are identical and scale as ~ , where represents the monomer size. For infinitely long-polymer chains, that is → ∞, matrix goes to ( / ) , where is the matrix unity. In this limit, to first order in , the general result (16) becomes ( ⃗) =

1− (

/

) ∑

(⃗ − ⃗ ) + (

The above equality can be seen as a Taylor series in

/

).

(28)

-variable.

5. CONCLUDING REMARKS We recall that the purpose of this paper is the investigation of correlations between points belonging to a supported membrane that is linked to a solid surface via long-flexible polymer chains. The presence of the connected macromolecules can be regarded as a new mechanism to bind the membrane for specific interests. This binding phenomenon depends naturally on the value of the molecular-weight of the connected polymer chains. High-molecular-weights have tendency to increase the shape-fluctuations amplitude. But, low-molecular-weights suppress these fluctuations. According to formula expressing the local fluctuations amplitude, the quality constitutes a pertinent parameter of discussion. In fact, the presence of a good solvent reduce Our predictions reveal that the fluctuations amplitude is local, and it is more important in regions of the membrane with low-grafting density. This means that such amplitude depends strongly on the distribution of the anchoring points. In our description, we assumed that the anchoring points on the inner monolayer of the supported membrane are immobile. In fact, these points free move since they are fixed on polar-heads of some phospholipids. Such a problem will be discussed elsewhere. For simplicity, we have ignored primitive interactions between the supported membrane and the wall, as van der Waals, hydration and shape-fluctuations interactions. Under certain physical circumstances, theses interactions can be neglected (rigid tethers…). Finally, questions such as dynamics, extension of study to more than one membrane and other developments are under consideration. REFERENCES 1. More details can be found in : M. Tanaka and E. Sackmann, Nature 437, 656 (2005). 2. E. Sackmann, Science 271, 43 (1996). http://dx.doi.org/10.1126/science.271.5245.43. 3. S. Terrettaz, M. Mayer and H. Vogel, Langmuir 19, 55675569 (2003). http://dx.doi.org/10.1021/ la034197v. 4. P.C. Gufler, D. Pum, U.B. Sleytr and B. Schuster, Biochimica et Biophysica Acta 1661, 154 (2004). http://dx.doi.org/10.1016/j.bbamem.2003.12.009. 5. V. Atanasov, P.P. Atanasova, I.K. Vockenroth, N. Knorr and I. Köper, Bioconjugate Chemistry 17, 631 (2006). http://dx.doi.org/10.1021/bc050328n. 6. L.J.C. Jeuken, R.J. Bushby and S.D. Evans, Electrochemistry Communications 9, 610 (2007). http://dx.doi.org/10.1016/j.elecom.2006.10.045. 7. P. Théato, R. Zentel and S. Schwarz, Macromolecular Bioscience 2, 387 (2002). http://dx.doi.org/ 10.1002/1616-5195(200211)2:83.0.CO;2-5. 8. R.-J. Merath and U. Seifert, Eur. Phys. J. E 23, 103 (2007), and references therein. http://dx.doi.org/ 10.1140/epje/i2006-10084-2. 9. P.B. Canham, J. Theoret. Biol. 26, 61 (1970). http://dx.doi.org/10.1016/S0022-5193(70)80032-7. 10. W. Helfrich, Z. Natureforsch 28c, 693 (1973). 11. T. Bickel, M. Benhamou, and H. Kaidi, Phys. Rev. E 70, 051404 (2004). http://dx.doi.org/10.1103/ PhysRevE.70.051404. 12. H. Kaidi, T. Bickel, and M. Benhamou, Europhys. Lett. 69, 15 (2005). http://dx.doi.org/10.1209/ epl/i2004-10305-4. 13. A. Bendouch, H. Kaidi, T. Bickel, and M. Benhamou, J. Stat. Phys.: Theory and Experiment P01016, 1 (2006).

Baku, Azerbaijan| 9

INTERNATIONAL JOURNAL of ACADEMIC RESEARCH

Vol. 5. No. 3. May, 2013

APPENDIX A The aim is to demonstrate that matrix , defined in Eq. (14), is symmetric and positive definite. To this end, ⃗ − ⃗ , with indices 1 ≤ ≤ and we introduce two matrices, and ̅ , of respective elements and 1 ≤ ≤ , and = 1, if = , and = 0, otherwise. It is easy to see that the matrix B expresses as follows = + ̅

,

(A.1)

where is the matrix unity. On the other hand, the matrix = ( + ̅

=

can be rewritten as

) .

(A.2) and ̅ are symmetric. Then, matrix

Such a form clearly suggests that = , since both matrices is symmetric. To show that is positive definite, we consider its inverse matrix

=

. Explicitly, we have

+ ̅ .

=

(A.3)

The associated matrix elements are [ To prove that matrix

]

=

⃗ −⃗ .

+

(A.4)

is positive definite, it will be sufficient to show that ≡∑

[

,

for any set of integer variables ( , … , integral representation of the bare propagator ⃗ −⃗

]

> 0,

), and

= 0, if only if ( , … ,

⃗. ⃗

⃗ )

= ∫(

(A.5) ) = (0, … ,0). Start from the Fourier

⃗

.

(A.6)

We show that =∑

⃗ )

+ ∫(

⃗. ⃗

∑

⃗. ⃗

∑

.

(A.7)

Here, ̅ denotes the complex conjugate of , and ̅ > 0 is its squared module. Since the elastic constants ( ̂ , ) of the fluid membrane are positive, > 0. The latter vanishes, if only if = 0 ( = 1, … , ). Matrix is then positive definite. Therefore, is a positive definite matrix. This ends the proof of the desired mathematical properties of matrix .

APPENDIX B To show formula (25), use is made of relation (A.4) of Appendix A. We start with identity ∑

,

∑

,

[

]

=

.

+(

)

(B.1)

According to Eq. (A.4), we have

A double summation over indices ∑

,

and =

=

.

(B.2)

gives =

(

) / 1+

We have Eq. (17). This ends the proof of formula (25).

10 | PART A. NATURAL AND APPLIED SCIENCES

( ) .

(B.3)

J Membrane Biol (2013) 246:383–389 DOI 10.1007/s00232-013-9544-9

The Quantum Casimir Effect May Be a Universal Force Organizing the Bilayer Structure of the Cell Membrane Piotr H. Pawlowski • Piotr Zielenkiewicz

Received: 1 November 2012 / Accepted: 29 March 2013 / Published online: 24 April 2013 Ó The Author(s) 2013. This article is published with open access at Springerlink.com

Abstract A mathematic–physical model of the interaction between cell membrane bilayer leaflets is proposed based on the Casimir effect in dielectrics. This model explains why the layers of a lipid membrane gently slide one past another rather than penetrate each other. The presented model reveals the dependence of variations in the free energy of the system on the membrane thickness. This function is characterized by the two close minima corresponding to the different levels of interdigitation of the lipids from neighbor layers. The energy barrier of the compressing transition between the predicted minima is estimated to be 5.7 kT/lipid, and the return energy is estimated to be 3.1 kT/lipid. The proposed model enables estimation of the value of the membrane elastic thickness modulus of compressibility, which is 1.7 9 109 N/m2, and the value of the interlayer friction coefficient, which is 1.9 9 108 Ns/m3. Keywords effect

Cell membrane Lipid bilayer Casimir

Introduction Casimir-Polder (Casimir 1948; Casimir and Polder 1948) forces are universal physical forces arising from a quantized field. They act even between two uncharged metallic P. H. Pawlowski (&) P. Zielenkiewicz Institute of Biochemistry and Biophysics, Polish Academy of Sciences, PAS, Pawinskiego 5a, 02-106 Warszawa, Poland e-mail: [email protected] P. Zielenkiewicz Plant Molecular Biology Laboratory, Warsaw University, Warsaw, Poland

plates in a vacuum, placed a few micrometers apart, without any external electromagnetic field. The idea that Casimir forces may play an important role in different biomembrane systems is quite new. Based on the fundamental work of Lifshitz (1956) dealing with retarded van der Waals forces between macroscopic bodies, first it was applied to the formation of cellular ‘‘rouleaux’’ (Bradonjic´ et al. 2009) and confined biomembranes (El Hasnaoui et al. 2010). In a natural manner, it supplements the theory of nonretarded van der Waals interactions in a lipid–water system (Parsegian and Ninham 1970), sometimes offering an interesting counterproposal. Thus, forces related to the zero-point energy of quantum fluctuations may play an important role in biology, and the analysis of these forces offers a new view into biological phenomena at the cellular level. Herein, we propose a simple second-quantized explanation for the fact that cell membrane bilayer leaflets slip (Otter and Shkulipa 2007) past one another, rather than penetrate each other. This membrane feature is of great importance as it determines the anisotropy of the membrane’s rheological properties. The relative freedom of movement of molecules along membrane leaflets and the relative restriction of displacement in the transverse direction account for the great lateral fluidity and the small perpendicular compressibility of a membrane (Evans and Hochmuth 1978). These dual mechanical properties, both fluid-like and solid-like, have an impact on the structure and function of the proteins embedded in the lipid bilayer matrix (Andersen and Koeppe 2007). This impact finally determines the status of the cell membrane as an active barrier, a natural organizer and an important participant in all processes of life. The proposed model of the interaction of the cell membrane bilayer leaflets considers the membrane interior to be a three-layer dielectric sandwich. The free energy of

123

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P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

the system is related to the electromagnetic field excitations in the ground state. The free energy depends on the varying thickness of the central layer, where lipid chains penetrate the opposite layers and where the density of lipid chains, and the dielectric permittivity, varies in space. The theoretical values of the two energy minima, the membrane elastic thickness modulus of compressibility and the interlayer friction coefficient, were set according to this model.

The Model The cell membrane interior was considered to be a threelayer dielectric sandwich, which consists of parallel slabs as in Fig. 1. The two lateral peripheral regions of this system are fully occupied by all the lipids of the local leaflet. The remaining central region contains the hydrocarbon tails that penetrate from the neighboring layers. The dielectric constant, ec, of this central layer differs from the dielectric constant, ep, of the other parts of the membrane. We assumed that the length of the lipids in each leaflet may vary within a narrow range, L ± dL/2, where L represents the average lipid length and dL represents the width of the length distribution. The distribution of the lipid lengths was assumed to be uniform. Thus, the perpendicular cross section of a membrane bilayer matrix resembles ‘‘the two overlapping combs with some broken teeth.’’

ep

L

ec

ep

δL

Fig. 1 Three-layer dielectric sandwich model of a cell membrane. Only the longest and shortest lipids are shown. L average lipid length, dL width of the lipid length range, ec dielectric constant of the central region, ep dielectric constant of the peripheral region

123

The thickness of the central region, dc, falls to a minimum value equal to dL in the configuration in which the total membrane thickness, d, equals 2L (Fig. 2a). This configuration we called ‘‘configuration MTCR’’ (minimal thickness of the central region). To clarify further considerations, configuration MTCR was treated as the reference configuration. In this configuration, dc may increase both with an increase (Fig. 2b) and with a decrease (Fig. 2c) in the total membrane thickness. Thus, the possible variation in the thickness of the central region in configuration MTCR is unidirectional. For convenience, we introduced the variable x to describe the difference between the actual total membrane thickness and the thickness of the membrane in configuration MTCR; that is, x = d-2L. Then to formalize the description of the central region, one may write dc = dL ? |x|, where |x| denotes the absolute value. The dielectric constant, ec, was assumed to vary in space within the central region due to variations in the lipid density. Variations along the direction perpendicular to the membrane plane were postulated. At first approximation, ec was simply characterized by the spatial average heci. In general, when the total membrane thickness differs from the thickness of the reference configuration by x, the average heci changes as described by the following formula: 8 jxj=dL > x\0 < 1 þ ep 1 1 þ 1þjxj=dL hec i ¼ ep ð1Þ x¼0 > : 1 þ e 1 1 x[0 p 1þx=dL For details, see Appendix. The plot of Eq. 1 indicates (Fig. 3) that the average heci increases with the compression of the membrane thickness (x \ 0) and decreases with membrane thickness extension (x [ 0). For the membrane in the configuration MTCR (x = 0), heci strictly equals ep. Quantum electrodynamic considerations reveal that two dielectric plates separated by a dielectric medium may be attracted as a result of the decrease in the zero-point energy of the quantum electromagnetic field excitations (Srivastava et al. 1985). In the case of the cell membrane, by approximating the dielectric constant of the central region with the average value heci, the free energy, F, of the field per unit area may be described by the following equation (Bradonjic´ et al. 2009): p2 ep hec i 2 hc pﬃﬃﬃﬃﬃﬃﬃﬃ F¼ 3 720 ep þ hec i hec iðdLÞ ð1 þ jxj=dLÞ3

ð2Þ

where h is the reduced Planck constant and c is the speed of light in a vacuum; the dependence of heci on x is described by Eq. 1.

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

385

d = 2L

d > 2L

d < 2L

MTCR

dc = δ L

a

b

dc > δ L

Fig. 2 Unidirectional variations in the thickness of the central region of the membrane. Only the longest and shortest lipids are shown. a Configuration MTCR. Total membrane thickness d = 2L. Central region thickness dc = dL. b Bilayer leaflets farther apart than in configuration MTCR. Total membrane thickness d [ 2L. Central

dc > δ L

c

region thickness dc [ dL. c Bilayer leaflets closer together than in configuration MTCR. Total membrane thickness d \ 2L. Central region thickness dc [ dL. L average lipid length, dL width of the lipid length range, MTCR minimal thickness of the central region

According to the model, in both cases, the thickness of the ˚) central region of the membrane (dc = 9.0 and dc = 7.9 A is larger than the assumed thickness in the configuration ˚ ). MTCR (dc = 5 A

εc 2εp - 1

εp

Discussion 1

-10

0

10

x/ δ L

Fig. 3 The space average of the dielectric constant in the central region, heci, as a function of the ratio x/dL. x the difference between the total membrane thickness and the membrane thickness in configuration MTCR, dL width of the lipid length range, ep dielectric constant in peripheral regions of the membrane

We assumed that the central region of the membrane may be treated as a kind of leaflet interface. One interesting question is how the interface is stabilized to avoid leaflet penetration and sticking. The proposed model may offer a simple explanation based on the Casimir effect in dielectrics. At reasonable values of the applied parameters (see ‘‘Results’’ section), the model predicts two free energy minima. For the membrane organized in the lower energy ˚ ), its leaflets are separated by 4 A ˚ minimum (x = 4 A

Results

0,03 0,01 2

F [J/m ]

Equations 1 and 2 show that, for ideally flat leaflets (dL = 0), Casimir forces couple membrane layers at zero distance (x = 0), with the free energy tending to negative ˚ , ep = 2), infinity. More reasonable calculations (dL = 5 A taking into account the heterogeneity of the lipid lengths (Blaurock 1982) and specifying the dielectric constant of the peripheral regions of the membrane (Huang and Levitt 1977) were also performed. These calculations indicate (Fig. 4) that there are two finite minima of free energy, both for the membrane in configurations close to the MTCR. The first (lower) minimum (F = -4.68 9 10-2 J/ ˚ ), and m2) is for leaflets being slightly separated (x = 4.0 A -2 2 the second one (F = -2.54 9 10 J/m ) is for leaflets ˚ ). being moderately pushed together (x = -2.9 A

0,05

-0,01 -0,03 -0,05 -0,07 -50

-30

-10

10

30

50

70

90

x [A] Fig. 4 The free energy, F, of the field per unit area as a function of a change, x, in the membrane thickness relative to the thickness of the MTCR membrane

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P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

greater than in configuration MTCR. At the assumed het˚ ), this result means erogeneity of the lipid lengths (dL = 5 A that 20 % of lipids (1-x/dL) penetrate into the neighboring ˚ (dL-x). leaflet (Fig. 5a) but by no more than 1 A According to the results presented in Fig. 4, transition ˚ ) requires into the second (higher) minimum (x = –2.9 A approximately 5.7 kT/lipid (calculated for T = 300 K and ˚ 2). In this minimum energy conarea per lipid a = 50 A figuration, the membrane leaflets are in closer contact. All ˚ lipids penetrate the opposite layer but no deeper than 7.9 A (Fig. 5b). The return to the other minimum requires 3.1 kT/ lipid. The revealed characteristic energies are several times higher than those estimated assuming the average tension c (0.003 mN/m) of living cells (Blanchard and Rauch 2012). The result ca kT indicates that the Casimir effect may be an important contributor to the membrane dynamics, along with the hydrophobic effect. To study the mechanical properties of the presented model membrane, the elastic thickness modulus of

a

100%

n1

-4.5

n2

-0.5 0.5

4.5

ξ [A]

3.95

ξ [A]

b

100%

n1

-3.95

n2

-1.05

1.05

Fig. 5 The area densities n1 and n2 of lipids belonging to a given leaflet as a function of a distance, n, from the membrane midplane. The range occupied by the longest and shortest lipids is shown above. ˚ . b For the a For the first minimum (lower) of free energy, x = 4 A ˚ second minimum (higher) of free energy, x = -2.9 A

123

compressibility, k, was numerically estimated using a differential second derivative of the free energy at the first minimum (k = d F00 ). Assuming a bilayer thickness of d = 5 nm (Kuchel and Ralston 1998), k = 1.7 9 109 N/ m2; i.e., this estimated value of k is of the same order as that measured using volume dilatometry of lipid bilayers (Srinivasan et al. 1974). This result indicates that the central region of the cell membrane resembles an ‘‘incompressible’’ core that does not allow lipid interleaflet penetration. The results of the presented model also enable estimation of the value of the interlayer friction coefficient, f (friction force per unit area and unit velocity). Approximating the end of the lipid penetrating the neighbor leaflet with a hemisphere of a radius r that experiences the action of the Stokes force from the liquid passing with the velocity 2v, one may obtain f = 6pgr/a, where g is the membrane shear viscosity. Taking the typical value of viscosity g = 0.1 Ns/m2 estimated from the diffusion coefficient of membrane-spanning proteins in phospholipid bilayers (Waugh 1982) and assuming that the membrane is ˚ , f = 1.9 9 108 Ns/ in the first minimum with r = 0.5 A 3 m , which is within the range of reported experimental values (Shkulipa et al. 2005; Otter and Shkulipa 2007). For the second minimum, one may expect an eightfold higher value because of a deeper penetration. According to the model, a membrane bilayer in a basic state (lower minimum) should possess relatively small interleaflet friction. The probability that, due to thermal fluctuations, some regions of the membrane reach second minimum is relatively small. At first glance, one may worry about some physical and mathematical problems with the proposed approach. It is obvious that some points require additional discussion, especially phenomena at a physical level neglected in our model. First of all, is it justified to assume that the lipids of opposite leaflets interpenetrate each other at all? Steric interactions seem to be the dominant suppressor of interdigitation. However, a spontaneous or induced interdigitated phase of bilayers consisting of double-tail lipids was confirmed in computer simulation and differential scanning calorimetry experiments (Kranenburg 2004; Kranenburg et al. 2004; Mavromoustakos et al. 2011). Moreover, a simple estimation below shows that the possible steric effect is not energetically dominant, as one may expect. Let us assume that during interdigitation the part p of lipids is compressed and their length decreases by an assumed certain value, k. Let us also assume that each of two lipid acyl chains contains the number b of C–C bonds characterized by a certain bond stiffness, g. Then, elastic deformation energy per lipid molecule may be calculated as pgk2/b. For typical conditions, p = 1/10 (meaning that 50 % of penetrating lipids in the lower minimum are

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

387

˚´ and b = 15; this compressed), g = 100 N/m, k = 0.5 A energy equals 0.8 kBT. These are only 7 % of the predicted value of the energy barrier in a lower minimum and, as such, may be neglected at first approximation. The next question is, how much does the derived shape of the free energy, F (Fig. 4), depend on the specific way that the interpenetration occurs? It was assumed that the distribution of lipid lengths was uniform and, in this way, interdigitation varied linearly with a distance from the membrane midplane (Figs. 5, 6). What will change if we assume the more spectacular variation? In extreme cases, when a single-point distribution (dL = 0) is assumed, one infinite minimum of energy at midplane will be obtained and the energy will increase with distance, like –1/|x|3. Thus, narrowing the distribution lowers minima and approaches them together. It should be stressed that the parameters heci, ep and dL in Eq. 2 may also effectively describe a more diverse system. Applied formula for the free energy per unit area, Eq. 2, assuming ep ? ? and heci ? ?, gives the famous Casimir result for the energy of attraction between ideal mirrors. This energy is a result of the change in the zero-point energy of an empty quantum vacuum. From the other hand, it is well known that real cell membranes are under the permanent influence of electrostatic interactions. Assuming a natural transmembrane electric potential V = 100 MV, a membrane thickness dm = 10-8, an effective membrane dielectric constant em = 2 and a vacuum permittivity e0 = 10-11 [F/m], it is easy to estimate the area density of the energy of an electric field, 0.5e0em (V2/dm). It is equal to 10-5 J/m2 . This result is three orders less than the energy of the considered Casimir effect.

np

n1

n2

ξ -L-x /2

−δ L/2-x /2

−δ L/2+x/2

0

δ L/2-x /2

δ L/2 +x/2

L +x/2

Fig. 6 The area densities n1 and n2 of lipids belonging to a given leaflet as a function of the distance, n, from the membrane midplane. The range occupied by the longest and shortest lipids is shown above. np total number of lipids belonging to a given leaflet per unit surface area, L average lipid length, dL width of the lipid length range, x = d - 2L, where d is the total membrane thickness

It is necessary to underline that the discussed formula is correct only within certain assumed constraints. One of them, zero temperature approximation (Landau and Lifshitz 1960) may be justified for considered conditions (kBT hc/dc). The second assumption, i.e., the same dielectric constant at all frequencies, is formally valid for ideal dielectrics or in the case of large separations. However, for values of constant permittivity not so far from unity (vacuum value), possible error for small distances is disregarded. Moreover, the bulk term in energy is disregarded, which may be dominant for large separations. As the ratio of energies of attraction of water slabs and lipid slabs (both estimated using Eq. 2) is as small as 1:10, for the sake of simplicity, the influence of water outside the membrane was not considered. Anisotropy in the dielectric constant is also beyond the scope of this article, and the model for the dielectric constant in the region of interdigitation is very simple, based on a linear superposition of dielectric constants in terms of the effective densities of tails. We realize that real membranes, especially cell membranes, are obviously complex, heterogeneous, nonideal dielectrics with complicated frequency responses and real conductivity. They are certainly not perfectly plain and smooth plates. Despite the above simplifications, the general predictions of our model, i.e., the magnitude of the free energy and the existence of two energy minima, seem to be still reasonable and wait for more detailed further investigation and confirmation. A lateral Casimir effect for corrugated planes or nonretarded local pairwise van der Waals forces might provide a better description of the physics there and provide a more accurate description. For example, the last one can replace an inverse-cube law, describing variation in energy with distance, by an inverse-square law. Using this method, a simple estimation of energy of so-called hydrophobic bonding, at a Hamaker function ´˚ equal to 7 9 10-21 J and a distance of 50 A (Parsegian and Ninham 1970), gives the value of the density of free energy close to 10-5 J/m2. This quantity is three orders less than the energy of lipid–lipid interactions and two orders less than the energy of water–water interactions estimated in our model. We think that future numerical brute-force simulations might make an important contribution toward a better understanding of the mentioned discrepancy. In light of these findings, it is evident that the Casimir effect may play an important role in many biological phenomena and may be a universal force that organizes the tensegrity structure of biological systems. Some scientists might expect spectacular ‘‘levitation’’ forces between the leaflets, but it appears that there is instead a ‘‘quantum trap’’ preventing membrane leaflet interdigitation and collapse as well as maintaining a significant gap between leaflets, which leaves molecules a plane with freedom of

123

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P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect

movement. Even without taking into account interlayer lipid collisions, hydrophobic interactions and the stabilizing role of proteins, Casimir forces may prevent lipid chain mishmash or molecular escape. This hypothesis appears to be fruitful and is worth experimental verification. Acknowledgements We thank Prof. Bogdan Lesyng and Dr. Marek Kalinowski from the University of Warsaw for inspiration and valuable discussion. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

8 > : np

ð6Þ Independently of the sign of x, the central region consists of three layers: 8 < dL=2 j xj=2 n\ dL=2 þ j xj=2 dL=2 þ j xj=2 n dL=2 j xj=2 ð7Þ : dL=2 j xj=2\n dL=2 þ j xj=2 Calculating the sum in Eq. 4, with the help of Eqs. 5, 6 in respective layers, depending on the sign of x, one can obtain: 8 n > np þ 2p 1 þ nx=2 > dL=2 > < x n ¼ np 1 dL > > > : n þ np 1 nþx=2 p dL=2 2

Appendix The Average Value of the Dielectric Constant in the Central Region of a Lipid Bilayer

L x=2 n\ dL=2 þ x=2 dL=2 þ x=2 n dL=2 þ x=2 dL=2 þ x=2\n L þ x=2

dL=2 jxj=2 n\ dL=2 þ j xj=2 dL=2 þ jxj=2 n dL=2 jxj=2

x\0

dL=2 jxj=2\n dL=2 þ j xj=2

ð8Þ Let variable n represent the distance from the membrane midplane; then, the central region may be defined as the membrane layer within the range dL=2 j xj= 2 n dL=2 þ j xj=2. Here, dL represents the width of the distribution of lipid lengths, and x = d - 2L, where d is the total membrane thickness and L is the average lipid length. In general, x may be positive (thickness expansion), negative (thickness compression) or zero (configuration MTCR) and falls within the range x [ dL - 2L. Assuming that the dielectric constant ec in the central region varies locally with n, ec depends on the number of lipids, n, that are passing through the unit area of plane n = constant. This dependence may be described as n ec ¼ 1 þ ðep 1Þ ð3Þ np where ep is the dielectric constant of the peripheral region of the membrane fully occupied only by the lipids belonging to a given leaflet and np is the total number of lipids belonging to a given leaflet per unit surface area. The area density n of lipids at distance n can be described by the equation n ¼ n1 þ n2

ð4Þ

where n1 and n2 are area densities of lipids belonging to a given leaflet. For a uniform distribution of lipid lengths, the constituent densities n1 and n2 (Fig. 6) can be described as follows: 8 > < np L x=2 n\ dL=2 x=2 np nþx=2 n1 ¼ 2 1 dL=2 dL=2 x=2 n dL=2 x=2 > : 0 dL=2 x=2\n L þ x=2 ð5Þ

123

n ¼ np

dL=2 n dL=2

8 np nþx=2 > 1 > dL=2 2 > < x n ¼ np 1 dL > > > : np 1 þ nx=2 2 dL=2

x¼0

ð9Þ

dL=2 x=2 n\ dL=2 þ x=2 dL=2 þ x=2 n dL=2 x=2

x[0

dL=2 x=2\n dL=2 þ x=2

ð10Þ The space average of n in the range dL=2 j xj=2 n dL=2 þ j xj=2 is calculated as dL=2þ R jxj=2

hni ¼

ndn

dL=2j xj=2

dL þ j xj

and according to Eqs. 8, 9 and 10 equals 8 j xj=dL > x\0 < np 1 þ 1þjxj=dL hni ¼ np x¼0 > :n 1 x[0 p 1þx=dL

ð11Þ

ð12Þ

Then, the space average of the dielectric constant ec according to Eqs. 3 and 12 can be described as 8 jxj=dL > x\0 < 1 þ ep 1 1 þ 1þjxj=dL hec i ¼ ep ð13Þ x¼0 > : 1 þ e 1 1 x[0 p 1þx=dL

References Andersen OS, Koeppe RE (2007) Bilayer thickness and membrane protein function: an energetic perspective. Annu Rev Biophys Biomol Struct 36:107–130

P. H. Pawlowski and P. Zielenkiewicz: Quantum Casimir Effect Blanchard A, Rauch C (2012) Membrane lipid asymmetry and permeability to drugs: a matter of size. In: Deavaux PF, Herrmann A (eds) Transmembrane dynamics of lipids. Wiley, Hoboken, pp 251–274 Blaurock AE (1982) Evidence of bilayer structure and of membrane interactions from X-ray diffraction analysis. Biochem Biophys Acta 650:167–207 Bradonjic´ K, Swain JD, Widom A, Srivastava YN (2009) The Casimir effect in biology: the role of molecular quantum electrodynamics in linear aggregations of red blood cells. J Phys Conf Ser 161:12035 Casimir HBG (1948) On the attraction between two perfectly conducting plates. Proc R Netherlands Acad Arts Sci 51:793–795 Casimir HBG, Polder D (1948) The influence of retardation on the London-van der Waals forces. Phys Rev 73:360–372 El Hasnaoui K, Madmoune Y, Kaidi H, Chahid M, Benhamou M (2010) Casimir force in confined biomembranes. Afr J Math Phys 8:101–114 Evans EA, Hochmuth RM (1978) Mechanical properties of membranes. Curr Topics Membr Transport 10:1–65 Huang W, Levitt DG (1977) Theoretical calculation of the dielectric constant of a bilayer membrane. Biophys J 17:111–128 Kranenburg M (2004) Phase transitions of lipid bilayers. A mesoscopic approach. Thesis, University of Amsterdam

389 Kranenburg M, Vlaar M, Smit B (2004) Simulating induced interdigitation in membranes. Biophys J 87:1596–1605 Kuchel PW, Ralston GB (1998) Schaum’s outline of theory and problems of biochemistry. McGraw-Hill, New York Landau LD, Lifshitz EM (1960) Electrodynamics of continuous media. Pergamon Press, Oxford Lifshitz EM (1956) The theory of molecular attractive forces between solids. Sov. Phys 2:73–83 Mavromoustakos T, Chatzigeorgiu P, Koukoulista C, Durdagi S (2011) Partial interdigitation of lipid bilayers. Int J Quantum Chem 111:1172–1183 Otter WK, Shkulipa SA (2007) Intermonolayer friction and surface shear viscosity of lipid. Biophys J 93:423–433 Parsegian VA, Ninham BW (1970) Temperature-dependent van der Waals forces. Biophys J 10:664–674 Shkulipa SA, Otter WK, Briels WJ (2005) Surface viscosity, diffusion, and intermonolayer friction: simulating sheared amphiphilic bilayers. Biophys J 89:823–829 Srinivasan KR, Kay RL, Nagle JF (1974) The pressure dependence of the lipid bilayer phase transition. Biochemistry 13:3494–3496 Srivastava Y, Widom A, Friedman MH (1985) Microchips as precision quantum electrodynamic probes. Phys Rev Lett 55:2246–2248 Waugh RE (1982) Surface viscosity measurements from large bilayer vesicle tether formation experiments. Biophys J 38:29–37

123

Colloidal aggregation in critical crosslinked polymer blends M. Benhamou∗ , F. Elhajjaji, K. Elhasnaoui, and A. Derouiche Polymer Physics and Critical Phenomena Laboratory Sciences Faculty Ben M’sik, P.O. Box 7955, Casablanca, Morocco

We consider a low-density assembly of small colloidal particles that are immersed in a critical crosslinked polymer blend made of two polymers of different chemical nature. We assume that, near the spinodal temperature at which the mixture exhibits a microphase separation, the particles preferentially adsorb one of the two polymers. The consequence is that, the beads aggregate in the non-preferred phase. The aim is an extensive study of the thermodynamical properties of the colloidal aggregation, under a change of a suitable parameter such as temperature. This phase transition drives the colloids from a dispersed phase (gas) to a dense one (liquid). To this end, we elaborate a new field theory that allows the determination of the effective free energy of the mobile colloids, as a functional of their density. From this effective free energy, we draw the phase diagram in particle volume fractiontemperature plane, and investigate all critical properties of the colloidal flocculation, from a static and dynamic point of view. Finally, the discussion is extended to impregnated crosslinked polymer blends in solution.

I. INTRODUCTION

The aggregation phenomenon within the traditional colloidal solutions are due to the van der Waals attractive forces, which generally originate from the fact that the particles possess a dipolar moment [1]. The same phenomenon may produces within the so-called critical systems with immersed colloidal particles, such as a ﬂuid near the liquid-gas critical

∗ Author

for correspondence; E-mail : [email protected]

1

point, a mixture of simple liquids (or polymers) near the consolute point, liquid 4 He near the λ-transition or liquid-crystals. For these systems, the critical ﬂuctuations of the order parameter generate long-range attractive forces between colloids, termed critical Casimir forces [2]. The word Casimir is attributed to the well-known Casimir eﬀect [3], according to which two perfectly conducting parallel metals attract each other, due to the vacuum quantum ﬂuctuations. The latter play the role of the critical ﬂuctuations for physical systems exhibiting a critical point. Physics of the colloidal aggregation in critical two-component liquid mixtures is a very exciting and rich problem, and has received a great deal of attention from a theoretical and experimental point of view. Theoretically, the critical Casimir eﬀect has been studied by several methods, such as conformal invariance in dimension 2 [4 − 8], ﬁeld-theoretical Renormalization-Group (RG) in dimension 3 [2, 9 − 15], and Monte Carlo simulation [16, 17]. Recently, the study was successfully extended to polymer blends [18]. Experimentally, the critical Casimir force was measured in a series of experiments [19 − 27]. Among these, we can quote certain experiments [19 − 23] concerning the behavior of silica beads of small diameter, immersed in a binary liquid mixture made of lutidine and water. In fact, the silica beads have tendency to adsorb rather lutidine, when one is in the vicinity of the consolute point of the mixture free from particles. This is the critical adsorption [28 − 41]. The experimental observation shows that, near criticality, the colloids undergo a reversible aggregation in the water-rich side. It was then natural to consider more complex systems such as crosslinked polymer blends (CPBs) conﬁned to two parallel walls [42] or with immersed colloids [43]. CPBs are made of two chemically incompatible polymers that are densely crosslinked in their one-phase region, by γ-irradiation [44], for instance. When the temperature is lowered, below some characteristic temperature depending on the reticulation dose, the CPB phase separates into mesoscopic phases alternatively rich in unlike polymers. This is the microphase separation (MPS). We note that the ﬁrst theory of MPS in CPBs was introduced by de Gennes [45], followed by several extended works [46]. The incorporation of a small amount of particles in a CPB may have some relevance for industry. Indeed, these particles can reinforce the mechanical properties of the considered

2

CPB. But, in most practical realizations, the immersed particles are instable and ﬂocculate under some special physical conditions. It is then useful to study their stability conditions. In fact, the presence of colloids inﬂuences the primitive segregation interactions between unlike connected polymer chains and their chemical potentials, and the internal rigidity of the polymer network. In this paper, we consider a low-density assembly of colloidal particles, immersed in a critical CPB made of two polymers A and B, of diﬀerent chemical nature. It is assumed that, near the spinodal temperature at which the CPB undergoes a MPS, the colloids preferentially adsorb one polymer, saying A. Thus, we are in the presence of a critical adsorption. The consequence is that, the particles aggregate in the non-preferred B-rich phase. For instance, the system may be a hydrogenated polyoleﬁn-deuterated polyoleﬁn mixture incorporating silicon particles, considered in some recent experiment by Wendlandt and coworkers [47]. The induced force responsible for the colloidal aggregation was computed in Ref. [43], at the spinodal point. The main result is that, the induced force decays as r−3 exp (−r/ξ ∗ ), where r is the interparticle distance and ξ ∗ ∼ an1/2 is the microdomains size (mesh size) [45]. Here, a denotes the monomer size and n is the mean number of monomers per strand (section of chains between consecutive crosslinks). The purpose is a study of the thermodynamical properties of the colloidal aggregation, from a static and dynamic point of view. This colloidal aggregation is a phase transition driving the colloids from a dispersed phase (gas) to a dense one (liquid). To this end, we ﬁrst elaborate a ﬁeld theory governing the ﬂocculation phenomenon. Combining this ﬁeld theory with the usual cumulant method [48, 49], we derive the expression of the eﬀective free energy of the moving colloids. The latter is naturally a functional of the volume fraction of particles. We assume that the colloidal system is disordered. The problem is then the speciﬁcation of the disorder nature. To be more realist, we choose the most diﬃcult one, that is a quenched disorder [50]. This means that one must average the logarithm of the partition function over the disorder. The opposite case is an annealed disorder, where the partition function itself is averaged over the external disorder variables. This disorder is relatively simple, and assumes that the colloids and polymers are at equilibrium. In fact, such an assumption is not reliable, since, in principle, the diﬀusion time of colloids is diﬀerent

3

from that of polymers. To do explicit calculations, we suppose that the disorder is rather quenched and follows a Gaussian distribution. After averaging over disorder, we ﬁnd that the eﬀective free energy of colloids is similar to that relatively to the traditional Flory-Huggins theory usually encountered in Polymer Physics [51, 52]. The obtained eﬀective free energy of the dispersed colloids allows the determination of the phase diagram shape in the particle volume fraction-temperature plane and all critical properties of the aggregation transition. Finally, the discussion is extended to troubled CPBs in solution. The remaining of presentation proceeds as follows. We derive, in Sec. II, the expression of the eﬀective free energy of the immersed colloids. The phase diagram is investigated in Sec. II. We discuss, in Sec. III, the static and dynamic scattering properties. Some concluding remarks are drawn in the last section.

II. EFFECTIVE FREE ENERGY

Consider M spherical colloidal particles immersed in a CPB. For simplicity, these particles are assumed to be small and of the same radius. Thus, we are concerned with a monodisperse system. The fundamental assumption is that, near the bulk spinodal temperature, the particles preferentially attract the polymer A. This means that one is in the presence of a critical adsorption, where the colloids are surrounded by the preferred polymer A. The consequence is that, the particles located in the non-preferred B-rich microdomains aggregate. In fact, this aggregation is due to a long-ranged attractive force experienced by the colloids, which was computed in Ref. [43]. To investigate the aggregation phenomenon, we introduce an order parameter or composition ﬂuctuation, ϕ, which is simply the diﬀerence of compositions, ΦA and ΦB , of the two polymers, that is ϕ = ΦA − ΦB . The order parameter ϕ (r) is then a scalar ﬁeld depending on the position vector, r, of the representative point of the mixture. To investigate the physical properties of the troubled CPB, we must precise the form of the Hamiltonian, which is a generalization of that introduced by de Gennes for studying MPS in pure CPBs [45]. In the absence of colloids, the de Gennes’ (bare) Hamiltonian writes [45]

4

Hb [ϕ] = a−d kB T

d

d r

t 2 a2 C ϕ + (∇ϕ)2 + P 2 2 2 2

.

(1)

The latter results from a competition between the usual phase separation and the elastic properties of the polymer network. In the above deﬁnition, the integration is performed over the d-dimensional inﬁnite euclidean space Rd . In the above deﬁnition, T stands for the absolute temperature, kB for the Boltzmann’s constant, and a for the monomer size. The notation t=

1 (χc − χ) < 0 2

(1a)

denotes the distance from the critical temperature, where χ is the standard Flory interaction parameter, which is inversely proportional to the absolute temperature T , and χc = 2/N is its critical value if the system is uncrosslinked. Here, N denotes the common polymerization degree of the polymer chains before they are crosslinked. Therefore, we are concerned with a monodisperse system. The ﬁrst two terms in relation (1) constitute the expansion of the traditional Flory-Huggins free energy [51, 52] around the critical composition. The gradient term accounts for the interfacial energy between A-rich and B-rich phases, and the third one describes the gel elasticity. The elastic contribution in this relation was ﬁrst introduced by de Gennes by analogy with the polarization of a dielectric medium. The internal rigidity constant, C, was related to the average number n of monomers per strand by [45] C ∼ a−2 n−2 .

(1b)

The quantity 1/n was interpreted as the reticulation dose [44]. In equality (1), the vector P denotes the average displacement between the centres of masses of A and B strands; it is the analog of the polarization in the dielectric problem. The monomers A and B play the role of the positive and negative charges of the dielectric medium, respectively. The charge ﬂuctuations correspond to the ﬂuctuations of composition. The quantities P and ϕ are not independent each other, but related by the Maxwell’s relation: divP = −ϕ (r). Such a relation enables us to transform the gel elasticity contribution as follows C C d 2 d rP → − dd rϕ∆−1 r ϕ , 2 V 2 V where ∆−1 r is simply the inverse Laplacian operator, deﬁned by 5

(1c)

∆−1 r ϕ (r)

=−

dd r′ K (r, r′ ) ϕ (r′ ) ,

(1d)

with the Green function K (r, r′ ), such that −∆r K (r, r′ ) = δd (r − r′ ) .

(1e)

It was found [45] that the system phase separates at the spinodal temperature, that is for 1/2 ts = −2 Ca2 ∼D ,

(1f)

where D = n−1 is the reticulation dose. This relation then deﬁnes a critical line in the (C, t)plane along which the CPB undergoes a MPS. For t > ts , the CPB is in the disordered phase, and for t < ts , it is rather in the ordered one. We recall that the microdomains size or mesh −1/4

size is as follows [45]: ξ ∗ = a (Ca2 )

∼ an1/2 .

For a CPB with immersed point-like colloids, the proposed Hamiltonian is M

M

H [ϕ] Hb [ϕ] ϕ2 (ri ) + C0 P 2 (ri ) . = + t0 kB T kB T i=1 i=1

(2)

Here, M denotes the total number of particles dispersed in the medium. The two last terms in the right-hand side describe the contribution of colloids to the Hamiltonian, where the positive quantities t0 and C0 measure the perturbation of the temperature parameter t and rigidity constant C due to the presence of the moving particles. For shapely particles, t0 and C0 express the interactions between the mixture and their surface. A simple dimensional analysis shows that: [t0 ] = L0 and [C0 ] = L−2 , where L is some length. It will be convenient to rewrite the above deformed Hamiltonian on the following form H [ϕ] Hb [ϕ] = + dd rρ (r) O (r) , (3) kB T kB T with the local particle density ρ (r) =

N

δd (r − ri )

(3a)

i=1

and local ﬁeld operator O (r) = t0 ϕ2 (r) + C0 P 2 (r) . 6

(3b)

Now, to determine the free energy of a CPB, F, where the particles positions, (r1 , ..., rM ), are ﬁxed in space, we shall need the expression of the partition function, Z. The latter is deﬁned by the following functional integral d 1 Z (r1 , ..., rM ) = Dϕe−Ab [ϕ]+ d rρ(r)O(r) , M!

(4)

with the bare action Ab [ϕ] = Hb [ϕ] /kB T . The above functional integral is performed over all possible conﬁgurations of ϕ-ﬁeld. The quantity M ! accounts for the usual symmetry factor. It will be useful to introduce the bulk expectation mean-value −1

Xb = Zb × DϕX [ϕ] e−Ab [ϕ] ,

(5)

with the partition function of the free CPB Zb = Dϕe−Ab [ϕ] .

(5a)

With these considerations, the partition rewrites as Zb d exp Z (r1 , ..., rM ) = d rρ (r) O (r) . M! b

(6)

We note that the bulk mean-value in the above expression can be computed using the cumulant method usually encountered in Statistical Field Theory [48, 49], based on the approximative formula

X 2 2 e b = eX b +(1/2!)[ X b −X b ]+... .

(7)

Then, we have Zb Z (r1 , ..., rM ) = exp M!

1 d rρ (r) O (r)b + 2 d

d

d r

d r ρ (r) ρ (r ) O (r) O (r )b,c + ... , d ′

′

′

(8) where

O (r) O (r′ )b,c = O (r) O (r′ )b − O (r)b O (r′ )b

(9)

accounts for the connected two-point correlation function constructed with the ﬁeld operator O (r). In relation (8), only one and two-body interactions are taken into account. High order terms describing the three-body interactions and more are then ignored. 7

To compute the desired free energy, we start from the standard formula F = −kB T ln Z and ﬁnd F [ρ] Fb Fint [ρ] = + − S/kB , kB T kB T kB T

(10)

where Fb = −kB T ln Zb represents the free energy of the host mixture, and Fint is the contribution of the eﬀective interactions that can be written as Fint [ρ] 1 d d = − d rρ (r) O (r)b,c − d r dd r′ ρ (r) ρ (r′ ) O (r) O (r′ )b,c + ... . kB T 2

(11)

Then, this formula is a combination of a one-body potential, O (r)b , and a two-body one,

O (r) O (r′ )b,c . We have

O (r)b,c = t0 ϕ2 (r) b + C0 P 2 (r) b ,

(12)

O (r) O (r′ )b,c = t20 ϕ2 (r) ϕ2 (r′ ) b,c + 2t0 C0 ϕ2 (r) P 2 (r′ ) b,c + C02 P 2 (r) P 2 (r′ ) b,c . (13) In equality (10), S accounts for the entropy of particles, and it is essentially the logarithm of the symmetry factor M !, to which the Stirling’s formula will be applied. Now, the aim is the computation of the eﬀective free energy of the trapped colloids. To do calculations, we ﬁrst assume that their positions in space are disordered at random, with a quenched disorder. The latter is more realistic than the annealed one, since the monomers and colloids are not in equilibrium. We denote by ρ0 = M/Ω the number density of particles, where Ω is the total volume occupied by the troubled CPB. According to the known central limit theorem, the random density ﬂuctuations around the mean-value ρ0 can be assumed to be governed by a Gaussian (or bimodal) distribution, that is 1 2 P [ρ (r)] = P0 exp − 2 [ρ (r) − ρ0 ] . 2ρ0

(14)

Here, the diﬀerence ρ (r) − ρ0 is the density ﬂuctuation and P0 is a normalization constant. Then, the ﬁrst and second moments of this distribution are as follows ρ (r) = ρ0 ,

(15)

ρ (r) ρ (r′ ) = ρ20 + ρ0 δd (r − r′ ) .

(16)

8

Therefore, we are concerned with a non-correlated disorder. For a quenched disorder, we have to average not the partition function Z but its logarithm ln Z = −F [ρ] /kB T , where F [ρ] is the free energy of expression (11). Then, we have Fb F [ρ] Fint [ρ] = + − S/kB , kB T kB T kB T

(17)

with 1 d d rρ (r) O (r)b,c − d r dd rρ (r) ρ (r′ ) O (r) O (r′ )b,c 2

ρ0 Ω ρ20 2 d d d = −ρ0 d r O (r)b,c − d r O (r) b,c − d r dd r′ O (r) O (r′ )b,c . (18) 2 2

Fint [ρ] ≃− kB T

d

To determine the architecture of the phase diagram of the colloidal system, we need the eﬀective free energy (per site), ∆F , that can be obtained from vF [ρ]/Ω by ignoring the constant Fb and terms proportional to density ρ0 . Here, v denotes the volume of colloids. In order to precise the form of ∆F , we ﬁrst introduce the dimensionless particle density (volume fraction of colloids) Ψ = ρ0 × v. After symmetrization (Ψ → 1 − Ψ), the entropy (per site) writes [51, 52] vS = −Ψ ln Ψ − (1 − Ψ) ln (1 − Ψ) . kB Ω

(19)

With these considerations, the eﬀective free energy (per site) is given by ∆F = Ψ ln Ψ + (1 − Ψ) ln (1 − Ψ) + uΨ (1 − Ψ) , kB T with the colloid interaction parameter 1 u= dd r O (0) O (r)b,c > 0 2v

(20)

(21)

that essentially presents as the spatial integral of the two-point correlation function of the ﬁeld operator O. The derived eﬀective free energy calls the following remarks. Firstly, its expression clearly shows its analogy with that relatively to the standard FloryHuggins theory (FH) [51, 52] describing the free polymer blends; u then plays the role of the Flory interaction parameter. Therefore, the present theory of free energy (20) is a lattice model, where a given site is occupied by one particle, with a probability Ψ. Thus, 1 − Ψ is 9

the probability to have an empty site. In fact, if Ψ is the composition of the dense phase (liquid phase), that of the dispersed one is 1 − Ψ. Secondly, formula (20) indicates that the critical ﬂuctuations of composition give arise to an eﬀective interaction energy between colloids that is responsible for their aggregation. Thirdly, we note that the dimensionless parameter u is temperature-dependent and always positive definite. This means that, above some critical value, u∗ , deﬁned later, one assists to a coexistence between dilute (gas) and dense (liquid) phases. Finally, we emphasize that the interaction parameter u is a linear combination of three bulk correlation functions (integrated over space coordinates), that is u=

1 2 t0 X00 + 2t0 C0 X01 + C02 X11 , 2v

with the coeﬃcients

X00 = dd r ϕ2 (0) ϕ2 (r) b,c , X11 =

X01 = X10 =

dd r ϕ2 (0) P 2 (r) b,c ,

dd r P 2 (0) P 2 (r) b,c .

Replacing P 2 by −ϕ∆−1 r ϕ and using the Wick theorem [48, 49] yields X00 = 2 dd r [ ϕ (0) ϕ (r)b ]2 , X01 = X10 = 2

X11 = 2

dd r ϕ (0) ϕ (r)b ϕ (0) ∆−1 r ϕ (r) b ,

−1 dd r ϕ (0) ϕ (r)b ∆−1 r ϕ (0) ∆r ϕ (r) b .

(22)

(22a)

(22b)

(22c)

(22d)

(22e)

Explicitly, we have X00 (T, C) = 2sd a

2d

∞

q d−1 dq , (a2 q 2 + t + C/q 2 )2

0

2d

X01 (T, C) = X10 (T, C) = 2sd a

0

2d

X11 (T, C) = 2sd a

∞ 0

∞

q d−3 dq , (a2 q 2 + t + C/q 2 )2

q d−5 dq . (a2 q 2 + t + C/q 2 )2

10

(22f)

(22g)

(22h)

Here, sd = 2π d/2 Γ (d/2) denotes the area of the unit sphere embedded in the d-dimensional euclidean space Rd , where Γ (z) is the Euler gamma function [53]. In mean-ﬁeld theory, the critical behavior emerges in the d → 4 limit, and we ﬁnd, near the spinodal point (t → t+ s ), the following expressions for the coeﬃcients Xij ’s X00 = 2π 3

a7 ξ ∗−3

X01 = X10 = π 3

X11 = π 3

a7 ξ ∗−1 (t − ts )

a7 ξ ∗ (t − ts )

3/2

t → t+ s ,

,

(t − ts )3/2

3/2

,

,

t → t+ s ,

t → t+ s .

Therefore, the colloid interaction parameter scales as

ts − t −3/2 u = u0 , t → t+ s , ts

(23a)

(23b)

(23c)

(24)

with the positive amplitude u0 = We have used the notation

a4 π3 2 2 2t + 2t C + C . 0 0 0 0 25/2 v 0 = C0 ξ ∗ 2 . C

(24a)

(24b)

The central result (24) call two remarks.

Firstly, as it should be, the colloid interaction parameter becomes more and more important as the spinodal temperature (at which the CPB undergoes a MPS) is reached (from above), and diverges at this same temperature. Secondly, all dependance on the perturbation parameters (t0 , C0 ), due to the presence of ﬁllers, is entirely contained in the positive amplitude u0 deﬁned in Eq. (24a). This dependence is quadratic in these parameters. In addition, this amplitude increases with increasing variables t0 and C0 . In Fig. 1, we draw the reduced colloid interaction parameter, u/u0 , upon the reduced (positive) temperature shift ∆t = (ts − t) /ts . Remark that this curve is universal, independently on the physical and geometrical properties of colloids and their interactions strength with the host CPB. 11

The constructed eﬀective free energy allows the determination of the phase diagram shape in the composition-temperature plane. This is precisely the aim of the following paragraph.

III. PHASE DIAGRAM

Now, the equilibrium volume fraction of particles solves the equation ∂∆F/∂Ψ = 0. Explicitly, we have

Ψ 1 ln . u= 2Ψ − 1 1−Ψ

(25)

This equation deﬁnes the coexistence curve in the (Ψ, u)-plane, along which the dispersed and dense phases coexist. On the other hand, the spinodal curve, along which the isotherm compressibility diverges, can be obtained solving ∂ 2 ∆F/∂Ψ2 = 0. Then, the spinodal curve equation is u=

1 . 2Ψ (1 − Ψ)

(26)

Finally, the location of the critical point, K, can be determined solving the two simultaneous equations ∂ 2 ∆F/∂Ψ2 = 0 and ∂ 3 ∆F/∂Ψ3 = 0. We ﬁnd that its coordinates (Ψ∗ , u∗ ) are such as Ψ∗ =

1 , 2

u∗ = 2 .

(27)

The critical value u∗ = 2 corresponds to a well-deﬁned critical temperature, T ∗ , which satisﬁes the implicit equation 1 2 t0 X00 (T ∗ , C) + 2t0 C0 X01 (T ∗ , C) + C02 X11 (T ∗ , C) = 2 . 2v

(28)

Combining the dominant parts of the coeﬃcients X00 , X01 and X11 we computed above with equality (28) yields the critical temperature, T ∗ , at which the colloids ﬂocculate u 2/3 0 ∗ t = ts 1 − . 2

(29)

We have ts < t∗ < 0, provided that u0 < 2. Notice that t∗ = (χc − χ∗ ) /2, with χc = 2/N and χ∗ ∼ T ∗ −1 . Here, the amplitude u0 is deﬁned in equality (24). The above relationship 12

then deﬁnes a critical surface in the (t0 , C0 , t∗ )-space on which the colloidal system presents a phase transition. Therefore, at T = T ∗ and Ψ∗ = 0.5, the colloidal system exhibits a second order phase transition from a dispersed phase to a dense one, at the critical temperature T ∗ , which is very close to the spinodal temperature Ts , since the perturbation parameters t0 and C0 are assumed to be small enough. Come back to Eq. (25) deﬁning the coexistence curve and try to rewrite it in the vicinity of the critical point. Close to this point, we obtain u = u∗ +

8 (Ψ − Ψ∗ )2 , 3

Ψ → Ψ∗ ,

(30)

or equivalently, 1 Ψ=Ψ ± 2 ∗

3 (u − u∗ )1/2 , 2

u → u∗ .

(30a)

Notice that u − u∗ ∼ T ∗ − T . The phase diagram of the colloidal system is drawn in Fig. 2, where the solid and dashed lines represent the binodal and spinodal, respectively. These lines meet at the critical point K = (Ψ∗ , u∗ ). The region between the two curves is a metastable domain. We note that the phase diagram architecture we investigated above has been determined using the FH approach that is a mean-ﬁeld theory (MFT), which underestimates the ﬂuctuations of the composition that are strong near criticality. To go beyond this theory, and in order to get a correct critical phase behavior, use will be made, in the next section, of the scaling approach that is a direct consequence of the RG theory [48, 49].

IV. SCATTERING PROPERTIES A. Static properties

Now, consider another main physical quantity that is the structure factor, S (q), which informs us on the critical properties of the colloidal system. This can be measured in lightscattering and neutron-scattering experiments. The structure factor is nothing else but the Fourier transform of the two-point correlation function G (r) = Ψ (0) Ψ (r)− Ψ (0) Ψ (r), which measures the ﬂuctuations of the particle composition Ψ. We write 13

S (q) =

dd q (2π)d

eiq.r G (r) ,

(31)

where q denotes the wave-vector, whose amplitude is given by |q| = (4π/λ) sin (θ/2), where λ is the wavelength of the incident radiation and θ is the scattering-angle. In particular, the zero-scattering-angle limit (q → 0) of the structure factor deﬁnes the thermal compressibility of the colloidal system, κT , that is S (0) = kB T κT .

(32)

This compressibility diverges as the critical temperature T ∗ is approached, and we have κT ∼ |T − T ∗ |−γt ,

(32a)

where γt is a critical exponent whose value depends only the space dimensionality d and not on the chemical details of the system. Other critical exponents can be deﬁned. For example, the order parameter ψ = Ψ − Ψ∗ behaves, around T ∗ , as 0 , T ≥ T∗ , ψ∼ (T ∗ − T )βt , T ≤ T ∗ .

(33)

with a second critical exponent βt , in the limit ∆µ = µ − µc → 0, where µ is the chemical potential and µc is its critical value [48, 49]. In term of the colloid interaction parameter u and composition Ψ, we have u − u∗ ∼ (Ψ − Ψ∗ )1/βt

(33a)

that is the equation of the coexistence curve close to the critical point. At T = T ∗ , the order parameter behaves as ψ ∼ |∆µ|1/δt , with a critical exponent δt . On the other hand, the size of domains rich in colloids or the thermal correlation length measuring the correlations extent, ξt , diverges at criticality according to ξt ∼ σ |T − T ∗ |−νt ,

(34)

with the critical exponent νt and an atomic-scale σ that may be the particle diameter. Also, the speciﬁc heat diverges at the critical point with some critical exponent denoted 14

αt . Finally, we deﬁne the last critical exponent, ηt , which characterizes the small-distance behavior of the correlation function, that is G (r) ∼ r2−d−ηt ,

r 0, where b0 is the covolume and a0 = −2π r0 r2 U (r), with the

primitive pair-potential U (r). Here, r0 is the unique zero of U (r), that is U (r0 ) = 0. In this case, the total colloid interaction parameter is then u + up . The presence of the primitive interactions naturally induces a change of the phase behavior. In particular, the location of the critical ﬂocculation temperature is shifted to its higher values. In other word, the critical ﬂuctuations of the host CPB have tendency to accentuate the colloidal aggregation. All conclusions concerning the scattering properties remain the same. Finally, this work must be regarded as a natural extension of a published work dealt

with the colloidal aggregation in uncrosslinked polymer blends with immersed colloids [59].

ACKNOWLEDGMENTS

We are much indebted to Professors Daoud for stimulating correspondences. One of us (M.B) would like to thank the Laboratoire L´eon Brillouin (Saclay) for their kinds of hospitality during his regular visits.

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FIGURE CAPTIONS

Fig. 1 : Variation of the reduced colloid interaction parameter, u/u0 , upon the temperature shift ∆t = (ts − t) /ts .

Fig. 2 : Phase diagram shape, where the solid line is the binodal and the dashed one is the spinodal. The two curves meet at the ”liquid-gas” critical point, K, of coordinates (Ψc = 0.5, u∗ = 2).

23

u/u0 20

15

10

5

0 0,0

0,2

0,4

0,6

0,8

1,0

∆t Figure 1

u 7 6 5 .

4 3

K ..

2 0,00

0,25

0,50

0,75

Figure 2

1,00

Ψ

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