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R) the equation is: 1 d_( 2 dy_\ = e^J fe_Mr)>| _ 2 r dr{ dr) zozD[ \ kT ) \
f # ) | kT )\
In the oil droplet (r < R) it has a similar form, but also includes the distribution coefficients:
These equations imply that concentrations of anions and cations in the aqueous phase are
c.(r) = « p [ S ^ ) ) and C , W = e x p ( - 2 ^ ) ,
(92)
correspondingly, while in the oil they are
«.o-(p/w+r,)eo.
(98)
The set of equations (94) and (95) should be supplemented with two boundary conditions. The first one is the condition of continuous potential at the water-oil interface: Vw(R) = yD(R).
(99)
The second one is the condition of electroneutrality of the total system, which can be presented by the integral: \cc(r)-ca(r)] o
r2dr = 0.
(100)
118
A.G. Volkov and V.S. Markin
Solution of equations (94) and (95) can be found as a combination of the modified Bessel functions of the order 1/2: = r- | / 2 [4/ 1 / 2 (K w r) + ^ I / 2 ( K w r ) ]
(101)
v|/o(r) = r- 1/2 [5!/ 1/2 (K w r) + 52Jf1/2(Kwr)] + v|/0.
(102)
¥^(r)
Both potentials should be limited and this immediately gives Af= B2 = 0. The remaining coefficients A2 and B\ should be found from the boundary conditions (99) and (100). Substituting functions (101) and (102) in (99) and (100) and performing integration, one obtains the following system of equations for coefficients A2 and B\.
e-K»* = Rf2-
\h^-
sinh{KDR) +WoR
(103)
2A2 p i ^ ± I e - - « + V r
(r+r)s,J-
(104)
1
AC0Sh(Koi?)_^_Sinh(K^)
=0
Solving this system and substituting coefficients A2 and B\ in (101) and (102) one finds the potential profile in the oil droplet and in surrounding aqueous media:
(105)
(106)
Electric Properties of Oil/Water Interfaces
119
The most interesting parameter is the potential in the center of the droplet. It can be found from Eq.105:
(107)
The expression in the square brackets is always less than 1. Therefore, the potential in the oil droplet is always less than \\i0 and the value of the distribution potential would be established if the oil was present in macroscopic amount. Dependence on radius. The potential (107) depends on the droplet radius R. Let us consider two limiting cases for very small and very large radii. For
(108) Therefore, in the very small droplet the potential disappears and becomes zero. Then again for R —> 0
(109)
In a very large droplet the distribution potential approaches its macroscopic value \|/0. The total dependence of the distribution potential on the droplet radius is presented in Fig. 10. There are two parameters that define this function. In this example, they are selected as: (110) The potential is normalized by \\i0 and the droplet radius is normalized by Debye parameter, KD . One can see the quadratic dependence of the potential on KDR at
120
A.G. Volkov and V.S. Markin
small radii. The half of the maximum potential is achieved when KDR « 2 and then the potential asymptotically approaches the maximum value of v|/0. Dependence on electrolyte concentration. In macroscopic phases the distribution potential does not depend on l:l-electrolyte concentration if one neglects the difference in activity coefficients of ion. The same is true for small droplets. It can be easily seen from Eq. (107) if one introduces a dimensionless radius p = KDR .
(111)
Debye parameters KD and KW depend on electrolyte concentration as 4c. However, these parameters appear in the equation only as a ratio of one to another. This ratio does not depend on concentration, and hence the distribution potential also does not depend on electrolyte concentration.
Fig. 9. Oil droplet (D) in aqueous solution of electrolyte (W). Radius of the droplet is R.
Electric Properties of Oil/Water Interfaces
121
Fig. 10. Distribution potential in the center of the droplet.
Potential at the surface of the droplet. The value of the potential at the droplet surface \\iw(R) = yD(R) = v|/(7?) can be found from Eq. (106):
(112)
Of course, it is only a fraction of the total distribution potential \|/ D (0). We have analyzed the case of small potentials, which is certainly justified for small droplet. We have also obtained analytical solution of the problem. If the droplet is not small and the potential exceeds kT/e0, then the solution can be found numerically.
122
A.G. Volkov and V.S. Markin
2. SURFACE POTENTIALS Mechanical and electrical properties of all interfaces - from pure water to elaborate surfaces of cellular membranes - depend on ion adsorption at these surfaces [25]. This phenomenon plays a very important role in colloid and physical chemistry, biology and medicine. The structure and stability of large biomolecules and membranes depend on the distribution of counterions in the aqueous phase. The transport of ions through ion channels in cellular membranes is determined by the surface potential generated by adsorption of ions from aqueous phase. Conduction of nerve impulses and effect of anesthetics strongly depend on this adsorption. Therefore, it is very important to know the structure of aqueous interfaces and distribution of ions in them. Other fields in which this knowledge is indispensable are seen in environmental aquatic chemistry and chemistry of the atmosphere. Chemical reactions involving aerosolized particles in the atmosphere are derived from the interaction of gaseous species with the liquid water associated with aerosol particles and with dissolved electrolytes. For example, the generation of HONO from nitrogen oxides takes place at the air/water interface in seawater aerosols or in clouds. Clouds convert between 50 and 80% of SO2 to H2SO4 which contributes to the formation of acid rain [26]. Another example is the release of chlorine atoms from sea salt aerosols reacting with the gaseous components. Molecular chlorine, a photolabile precursor of Cl atoms, is a product arising from a heterogeneous reaction of ozone with Cl" ions. Measurements of inorganic chlorine gases indicate the presence of reactive chlorine in the remote marine boundary layer; reactions involving chlorine and bromine can affect the concentrations of ozone, hydrocarbons and cloud condensation nuclei. The heterogeneous reaction between HOBr and HCl converts significant amounts of inactive chlorine (HCl) into reactive chlorine (CIO). These are only a few examples of global processes in environmental chemistry occurring at the air/water interface. In all these cases, the adsorption of simple inorganic ions is especially interesting. In the beginning of the last century, Heydweller [27] found that surface tension of water increases with addition of inorganic electrolytes having relatively small ions. Some organic molecules (sodium formate, glycine, amino acids at the isoelectric point when both the amino and the carboxyl groups are ionized in the form of zwitterion) also increase surface tension. The surface tension of solutions increases almost linearly with increasing concentration of salts, and for inorganic salts there is an empirical equation: y = y0 + be, where b = 1.7 mNttV'M"1 for LiOH, 1.8 niNn^'M"1 for NaOH and KOH [28], 1.63 mNm' 'M"1 for NaCl and 1.33 m N m ' V for KC1 [29]. For organic tetraalkylammonium salts the constant b decreases with increasing of cation
Electric Properties of Oil/Water Interfaces
123
radius: for tetramethylammonium chloride b is positive, for tetraethyl ammonium chloride b is about zero, and for bigger cations b is positive. This phenomenon was explained as a result of negative adsorption of small ions. Wagner [30] was the first to describe this phenomenon as an electrostatic effect of image forces. Onsager and Samaras [31] derived equations, which predicted the increase of surface tension with increase of electrolyte concentration, but only for the very dilute solutions. Later Buff and Stillinger [32] presented the statistical mechanical formulation of the same problem by also employing the conventional model for very dilute solutions. They calculated the molecular distribution functions from a generalized form of Kirkwood's integral equation and found the surface tension directly from the molecular theory for this thermodynamic parameter. These approaches were quite successful, although they used a simple theory for point charges and did not take into account effects of ion radii. It was Frumkin [33,34] who indicated that the increase of surface tension and surface potential of aqueous solutions of inorganic salts depended on ionic radii. However, it was believed for a long time that equivalent concentrations of different salts gave very similar variations of surface tension^ at least at small concentrations [35]. Later it was found that the differences are rather pronounced even at small concentrations [36]. At higher concentrations, different ions exhibit specific differences: the higher the ion hydration, the higher the surface tension increment. The contributions of two ions of the electrolyte to the surface tension appear almost additive. There was a series of attempts to advance the theoretical explanation of the surface tension given by Onsager and Samaras [31], but without much success. A critical discussion of these attempts can be found in Randies [37]. Krylov and Levich [38] considered the discreetness of the charge of adsorbed ions, but found this effect to be rather small. Having in mind all these improvements, we still have to stress that the original works of Wagner [30] and Onsager and Samaras [31] correctly identified the major players determining the surface tension of the aqueous electrolyte - image forces at the interface and negative adsorption of ions. Using the well-known Gibbs adsorption equation, they calculated the value of surface tension without ascribing too much importance to the difference between anions and cations [30, 31]. The surface tension increment was explained at least for the small concentration of electrolytes. In contrast to this, the situation with the surface potential was completely hopeless. In Onsager-Samaras approach, if there is no difference between anions and cations (beyond the charge sign), there is no reason to expect that they would generate any surface potential because they would equally be negatively adsorbed at the interface and their charges completely compensate each other. Therefore, the whole effect of change of the surface potential of water in the presence of electrolytes originates from the
124
A.G. Volkov and V.S. Markin
difference between anions and cations. In this sense, this effect is more subtle and elusive than surface tension. The theoretical explanation of this phenomenon was published recently by Markin and Volkov [39]. The explanation of the surface potential can be found in the difference in ionic radii and their manifestations at water surface. Because of finite radii, different ions can interact differently with interfaces. This results in a different interfacial distribution of energy, concentration and charge density of these ions. We shall use the model of ions with finite radii to describe both surface tension and surface potential at the wide range of concentrations. 2.1. The model: Ions of finite radii at the air/water interface Let us consider an air/water interface as depicted in Fig. 11. Coordinate x is directed from the interface to the bulk of water. The energy of an ion with radius a at the interface was calculated by Kharkats and Ulstrup. They accounted for the Born solvation energy and the energy of interaction with the image. When x > a, the energy of an ion can be presented as WKU{x) = (zeof \ 327te0E(ra|
Uw-zA\la
Uw-zA~J\
\KEW+EA)X
\EW+EA)
2 [_\-{2xlaf
^a^lx 2x
+ a^ 2x-a_\
(H3)
The first term in curls in (113) represents the Born solvation energy, and the last two terms give the interaction with the image. In the region 0 < x < a:
327iEoSw,a [V
\\ + xla){\-2xla) \ + 2xla
|
a)
a
f
\zw+zA)\ |
2x \
2xVjj a)\\
a) |
{EW+EAJ
(zeof f 2s, Y ^ x~\ \6%zazAa\zw +e 4 J ^
a)
When the center of the ion is right at the interface (x = 0), the electrostatic Gibbs energy is equal to WKU(x = 0) =
(ze Y K -^
4mo(Ew+sA)a
.
(115)
Electric Properties of Oil/Water Interfaces
125
Fig. 11. Distribution of water and ion concentration at distance* from the air/water interface; XG is the position of the Gibbs dividing surface.
When the ion is located in the air, the electrostatic free energy is obtained from equations (113) and (114) by exchanging Zw %AThe image energy can be derived from eqs. (113) - (115) by subtracting Born energy W^om = JFKu(+GO) • One can see an important difference: the energy of a point charge becomes infinite at the interface, while the Kharkats-Ulstrup function eliminates this feature and makes the curve continuous. Equations (113)-(114) give the energy of a lone ion interacting only with its image. However the presence of other ions in the solution will modify this interaction due to Debye screening of electrical field. Therefore the ion/image interaction should decay much faster than in equations (113) - (114). To take this effect into consideration, we shall introduce the screening function J[x) as follows: (116)
126
A.G. Volkov and V.S. Markin
with Debye constant K and ion radii aan and acat. For binary electrolyte the Debye constant is given byK= jegCg^zf /zoekT . This definition consistently takes into consideration the model of ions as hard spheres so that cations (cat) and anions (an) cannot approach each other closer than at aan + acal. Therefore the image energy can be presented as
Wbnv(x) = f(x)[WKU{x)-WKU(^)].
(117)
This function would work rather well for calculations of the surface tension. However, for calculation of the surface potential one would need to take into account more subtle effects that help to distinguish between cations and anions. One of them is the solvophobic effect that takes into consideration the work of creating the cavity to place the ion into the solvent. If one assumes that there is surface tension ycavily in this cavity, then in the bulk of the solvent the solvophobic energy is 4na2ycavity. If the ion crosses the interface from right to left (Fig. 11) the area of the cavity decreases and with it the solvophobic energy. If the ion center is positioned at x, then in the range -a < x < a the energy is 2na(a + x)ycmily; at x l O J-S&l. dx
sos i
\_
(120)
kT
where dielectric constant 8 assumes the value eA or ew depending on coordinate x. We assume that the aqueous phase contains a binary 1:1 electrolyte with bulk concentration c0. Energy W\ is given by equation (96) and summation includes both cations and anions. For boundary conditions, we assume that the electrical potential
(122)
l6m0Ewx where e0 designates the charge of the ion, s0 - the permittivity of free space, sw - the dielectric constant of water, K - Debye constant. This function is presented in Fig. 12 by the dashed line. The surface excess of ions (/) with respect to water (w) was calculated as CO
,(W) = o J W h W(XV kT] - l)dx .
r
c
(123)
o Using surface excesses of ions, Onsager and Samaras calculated the surface tension as a sum of a series and tabulated this sum [31]. For very dilute solutions of 1:1 electrolytes, they found an analytical expression for the surface tension, which they called the limiting law: Ay = y-y 0 =1.012clog(1.467/c)-
024)
This equation is presented in Fig. 12 (dashed line), together with experimental data for NaCl. As one can see, the limiting law coincides with the experimental data only at concentrations smaller than 0.1 M, after which the discrepancy becomes very large. The tabulated values improve the situation only a little. And of course, the limiting law does not distinguish between different ions describing them by the same equation. The main drawback of previous approaches was the absence of an appropriate model of ions at the air-water interface. To obtain a radical improvement of the theory of surface tension in the presence of simple electrolytes, we shall use the new expression for energy (119) that accounts for both finite radius of ions and solvophobic effect. We shall begin with calculation of the surface tension, because surface potential at low electrolyte concentrations has some peculiarities and cannot be found directly from the standard Poisson-Boltzmann equation. Fortunately, this does not influence the calculation of surface tension because at low
Electric Properties of Oil/Water Interfaces
129
concentrations the potential is small and has no effect on the distribution of ions that are determined by image forces and solvophobic effect only. Let us calculate surface excesses of ions i, with respect to water Fj(W) . The Gibbs dividing surface corresponding to zero-excess of water (Fw = 0) does not necessarily coincide with the surface of water as demonstrated in Fig. 11, where its coordinate is designated xG . If the bulk concentration of ions in the air and water are correspondingly 0 and CQ, then the ion surface excesses [39] are
F, ( w ) = c ;|exp[-^lj dx + c 0 + |[exp[-^ll-l \dx.
(125)
Position of the Gibbs dividing surface, xG, was found in the following manner. Water concentration in solutions of simple salts depends virtually linearly on salt concentration as one can see in Fig. 13 for solutions of NaCl and KC1. The data for these examples were taken from CRC Handbook of Chemistry and Physics [29]. The trend lines of these dependences can be presented as cwaler = 5 5 . 6 - a csah.
(126)
The slope a was found for NaCl to be equal to 1.18 and for KC1 equal to 1.73. One has to have in mind that at the far right end of the concentrations scale (about 3 M) the data taken from CRC Handbook of Chemistry and Physics [29] probably are not very reliable because of the saturation effect. However, at smaller concentrations, the data ideally fit a straight line and this is the only result we take from these data. Besides, as indicated later in the text, the effect of the displacement of the Gibbs dividing surface is rather small. Therefore, any small inaccuracy resulting from saturation effect will become negligible in the final result. Because concentrations of anions and cations are slightly different at the interface, we shall define the salt concentration as the mean of the two. Then the spatial distribution of water concentration in the solution can be presented by
c,^(x) = 55.6-^S> jexp " ^ ^
+«xP - ^ ^
j.
(127)
Assuming that the actual surface of water is located at coordinate xD = 0, the water excess at this dividing surface is
130
A.G. Volkov and V.S. Markin
r».,,,« ) =|]{2-expf-2^W] + e x p [ - ( ^ ] } ,
(12S)
and the position of the dividing surface corresponding to zero water excess is
Fig. 13. Water concentration in aqueous solutions of NaCl and KCI. Experimental points are taken from ref. [29].
Fig. 14. Dependence of the surface tension of aqueous solution of KCI on concentration. Experimental points are taken from ref. [29]. The solid line is calculated using Eq. 131.
Electric Properties of Oil/Water Interfaces
131
However, position of this Gibbs dividing surface is different from 0 only at very high salt concentrations. In the isothermal case the surface tension is related to the surface excesses by the Gibbs adsorption equation:
rfy = - t r w 4 .
(130)
In the simple case of a uni-univalent electrolyte the increment of the surface tension can be presented as C
j
Ay = y - Yo = - j(r flmon(w) + rcalmniw))—. o
(131) c
To calculate this quantity we have to solve the Poisson-Boltzmann equation (123) and find the integral in Eq. (131). We carried out numerical integration for a number of different electrolytes and compared them with experimental data. Results for KC1 are presented in Fig. 14. One can see that the present model provides radical improvement comparative to the Onsager limiting law, although at very high concentrations there is a certain discrepancy between experimental and theoretical data. 2.3. Surface potential The elegant theory by Onsager and Samaras for surface tension predicted a zero surface potential because it did not envision any difference between the way cations and anions interacted with the interface. However, experimental observations clearly demonstrated rather pronounced surface potential depending on electrolyte concentration. The sign of this potential with respect to the bulk of water very often is negative, although some salts give positive potential. In the absence of a rigorous theory, a number of semi-empirical attempts were made [37, 43]. The increase of the surface tension with electrolyte concentration was interpreted as the presence of a solute-free layer 4-5 A thick at the surface. Randies [43] assumed that this surface layer is completely inaccessible to cations in solutions of all alkali metal salts, but anions can penetrate more closely to the surface. In other words, Randies postulated two different planes of closest approach for anions and cations. The anions built up negative charge at the surface that was balanced by positive charge of cations deeper in the bulk and hence a negative surface potential was established. Using this simple model and the standard kinetic theory of the electrical double layer,
132
A.G. Volkov and V.S. Markin
Randies calculated concentration dependence of the surface potential. However, comparing his theoretical results with experimental data he had to admit that his model was quite unable to account for the observed surface potentials. The problem was not simply in numbers: theoretical predictions were qualitatively wrong. The Randies theory predicted that the surface potential at low concentrations should increase as a square root of concentration, while experimentally this dependence was either linear or even superlinear. Therefore the theoretical and experimental lines had opposite curvature which was unacceptable. This finding seemed rather strange because the model qualitatively cannot be wrong: there is definitely spatial separation of positive and negative ions at the interface that creates surface charge and this charge should be balanced by the other part of the electrical double layer. At low concentrations the electrical potential is small and does not influence the ion distribution at the interface. Therefore one can conclude that the surface charge Q should grow linearly with the electrolyte concentration: Q~c0 . If the double layer capacitance is CDL , then the surface potential is (? = Q/CDL. According to Gouy-Chapman theory, at low concentrations the capacitance is equal to the capacitance Cdiffuse of the diffuse part of the layer and hence is proportional to the square root of the electrolyte concentration: CDL
~ Cd,ffuse ~ V C 0 •
(132)
That is why Randies came up with the result
- A ^ , ) / 2RT]
( s P K 3 / s a K a ) + exp[-F(A a p (t)-A a p^)/2i?r]
(145)
146
A.G. Volkov and V.S. Markin
=
"
RT (£ K K a /E p K p ) + e x p [ F ( A ^ - A > J / 2 / ? r ] _ F n (E a K a /s [ 5 K p ) + exp[-F(Aap(x')dx' l U
(163)
x-d
We find the position of the minimum in the C vs. Aap"=Aap(t);'+(t)«"+tf"=O.
(164)
It follows from the Gouy-Chapman diffuse-layer theory that if qa= q$= 0, then((|>°)" = (x^,z
(|)" = ( K J 2 ( ) )
= zorg: V = (Karg)\ =u°Q + R T l r i $
(180)
and
ufs = u RS + R T lriXS.
(181)
In the bulk phase we have
A = u ? + R T l n XhA,
(182)
uh0 = u ? + RTln X^ ,
(183)
u^ = u° w b +RTln X^ .
(184)
Electric Properties of Oil/Water Interfaces
157
In all these equations X designates the mole ratio of corresponding substances. Substituting these equations into (175), one obtains: p& + ry?* - n 0 / - p\x°f - pnywb + RT In
/
X
^
= RT In -%-
(185)
Using the standard Gibbs free energy of adsorption:
AIG° = ft - rtf + ptf + pny* - n%
(186)
one obtains the adsorption isotherm: J^=(^l exp(-^) (X^)" (X*)"(Xt)'"1" RT
(187)
We considered the 2-D solution of surfactant B in the solvent of quasi-particles Q, in which the mole ratios were defined as N
Y .«
N
.y»
B
X =
Q
X
' V7iF-' °-N-B+Nl
0 88) (188)
Some authors prefer another set of definitions when real particles in the interface are considered. The equation for this state with real particles A, O, W becomes:
X
>^fe'
Xs
K
Q
N'A+N'O
+
K'
(190) x,
_
" (191)
K N'A+N^+N'W'
and we can obtain:
O89)
158
A.G. Volkov and V.S. Markin
X;=—^—;X!;=—^—.
(192)
The adsorption isotherm can then be presented in the form
-^MX*
+XT1
/X'^
=
s
b
{^ Qy
exp(-^^)
b
(x onx H,r»
pv
RT
(193)
In the past the adsorption isotherm was presented in terms of the fraction 6 of the surface actually covered by the adsorbed surfactant. If we introduce r| as the ratio of areas occupied in the interface by the molecules of surfactant and oil, the mole fractions in surface solution can be presented as follows: B
© 0 + n(i-0)
s =
°
nO-Q) ©+Ti(i-©)
(194) V ;
The adsorption isotherm takes the form: 0 p
n C\ —Pi\p V \l U 7
[Q + , ( l - e ) r
=
/ ^ ^ (Y \P(Y V"' \^o) y-^w)
exp(-^ RT KI
(195)
In this isotherm the mole fractions X*,X*,X* of the components in the bulk solution are presented. In the general case, they must be substituted with activities: 0
npC\—CV\p
[0+ ,(l- Q ) r- = y*\
exP(-^ RT
K1 J \ao> \aw) (196) If the molecules B can interact as pairs in the adsorbed layer and the energy of each new particle is proportional to its concentration, then their chemical potential, \x*B, instead of equation (181), should be presented as:
V U
U
(nb\p(nb\p"'-
VSB = \xf + RT In X-2aRTX^
( 197 )
where a is so called attraction constant. Then after some algebra we obtain, instead of Eq. (196), the isotherm [61, 88]:
Electric Properties of Oil/Water Interfaces
F
^(l-©)"
'
(x*)"(x*f)p"«
159
i?r
v
'
Recall that ri was introduced as the ratio of areas occupied in the interface by the molecule of surfactant to the same of oil and p was introduced as the number of columns of oil, which could be supplanted with one molecule of surfactant. Therefore, p is a relative size of the surfactant molecule in the interfacial layer. It is reasonable to suppose that: T\=P
.
(199)
If the concentration of surfactant in the solution is not high and the mutual solubility of oil and water is low, then we can use the approximation x ! = x * = I s o that the general equation (196) simplifies to: 0 [ / 7
-^"l)0]
ex P (-2a0) = (X*)' e x p ( - ^ - )
(200) This is the final expression for the isotherm that we will call the amphiphilic isotherm. It is straightforward to derive classical adsorption isotherms from the amphiphilic isotherm (200): 1. The Henry isotherm, when a — §,r=\,p=\,
0 « I:
© = X*flexp(-^). 2. The Freundlich isotherm, when a = 0, p= I,® 0 =(X*)rexp(-^).
(201) «l: (202)
Kl
3. The Langmuir isotherm, when a = 0, r = \,p= I: - ^ - =X*exp(-^). 1-0 RT 4. The Frumkin isotherm, when r— l,p= 1:
(203)
160
A.G. Volkov and V.S. Markin
-^— exp(-2a©) = X"a exp(- ^ - ) .
(204)
Therefore, the amphiphilic isotherm (200) could be considered as a generalization of the Frumkin isotherm, taking into account the replacement of some solvent molecules with larger molecules of surfactant. Of course, the amphiphilic isotherm includes all the features of the Frumkin isotherm and displays some additional ones. To elucidate them, it will be convenient to change the variable x* to the relative concentration y= X* / X ^ ( 0 = 0.5), where x*(0.5) is the concentration corresponding to the surface coverage 9 = 0.5: y=
:
exp(a-2a&)
(205)
This equation gives the coverage fraction 8 as a function of relative concentration y, while a and p are the parameters of this isotherm - the first being the attraction constant and the second, the size of surfactant. These parameters play an important role because their effect on the shape of amphiphilic isotherm is very strong. Amphiphilic isotherm (200) analysis can be used for the determination of the interfacial structure. An amphiphilic molecule, which consists of two moieties with opposing properties such as a hydrophilic polar head and a hydrophobic hydrocarbon tail, should be used as an analytical tool located at the interface. Pheophytin a is a well-known surfactant molecule that contain a hydrophobic chain (phytol) and a hydrophilic head group. The value of p less than 1.0 indicates that adsorbed molecules of w-octane are parallel to the interface between octane and water. Substitution of one adsorbed octane molecule requires about 4-5 adsorbed pheophytin or chlorophyll molecules. These are supported by molecular dynamic studies in the systems decane/water, nonane/water and hexane/water. The structure of both water and octane at the interface is different from the bulk. Adsorbed at the interface octane molecules have a lateral orientation at the interface. 3.7. Image forces Any charge near phase boundary interacts with the interface due to the different polarization of two phases. The force of this interaction can be found by solution of the Laplace equation, but it is more convenient to use the method of images. This method employs a fictitious charge {image) which, together with
Electric Properties of Oil/Water Interfaces
161
the given charge creates the right distribution of electric potential in the phase [8, 90]. All ions interact with all images giving rise to the so called image forces. The image forces are largely responsible for both positive and negative adsorption. Important contribution to this effect is given by the change of the size of solvation shells of ions in the boundary layer. This has an impact on the planes of closest approach of ions and dipolar molecules to the phase boundary. The electrostatic Gibbs free energy for an ion in the vicinity of a boundary between two liquids with dielectric constants sj and E2 is determined by the Born ion solvation energy and by the interaction with its image charge [8]. Dealing with the energy of image forces and with the interactions of charges at the oil/water interface, approximate models of the interface are often employed, which are based on the traditional description of the interface between two local dielectrics. In the oil/water system, the force of attraction (or repulsion) of charge in the organic (P) phase with its image in the aqueous phase (a) sitting on the same axis perpendicular to the dividing plane is given by: F(h) = -^^S.-^-j,
(206)
where h is the distance from the interface. If ea > sp, the charge in the nonpolar phase is attracted to its image, but if za < 8p there is repulsion between the charge in an organic phase and its image. From equation (206), it follows that charges in the organic phase are attracted to the oil/water interface. Image forces attract the diffuse layer on the organic side, making it thinner, and repel, thus thickening the aqueous diffuse layer. The Kharkats-Ulstrup model [40, 41] incorporates the finite radius of an ion a, which is assumed to have a fixed spherically uniform charge distribution and can continuously pass between the two phases. For a point charge at long distances from the interface, electrostatic Gibbs free energy can be written as
AG =
j £ ^ L f 4 + 5iZft2£l 327ieoeaa^
(207)
ea+sp h )
where a is an ionic radius, h is the distance from the interface and ze is the charge of an ion. The potential j for the spherically-symmetrical charge distribution is:
162
A.G. Volkov and V.S. Markin
Z6
°
+—^
4jiE0ear1
£g l
~£p
(in region a)
4jre 0 E o #' 1 e a + Ep
a AG
&of \ 327i808aa|
UaSi\2a Ua-z^\ 2 [ s a + e j / ? [e a +E p J \\-(2hlaf
^ ^ 2 / z + alj- (209) 2h 2/z-aJj
The first term in (209) is the Born solvation energy, and the second is the interaction with the image. For a charge located in region p, the electrostatic free energy is obtained from (209) by exchanging s a ep Kharkats and Ulstrup [40, 41] obtained the expression for the electrostatic free energy in region 0 < h< a:
AG=325T£ ^ £ a {[V f 2a)3 l Vs + ^+EYJV4 - ^a)+ 0 a
a
(za-zA2U\+h/a)(l-2h/a) E
Ua+ JL
\ + 2h/a
p
^ a Ju2hVJ 2h V
a) \\
(ze0)2 f 2sp Yf
hV
167i£0Epa l v s a +E p J V a)
If h - 0 and center of an ion is at the interface, electrostatic Gibbs energy will be equal to (ze \2 AG(h = 0) = ^ ^ . 47i8 0 (s a +s p )a
(211)
Image forces play a significant role in electric double layer effects. The excess surface charge density is: K
q = ec0 ]{exp[-*g(h)/kBT]-l}dh, K
(212)
Electric Properties of Oil/Water Interfaces
163
where Ag(h) = AG{h)--^—.
(213)
AG{h) is the electrostatic contribution to the Gibbs energy of solvation, and hd and hc are the distances of closest approach of the anion and the cation. If image forces are taken into account, the diffuse layer capacitance can be calculated by the equation Cd = Cf exp[Ag(h,)/2kBT),
(214)
where C'f is the Gouy-Chapman diffuse layer capacitance without image terms. As noted by Kharkats and Ulstrup [40], equation (214) always gives diffuse layer capacitance corrections toward higher values. The first step to the statistically correct Gouy-Chapman theory for the diffuse double layer uses a restricted primitive electrolyte model. This model considers ions to be charged hard balls of identical radii in a structureless dielectric continuum with constant dielectric permittivity. There are three main approaches to creating a statistical theory with this model. The first approach uses the Gouy - Chapman theory, there the average electrical field in the diffuse layer is calculated from the Poisson equation. The structure and physical properties of the double layer are then calculated from the restricted primitive electrolyte model. As a result, the modified Poisson-Boltzmann theory (MPB) was developed. The second statistical approach incorporates correlation functions and integral equations from the theory of liquids. In this case, the well-known uniform Ornstein-Zernike equations were modified to calculate the ion distribution function near a charged interface. Modified in this way, the Ornstein-Zernike equations became nonuniform and were solved using hyperneted chain approximations (HCA) incorporating a mean spherical approximation for correlation functions (HCA/MSA). A third statistical approach used to describe the electrical double layer was achieved by using the integral Born-Green-Ivon equations. These correlation function approaches resulted in the theory of ionic plasma in semi-infinite space. The next advance in describing the double layer was achieved by upgrading the restricted primitive electrolytes model to a "civilized" or "non-primitive" model. This was accomplished by addition of hard spheres with embedded point dipoles to the restricted "primitive" model.
164
A.G. Volkov and V.S. Markin
This theory takes into account the presence of long-range Coulomb interactions, image effects and external electrical fields. The new "non-primitive" model became the basis for the theory of ion-dipole plasma. The serious mathematical difficulties encountered in describing the "non-primitive" models necessitated the development of less statistically rigorous intermediate models. In these intermediate models, the first monolayer of solvent molecules at the charged interface was considered as a complex of discrete particles and an electrolyte outside this layer was described by the Gouy-Chapman theory. This intermediate approach was able to provide good theoretical agreement with experimental results. The concept of constant dielectric permittivity in the double layer is also of concern. Different factors such as high ionic concentration or strong electric fields can affect the dielectric permittivity. In addition, description of the dielectric properties of solutions by a single parameter, the dielectric permittivity, also came under criticism and led to the development of nonlocal electrostatics. Non-local electrostatics takes into consideration discrete properties of solvent molecules. It assumes that fluctuations of solvent polarization are correlated in space. This means that the average polarization at each point is correlated with the electric displacement at all other points and, therefore, uses the solvent dielectric function which couples polarization vectors throughout the solvent. 3.8. Modified Poisson-Boltzmann (MPB) model The popular Poisson-Boltzmann equation considers the mean electrostatic potential in a continuous dielectric with point charges and is, therefore, an approximation of the actual potential. An improved model and mathematical solution resulted in the modified Poisson-Boltzmann (MPB) equation [91]. This equation is based on a restricted primitive electrolyte model that considers ions as charged hard spheres with diameter d in a continuous uniform structureless dielectric medium of constant dielectric permittivity s. The sphere representing an ion has the same permittivity s. The model initially was developed for an electrolyte at a hard wall with dielectric permittivity ew and surface charge density a. The charge is distributed over the surface evenly and continuously. This theory takes into account the finite size of ions, the fluctuation potential, and image forces in the electrolyte solution next to a rigid electrode, but it still an approximation. The MPB theory begins with the Poisson equation for the mean electrostatic potential \\> in solution:
Electric Properties of Oil/Water Interfaces
d2y
-*
r =
1
v
165
0
-wZ**»,g«W.
(215)
where s0 is the electrical constant, z\ is the charge number of ion i, e0 is the elementary charge, n° is the bulk number density, and go\(x) is the wall-ion distribution function representing the ratio of the local ion density to the ion density of the bulk electrolyte solution. The distance x is measured from the electrode so that the distance of closest approach of an ion / is d/2. According to Kirkwood the distribution function is given by the following equation:
go,X*) = S,X*)exp[- —z,.e o v|/(*) + T|,-].
(216)
The factor tfa) accounts for the excluded volume of the ion and r\, is the fluctuation potential that takes into consideration the ionic atmosphere around an ion created by other ions. The fluctuation potential can then be presented as n =
' ~~kf -T W * ' -^.b)d(z,eo),
(217)
where
*:=i-7
£ l 5 L
-,
(218)
| j is the fluctuation potential created by the ion / and its atmosphere, r is the distance from the center of the ion. The difference |* is the potential created by the ionic atmosphere only and *tf is the value of this potential in the bulk electrolyte solution. There are several methods for calculating §; , and if the image effect is taken into account, it can be presented as ,. = fl0 - e x p ( - K r ) + — e x p ( - K r * )
}
ifr>^,x>0,
(219)
where/is the image factor determined by the difference between two dielectric constants of the electrolyte solution and the electrode:
166
f
A.G. Volkov and V.S. Markin
£
~ Sw
/ =
.
(220)
The local Debye-Huckel parameter at a given point is:
£Q£/C-/
,
In equation (219), 5 0 is a constant and r* is the distance from the mirror image in the interface. Hence, equation (219) takes into account both the ionic atmosphere and image forces. If r < d and x > 0, the equation for the fluctuation potential should be written as: d2\\>
vy2,
Ib^'
r0.
(222)
The term Cfe) can be presented as a power series expansion in the bulk density:
;,(x)=i+5(x)+^-5>;,
(223)
where X+
\[(x'-X)2-d2]goj(x
The solvation energies calculated by the Born formula differ noticeably from experimental values [48, 98]. Since, in most of the cases, the resolvation energy is the difference between two comparatively large solvation energies, even a relatively small error in each energy may give rise to considerable error in the resolvation energy, even an incorrect sign. For example, equation (219) implies that if ea > sp there is a higher probability for the ion to reside in solvent a, irrespective of ion size. In practice this is not always the case. It is known that small ions of radius a < 0.2 nm reside mainly in a polar solvent of high permittivity, while large organic ions reside preferably in the hydrophobic
Electric Properties of Oil/Water Interfaces
173
phase. Data on the partition coefficients, extraction, solubility, and currentvoltage characteristics show that the standard free energy of ion transfer from water into a less polar solvent is positive for small radius ions and negative for large radius ions, while equation (242) implies that the sign of this energy does not depend on the ion radius. 4.3. The Non-local electrostatics method Another rapidly developing semi-macroscopic approach is to calculate the electrostatic part of the solvation energy using the non-local electrostatics method. Non-local electrostatics takes into account that fluctuations of solvent polarization are correlated in space, since liquid has a structure caused by quantum interaction between its molecules. This means that the average polarization at each point depends on the electric displacement at all other points of the space correlated with a given point [98]. In non-local electrostatics the electric displacement D and electric field E are related by the tensor emn(r):
£ m (r) = ZJ*'£ o e m n (r-r')£"(r'), (m,n = x,y,z).
(243)
n
It should be noted that, although, this relation is spatially complicated, it is linear. Further calculations are carried out in terms of the Fourier transform of the tensor emn(r), which is called the static dielectric function e(k):
eflO = YMr\ m,n
d(r - V )e~lk^\
K
mn(r -
r').
(244)
The potential produced in a medium by a charged sphere of radius a at a distance r from its center is given by:
ze0
"t dk sin kr sin ka
Hence, we can easily find the electrostatic contribution to the solvation energy:
174
A.G. Volkov and V.S. Markin
Polarization of the medium can be divided into three main groups: optical, vibrational and orientational. To evaluate equation (246), one has to specify the function s(k). This can be done, for example, by subdividing fluctuations of the medium polarization into three modes relating to different degrees of freedom: namely, (1) electronic or optical; (2) vibrational or infra-red; (3) orientational or Debye. If the radius of the correlation of the zth mode of a fluctuation is Xx, then: j . ^ )
j =
1
" S
_ O
/
i (
B
O
i ^ B
i )
2
1
+
^
i _ i + ( 2
S 2
" B
i )
3
1 + ^
2
3 -
( 2 4 7 )
In this expression the correlation length of the electronic mode is set equal to zero. The exact values of the correlation lengths X2 and X,3 cannot be determined a priori, but they can be approximated from physical considerations. In the infra red region, the length ^3 depends on liquid type. In the case of non-associated liquids the correlation length for orientational vibration is approximately equal to the intermolecular distance, while for associated liquids (water for example) A,3 is equal to the characteristic length of the hydrogen bond chain, i.e. 0.5 - 0.7 nm [48, 98]. Integration of equation (246) with (1 - l/s(k)) expressed by equation (247) yields:
A—cm=T^ ( • - - • ( - - - ) #)•{ Snsoa[ where: \\)(x) = l-(l-e~x)-
sopt
.
\sopl
sj
\A2J
L \s2
- -Mr)L s3j
V/Vj'v
(249)
The agreement between theory and experiment obtained by Kornyshev and Volkov is more than satisfactory [48]. Yet, the question arises whether the results obtained are sensitive to the fitting parameters Xt and whether such a choice of X\ is justified. In addressing this question, one can conclude that the results are most sensitive to parameter A,2 and depend very little on A,3. A detailed analysis of this situation is presented in [48]. The non-local electrostatics method refers to the continuum models, but the effective parameters needed for calculation are chosen by analyzing the solvent structure. This method gives rather accurate values for the solvation energy for small ions, and also permits calculation, by virtue of equation (248), of the resolvation energy. For large ions
Electric Properties of Oil/Water Interfaces
175
there remains considerable discrepancy between theory and experiment, which makes it necessary to take into account other effects. In our case, these are the work done in the formation of a cavity in the solvent and the solvophobic effect. The largest value for the solvation energy is obtained in the Born limit A.2 = A,3 = 0, i.e. in a structureless solvent. The major disadvantage of the continuum and semi-continuum approaches to the solvation problem lies in the solvent model itself. An allowance for dielectric saturation or dipole correlation is an attempt to partially describe the discrete properties of the solvent within the framework of the continuum model. Other analogous approaches attempt to take into account the unknown effect of the solvent molecular structure on the thermodynamic properties of a system. However, the problem can only be solved correctly using a statistical model of the solvent and taking into account its discrete structure. 4.4. Statistical solvent models A consistent approach to the solvation problem is to develop a statistical theory that will consider both ions and solvent molecules as discrete species, though such an approach demands the introduction of some model potentials for particle interaction. Unlike "primitive" electrolyte models which use continuum descriptions of the solvent, the modern statistical analysis employs a "nonprimitive" or a so-called ion-dipole plasma model in which the electrolyte solution is considered as a system of hard spheres of radius a and aQ each carrying at their center a point electric charge or a constant point dipole, respectively. Ordinary electrostatic interactions are assumed to exist between the charges and dipoles. The system is described in terms of many-particle correlation functions satisfying the corresponding equations [100, 101]. To solve these equations, one has to introduce various simplifications, the mean-spherical approximation being the most widely used [102]. The ion-dipole plasma model permits calculation of the ion solvation energy, and also the properties of the pure dipole liquid. Wertheim [101] calculated the permittivity of the dipole liquid in the mean-spherical approximation: e = q(2Q/q{-Q,
(250)
where:
]-
(260)
Generally speaking, the dipole moment, |^, can vary from solvent to solvent. The concentration of dipole molecules in an oil phase can be calculated from the estimated free Gibbs energy of a dipole molecule transfer from water to the hydrophobic phase: cot, = cwcxp(-AGlr/RT).
(261)
The partition coefficient of an ion or dipole Kj can be also calculated from the free Gibbs energy of transfer: Kt = exp(-AG? (tr) IRT)
(262)
Equations (260) - (262) are very useful in the theory of extraction, partitioning and passive transport of small neutral molecules across liquid membranes. The partition coefficient Kj is also important in the formulation and delivery of pharmaceutical agents because it describes relative lipophilicity and solubility in lipid/water systems. For many drugs the inequality can be written: 0 < log Kj < 4.
(263)
180
A.G. Volkov and V.S. Markin
The choice of a given drug delivery method can be guided by the partition coefficient of the agent. For instance, a drug with a low partition coefficient will likely be given by injection, while a skin patch or ointment could be used for drugs with high partition coefficients. Oral consumption would be appropriate for drugs with coefficients in the intermediate range. REFERENCES [I] V.S. Markin and A.G. Volkov, J. Colloid Interface Sci, 131 (1989) 382. [2] V.S. Markin and A.G. Volkov, Russ. Chem. Rev., 57 (1988) 1963. [3] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 23 (1987) 1405. [4] V.S. Markin and A.G. Volkov, Adv. Colloid Interface Sci., 31 (1990) 111. [5] E. Lange and K.P. Miscenko, Z. Phys. Chem., 149 (1930) 1. [6] F.G. Donnan, Z. Electrochem., 17 (1911), 572. [7] W. Nernst, Z. Physik. Chem., 4 (1889) 129. [8] A.G. Volkov, D.W. Deamer, D.J. Tanelian and V.S. Markin, Liquid Interfaces in Chemistry and Biology, J. Wiley, New York, 1998. [9] Z. Samec, in: Liquid-Liquid Interfaces. Theory and Methods (A.G. Volkov and D.W. Deamer, eds.) pp. 155-178. CRC-Press, Boca Raton, New York, London, Tokyo, 1996. [10] A.G. Volkov, J. Electroanal. Chem., 205 (1986) 245. [II] A.G. Volkov and Yu.I. Kharkats, Chem. Phys., 5 (1986) 964. [12] A.G. Volkov and Yu.I. Kharkats, Kinetics and Catalysis, 26 (1985) 1322. [13] Yu.I. Kharkats and A.G. Volkov, J. Electroanal. Chem., 184 (1985) 435. [14] Yu.I. Kharkats and A.G. Volkov, Biochim. Biophys. Acta, 891 (1987) 56. [15] I.M. Kolthoff, Pure Appl. Chem., 25 (1971) 305. [16] Z. Koczorowski, in: Liquid-Liquid Interfaces. Theory and Methods (A.G. Volkov and D.W. Deamer, eds.) pp. 375-400. CRC-Press, Boca Raton, 1996. [17] B.S. Gourary and F.S. Adrian, Solid State Physics, 10 (1960) 127. [18] R.A. Robinson and R.H. Stokes, Electrolyte Solutions, Butterworths, London, 1959. [19] L.Q. Hung, J. Electroanal. Chem., 115 (1980) 159. [20] L.Q. Hung, J. Electroanal. Chem., 149 (1983) 1. [21] L.Q. Hung, in: Interfacial Catalysis, A.G. Volkov (Ed.), pp.83-112, M. Dekker, New York, 2003. [22] T. Kakiuchi, in: Liquid-Liquid Interfaces. Theory and Methods (A.G. Volkov and D.W. Deamer, eds.) pp. 1-18, CRC-Press, Boca Raton, 1996. [23] T. Kakiuchi, in: Liquid Interfaces in Chemical, Biological and Pharmaceutical Applications, A.G. Volkov (Ed.), pp. 105-121, M. Dekker, New York, 2001. [24] A.G. Volkov and D.W. Deamer (Eds.) Liquid-Liquid Interfaces: Theory and Methods, CRC-Press, Boca Raton, 1996. [25] A.G.Volkov, Sov. Electrochemistry, 23 (1987) 90. [26] A.G. Volkov, J. Mwesigwa, A. Labady, S. Kelly, D.J. Thomas, K. Lewis and T. Shvetsova, Plant Sci., 162 (2002) 723. [27] A. Heydweller, Ann. Phys., 33 (1910) 145. [28] J.J. Bikerman, Physical Surfaces, Academic Press, New York, 1970. [29 D.R Lide, Ed., CRC Handbook of Chemistry and Physio?, 77th Edition, CRC-Press, New York, 1996. [30] C. Wagner, Physik. Zeitschr., 25 (1924) 474.
Electric Properties of Oil/Water Interfaces
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[31] L. Onsager and N.N.T. Samaras, J. Chem. Phys., 2 (1934) 528. [32] F.P. Buff and F.H. Stillinger, J. Phys. Chem., 11 (1955) 312. [33] A.N. Frumkin, Sbornik rabot po chistoy i prikladnoy khimii, Scientific Chemical Technical Publishing House, Petrograd, pp. 106-126, 1924 (in Russian). [34] A.N. Frumkin, Z. Phys. Chem., 109 (1924) 34. [35] R.M. Suggitt, P.M. Aziz and F.E.W. Wetmore, J. Amer. Chem. Soc, 71 (1949) 676. [36] N.L. Jarvis and M.A. Scheiman, J. Phys. Chem., 72 (1968) 74. [37] J.E.B. Randies, in: Advances in Electrochemistry and Electrochemical Engineering, P. Delahay, Ed., Vol. 3, pp. 1-3 0, Interscience, New York, 1963. [38] V. S. Krylov and V.G. Levich, Doklady Acad. Nauk SSSR, 159 (1964) 409. [39] V.S. Markin and A.G. Volkov, J. Phys. Chem. B, 106 (2002) 11810. [40] Yu.I. Kharkats and J.J. Ulstrup, J. Electroanal. Chem., 308 (1991) 17. [41] J.J. Ulstrup and Yu.I. Kharkats, Russ. J. Electrochem., 29 (1993) 299. [42] R.C. Tolman, J. Chem. Phys., 17 (1949) 333. [43] J.E.B. Randies, Discuss. Faraday Soc, 24 (1957) 194. [44] R. Parsons and F.G.R. Zobel, J. Electroanal. Chem., 9 (1965) 323. [45] Z. Samec, V. Marecek and D. Homolka, J. Electroanal. Chem., 187 (1985) 31. [46] T. Wandlowski, K. Holub, V. Marecek and Z. Samec, Electrochim. Acta, 40 (1995) 2887. [47] A.G. Volkov, D.W. Deamer, D.I. Tanelian and V.S. Markin, Progress Surf. Sci., 53 (1996) 1. [48] A.A. Kornyshev and A.G. Volkov, J. Electroanal.Chem., 180 (1984) 363. [49] B. Sisskind and J. Kasarnowsky, Zh. Fiz. Khim., 4 (1933) 683. [50] H.H. Uhlig, J. Phys. Chem., 41 (1937) 1215. [51] V.S. Markin and A.G. Volkov, J. Electroanal. Chem., 235 (1987) 23. [52] L.S. Bartell, J. Phys. Chem. B, 105 (2001) 11615. [53] T.V. Bykov and X.C. Zeng, J. Phys. Chem. B, 105 (2001) 11586. [54] V.S. Markin and A.G. Volkov, Electrochim. Acta, 34 (1989) 93. [55] W.D. Harkins and E.C. Gilbert, J. Am. Chem. Soc, 48 (1926) 604. [56] W.D. Harkins and H.M. McLaughlin, J. Am. Chem. Soc, 47 (1925) 2083. [57] L.I. Daikhin, A.A. Kornyshev and M.I. Urbakh, J. Electroanal. Chem., 483 (2000) 68. [58] M. Wilson and A. Pohorille, J. Chem.Phys., 95 (1991) 6005. [59] V.S. Markin and A.G. Volkov, Progress Surf. Sci., 30 (1989) 233. [60] V.S. Markin, A.G. Volkov, Electrochim. Acta, 34 (1989) 93. [61] V.S. Markin and A.G. Volkov, in: Liquid-Liquid Interfaces. Theory and Methods (Eds.: A.G. Volkov and D. W. Deamer), CRC-Press, Boca Raton, pp. 63-75, 1996. [62] V.S. Markin and A.G. Volkov, in: The Interface Structure and Electrochemical Processes at the Boundary between Two Immiscible Liquids (V. E. Kazarinov, Ed.) pp. 94-130, Vol. 28. VINITI, Moscow, 1988. [63] A.G. Volkov, M.I. Gugeshashvili and D.W. Deamer, Electrochim. Acta, 40 (1995) 2849. [64] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 24 (1988) 318. [65] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 24 (1988) 325. [66] V.S. Markin and A.G. Volkov, Progress Surf. Sci., 30 (1989) 233. [67] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 24 (1988) 478. [68] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 24 (1988) 579. [69] V.S. Markin and A.G. Volkov, Russ. Chem. Rev., 57 (1988) 1963. [70] V.S. Markin and A.G. Volkov, Electrochim. Acta, 35 (1990) 715. [71] M. Planck, Ann. Phys., 44 (1891) 385. [72] M. Planck, Ann. Phys. Chem.N. F., 40 (1890) 561.
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[73] F.O. Koenig, J. Phys. Chem., 38 (1934) 111. [74] D.C. Grahame and R.W. Whitney, J. Amer. Chem. Soc, 64 (1942) 1548. [75] E.J.W. Verwey and K.F. Nielsen, Phil. Mag., 28 (1939) 435. [76] G. Gouy, Compt. Rend. Acad. Sci., 149 (1910) 654. [77] D.L. Chapman, Phil. Mag, 25 (1913) 475. [78]. A.G. Volkov (Ed.) Liquid Interfaces in Chemical, Biological, and Pharmaceutical Applications, M. Dekker, New York, 2000. [79] C. Gavach, P. Seta and B. d'Epenoux, J. Electroanal. Chem., 83 (1977) 225. [80] R.S. Hansen, J. Phys. Chem., 66 (1962) 410. [81] A.G. Volkov, Langmuir, 12 (1996) 3315. [82] A.N. Frumkin, Zero Charge Potentials, Nauka Publ., Moscow, 1979. [83] Z. Samec, V. Marecek and D. Homolka, J. Electroanal. Chem., 187 (1985) 31. [84] T. Kakiuchi, M. Kobayashi and M. Senda, Bull Chem. Soc. Jpn, 60 (1987) 3109. [85] C. Yufei, VJ. Cunnane, D.J. Schiffrin, L. Murtomaki and K. Kontturi, J. Chem. Soc. Faraday Trans., 87 (1991) 107. [86] Freundlich, H. (1926). Colloid and Capillary Chemistry, Methuen, London. [87] A.N. Frumkin, Electrocapillary Studies and Electrode Potentials, Saposhnikov Publ. House, Odessa, 1919. [88] V.S. Markin and A.G. Volkov in: Encyclopedia of Electrochemistry, Eds. E. Gileadi and M. Urbakh, Vol. 1, pp. 162-187, Wiley-VCH, Weinheim, 2002. [89] A.R. Vanbuuren, S J. Marrink and H.J.C. Berendsen, J. Phys. Chem, 97 (1993) 9206. [90] L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media, 2nd Ed, Pergamon, New York, 1984. [91] C.W. Outhwaite, L.B. Bhuiyan and S. Levine, J. Chem. Soc, Faraday Trans. II, 76 (1980) 1388. [92] F.H. Stilinger and J.G. Kirkwood, J. Chem. Phys, 33 (1960) 1282. [93] G.M. Torrie and J.P. Valleau, J. Electroanal. Chem., 206 (1986) 69. [94] Q. Cui, G.Y. Zhu and E.K. Wang, J. Electroanal. Chem, 383 (1995) 7. [95] V.S. Markin and A.G. Volkov, Sov. Electrochemistry, 23 (1987) 1105. [96] V.S. Markin and A.G. Volkov, Russian Chem. Rev, 56 (1987) 1953. [97] V.S. Markin and A.G. Volkov, J. Electroanal. Chem, 235 (1987) 23. [98] A.G. Volkov and A.A. Kornyshev, Sov. Electrochemistry, 21 (1985) 814. [99] M. Born, Z. Phys, 1 (1920) 45. [100] D. Henderson, Progress Surf. Sci, 13 (1983) 197. [101] M.S. Wertheim, Ann. Rev. Phys. Chem, 30 (1979) 471. [102] D.J.C. Chan, D. Mitchell and B.W. Ninham, J. Chem. Phys, 70 (1979) 2946. [103] A.K. Govington and K.E. Newman, in: Modern Aspects of Electrochemistry, Eds. J.O'M. Bockris and B.E. Conway, Vol. 12, pp. 41-129, Plenum Press, New York, 1977. [104] G. Antonow, J. Chim. Phys, 5 (1907) 372. [105] R.P. Bell, J. Chem . Soc, 32 (1931) 1371.
Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.
Chapter 5
Deformation of fluid particles in the contact zone and line tension V. Starov Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK 1. TWO DROPS/BUBBLES AT EQUILIBRIUM IN A SURROUNDING LIQUID The hydrostatic pressure in thin liquid films intervening between two drops/bubbles differs from the pressure inside the drops/bubbles. This difference is caused by the action of both capillary and surface forces. The manifestation of the surface forces action is disjoining pressure, which has a special S-shaped form in the case of partial wetting (aqueous thin films and thin films of aqueous electrolyte and surfactant solutions). Disjoining pressure solely acts in thin fiat liquid films and determines their thickness. If the film surface is curved then both disjoining and the capillary pressure act simultaneously. Disjoining pressure (DP), Tl(h), is a manifestation of colloidal forces action. In Fig. 1 S-shaped DP isotherm is presented, which correspond to a sum of three component: dispersion, electrostatic and structural [1]. DP pressure isotherms can be directly measured, however, not in the whole range of film thickness but only those where Tl'(h)(V-,0)
Let us compare the latter expression with the solution for solid spheres. Their equilibrium state (Fig. 7b) can also be characterized by the interaction region of radius rn, within which the surface forces exert their effect. The profile of the solid sphere can be represented as h=hQ+2R- \—^l-(r/R.)
. At r0 «R,
its
approximate form may be used: h = ho+{r1IR)
(41')
The minimum distance, h0, between the solid particles can be determined from the same condition (7) and the known Derjaguin's approximation [1]:
(79)
Eq.(78) can be integrated once using boundary conditions (79), which yields:
h id=
' i~P~~l
(80)
V '-'id
where Lld(hld) = P(H-h,d). Line tension r can be expressed using Eqs. (64) and (65) as:
r = J ly^X + h'1 -1) + ]n(h)dh - )ll{h)dh+ 2P-(h-hJ\ dxo|_
2/i
Ih,
J
- f 2 ^ 1 + fc)2 -\]-"\n{h)dh+2P\hd -h^dx Using Eqs. (70) and (80) we can rewrite the latter equation as
T =2)[yJl + h'2 -L(h)]dx-2)[r^
+ {h'J -4,(AjJfc
(81)
x,,
0
Using Eqs. (70) and (80) we can switch from integration over x to integration over thickness. The latter transformation of Eq. (81) gives:
x = 2{(Vy2-Z2(/z) - V Y 2 ~ 4 W ) dh
(82)
K
The similar equation was obtained earlier [20]. Eq. (82) can be rewritten as \2
/„ m 2P(H
t =- 1
,
- h) \u{h)dh -
\l\(h)dh 2
. ^"
^ dh
(83)
In the case of h«H inside the whole transition zone expressions for L(h) and Lid(h) can be rewritten as
210
V. Starov
/ \ L(h) = r\cos0f-e(h)\
L(h)
1 °° e(h) = - \U(h)dh
= ycosd Using the latter expression Eq. (83) can be rewritten as 2 2 cos 6,e{h)-E y{h) ' y ' ' dh 2 2cosdfs(h)-s (h)
2y " r =- = 2 — { sin#, K •
i
i+
+
(84)
1— 2
"y
sin 6f
In the case of small contact angles s ( h ) « l and the latter equation takes the following form: r =- ^ - f tan 0f I
,
£{k)
dh
(85)
2cos$fe(h) l+ + 2
f
sin ^
It is possible to show that 0.5 < 1+
, 1. For both cases the solution of the integral equation is expressed as a power series in (ejl):
(2.39) where J',(f) might be found in the Appendix (Eq.(A.3)).
232
E. Mileva andB. Radoev
(2) sjl » 1. Insofar as one has always e « 1, this is possible only for high viscosity in the droplet. Thus, the tangential velocity is suitably presented as a power series in (sjl)"' or (Mo) :
vrrf0[r,^(r)] = -4[^(r)+^s)]+ 4f4:Tl \o [r. h " (F)] = MO[J'3 (F) + O{z )] + o\Mo)2 J
(2.40)
with J'3(r) given in the Appendix (Eqs.(A.4)). For the two major cases of low and high droplet viscosities the drag-force expression acquires the form of [30]: Fvhere=-^fVf^-{l
+ ^[Q2+^)]}
for
efi«7
(2.41)
ej!»7
(2.42)
and F^=^'V'^\I--L[R2+C?V^
Fsphere=-2izWVf^{l-Mo[R2+a{z)}}
or
for
The numerical quotients, g 2 and R2, are shown in the Appendix (Eqs.(A.5-6)). The force expressions (2.41-42) need some additional comments. The essential functional dependence on the parameters of the system is totally determined by the presence of (at least) one tangentially immobile boundary in the thin liquid layer. If all the rest conditions (namely Vf ,1, \if ,\xd,R) are the same, the drag force has a maximum value when the mobile interface is flat. The 'bending' of this interface into a droplet surface, results in a relative decrease of the shear stress on it. This effect (might be termed 'an opening of the gap' [30]) is characteristic of the lubrication-theory cases. It manifests itself regardless of whether the adjacent phase is gaseous, liquid or solid. The ratio of the factors outside the brackets in the drag force expressions is constant, irrespective of F • Fplam/Fsphere ~ 4 (see (2.35) and (2.41); (2.36) and (2.42)) [29,30,36,48].
Hydrodynamic Interactions and Stability of Emulsion Films
233
If one neglects this purely geometric effect and examine more closely the structure of the terms related to the viscosity in the contiguous phase, two effects are clearly outlined. The first effect is 'the pure influence of the viscosity in the droplet' and it is found in both asymptotic models (3.57 in (2.35) against Q2 in (2.41); 0.66 in (2.36) against R2 in (2.42)). The second effect (~ 6>(z)) appears only in the case of a spherical droplet (2.41-42). It is related to 'bending of the streamlines' inside the droplet, as compared to those in the semi-infinite liquid space. Insofar as this effect results in additional dissipation of energy, it leads to a relative (slight, however) increase in the drag force, as compared to the flat interface case. Thus, the main conclusion, to be drawn from the presented model investigations, is that at small gap widths between the particles (s «1), the effective flow zone inside the droplet shrinks uniformly around the symmetry axis and to the surface bordering the emulsion film. Therefore, the fluid flow interior the droplet could be regarded as a motion in a semi-infinite space (Fig.2a) [7,29-30]. This model situation constitutes the true specification of the 'primitive model of emulsion hydrodynamics', always when the flow in the emulsion film is of lubrication type. 3. ROLE OF SURFACTANTS The surfactants have a profound influence on the tangential mobility of the liquid interface [38]. The geometric anizodiametry of the thin layer between the droplets ( s « l ) results in a specific coupling of their mass transport and the hydrodynamics of the system [3]. One manifestation of this coupling is the relationship between the kinetic stability of the emulsion films and the preferential solubility of the surfactant species either in the film, or in the contiguous phase. As already mentioned, this effect is a firm experimental fact and is named a Bankroft rule [32]. An attempt to link it to hydrodynamics of emulsion films was first made by Lee and Hogdson [3], and was extensively studied for different regimes of the flow interior the droplet [11,12,25,31,33-35]. The essence of the mutual influence of fluid motion and surfactant mass transfer in the adjacent phases is incorporated in the boundary conditions (2.2). They are specified to include the Marangoni stress component that arise due to the interfacial tension gradient, and the surface viscosity effect [31,38,49]: (3.1) where y is the interfacial tension. In what follows, the issue of surface viscosity [Is [50,51] is neglected. The latter might routinely be included in our analysis. It
234
E. Mileva and B. Radoev
was shown in [49], that this effect results in a slight modification of the interface mobility expression. Upon the application of the potential theory approach an integrodifferential equation is obtained for the interface mobility, instead of the basic integral equation (3.27) (see Ref.[49] for details). As has been mentioned also in [31], for emulsion films, this surface viscosity effect is usually small compared to the other terms in (3.1). So, it won't be dealt here further on. Another important boundary condition is the surface mass balance of the surfactant species:
v t (rv/)-DX 2 r=y n
AT-) with jn= M^f)
(3.2)
-'' =-/„
(3.2')
Only the case of diffusion controlled fluxes towards the droplet's surface is presented. It was established that this possibility is of utmost importance for the interface mobility when thin liquid layer is formed in the gap between the fluid particles [21,22]. Depending on the preferential solubility of the surfactant in the respective cases, jn might acquire any of the forms in (3.2'). F is the surface concentration of the surfactant; Ds,Df,Dd are the surface and the bulk diffusion coefficients; cf ,cd are the bulk concentrations of the surfactant in the phases and they obey the steady mass transfer equations [38]: f
_ / / d dcf d2cf) f vf + v{ =D\ r + r dr dz \r dr dr dz~ J fdc
dr
f
fdc
dz
yr dr
dr
dz J
(3.3)
(3.4)
3.1. Scaling analysis The analysis, already presented in subsection 2.2, has to be modified to include additional scaling parameters related to the surfactant properties, as well
235
Hydrodynamic Interactions and Stability of Emulsion Films
Fig. 3. Mass transfer fluxes: (a) and (b) for a surfactant, soluble in the film; (c) for a surfactant, soluble in the droplet phase; c" - surfactant concentration in the meniscus; +f"d
(3.16)
where the concentration factor is:
r" 5 ^ ^ (Rflbd)2+PeJ ff = V1***'*'* 7+ P ^ +
and
Ds (bd\dT Da'Rf[Fj'd?0
(Rflhd)2 + Ped 1 + Ped
P*J™-,
5 - = - ^ - - ^
(3.18)
Note that unlike the case of Pef < 1 when the maximum value of the concentration scaling normally to the film interface is 8f ~Hf, here 5 d ~ Rf{~ Rd ~ Hd) for Ped < 1 (compare Fig. 3a and Fig. 3c). Then:
r " ay d=_v/D?_dc\
Ds
F"
//=
for
dy
s
*'»'*'o'Jrt ;+
ped<J
dT
J
^
5 r
for
^
(3.20)
> 7
£
Alternatively, the interface velocity scaling (3.16) might be expressed via an interface mobility factor Mod: UfMod
TTd
U
.t, -T
with
d
.,
U MoMod
1
d
=
-r =
fh+ff
d
c
—r
Mo + Mo
d
c
and
1
— -= f
1 + Mo Mo
Af/
T
(3.16')
d
1 + Mo 1
Moc - —j.
f
(3.1V)
240
E. Mileva and B. Radoev
with Mo given by (2.10a'). Therefore, the criterion (2.12) is specified as
T^>z'
h) [17,64], the hydrodynamics of the unstable waves is adequately approximated with the already considered lubrication approximation (see Subsection 2.1, Eq.(2.10'); see also [67]): ^ = -^T(l dt 12\iJ
+ Mo)V2Ap
(4.4)
where Ap is the driving pressure (4.2'), dC, Idt is the velocity of deformation and Mo is a coefficient of the type already introduced in the previous subsections, accounting for the mobility of the film interfaces. Analogous mobility coefficient is introduced also in [70,71], where miscellaneous effects influencing the surface mobility of thin films are commented. Here we shall focus on the effects that are typical for emulsion systems. As is already shown in previous sections of this chapter, the surface mobility is controlled by the viscosity in the droplets and by the Marangoni effect (see e.g. Eqs.(2.10) and (2.10a'); Eqs.(3.12') and (3.17'); Eq.(3.37)). Note that both, hydrodynamic and Marangoni effects, might lead to Mo=0, i.e. practically to immobilization of the surfaces. The evolution of the perturbation C,(t) follows directly from the solution of (4.4), which, as we have found (see (4.2")) is suitably presented through Fourier transformation. Combining (4.4) with (4.2') yields the equation (4.5):
Hydrodynamic Interactions and Stability of Emulsion Films
^--^{l
+ MoWU-l^-V
12\iJ
dt
y
251
(4.5)
dh J
The solution of (4.5) poses two problems for thinning films: the initial condition ^{q.t = 0) and the time dependence h(t). The specificity of the initial condition is related to the fluctuation (random) character of the perturbation waves. In [65] this problem is reduced to the solution of the respective FokkerPlanck equation of the kind (see also Ref. [72]):
P^fy-4—1^W^4 \
dt
q- dh ) 8C,
(4.6)
DC,
where XCO is a distribution function (probability density) of the process, normalized on f a s J/(£ ,t]dC, = 1. In (4.6) p = 12\if jh\l + Mo)q4 is the drag coefficient, kB is Boltzmann constant, T is temperature. Since the fluctuation mean value is zero, i.e. (£) = 0, the analysis should be carried out on its dispersion (C,2(k,t)). Applying the standard stochastic methods, from (4.6) one obtains for the dispersion of the surface wave amplitudes the following evolution equation [65]:
dt
y
q dh p
'
with zero initial condition (C 2(k,t = 0)) = 0. Note that in contrast to (4.4) where zero initial condition is equivalent to trivial (zero) solution, Eq. (4.6) has non-zero solution due to the inhomogeneous term kgT. Thus, for example, for a steady state film (/?=const), where all factors in (4.6) are constant, its solution takes the form:
Eq. (4.7) demonstrates explicitly the conclusion from (4.2") that the long wave spectrum {qqc) is stable (£2{k,tJ) (decreasing with time). It is worth pointing out that initially, i.e. .at
\f - 2(dn /dh)/q2
t«l,
the
process for both spectral branches acquires the character of Brownian motion: ((,2{q,t^0)) =
k
-^t.
(4.7')
For the kinetics of the process, the resulting equation (4.7') means that the longest time is spent around the initial state, followed by an exponential growth (or recession). For emulsion systems, the important question is whether coalescence would occur at the collision of two droplets. Translated into the language of the thin-layer model this is equivalent to the problem of the wave evolution in a thinning film. As we already pointed out, the solution of (4.6) for these cases is complicated because of the dependence of /? and IJ on the film thickness h, which in its turn is a function of time, h{t). This problem finds an elegant solution within the frame of the linear perturbation approach, i.e. in the absence of coupling between the film dynamics and the perturbation waves. Indeed, in this case the film thickness h does not depend on the perturbation £"(0 and, as a unique function of time h(t), provides the correctness of the substitution: d/dt = Vfd/dh, where Vf --dh/dt is the velocity of film thinning (Eq. (2.3), Section 2.1) (see e.g. Ref.[64]). By means of this substitution (4.6) becomes:
dh
\
q an y
'
In Eq. (4.8) time is eliminated and the amplitude of perturbation is presented as a function of the film thickness (C2\(h), which is convenient for obtaining an explicit solution for the thickness of rupture h=hrupt. Following Vrij [39], the formal condition of rupture, reads: ^2)(h
= Km) = hnipt/2
(4.9)
Here, however, comes another problem related to the random nature of the surface waves. It can be seen that the direct inverse Fourier transform of
Hydrodynamic Interactions and Stability of Emulsion Films
253
(4.8.) is divergent and the reason for this is the formal infinite correlation of the surface perturbation £(r). The correct approach requires taking into account the finite spatial correlation when estimating the mean square amplitude (f,2) (t, (r)C, (r,)). Already Vrij refers to the presence of spatial correlations of the fluctuation waves [39]. More detailed analysis is presented in [73,74]. There the term 'correlated subdomain', i.e. the region in which ((^(r)^(r ; )):£0, is defined, as well as the important for the film quantity number of uncorrelated subdomains. The main result of the stochastic analysis in the above cited studies is that the larger (by area) films contain larger number of uncorrelated subdomains, which leads to shorter life-time and, respectively, to larger thickness of rupture hrupt. Besides, these results confirm the well-known fact that the larger (macro-) systems are less stable. The authors in [73] give a numerical solution for the particular case, which corresponds to the approximate relation: K,,P,~R'fiL
(4-10)
For the traditionally studied microscopic films (film diameter from 0.1 to 1.0 mm, see Refs. [65,75]) the already commented stochastic effects, i.e. the effects of correlation, are negligible with respect to the effect of the so-called most rapidly growing wave [39]. The reason is the sharp maximum of the factor (~ I2 v^2 ~ 2{d£l /dh)\) in Eq. (4.6), which determines the most rapidly growing wave in the entire Fourier spectrum causing the film rupture. The literature abounds of formulae for estimating the thickness of rupture, based on the above considerations, whose results do not differ significantly [64,67,76]. An example of an excellent agreement with the experiment is the result obtained in [65]: *.(*!/**«, yu7
(4
.n)
It is worth pointing out that the relation (4.11) does not require the use of a thinning-film model and experimental data can be directly applied instead. The latter is a great advantage of (4.11), as with larger films, the kinetics of thinning is rather complex and its simplification within the scheme of a given model increases the danger of erroneous hrup{ estimate [77]. The fact that fluctuation wave growth competes with film thinning, i.e. that the wave velocity DC, /8t is scaled with the film velocity Vf (see (4.8)), is of a decisive importance for the role of surfactants in the film stability. As we have pointed out above, the influence of the surfactant on the wave kinetics is
254
E. Mileva and B. Radoev
accounted for by the mobility factor Mo (see (4.4)), but the mobility of the film surfaces affects to the same extent the film velocity Vf, as well. Moreover, since Vf ~ 7/p ((3 is proportional to the factor inside the brackets on RHS of Eq. (3.39), Section 3.2.2), the product F7(3 in Eq.(4.8) is independent of Mo. An important consequence of this theoretical result is that hrupt must be independent on the surface mobility. From the experimental viewpoint, the rupture thickness independence on Mo is equivalent to its independence of the surfactant concentration. This, however, is not consistently supported by the experimental results [64,78]. Such a discrepancy between theory and experiment is still an open question. Most frequently, its solution is sought in non-linear effects [76,79], a hypothesis that has not found convincing evidence so far. Another possibility for such an influence of the surfactant on hrupl is the effect of macroscopic heterogeneities in the film thickness caused by hydrodynamic factors. At higher drainage rates Vf, these non-homogeneities are more pronounced [65,77]. Since lower surfactant concentrations correlate with higher Mo-coefficient or lower coefficient fc (see Eqs. (3.19),(3.33),(3.35), (3.37),(3.39)), i.e. with higher Vf, it is to be expected that generally, lower surfactant quantities will incite stronger (macro-) heterogeneity. On the other hand, more and larger non-homogeneities are registered as higher mean film thickness [65,77]. This may be the true reason for obtaining higher than the actual values oihrupl, and not the effect of decreasing surfactant content [80]. 5. CONCLUDUNG REMARKS One very important dynamic characteristic feature of the emulsion systems is the mobility of the interfaces of the droplets. The treatment in the present chapter shows that it is determined by two major factors: (1) the relative viscosity of the droplets and the continuous phase; (2) the presence of surfactants. The viscosity factor fT was first introduced by Rybczinski and Hadamard [38]. At close separations between the fluid particles, this effect is modified due to the hindered outflow in the thin liquid layer. In flow characteristics, it appears always in a combination with the geometry of the emulsion film in the form of the hydrodynamic factor fh. The impact of the surfactant mass-transfer on the hydrodynamics of emulsion systems is related to the onset of flow-driven surface-tension gradients that tend to immobilize the interfaces, and to the responding surfactant fluxes. At small separations, the coupling of the surfactant balance and the fluid motion is modified by the formation of thin emulsion layers, thus leading, in some particular cases, to an enhanced suppression of the tangential mobility of the
Hydrodynamic Interactions and Stability of Emulsion Films
255
surfaces. This effect might be accounted for as a specification of the interfacial mobility scaling by introducing the so-called Marangoni factor: fc. For the majority of the cases of correlation between the flow motion and the surfactant fluxes, both effects, fh and fc act in an autonomous additive manner. Their overall impact on the emulsion film hydrodynamics might be accounted for via a unified parameter termed mobility of the fluid surfaces (Eqs. (3.12'),(3.17'))As for the overall stability of the emulsion systems, it is determined both by the capillary forces and the surface force interactions, exactly as by the foam systems. The interfacial mobility of the droplets is important, insofar as it ensures sufficient slowdown of the emulsion film drainage. For higher interface mobility, the drainage velocity is also increased. This increase is related to the onset of well-defined non-homogeneities of the film thickness, and therefore, in a more rapid coalescence. As a rule, higher surface mobility is observed when surfactants are solvable in the droplets, or almost equally solvable in the contiguous phases.
REFERENCES [I] J. Sjoblom, Emulsion and Emulsion Stability, Marcel Dekker Inc., NY, 1996. [2] T. Gurkov and E. Basheva, in Encyclopedia of Surface and Colloid Chemistry, A. Hubbard (ed.), Marcel Dekker Inc., NY, 2002. [3] J. Lee and T. Hodgson, Chem. Eng. Sci., 23 (1968) 1375. [4] S. Haber, G. Hetsroni and A. Solan, Int.J.Multiphase Flow, 1 (1973) 57. [5] E. Rushton and G. Davis, Int.J.Multiphase Flow, 4 (1978) 357. [6] A. Sharma and E. Ruckenstein, Coll&Polym Sci., 266 (1988) 60. [7] E. Mileva and B. Radoev, Theor.&Appl. Mechanics, Bulg. Acad. Sci., 12 (1981) 49. [8] E. Mileva and B. Radoev, Coll&Polym.Sci., 263 (1985) 587. [9] E. Mileva and B, Radoev, Coll&Polym.Sci., 264 (1986) 823. [10] E. Mileva and B. Radoev, in Particulate Phenomena and Multiphase Transport, T. N. Veziroglu (ed.) Vol.4, Hemisphere Publ.Corp., (1988) 261. II1] E. Klaseboer, J. Ph. Chevaillier, C. Gourdon and O. Masbernat, J.Coll.Interface Sci., 229 (2000) 274. [12] L. Y. Yeo, O. K. Matar, E. S. P. de Ortiz and G. F. Hewitt, J.Coll.lnterface Sci., 257 (2003)93. [13] S. Jeelani and S. Hartland, J.Coll.lnterface Sci., 156 (1993) 467. [14] S. Jeelani and S. Hartland, J.Coll.lnterface Sci., 164 (1994) 296. [15] A. Chester and I. Bazhlekov, J.Coll.lnterface Sci., 230 (2000) 229. [16] A. Scheludko, G. Dessimirov and K. Nikolov, Ann.Sof.Univ., 49 (1954/55) 127. [17] A. Scheludko, Proc. Konikl. Ned. Akad. Wetenschap., B65 (1962) 87. [18] A. Scheludko, Adv.Coll&Interface Sci., 1 (1967) 391. [19] B. Radoev, E. Manev and I. Ivanov, Kolloid Z., 234 (1969) 1037. [20] I. Ivanov, D. Dimitrov and B. Radoev, Colloid Journal, 41 (1979) 36 (in Russian). [21] B. Radoev, D. Dimitrov and I. Ivanov, Coll&Polym.Sci., 252 (1974) 50.
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[22] I. Ivanov and D. Dimitrov, in Thin Liquid Films, I. Ivanov (ed.), Marcel Dekker Inc., NY (1988) 379. [23] D. Exerowa and P. Kruglyakov, Foams and Foam Films, Elesevier, Amsterdam, 1998. [24] R. Sedev and D. Exerowa, Adv.Coll&Interface Sci., 83 (1999) 111. [25] L. Y. Yeo, O. K. Matar, E. S. P. de Ortiz and G. F. Hewitt, J.ColI.Interface Sci., 241 (2001)233. [26] K. Jansons and J.Lester, Phys.Fluids, 31 (1988) 1321. [27] S. Yiantsios and R. Davis, J.ColI.Interface Sci., 144 (1991) 412. [28] R. Davis, J. Schonberg and J. Rallison, Phys.Fluids, Al (1989) 77. [29] E. Mileva and B. Radoev, Coll&Polym.Sci., 266 (1988) 368. [30] E. Mileva and B. Radoev, Coll&Polym.Sci., 266 (1988) 359. [31] T. Traykov and I. Ivanov, Int. J. Multiphase Flow, 2 (1976) 397. [32] J. Davies and E. Rideal, Interfacial Phenomena, Academic Press, NY, 1960. [33] T. Traykov, E. Manev and I. Ivanov, Int. J. Multiphase Flow, 3 (1977) 485. [34] E. Mileva and B. Radoev, Coll&Polym.Sci., 264 (1986) 965. [35] E. Mileva and B. Radoev, Coll&Surf., A74 (1993) 259. [36] E. Mileva and L. Nikolov, Coll&Surf., A74 (1993) 267. [37] E. Mileva and L. Nikolov, in Proc. First World Congress on Emulsions, Paris, France, Vol 2 (1993) 310.1-6. [38] V. Levich, Physicochemical Hydrodynamics, Prentice Hall, Engelwood Cliffs, 1962. [39] A. Vrij, Disc. Faraday Soc, 42 (1966) 23. [40] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics Prentice Hall, Engelwood Cliffs, 1965. [41] L. Sedov, Methods of Similarity and Dimensional Theory in Mechanics, Nauka, Moscow, 1977 (in Russian). [42] N. Gunter, Die Potentialtheorie und ihre Anwendungen auf Grundlagen der Mathematischen Physik, Leipzig, 1957. [43] V. Volterra, Theory of Functionals and of Integral Equations, Dover, NY, 1959. [44] F. Odquist, Math. Z., 32 (1930) 329. [45] O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, NY, 1969. [46] S. H. Lee and L. G. Leal, J.ColI.Interface Sci., 87 (1982) 81. [47] K. Schroder, Math. Z., 49 (1943) 110. [48] V. Beshkov, B. Radoev and I. Ivanov, Int.J.Multiphase Flow, 4 (1978) 563. [49] E. Mileva and D. Exerowa, in Proc. Second World Congress on Emulsion, Bordeaux, France, Vol.2 (1997) 2-2-235/1-10. [50] L. Scriven, Chem.Eng.Sci., 12 (1960) 98. [51] T. S0rensen, in Dynamics and Instability of Fluid Interfaces, T. S0rensen (ed.), SpringerVerlag, Berlin (1979) 1. [52] E. Mileva and B. Radoev, Commun.Dept.Chem., Bulg.Acad.Sci., 24, (1991) 513. [53] J. Israelashvili, Intermolecular and Surface Forces, Academic Press, London, 1992. [54] B. V. Derjaguin, Kolloid Z., 69 (1934) 155. [55] B. V. Derjaguin and L. Landau, Acta Physicochim. URSS, 14 (1041) 633. [56] E. J. W. Verwey and J. Th. Overbeek, Theory of Stability of Liophobic Colloids, Elsevier, Amsterdam, 1948. [57] B. V. Derjaguin, Theory of the Stability of Colloids and Thin Films, Consultants Bureau, New York, 1989. [58] R. S. Allan, G. E. Charles and S. G. Mason, J.ColI.Interface Sci., 16 (1961) 150. [59] D. Platikanov, J.Phys.Chem., 68 (1964) 3619.
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K. A. Burrill and D.R. Woods, J.Coll.Interface Sci., 42 (1973) 15. L. Mandelstamm, Ann.Phys., 41 (1913) 609. A. Vrij, J.Coll.Interface Sci., 19 (1964) 1 J. G. H. Joosten in Thin Liquid Films, Surfactant Science Series, vol. 29, Marcel Dekker Inc., NY, 1988. [64] I. Ivanov, B. Radoev, A. Scheludko and E. Manev, Trans.Faraday Soc, 66 (1970) 1262. [65] B. Radoev, A. Scheludko and E. Manev, J.Coll.Interface Sci., 95 (1983) 254. [66] K. W. Stockelhuber, B. Radoev, A. Wenger and H. J. Schulze, Langmuir, 20 (2004) 164. [67] A. Vrij and J. Th. Overbeek, J.Am.Chem.Soc, 90 (1968) 3074. [68] D. Exerowa, A. Nikolov and M. Zacharieva, J.Coll.Interface Sci., 81 (1981) 419. [69] J. Lucassen, van den Terapel, A. Vrij and F. T. Hesselink, Proc. Konikl. Ned. Akad. Wetenschap., B73 (1970) 108. [70] R. Tsekov, H. J. Schulze, B. Radoev and Ph. Letocart, Coll&Surf, A142 (1998) 287. [71] R. Tsekov and B. Radoev, Int.J. Min. Processing, 56 (1999) 61. [72] E. W. Montroll and J. L. Lebowitz (eds.), Fluctuation Phenomena, North-Holland, Amsterdam, 1989. [73] R. Tsekov and B. Radoev, J.Chem.Soc.Faraday Trans., 88 (1992) 251. [74] R. Tsekov and B. Radoev, Adv.Coll&Interface Sci., 38 (1992) 353. [75] A. Scheludko and E. Manev, Trans Faraday Soc, 64 (1968) 1123. [76] D. Valkovska, K. Danov and I.B. Ivanov, Adv.Coll&Interface Sci., 96 (2002) 101. [77] E. Manev, R. Tsekov and B. Radoev, Disp.Sci.Technol., 18 (1997) 769. [78] E. Manev, A. Scheludko and D. Exerowa, Coll&Polym.Sci., 252 (1974) 586. [79] A. de Wit, D. Gallez and C. Christov, Phys.Fluids, 6 (1994) 3256. [80] J. Angarska and E.Manev, Coll&Surf., A190 (2001) 117.
APPENDIX
jfrh)f^U'%)V°SV^'t 2
2
oo \r +r,
-2rrlcos(pl)
cos
's&h-Yi^Nl, J',(r)= "fc L A
II
* **>**>v/2
^
\FdF)
(A.2) . _ Yfi - r, cosy, )Vr;U° (r, frdr.dy, J
2\r)-}}
rr2 oo
z~2 i n
^A-^
W3
[r +r, - JrrjCOSip,)
i (-\-TflTrfc ^''"JJlJJ no [no
~T'cosW-)Pl{z,H')dZ.
(29)
o
where u^z,!!) is the "bare" interaction between fluid species / on the distance z from confining surfaces fixed at separation H. Very often the force / ( / / ) is referred to as the solvation force; f(H = <x>) corresponds to the bulk fluid pressure p , i.e., pressure of a homogeneous phase of suspending fluid that is in an equilibria with film fluid. The solvation force measured relative to the bulk pressure defines the so-called disjoining pressure, Tl(H) = f(H)-p.
(30)
268
D. Henderson, A.D. Trokhymchuk and D. T. Wasan
In practice, the disjoining pressure can be measured by displacing one of the surfaces a distance dH and calculating the work per unit area to bring the surfaces from the separation H to infinite separation. Thus, the film energy per unit area, E(H), can be calculated as CO
E{H)=
\U{H')dH'. //
(31)
Alternatively to the HAB equation, Eq. (28) usually serves to obtain the interaction energy between a pair of giant spheres. In particular, applying the HNC closure (12) to describe the correlations between a pair of suspended giant spheres, Eq. (28) becomes OO
In g(x) + pu{x) = TTD^PJ j
CO
\hj{s)ds \Cj{t)dt, 0
(32)
x-s
where u(x) is the "bare" interaction energy of the two giant spheres at the gap width x. Taking into account relation (20), the effective interaction between the two giant spheres mediated by a suspending media, has form QO
00
fiW{x) = pu(x) + «£>£PJ \ hj (s)ds jCj (t)dt. j
(33)
X-S
-OO
It is useful to note that Eq. (28) and consequently Eqs. (32) and (33) can be easily differentiated to give d\h(x) - c(x)]
n *^ = ~~zDTPj
'
^
^
j
"r. , , ,
, ,
\hj (s)cj(*"
s ds
)
.. .. (34)
•
-00
Therefore, the force F(x) between the two giant spheres separated by gap distance x is given by „ „
Fix) =
dW(x)
du(x) n -^
%, . . ,
.,
- ^=
^ - -£>£pj \hj(s)cj(x - s)ds.
dx
dx
2
, J
._,.
(35)
J -00
Thus, the HAB equation, (24), in conjunction with Eqs. (29)-(31), provides a description of a slit-like geometry while Eq. (28) combined with Eqs. (32)-
Structure and Layering of Fluids in Thin Films
269
(35) provides a route to the sphere-sphere geometry. The connection between these two geometries can be obtained from the Derjaguin construction [23],
nD
nD
dx
which gives the energy per unit area of two flat surfaces in terms of the force per radius between two giant spheres. We recall that here x = R-D is the gap width between two spherical surfaces and H is the separation between two plane parallel surfaces. Another form of the Derjaguin approximation concerns the disjoining pressure between two flat walls can be expressed through the second derivative of the potential of mean force between two giant spheres
dH
nD
dx2
Equations (36) and (37) are obtained by regarding the giant spheres as being composed of a collection of planar integration elements and integrating. If we suppose that the same approximation has been used in the set of OZ equations (16)-(18), then Eqs. (36) and (37) can be used to validate the Derjaguin construction. Summarizing this section, we wish to make some general and useful comments. First of all, we note that the initial set of the three OZ equations (16)-(18) form a hierarchy. The homogeneous equation, (16), can be solved without reference to Eqs. (17) and (18) that involve the presence of the giant particles. Equation (17) requires as input the DCFs of the homogeneous fluid either from Eq. (16) or some other source, such as a computer simulation. Equation (18) requires as input the correlation functions (or local densities) for the suspending fluid near a giant particle, either from Eq. (17) or some other source. Both Eqs. (16) and (17) can be solved numerically using an iterative algorithms. In contrast, a numerical solution of Eq. (18) does not require an iterative algorithm. Secondly, in applications to a particular problem, it is not necessary to use the same closure for each equation of the set (16)-(18). For example, in electrochemical applications it is often convenient to use the MSA closure for Eq. (16) and the HNC closure for Eq. (17). This procedure is called the HNC/MSA theory. Finally, in Eq. (18) the correlation functions for giant particles appear only on the LHS of this equation. Thus, applying a particular closure for the giant particle correlations need not relate to the closures used for the correlation functions of the suspending fluid on the RHS of Eq. (18). For example, using
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D. Henderson, A.D. Trokhymchuk and D. T. Wasan
the PY closure (10) to describe the correlations between giant spheres, Eq. (18) becomes CO
y(x)-l
= nD^PJ j
GO
\hj(s)ds -oo
Jc,(t)dt.
(38)
x~s
By comparing Eq. (38) with Eqs. (32) and (33), we obtain /3W(x) = -\ngHNC(x)
= -hPY{x),
for x>0,
(39)
where use has been made of the fact that for hard spheres y(R) = g(R) outside the hard core. The result (39) is consistent with the energy of interaction W{x) being proportional to the diameter D of the giant spheres. Thus, the HNC approximation predicts correctly that W{x) is proportional to D whereas both the PY approximation and MSA lead to the incorrect prediction that W(x) is proportional to the logarithm of D. This is because both the PY approximation and MSA are linearized versions of the HNC approximation. If the PY approximation or the MSA are used for the giant particle correlations then the ansatz, /3W(x) = -h(x),
(40)
should be applied. This ansatz is consistent with Eq. (39) and is equivalent to using the HNC for the correlation between giant spheres. In this case, using the MSA or PY closures for Eqs. (16) and (17), yields the HNC/MSA/MSA or HNC/PY/PY approximations, respectively. 3. UNCHARGED FLUIDS NEAR SURFACES 3.1. Simple hard-sphere fluid The simplest system to which the equations of the previous section can be applied is giant hard spheres dispersed in a one-component (or simple) hardsphere fluid. Wertheim [24] and Thiele [25] have solved analytically the OZ equation (16) with the PY closure (10) for a homogeneous hard-sphere fluid and obtained the DCF and thermodynamic properties. Subsequently, Wertheim [26] obtained the Laplace transform (LT) of the RDF for this fluid. Baxter [27] has given an alternative, and in some ways simpler, solution in terms of FTs and contents himself with the DCF and thermodynamic functions. Barker and Henderson [18] used Baxter's method to obtain the RDF in real space for d < R < 2d (d is fluid particle diameter) analytically. This method can be
Structure and Layering of Fluids in Thin Films
271
extended to greater values of R as has been done numerically by Perram [28]. Alternatively, Smith and Henderson [29,30] have inverted analytically the Wertheim's LT of RDF for d < R < 6d using a zonal expansion technique. However, our main interest here is not a homogeneous hard-sphere fluid but the suspending hard-sphere fluid with the presence of giant particles. The starting point for such study is a binary mixture of hard spheres. Lebowitz [31] has extended Wertheim's analysis to obtain, within the PY approximation, the LTs of RDFs and thermodynamics of a mixture of hard spheres. Using Lebowitz's result, Henderson [14] has obtained the LTs of RDFs for the system composed of giant spheres diluted in a single-component (call it species 1, since M = 1) hard-sphere fluid. Together with Wertheim's result [26], these LTs are
and
£|#)1,'"|!'"l-"A'!';-M'ti|lO, -Lx(s)e~sd' +S(s) where
(43)
variable s is the LT variable and 77, = npx 16. The fluid particle
diameter is dx=d, and t]xd\ = is the volume fraction of the suspending fluid. The functions L\(s) and S(s) will be defined shortly. The inverse LTs (41)-(43) can be evaluated by means of a line integral in the complex plane, evaluated analytically usually by residues. Since the giant spheres are present in extreme dilution (pG = 0), the basic behavior of all correlations in the system must be that for a suspending fluid. This yields the result that the denominator in LTs (41)-(43) is the same. Henderson [14] exploited this fact and has related all three LTs to a single LT that he inverted analytically using a zonal expansion. These results are plotted in Fig. 1. Indeed, the profiles of all three curves are quite similar having an exponentially
272
D. Henderson, A.D. Trokhymchuk and D. T. Wasan
decaying oscillatory shape with the same "periodicity" and decay length but with a different magnitude and phase. As is seen in Fig. lb, the fluid density near the giant sphere (or, for that matter, a surface) is stratified or layered just as it is in the vicinity of a fluid sphere in Fig. la. These results, first given in Ref. [20], were perhaps the earliest indication of layering near a surface. Later, this has been observed experimentally [7,11,32]. An alternative, easier but numerical, method to calculate the correlation functions in real space can be developed by converting the LTs into FTs that can then be inverted numerically by simple quadrature. With this method one can obtain numerical results for any value of R. An advantage of the inversion by means of residue theorem is that it provides one with simple analytical results for the correlation functions at short distances as well as in the asymptotic regime of large R. In particular, for the correlation function between a pair of giant spheres at their contact, x = 0, we obtain h(0)=^^.D,
(44)
2(1 -f d while at large distances x the same correlation function behaves as [15]
h(x) ~ 2 | yHS | cos{coHSx + arg{yHS})exp(-/cHSx),
(45)
where coHS and KHS are the real and imaginary parts of the pole of c[g{R)\ with the smallest imaginary part, and yHS is the residue of £.[g(R)] at the same pole. The latter result (45) is plotted in Fig. lc as well and we can see that the agreement between a leading asymptotic term (45) and a complete FTNC/PY/PY result (43) is rather good except for short distances of about one layer of suspending fluid, where an asymptotic formulae should not be expected to be applicable. The results of Eqs. (44) and (45) are valuable, since, in accordance with ansatz (40), both provide information on the effective interaction between the giant spheres mediated by the suspending fluid. Recently, Roth et al [33] exploited the fact that the asymptotic term (45) describes accurately the oscillatory structure of the total correlation function and have developed a simple parametrization for the interaction energy W(x) between two macrosurfaces suspended in an ordinary hard-sphere fluid. Their parametrization provides an accurate fit to the density functional theory (DFT) results as well as to the existing computer simulation data.
Structure and Layering of Fluids in Thin Films
273
Fig. 1. Correlation functions for a pair of giant spheres of diameter D suspended in a simple hard-sphere fluid. The suspending fluid consists of spheres of diameter d whose volume fraction is <j> = 0.35 . Part a gives the RDF of a bulk phase hard-sphere fluid. Part b gives the fluid-sphere RDF, which is the normalized density profile of the suspending fluid near a giant sphere or a flat surface. Part c gives the TCF of the pair of giant spheres which is related to effective interaction or mean force potential; the dashed line represents the calculations using a leading asymptotic term (45).
In Fig. 2 we plot the interaction energy, W{x), obtained from Eqs. (40) and (43) together with DFT-based results of Roth et al [33]. Note that the energy is oscillatory and exhibits stratification. As the two giant spheres are brought closer together, layers of suspending fluid species are ' squeezed' out. At very close separations, all of the suspending fluid species have been squeezed out. When the gap between the pair of giant spheres is depleted of the species of suspending fluid one may speak of a depletion force that is attractive. This effect was first noted by Asakura and Oosawa (AO) [35], who examined the depletion force at low densities of the suspending fluid and found W(0) = -3D(#/2J. We can see that in the low density limit, expression (44) reduces to the AO result [35]. This is not surprising since the PY theory is known to be correct at low densities.
274
D. Henderson, A.D. Trokhymchuk and D. T. Wasan
Fig. 2. Interaction energy W between two giant spheres whose diameter is D, and disjoining pressure n , between two plane parallel surfaces suspended in a simple hard-sphere fluid. The suspending fluid consists of spheres of diameter d whose volume fraction is <j> = 0.314 . The dashed lines are the results of the HNC/PY/PY approximation while the solid lines represent the parametrization of Roth et al [33] for the interaction energy, and that of Trokhymchuk et al [34] for the disjoining pressure. The symbols for the disjoining pressure correspond to the MC data of Wertheim et al [38],
The agreement of the HNC/PY/PY result [ Eqs. (40) and (43) ] with the DFT-based result is remarkably good for all range of separations except for those separations near contact at x = 0 , where the agreement is only qualitative. The result of Eqs. (44) is larger in magnitude by roughly a factor (1 - £,y2. This is the penalty for using the PY approximation to treat the fluidfluid and fluid-giant sphere correlations. Beyond contact, as the gap width between macrosurfaces increases, the interaction energy calculated within HNC/PY/PY approximation approaches the DFT-based data. Particularly, as we can see from Fig. 2, beyond the depletion region, defined by the position of the main repulsive maxima in the energy profile, both results are practically indistinguishable. This is an expected result, since, as we already noted [36,37], both approaches have common roots. Trokhymchuk et al [34], using the asymptotic term (45) for the interaction energy W(x) and Derjaguin construction (36) and (37), have given simple analytical expressions for the film energy per unit area E(H) and disjoining pressure Tl(H) of a hard-sphere fluid between two flat surfaces, as a function of suspending fluid density. These expressions are parametrized to satisfy with some known exact relations for a confined hard-core fluid and are designed to be easily implemented in the calculations of film properties in the various
Structure and Layering of Fluids in Thin Films
275
applications of the type reported recently by Wasan and Nikolov [3]. Figure 2 shows the disjoining pressure Tl(H) exerted by a hard-sphere fluid film that is calculated from parametrization of Trokhymchuk et al [34] and compared with MC data for the same system studied by Wertheim et al [38]. 3.2. Binary and many-component hard-sphere fluid and size polydispersity It is interesting to generalize Eq. (39) to an arbitrary number of components in the suspending fluid. Here we present such a generalization that is intuitive but not unreasonable. The result [37] is 3^?]le-sd!-(3S2s2/2-3S]s
£fe(*)] = ^
sld+d)
^
i<j
+ 3S0) d
D,
(46)
i
where
hiJ=36TJiTiJ(di-dJ)2,
(48)
Li:(s) = 12f7,.[(1 + ^3 /2> 2 J,. + (1 + 2S3)s] ^ , » + £ [ 1 8 7 ^ / 4 ( 4 " dj)s2 + ^ ( 1 - sdtJ)]
(49)
j*i
and S(s) = \25Q(l + 2S3)s-\SS22s2
-6S2(\-52)s3
-(l-S3)2sA (50)
+ ^dhiJ[l-S(di+dj)] The definitions of L^s) and S(s) are based on the definitions of Lebowitz, changed slightly for an easier generalization to an arbitrary number of components, and differ somewhat from the definitions used by Wertheim. Because a large number of components leads to a large number of zones, Eq. (46) is not suitable for a zonal expansion. However, numerical inversion is
276
D. Henderson, A.D. Trokhymchuk and D. T. Wasan
not difficult. The energy and force in a case of binary suspending fluid (M = 2) with a size ratio, small-to-large 1:10, are plotted in Fig. 3. Because of the large size asymmetry, the depletion interaction scenario observed for two giant spheres in a one-component suspending fluid composed of only large particles, is significantly affected when the fine species of suspending fluid are taken into account. The gap between two giant spheres, that is depleted of the large particles in a one-component fluid, becomes filled by the fine particles in the case of a two-component fluid. Now the small species of diameter dx = ds exhibit stratification since they are confined by giant spheres of diameter D and the large suspending fluid species of diameter d2= d. This changes qualitatively the shape of both the interaction energy and force profiles that now show fine oscillatory structure at separations near the contact of the giant spheres; the oscillations are governed by the diameter of small species. To a reasonable approximation, the energy and force between two giant spheres suspended in a bidisperse fluid can be viewed as a superposition of single-component results obtained using the diameters and, perhaps, effective densities of the both fluid components. Similar trends were seen previously for a bidisperse colloidal suspension between two flat parallel walls [39]. This, of course, is to be expected since energy per unit area, E(H), and disjoining pressure, Yl(H), for a pair of flat parallel surfaces are related simply, by means of Eqs. (36) and (37), to W(H) and its derivatives.
Fig. 3. Interaction energy, W, and corresponding force, F, between two giant spheres whose diameter is D, and dissolved at infinite dilution in a two-component hard-sphere fluid thatconsists of the small spheres of diameter d\ = ds and volume fraction ^ =0.15 and the large spheres of diameter d2 = d = \0ds and volume fraction ^ 2 = 0-20.
Structure and Layering of Fluids in Thin Films
277
To a reasonable approximation, the energy and force between two giant spheres suspended in a bidisperse fluid can be viewed as a superposition of single-component results obtained using the diameters and, perhaps, effective densities of the both fluid components. Similar trends were seen previously for a bidisperse colloidal suspension between two flat parallel walls [39]. This, of course, is to be expected since energy per unit area, E(H), and disjoining pressure, Yl(H), for a pair of flat parallel surfaces are related simply, by means of Eqs. (36) and (37), to W(H) and its derivatives. Similar results are plotted in Fig. 4 for a four- and ten-component suspending fluids. Each of these multi-component fluids include the fine particles as in the case of a binary fluid. Again, there is layering but the "period" of the layering reflects the sizes of all the constituent species of suspending fluid. In general, an increase of the number of components of the suspending medium shows a tendency to destroy the fluid layering. As the giant spheres are brought together, the interfacial region becomes depleted of the larger fluid particles but can still be filled with smaller particles. Thus, oscillations are present at a wide range of separations determined by the size of the smallest and the largest suspending fluid particles.
Fig. 4. Interaction energy, W, and corresponding force, F, between two giant spheres whose diameter is D, and dissolved at infinite dilution in a four and ten-component hard-sphere fluid. In each case suspending fluid consists of fine species of diameter d\ = ds and volume fraction \ = 0.15 . In the four-component case (solid line) the other species have diameters d2 = d = \0ds , and (\±0.5)d diameters are d =\0ds, (\±Q.2)d
whereas in the ten-component case (dashed line) the , (]±0A)d
, (1 ± 0.6)rf , and (l±0.8)J . In both cases,
the total volume fraction of the suspending species, other than fine component 1, is 0.2.
278
D. Henderson, A.D. TrokhymchukandD.T. Wasan
At this point we wish to show how our results for a many-component hardsphere suspending fluid can be extended to describe the polydisperse hardsphere suspending fluid with a continuous size distribution /(<x) that is normalized to be a probability density, i.e., \f{cr)d(j = 1. In a rigorous way the concept of polydispersity has been discussed by Salacuse and Stell [40] as well as others and applied to the polydisperse fluid of hard spheres by Blum and Stell [41]. Following their recipe, the introduction of probability density /(cr) projects a countable infinitude of the fluid components into a continuous distribution of sizes, replacing the species number densities pt by the average density //(cr,) where p is the fluid total number density. Then Eq. (46) has the form £[g(R)] =
5
, , , 3?7\f{x)e~sxdx - (\S2s2 - 3S]S + 3S0) 2*12 \\f{x)f{y)h{x,y)e-s{x+y)dxdyr,\f{x)L{x;s)e-sxdx
,
(51)
+ S{s)
where Sn=tjjf(x)xadx,
(52)
h(x,y) = 36(x-y)2,
(53)
L(x; s) = 12[(1 + —> 2 x + (1 + 2 S3 )s] 2
(54) 2
+ n J/O0[l W (x ~y)s + h{x, y){\ - s
*^)]dy,
S(s) = US0(\ + 2S3)s - 18£2 V - 652(\ - S3)s2 - (1 - 53)2s4 + \l2 \\f(x)f(y)h(x,y)(\ In these equations rj = npl6.
- s[x + y])dxdy
Structure and Layering of Fluids in Thin Films
279
Figure 5 shows the effect of polydispersity on the interaction energy between two giant spheres suspended in a fluid with the Shultz size distribution
( V+z
where (a) is the average diameter and the degree of polydispersity is controlled by parameter z. For the sake of comparison, the results of AO polydisperse model at a low density are shown as well. The AO results are calculated from oo
pWA0{x)=7^pd\f{G){(T-xfda,
(57)
x
where x = R-D. We can see that results of both, Eqs. (51) and (57), are quite similar showing that an increase of the polydispersity at low volume fraction of the suspending medium leads to an increase of the range of depletion attraction.
Fig. 5. Effective interaction energy, W, between two giant spheres whose diameter is D, and dissolved at infinite dilution in a hard-sphere suspending fluid with different degrees of size polydispersity. Part a shows results at a low volume fraction 0.01; the dashed lines represent the results of the AO polydisperse model. Part b shows results at a high volume fraction 0.3925.
280
D. Henderson, A.D. Trokhymchuk and D. T. Wasan
As we already learned from the results of Fig. 2, a high volume fraction of monodisperse suspending medium promotes an oscillatory structural interaction between suspended species. The results presented on part b of Fig. 5 shows that introducing size polydispersity at a fixed volume fraction results in a decrease of both the attractive well at x - 0 and structural repulsive peak as the polydispersity goes up. Eventually, this leads to a smoothing the oscillatory features of the interaction energy profile. The effects of size polydispersity, similar to those of Fig. 5, have been observed by Walz and Sharma [42]. 3.3. Fluid of dumbbells, flexible linear chains and more complex aggregates The equations in the previous subsections are appropriate for a suspending fluid composed of spherical hard-core species with different diameters. An extension of the analysis to the suspending fluids comprising non-spherical particles can be provided by generalizing the set of OZ equations (16) - (18). One of the ways to do this is by using the ideas and models from the theory of site-site associating fluids. We will illustrate this by considering three models of a suspending fluid: a fluid formed from dimerizing hard spheres or dumbells, a fluid of flexible linear chains and a network-forming fluid. To formulate the fluid models we assume that the hard-sphere particles that compose a suspending fluid are of the same diameter d and each may have one, two or four attractive sites placed on the particle surface. The potential of interaction between two fluid particles for such model fluids becomes angulardependent and consists of a hard-core repulsion as in an ordinary hard-sphere fluid, plus a contribution describing a highly directional associative (AS) attraction between the sites of two different fluid particles,
fljn
^
\~£ab>
I 0,
x
ab
<x
c
r,-Q\
xab>xc
where 1 and 2 denote the position and orientations of the two fluid particles while a and b are the notations for sites on the particle 1 and 2, respectively. The variable xab is the distance between the sites a and b of the particles 1 and 2; an associative bond between two particles is formed if their two sites are within the distance xc. The parameter sah reflects the strength of an associative interaction; when sab - 0, the models converge to the ordinary hard-sphere fluid. Models with one and two sites describe the dimerizing and polymerizing fluids, respectively [43-45]. The four-site model mimics symmetrical and nonsymmetrical network-forming fluids that may consist of very complicated
Structure and Layering of Fluids in Thin Films
281
aggregates, e.g. linear branches, loops, crossing rings, etc [46]. Therefore, the four-site model is able to reproduce various kinds of macromolecules and a network of bonds by appropriate choices of the energy parameter sah . The properties of these model fluids in a bulk phase have been investigated intensively during the last decade [45-49] using a multidensity version of the OZ equation (16) that is usually called the Wertheim OZ (WOZ) equation [43,44]. For a simplest case of a dimerizing fluid, i.e., with only one site per fluid particle, WOZ reads
hf{\,2) = cf (1,2) + X \Kr {\,l)PrEcf (3,2) [(e-M-^y
1] = K8(R - d),
(63)
where K is the parameter uniquely determined by the strength of associative interaction between sites a and b. The calculations are simplified further by the assumption of energetic equivalence of attractive sites, i.e., sab = s. To show the effect of the non-spherical nature of suspending fluid on stratification phenomena, in contrast to preceding subsections 4.1 and 4.2, here we explore the route utilizing Eq. (17), i.e., route based on the HAB equation. The route that is based on Eq. (18) has not been elaborated for the associating fluids so far. The associative version of hard-sphere HAB equation has been proposed by Holovko and Vakarin [47] and for dimerizing fluid reads
Ko(Rn) = c%(Ru) +1 K f (Rn)pPrcJ?(h2)dr,,
(64)
Pr where hjG°(R) and cfg(R) are the partial total and partial direct fluid speciesgiant sphere correlation functions, respectively. The superscript 0, that corresponds to giant sphere subscript G, indicates the absence of any bonded states at the surface of giant spheres if an associative interaction of the type of (58) between fluid particles and giant sphere is not considered.
Structure and Layering of Fluids in Thin Films
283
Recently, Duda et al [50] have applied the associative HAB equations of the type of Eq. (64) to evaluate the local density p(z,H) = phlG(z,H) for the suspending fluid composed of dumbells, flexible linear chains and network aggregates confined to a slit pore. The fluid-wall total correlation function hiG{z,H) was calculated as a linear composition of the partial fluid-wall functions hiG(z,H) and h]G{z,H) in the way similar to the bulk fluid case, Eq. (61). Some representative density profiles are plotted in Fig. 6 as a function of fluid particle distance from the confining surfaces. To mimic the reality where confining surface can be lyophilic or lyophobic, the fluid species-giant sphere interaction, ujG(z,H), was used as a (9,3) Lennard-Jones potential with and without attractive term, respectively. For the comparison, the local densities of an ordinary hard-sphere fluid are shown as well.
Fig. 6. Density profiles p(z,H)of a simple hard-sphere fluid (thin solid line), fluid consisting of dumbbell-shaped particles (dashed line), linear flexible chains (short dashed line) and network-forming fluid (thick solid line) in a slit-like film. The film thickness is H = 8d.
284
D. Henderson, A.D. Trokhymchuk and D. T. Wasan
The quite tall and narrow peaks that are seen in Fig. 6 next to the confining surfaces reflect the well defined surface layers for a hard-sphere fluid. These peaks decrease for a fluid of dimers, decrease and widen for polymer chains, and bifurcate and almost disappear for fluid composed of network aggregates, especially in the case of the repulsive fluid-surface interaction. As the distance from the substrate increases, the heights of the peaks decrease and their widths increase. Nevertheless, we still can observe at least three well-defined layers even for flexible linear chains. At the same time, it is worth noting that the left shoulder of the double first peak of the network fluid density profile, that we observe near the attractive surfaces, totally disappears near the repulsive surface. It means that the fluid comprised of species of irregular shape tends to segregate from the confinement diminishing the effect of stratification. To understand the qualitative trends in the film properties initiated by the shape of suspending fluid species and different level of fluid stratification, in Figs. 8 and 9 we show the results for the force between film surfaces that is mediated by suspending fluid. Experimentally the force/radius between the two crossed cylinders of radius a-D/2 can be determined directly in the method of Israelachvili [7]. To obtain results that are comparable with measurements we have added to the fluid mediated force the long range van der Waals contribution [the so-called Hamaker force] that arises from the direct interaction between the surfaces. This interaction is obtained by integrating the - R~6 term in (say) the Lennard-Jones interaction (a four-fold integration) over the volumes of the two interacting bodies. The result is complex. However, if the bodies are giant, the result is quite simple and in the case of the two crossed cylinders equals - ADlYlx. Thus, the calculated force, which can be compared with the observed data, has the form [see, e.g., Eqs. (31) and (36)]
F
^
a
=
^
\2x2
+ n [U(H )dH ,
(65)
I
where A is the Hamaker constant and H is the gap width between confining surfaces. Figure 7 shows, by a dashed line, the force/radius calculated according to Eq. (65) between a pair of macrosurfaces that are suspended in a hard-sphere fluid of density pd3 =0.85. Taking a hard-sphere diameter d =0.9nm such fluid roughly simulates octamethylcyclotetrasiloxane (OMCTS), which is the frequently investigated experimental system [7]. The inset shows the measured force law between two cylindrically curved mica surfaces in OMCTS [6]. OMCTS is an inert silicone liquid whose non-polar molecules are quasispherical in a shape. To take into account the effect of non-sphericality, we
Structure and Layering of Fluids in Thin Films
285
assumed that the OMCTS molecule is slightly elongated in one direction and this shape can be represented by a dumbbell composed of two fused hard spheres, each of diameter dd. Choosing the diameter dd from the condition that the volume of the dumbbell-shaped molecule is the same as the volume of the spherical molecule of diameter d = 0.9nm, we performed calculation of the force in a dumbbell fluid with diameter dd =0.73nm, maintaining fluid volume fraction the same as in a monomer hard-sphere fluid. We can see that two calculated force profiles differ with the dumbbell model improving an agreement with experimental data.
Fig. 7. Force between two macrosurfaces suspended in a hard-sphere fluid of diameter d = 0.9 nm [dashed line] and in a dumbbell fluid of diameter dd =0.73nm [solid line]. The dotted line shows the van der Waals force. The fluid volume fraction in both cases is the same, pd
= 0.7. The inset shows the measured force law between two cylindrically curve
mica surfaces in OMCTS with average molecular diameter - 0 . 9 nm. The inset is adapted from Fig. 5 of Ref. [6].
286
D. Henderson, A.D. Trokhymchuk and D. T. Wasan
Figure 8 shows the predicted results for the force between two macroscopic surfaces in a chain and network-forming fluids. These two results can be compared with measured force laws between mica surfaces observed by Christenson et al [51] across straight-chained liquid alkanes such as ntetradecane and n-hexadecane, and by Gee and Israelachvili [52] measured in a branched alkane [iso-parafin] 2-methyloctadecane. We see, that the force in linear chain fluid still exhibits oscillatory behavior, similar to the film of spherical molecules, but is shifted to the attraction region. The force mediated by a network fluid is completely different: it is repulsive until a thickness of about 1 nm [of order of two-to-three single bead diameters] and then becomes attractive and almost totally monotonic. The overall qualitative agreement between the calculated and experimental results is quite good.
Fig. 8. Force between two lyophilic macrosurfaces in a fluid of linear chain (thin solid line) with d - 0.4 nm, and in a network-forming fluid (thick solid line) with d = 0.45 nm. The density of both fluids is pd =0.85 .The dotted line shows the van der Waals force. The inset shows the measured force laws between mica surfaces in straight-chained liquid alkanes such as n-tetradecane and n-hexadecane (molecular width —0.4 nm), and in the branched alkane (iso-paraffin) 2-methyloctadecane. The inset is adapted from Fig. 13.5 of Ref. [7], page 272.
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287
The repulsive force between lyophilic surfaces at small separations originates from the first adsorbed layer of suspending fluid and extends on the separations of order of the thickness of this layer. The fluid layering near substrate can be identified from the oscillations of local density [see Fig. 6]. Two density maxima that we observe in a network-forming fluid next to the surface can be treated as the spliting of a rather thick first layer, which remains disordered or amorphous. The local density in such layer is always higher than the corresponding bulk density, and the force exerted by the fluid on the surfaces is repulsive. The fact that the chain structure of the suspending fluid can result in an attractive or repulsive contribution to the solvation force has been pointed out by Israelachvili [7] who calls this effect a positive or a negative solvation force. 4. CHARGED FLUIDS NEAR SURFACES In electrochemistry and colloidal science, the suspending medium is usually an electrolyte containing charged species - ions and polar solvent molecules. Further, the confining surfaces - electrode and/or the colloidal particles are charged. The MSA can be applied to such complex systems. 4.1. Bulk electrolytes The simplest model of an electrolyte is a system of charged hard spheres (the ions) in a continuum dielectric medium whose dielectric constant is s (the solvent). To keep the model simple we assume that the dielectric constant inside the charged hard spheres is the same dielectric constant as that of the electrolyte so that we need not worry about induced charges on their surface. This model is called the primitive model (PM) of an electrolyte. If the charged hard spheres all have the same diameter, this model is usually called the restricted primitive model (RPM). Waisman and Lebowitz [53] have solved the OZ/MSA equations analytically for the RPM. Their results have been simplified by Blum [54], who has also solved the OZ/MSA equations for the more general case of differing diameters, i.e., for the PM. In the case of the bulk RPM, the ion-ion RDFs are given by
J
s(\ + Yd)2
R
where d is the ion diameter. For the moment, all ions are considered to have the same diameter. The function gHS(R) is the hard-sphere RDF, f(R) is a
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D. Henderson, A.D. Trokhymchuk and D. T. Wasan
function that will be defined shortly, and F is a parameter that is related to the Debye screening length, K , defined by
-2 =4 fl>,V,
(67)
through the relation K = 2Y(l + Td).
(68)
Note that at low concentrations, when K is small, F = K/2 .
The LT of f(R) is given analytically, L[f(R)] = -7
,
•
(69)
Henderson and Smith [30] have inverted this LT and obtained f(R) as an infinite sum that involves spherical Bessel functions. From Eq. (69), we see that for small K (low concentrations) L[f(R)] = — -
(70)
S+K
So that f(R) = e~KR.
(71)
Thus, at low concentrations, the MSA result for the ion-ion RDFs becomes fin n
a-K(R-d)
If we make the approximation, valid at low concentrations,
{\ + Ydf
( d\2 K =[\ + =l + xd,
(73)
Structure and Layering of Fluids in Thin Films
289
neglect d in the argument of the exponential, and use gHS(R) = l,
(74)
the MSA result becomes the Debye-Huckel (DH) result [55]. We have obtained the MSA result from an integral equation whereas the DH approximation is usually obtained from a differential equation [often called the Poisson-Boltzmann (PB) equation]. The DH or PB approximation is the zero-diameter version of the MSA for charged hard spheres. In the DH/PB approach, Eq. (15) is assumed to be valid for c(R) for all distances. The MSA is a systematic generalization of the DH theory (strictly speaking in its linearized version). The factor 1 of the DH/PB approach becomes gHS(R), the exponential function becomes f(R), and (1 + KT/) becomes 2 (1 + Yd) . We will see that the factor (1 + Yd)2 overcomes the asymmetry that is a problem in the DH/PB theory. In the DH/PB approximation, only the central ion is given a size. In the MSA theory, all ions are treated consistently. The appearance of g (R) leads to oscillations that reflect the stratification phenomenon. If the PM is used, the density of the hard spheres in an electrolyte is low and these oscillations are correspondingly small. However, if an explicit molecular model of the solvent is used, the hard-sphere oscillations become important. The function f(R) also oscillates but this is not very important as the concentration of ions is small, whether or not an explicit model of the solvent is used. The oscillations in f(R) become more important with divalent ions. In a molten salt, the oscillations of f(R) would be of even greater importance. The renormalized screening parameter, 2F, is smaller than K SO that the system does not become overscreened. In the DH/PB theory \lK can become smaller than d. This is an unphysical situation because it would mean that the screening atmosphere is inside the central ion. We have commented that the MSA is a generalization of the linearized DH or LDH theory. The HNC is an appropriate generalization for the nonlinear DH theory [which does not yield an analytic solution]. Our attention here is restricted to the MSA and LDH approximation. The generalization of Eq. (66) to the case where the electrolyte ions have different diameters is difficult. However, considerable progress has been made by Blum [54]. As long as most of the ions have the diameter d and there are only a very small number of ions with different diameters, the ion-ion RDF is given by
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D. Henderson, A.D. TrokhymchukandD.T. Wasan
where gjj (R) is the RDF of a hard-sphere mixture. The symmetric treatment of the ions [lacking in the DH/PB theory] is apparent in Eq. (75) and is of particular importance in applications to charged colloidal suspensions. To progress beyond the PM, we must include a molecular model of the solvent. A particularly simple model of the solvent is obtained using a system of hard spheres together with the dielectric background. This model has been called the solvent primitive model (SPM). At first glance, this would seem to be an unpromising model of a polar solvent. However, this model does recognize that the solvent molecules occupy space. Within the MSA, results obtained for a bulk electrolyte using the SPM are similar to the results obtained above with the PM. The solvent molecules are just another species of charged hard spheres that happen to have zero charge. Thus, Eq. (75) still applies but the density of hard spheres now includes the density of solvent and is large and g{fs(R) has pronounced oscillations. The second term in Eq. (75), that controls the electrical response of the system, is unaffected by SPM. Of course, this separation of the charged and non-charged terms is a feature of the linearized nature of the MSA. It would not be occur in a nonlinear theory, such as the HNC approximation where the effect of the SPM would propagate to the charged terms. A somewhat more sophisticated model of a solvent is a system of dipolar hard spheres. Now there is no dielectric background. The dielectric constant is the result of the non-zero dipole moment, /z, embedded into the solvent spheres. Wertheim [56] has used the MSA to obtain the properties of the dipolar hardsphere model solvent. He found \6s = A2(l + A)4
and
(76)
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291
where ps is the number density of the solvent. Note that Eq. (76) is a cubic equation in X1 that can be solved analytically. For s=\, X=\ and for £-78, X-2.65. Thus, X is a weak function of s. These equations give the relation between dipole moment /j and s. Although approximate, this relation is an improvement over the earlier Onsager formula. The value of /J , corresponding to the dielectric constant of water, is greater than the dipole moment of water vapor molecules. The usual view is that at high densities Eqs (76) and (77) give an effective dipole moment that accounts for such effects as polarization. Blum [54] has solved the MSA for a mixture of dipolar and charged hard spheres [loosely called the ion-dipole model] and has obtained results for the thermodynamic properties for this model electrolyte. The MSA results for this ion-dipole model will be discussed further in the next section. 4.2. Electrolyte near an electrode Equation (75) can be applied to an electrolyte near an electrode. Many electrodes are planar but some, for example a hanging drop mercury electrode, are non-planar. The planar electrode can be regarded as a giant sphere. The RDF for a pair consisting of a giant sphere of charge Q and radius a - D/2 and an ion of charge qt and diameter d, or in other words the normalized density profile gt(R) = pt(R)f p of the ions near an electrode is
M w-a w- n ?*L,/(*-f-f) a s{\ + Td)2Ta
v
™
2 2J
or gi(z)
=g f ( z ) - ^ . O o / ^ - 0
(79)
where z = R-D/2 and % =- ^
(80)
is the electrostatic potential at the giant sphere. Equation (68) has been used to obtain Eq. (79) which can be rewritten as g,(z) =g -(z)-^/z-*a £K \ 2
(81)
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D. Henderson, A.D. Trokhymchuk and D. T. Wasan
where E is electric field at the electrode. The electrostatic potential of the electrode, V, is <J>0 plus the potential difference, Ed I'2s, across the region of thickness d 12 between the electrode and the distance of closest approach of the ions. If the PM is appropriate, the PB theory is called the Gouy-Chapman (GC) theory [57], which actually predates the DH theory. If the GC/PB theory is applicable, Ed
ir
.
Ed
E
^ = ^ + ^o = — + —• 2s 2s SK
(82)
Using the MSA, gives
v
( 83 )
= 4^'
s2F and with Eq. (68),
1 1+IW 1 d — = = - + - + ... 2T
K
K
(84)
2
the GC/PB theory is recovered. The region near the macrosurface 0 < z d 12, this is described by the GC/PB or MSA theories, is often called the diffuse layer. The whole system is called a double layer because there is a layer of charge on the electrode and a compensating layer of charge in the electrolyte. In this picture, the potential difference across the double layer consists of the potential difference across the inner layer and the potential difference across the diffuse layer. The double layer acts similarly to two capacitors in series. Indeed, there is some seemingly convincing experimental evidence [58] supporting C"1 =C-t}+C~D\
(85)
where CD is the diffuse layer capacitance, as predicted by the GC/PB theory, and CH is the inner, or Helmholtz or Stern, layer capacitance that is independent of the concentration and other properties of the ions. As will be seen shortly, this picture is overly simplified. As seen in Eq. (84), two terms appear in the MSA expression for V only if the expansion is truncated at two terms.
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293
To bring Eqs. (82) and (85) into agreement with experiment, the properties of the inner layer must be adjusted. One popular scheme is to say that the dielectric constant, s *, in the inner layer is different from the bulk value. Thus, Eq. (82) becomes
2s*
SK
A value s * =3~6 seems appropriate. A reduced value for s * seems reasonable since the solvent is expected to be less polarizable near the electrode. However, this empirical approach is less than satisfying. Is it reasonable to confine the interfacial region of the solvent molecules to a region right at the electrode? Is a sharp dielectric boundary reasonable? Maxwell's equations require a polarization charge at a dielectric boundary. This effect has been satisfied only in part because no attempt has been made to ensure that the induced charge does not disturb the GC/PB ionic profiles. For that matter, induced charges have been ignored at the electrode surface also. However, this latter point may be reasonable there since 78 is well on the way to infinity [the value of the electrode dielectric constant, assuming the electrode to be metallic]. Since the value of s* is empirical, one might argue that the effect of the induced charges has been included in the value of s *. For this to be the case one would need to establish that the induced charge at the interface is independent of concentration [or at least only weakly dependent].
Fig. 9. Ion density profiles pt(z) calculated from the GC/PB theory (solid lines) compared with MC simulation data (symbols). Part a gives typical results for a monovalent salt. Part b gives typical results for a divalent salt.
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D. Henderson, A.D. Trokhymchuk and D. T. Wasan
As was the case for a bulk electrolyte, the PB theory is the point charge [or low concentration] limit of the MSA theory. For the double layer, the PB theory is called the GC theory. In contrast to the homogeneous version of the PB [the DH theory], the GC/PB equations can be solved analytically even when the electrode charge is large and the equations are nonlinear. The nonlinear version of the MSA theory, the HNC approximation, requires a numerical solution. Linear or nonlinear, the GC/PB results for concentration, charge, and potential profiles are monotonic and the term double layer is sensible. In contrast, in the MSA or HNC approximations these profiles can oscillate, again reflecting the presence of stratification phenomenon. As is seen in Fig. 9 where a comparison of the GC/PB ion density profiles with MC simulation data is shown, this is particularly apparent for divalent salts where the oscillations in the density profile are large enough that there is charge inversion where the density of co-ions can exceed the density of counter-ions. In this case the charge in the layer of counter-ions near the electrode exceeds the magnitude of the electrode charge and there is a further layer of co-ions. The total charge in the profiles is still equal and opposite to that of the electrode as overall charge neutrality requires. However, rather than a double layer, there is a triple layer of charges. The oscillations seen in Fig. 9 are relatively mild. Much more pronounced oscillations would be seen if a molecular model for solvent were used. One result of the GC theory is kTYpAd/2)
= kTp +
F2
,
(87)
where p = ^ pt is the total density of the ions. This is a special case of the exact, within the PM or SPM, result due to Henderson et al [59] kT^Pi(d/2) = p + ^ ,
(88)
where p is the bulk pressure (including both hard sphere and electrostatic contributions) of the suspending electrolyte. The term on the LHS in Eq. (88) is the momentum transfer to the electrode. This must equal the sum of the pressure and the second term on the RHS, which is the electrostatic (or Maxwell) stress. The GC/PB result is a special case of the exact result for the case where p = pkT (i.e., uncharged point particles with zero diameter). Neglecting terms that are higher order in E than linear, Eq. (88) becomes
Structure and Layering of Fluids in Thin Films
295
kTy£dPi(d/2) = p.
(89)
In contrast, the MSA and HNC theories give, instead of p, a geometric or arithmetic mean of the hard-sphere values of p and p/3(dp/dp), in Eqs. (89) and (88), respectively. At low concentrations and not too large values of the dimensionless charge parameter, flq^jlsd, the difference with p is small. However, if the concentration is significant, the valence of the salt is large or if the temperature, dielectric constant, or diameter are small, p can be large or, more usually, small, resulting in large or small contact values for the ion density profile. In contrast, the GC/PB, MSA and HNC results for the contact values of the ion density can only equal or exceed the bulk density. Thus far, only the PM and SPM have been considered. Carnie and Chan [60], Chan et al [61] and Blum and Henderson [62] have obtained an analytic solution of the MSA equation for the ion-dipole model of suspending fluid. For ions of diameter d and solvent dipolar hard spheres of diameter ds that all of the same size ds - d, the normalized ion density profiles have the form gl(z) = gHS(z) + Ahi(z)
(90)
and the normalized solvent density profile has the form gs(z,0) = gHS(z)+ l3Ahs(z)cos0, where gHS(x)
(91)
is the normalized density profile for neutral hard spheres (ions
plus solvent molecules) near a hard wall. Although analytic, the MSA expressions are not explicit. However, at low ionic concentrations an expansion may be made and explicit results can be obtained. For example, the contribution to the potential resulting from an integration of gt(z) is VI =E +Ed +..., (92) K 2 and the contribution to the potential resulting from an integration of gs (z, 0) is
K,=[^(l-I]l^ + ....
(93)
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D. Henderson, A.D. Trokhymchuk and D. T. Wasan
The total potential is
K . £ + flf, + £rl) + .... SK
2s V
(94)
1 J
For the more general case of differing diameters where ds •*• d, Carnie and Chan [60] have shown that Eq. (94) becomes
V = - + ^-[d + ^ d , ) SK
2s\
A
+
....
(95)
)
It is to be noted that there has been a cancellation between terms of order 1 and (1 - s) I s to produce a term of order 1 / s. This is no problem with analytic expressions but might be a problem in numerical calculations when s -80 as a small error in either term might result in an appreciable error in V.
Fig. 10. Values of Ahs(z) electrode.
from the MSA for a 0.1M monovalent electrolyte near an
The parameter Cs is a normalization constant that is not relevant to our
discussion. The solid curve is the MSA result, calculated using the ion-dipole model, whereas the dashed curve is a plot of exp[-xr(z - d 12)]. Adapted with permission from Schmickler and Henderson [63].
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297
When the higher order terms are neglected, Eqs. (94) and (95) are formally identical to Eq. (86). However, the interpretation is quite different. The potential consist of two terms only if the expansion is truncated at two terms. Further the "inner layer" term in Eqs. (94) and (95) results from an integration over all space. The interfacial region for the solvent molecules is as diffuse as that of the ions. As long as the screening of the ions is incomplete, there is an electric field and, as long as there is an electric field, the solvent molecules will be oriented. We do not have two capacitors in series. If anything, we have capacitors in parallel. Note in Eq. (95) that ds is weighted by the factor (s-\)/A. that for an aqueous solution is about 30. This means, that in agreement with experiment, the "inner" layer term is not affected appreciably by the nature of the ions in the electrolyte. The MSA function Ahs(z) is plotted in Fig. 10. Note that Ahs(z) oscillates and ultimately decays exponentially, which is consistent with our remark that the ion and solvent interfacial regions are of similar thickness.
Fig. 11. Potential profile for a 0.01M monovalent electrolyte near an electrode. The solid curve gives the MSA result, calculated using the ion-dipole model whereas the dashed curve gives the linearized GC result. For the region, 0 < z < d 12, the lower and upper curves give the GC result calculated using s*= 80.5 and 2.5, respectively. Adapted with permission from Carnie and Chan [60].
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D. Henderson, A.D. TrokhymchukandD.T. Wasan
The MSA potential profile is plotted in Fig. 11 and is also seen to be oscillatory. In contrast, the linearized GC/PB profile is monotonic and near the electrode contact, z = d 12 is quite different from the MSA result. We have commented that the GC/PB capacitance can be brought into agreement with experiment by empirical adjustment. Without empirical fitting the MSA iondipole result is in fairly good agreement with experiment. By taking into account [63] the polarization of the metallic electrons in the electrode, the MSA ion-dipole prediction is in very good agreement with experiment not only for a mercury electrode but for a wide variety of metal electrodes. It is not just that the lack of any need for an empirical fit that makes the MSA theory preferable to the GC/PB theory. We see from Fig. 12 that even if s* is adjusted so that the capacitance, a macroscopic quantity, agrees with experiment, the GC/PB values of the microscopic potential near the electrode are quite wrong. As a result, any description of electrochemical reactions that is based on the GC/PB theory, with or without empirical adjustment, will be misleading at best.
Fig. 12. Inverse capacitance of a monovalent electrolyte at the potential of zero charge as a function of the inverse diffuse layer capacitance obtained from the GC/PB theory. The points give the experimental results of Parsons and Zobel [58]. The light line gives the low concentration MSA results and the heavy line gives the full MSA results. Adapted with permission from Schmickler and Henderson [65].
Structure and Layering of Fluids in Thin Films
299
We have already pointed out Eq. (85) is thought to be in good agreement with experiment. This, after all, is the main justification for the division of the electrochemical interface into inner and diffuse layers. Figure 13 shows the full MSA result [65] for the ion-dipole model which is compared with experiment. We see that there is a small departure from linearity in the MSA results that is not predicted by the conventional theories. Note that the experimental results also show this departure from linearity, not just qualitatively but quantitatively. In summary, the conventional description of the electrochemical interface is misleading. There is no artificial "inner" layer. The interfacial region for the solvent molecules is as diffuse as the interfacial regions for the ions. The electrostatic potential close to the electrode is quite different from that given by the conventional picture. Finally, the density and potential profiles are not monotonic but oscillatory or stratified. 4.3. Charged colloidal suspensions Returning to Eq. (75), the TCF between a pair of charged giant spheres, i.e., colloids in this case, both of radius a=D/2, is
h(R-D) = hHS(R-D)
^ f(R-D) s{YD)2D
= h»s{x)--^f(x)
,
(96)
where x = R- D is the gap width between a pair of colloids. It is useful to note that we cannot obtain Eq. (96) from the DH/PB theory even with f(x) ~ e'** because of the asymmetric treatment of the spherical cores in the DH/PB theory, where only the central ion is given a nonzero size. Because the MSA is a linearized theory, we must use the ansatz (40) to obtain the interaction energy W{x) between colloid particles. Thus, the electrostatic and excluded volume contributions to W(x) are
J3W(X)
= -hHS (x) + ^
^ af(x).
(97)
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D. Henderson, A.D. Trokhymchuk and D. T. Wasan
In addition to these two contributions, in practice there is a short range dispersion interaction between the colloidal spheres, - Aa/\2x, where A is the Hamaker constant that can be measured experimentally. Thus,
™ = V ^ - - f + ^l /w . a
a
\2x
2
(98)
We have seen already [e.g., Eqs. (43) and (44)] that the hard-sphere correlation ETC"
function h between two giant spheres is proportional to the radius a. Hence, Wla is independent of a. Some years ago Derjaguin and Landau [4] and Verwey and Overbeek [5] proposed a theory (DLVO) for colloidal interactions. Their result is
^ a
=_
A+ ^ exp( _ ra) . \2x
m
2
Thus, the DLVO result is Eq. (98) in the limit of zero diameters for the ion species. The DLVO theory is another version of the PB theory. Of course, the derivation [4,5] differs from that given here. They solved the PB equation for an electrolyte between two plane parallel surfaces and then used the Derjaguin construction [15] to obtain the interaction between two giant spheres whereas we have obtained Eq. (98) from the OZ equation for two giant spheres using the HNC/MSA/MSA approximation. In the DLVO/PB theory, the interaction between colloidal particles is the superposition of an electrostatic repulsion that dominates at large separation and a dispersion attraction that dominates at small separations with a maximum at intermediate separations. This gives a pleasing account of colloidal stability. If the maximum is less than the thermal energy, the colloids flocculate whereas if the maximum is greater than the thermal energy, the colloids are stable. The assumption of zero ion diameters and the replacement of f(x) by an exponential that yields the electrostatic part of the DLVO theory are not too bad. However, neglecting the excluded volume contribution to the colloid-colloid interaction, -hHS(x), is a very poor assumption. The number concentration of the ions may be small but the density of the solvent molecule is not small. The presence of h (x) reflects the oscillatory structural force that results from the suspending fluid stratification in the gap between two colloids. In applications to colloids in an aqueous electrolytes an additional term, called as a dipole alignment contribution, -kThDW(x)la, should be added to Eq. (98) because of the reduced polarization of the water molecules near the
301
Structure and Layering of Fluids in Thin Films
colloidal particles. Henderson and Lozada-Cassou [4] have made a crude estimate of this term. Recently, Trokhymchuk et al [15] have made a more sophisticated derivation based on the MSA results for the ion-dipole fluid. Particularly, it has been obtained that similar to the excluded volume hardsphere contribution [see Eq. (45)], the decay of dipole contribution, h (x), has the same functional form
hDlp(x)~21
yDlp | c o s ^ x + arg{/D//>})exp(-/cZ)//>x),
where the coefficients coDIP, KDIP
and yDIP
(100)
are defined in the same way as
their hard-sphere counterparts a>HS, KHS and yHS but depend on both solvent density and dipole moment. The results of such calculation are presented in Fig. 13.
Fig. 13. The force between two giant charged spheres suspended in a 10
M aqueous
electrolyte. Part a shows the electrostatic contribution only: PB results [dotted line], HLC semiempirical approximation [long dashed line], MSA dipole alignment complete result [solid line] and asymptotic Eq. (100) [short dashed line]. Part b shows the total force (solid line) calculated according to Eq. (98) plus dipole alignment term, - kTh
I a.
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D. Henderson, A.D. Trokhymchuk and D. T. Wasan
Similarly to the excluded volume contribution, the dipole contribution has an oscillatory behavior and increases the total electrostatic repulsion at short separations, improving the quantitative agreement with experiment. The dipole alignment contribution extends until about 3-4 solvent layers near the surface and this region can be associated with an assumed region of lower dielectric constant [the inner layer] that has been discussed above to account for the observed capacitance of electrodes in electrochemical measurements. 5. FILMS FORMED FROM COLLOIDAL SUSPENSIONS Recently, considerable attention has been concentrated on the phenomenon of the layering of the nano-sized or submicroscopic colloids themselves when they are added to macrodispersions such as foams, emulsions, etc. The thin films formed from colloidal particles in such systems can be used as a tool to probe the stratification phenomenon within a colloidal substance [11]. It has been observed that thin colloidal films stratify, become thinner, in a regular step-wise manner. In this section we present some results obtained in the way when the OZ based formalism is combined with computer simulations to make a progress in studying a complex colloidal system. Colloidal films represent a complex system with the superposition of few (not just two as in the case of regular colloidal suspensions) distinct length scales. Such complex systems can be studied by an approach outlined in preceding sections using a many-component version of the OZ equations assuming that the submicroscopic colloids now are a part of a suspending medium. Proceeding in this way, we have found [66] that the hard-sphere colloidal suspension tends to be ordered in a monolayer structure next to a macrosurface. It has been argued that such enhanced layering or stratification is driven by the excluded volume forces that are entropic in origin and can be revealed only if the molecular nature of the primary suspending fluid, i.e., molecular solvent is taken into account. As we can see from Fig. 14, only 1% of the colloid particles dispersed in a hard-sphere solvent comprised of 15% of the fine (solvent-to-colloid size ratio is 1:10) species clearly indicates the formation of a surface-localized monolayer of colloid particles. This is a strong evidence of the prominent role that the fine particle medium plays in colloidal dispersions to enhance a structural forces both within the colloid particles and between the colloid particles and a macrosurface. We turn our attention to this fact, since no oscillations in correlation functions, i.e., no stratification, are expected when a colloid primitive model of a hard-core colloidal suspension at a low bulk volume fraction, such as 1% is used.
Structure and Layering of Fluids in Thin Films
303
Fig. 14. Normalized local density distribution of a suspension of hard-core colloids next to a film surface and its pictorial two-dimensional interpretation.
As the colloid volume fraction in a bulk phase increases, a well-defined second monolayer of the colloid particles next to a film surface also is observed; it is composed by the colloids that are adsorbed on a surface layer of the solvent species. In such an environment, the fine species prefer to be adsorbed on the surface of the colloids and fill the cavities made by the colloidal particles and the film surface, forming an effective film surface coverage. In practice, most colloidal suspensions are composed of charged colloidal particles or macroions. The explicit many-component modeling of such a system becomes progressively more complicated since besides the different length scales, similarly to non-charged colloidal suspensions, the total number of components present in suspension is increased: the molecular solvent is replaced by an electrolyte solution that additionally consists of cations and anions; plus there are counterions to maintain the electroneutrality of the system. Although the explicit many-component OZ approach still can be applied to such a system [15], here we wish to elaborate an approach that is based on the concept of mean or effective interaction [67-70]. This concept exploits the fact that essential forces experienced by colloidal objects in a suspending medium are those mediated by medium. Thus, the colloidal suspension can be viewed as a fluid of macroions interacting via the effective potential obtained by an averaging out the macroscopic degrees of freedom due
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D. Henderson, A.D. Trokhymchuk and D. T. Wasan
to suspending medium. Usually the DLVO potential (99) is used as an effective potential between colloids. The effects of the solvent and electrolyte ions are incorporated in the DLVO potential via a continuum approximation, i.e., they are present in determining only the screening length. The main goal of the study discussed here is to take into account the excluded volume effects that result from both the finite size of the simple electrolyte ions and from the molecular nature of the solvent. The basic theory of interactions between spherical macroions suspended in an electrolyte solution has been discussed in the preceding section. Due to this, we use the one component fluid model with an effective interaction between two macroions that includes both electrostatic and excluded volume interactions in accordance with Eq. (97). Trokhymchuk et al [71] have made such a study by applying canonical Monte Carlo method that was combined with a simulation cell where the colloid film region and a colloid suspension bulk phase are connected as has been first suggested by Gao et al. [72].
Fig. 15. Monte Carlo data for the normalized local density distribution of the macroions in a film formed at 0.049 volume fraction. The thick vertical lines mark the film left and right boundaries. The film contains four layers and has the thickness that corresponds to seven and half hard-core diameters of macroions. The dashed line shows the Monte Carlo data obtained from the simulation with DLVO forces only. The macroion charge number is 30 in both cases.
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Figure 15 displays MC data for the normalized local density distribution of charged macroions (presumably micelles) across the film region of thickness H. To reveal the role that the excluded volume forces due to suspending fluid are playing, the dashed line shows MC data for the macroion density distribution in a film formed under the same bulk conditions but assuming that the colloids interact only via the DLVO potential. Both models show that the colloid particles are forming the film that has a thickness between seven-to-eight hardcore colloid diameters and are organized into four well-defined layers. In both cases there is an evident difference between the layers next to the film surfaces and those in the middle of the film. Each surface layer is rather narrow with the higher density of the particles condensed directly on the film surface that steeply decreases going to the layer boundary. In contrast to the surface layers, the middle-film layers are thick and more diffusive, i.e., less organized. The thickness of the each of two middle-film layer in Fig. 16 is around two colloid particle hard-core diameters while the surface layer thickness only slightly exceeds the one particle diameter
Fig. 16. Snapshots of MC generated configurations of macroions at a bulk volume fraction 0.049 that corresponds to a high surfactant concentration, 0.10 mole/1. Number of charges carried by macroions is 30. The film with three layers has a thickness of five hard-core diameters of the macroions while film with two surface layers has a thickness of three and half hard-core diameters.
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The main differences between the model with the suspending fluid contribution and the DLVO-like model are found in the surface layers. In the case of a model with a suspending fluid taken into account, the surface layers themselves show structuring with respect to the film surfaces. This results in each surface layer consisting of a well-defined mono-layer in the immediate vicinity of the film surface and one or two similar sub-layers that are less pronounced and are separated by an "effective" layer of a suspending solution. The shape of the density profiles of the surface sub-layers in this case has a 5 - like form indicating that surface sub-layers are the quasi-two-dimensional monolayers. The surface layers formed within the DLVO model, although thinner than the middle-film layers, are still far from being monolayers. As a result, the segregation of the middle-film particle layers from the surface layers is not observed in this case. As for the middle-film layers for both models only some quantitative differences in the particle local density distribution are observed. When the separation between surfaces decreases, we find that the next two films that provide the local minima of the film energy (per film particle) have a thickness around five and three colloid hard-core diameters, respectively. In particular, one of the films shown in Fig. 16 has thickness H - 5D and contains three layers of colloid particles, i.e. has one layer less than in the film discussed in Fig. 15. Again we observe that the middle-film layer is almost completely separated from the surface layers. The thickness of the middle-film layer decreases slightly when the separation between film surfaces decreases. In contrast, the surface layers do not change notably. This quasi-stability of the surface layers becomes even more evident by analyzing the local density distribution in the film that has thickness around three times the micelle hardcore diameter, H = 3.5D, and contains two surface layers only (Fig. 16). We conclude that the surface layers for three considered films remain largely unaffected and are almost identical. It follows that during the film thinning process the film changes its thickness by squeezing out one middle-film layer of particles (the so-called "squeezing layer" mechanism ). Then the height of the step-wise layer-by-layer thinning will depend on the effective thickness of the squeezed layer. To verify this assumption, the effective thickness of the squeezed layers have been calculated as the difference between the metastable thickness of the films containing four and three, three and two particle layers for both the low and high surfactant concentrations and compared with observation for the heights of the thinning steps. The film containing two surface layers is mostly stable and its thickness corresponds to the final film thickness in the film thinning process at the surfactant concentrations considered in the present study. These also agrees qualitatively well with the force measurements of Richetti and Kekicheff [8].
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6. SUMMARY A statistical mechanical approach to study the layering or stratification of the species of suspending fluid that occurs in a vicinity of suspended macrosurfaces has been presented and discussed. This approach is based on the solution of the system of Ornstein-Zernike (OZ) equations that relate the direct and total correlation functions of the giant solutes and the species comprising the suspending medium. In a self-consistent way this approach provides with the bulk phase properties of a discrete suspending medium, its inhomogeneous properties, i.e., local density distributions in an external field of the suspended species, and with the interaction energy between a pair of giant species that is induced by a discrete suspending medium. There are two main driving forces for suspending fluid stratification excluded volume, or entropy, and electrostatics. Excluded volume forces rely on the volume fraction occupied by the species comprising suspending medium, i.e., on their sizes while electrostatic forces contribute by the number of charges carried by suspending species. Within the mean spherical approximation (MSA) excluded volume and entropic contributions are additive. Using the MSA solution we have shown the way in which phenomena of the suspending fluid stratification are related and affect the properties of electrochemical interfaces such as double layer and electrostatic potential, as well as the stability of colloidal suspensions. In a transparent manner we have obtained that the conventional theories of suspending electrolytes such as due to Debye and Hiickel, Gouy and Chapman, and Poisson-Boltzman and DLVO theories all are zero-diameter limits of the suspending fluid species within the MSA theory. Although the assumption of zero ion diameters seems to be reasonable for electrolyte ions, the density of the solvent molecule is not small. Consequently, all these traditional theories neglect the excluded volume contribution to the stratification that results in a series of drawbacks related with their application. In particular, the conventional description of the electrochemical interface is misleading. There is no artificial "inner" layer. The interfacial region for the solvent molecules is as diffuse as the interfacial regions for the ions; the electrostatic potential close to the electrode is quite different from that given by the conventional picture. The excluded volume contribution to the effective interaction between two giant spheres in the case of a simple suspending fluid is characterized by (i) a monotonic depletion attraction for separations between confining surfaces that roughly are smaller than half of the fluid particle diameter, (ii) a repulsive maximum, located at about three quarters of the fluid particle diameter, (iii) a secondary minimum just after a separation of one diameter of the suspending fluid species, and (iv) has a shape of an oscillatory structural repulsion for separations larger than one suspending fluid particle diameter. In the limit of
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low density of the suspending fluid this results are reduced to the AsakuraOosawa depletion interaction. We have shown that using a one-component, two-component and manycomponent modeling of a suspending medium there is a possibility to mimic the whole spectrum of colloidal dispersions from the submicroscopic to macroscopic scales. The real objects to which such approach can be applied include complex colloidal dispersions composed of solid surfaces, emulsion droplets, etc., all dissolved in aqueous or non-aqueous suspensions of colloidal particles, surfactant micelles, i.e. fluid systems where the constituents with a few competing length scales are involved. Numerical calculations have been performed for one-, two-, four- and ten-component model systems. A onecomponent model of suspending medium is essential and represents a crucial necessary step that allows one to move beyond the primitive modeling of colloidal suspensions, fn particular, it provides with an effective interaction energy between two mesoscopic surfaces in a molecular solvent. Proceeding in this way we have shown the importance of taking into account the molecular nature of suspending fluid that reveals the primary molecular solvent stratification in the vicinity of a single surface and in the film confinement formed by a pair of macrosurfaces. This theoretical picture is supported by the surface force measurements conducted by Israelachvili and his colleagues [7]. An extension of the simple suspending fluid to a bidisperse solution comprising species with highly asymmetric sizes, i.e., both nanosized colloids and species of molecular solvent, has revealed some new qualitative features for which the stratification phenomenon is responsible. The most notable is that the effective interaction between a pair of giant spheres in a two-component suspending fluid of colloid and fine species is no longer monotonic in the gap region depleted by colloid particles but shows an oscillatory repulsive behavior, reflecting the filling of this region by the fine species. This modeling prediction agree well with observed layering of water molecules known as a repulsive hydration forces [7]. Taking into account the presence of the molecular solvent component also affects the layering of colloid particles near the surface of giant sphere. It is interesting to note, that due to the large asymmetry in the particle size in a bidisperse suspending fluid, the contribution of each of the component to the interaction energy is split on the length scale. Due to this, it seems that to some extent the effect of each component with a competing length scales associated with particle diameters can be treated as a superposition of the results for two monodisperse fluids with corresponding particle size and, probably, with an appropriate effective density. Another interesting finding is that the contribution to the effective interaction between a pair of colloids due to the fine particles of bidisperse fluid medium seems to acquire characteristics that are similar to an adhesive sticky potential when compared to the
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contribution of the larger particles. This suggests that it may be possible to mimic the presence of the fine species in a simple manner by means of a sticky potential. Colloidal suspensions invariably contain particles with size polydispersity ranging from several percent in the very carefully controlled polymerization process to hundreds of percent in a typical emulsification process. An increase in the number of species that represent the colloid particles with different but similar diameters (four- and ten-component fluid models), i.e., a primitive attempt to explore the role of colloid particle size polydispersity, shows a tendency to diminish the effects observed when the colloid particle component is monodisperse; however, this does not affect the features introduced by the taking into account a monodisperse fine ("solvent") component. Additionally, we have presented the way in which continuous size polydispersity can be treated. Finally, we have combined the OZ/MSA approach with computer simulations techniques to study the complex colloidal systems involving more than two distinct length scales that is a case of emulsions and foam systems. In particular, we have employed an effective interaction energy between two giant spheres in a bidisperse suspending fluid, i.e., in a colloidal suspension in a MC study of the ionic micelle stratification. We have shown that the forces operating at the submicroscopic scale between colloidal particles are governing the stability of both micro- and macrodispersions. ACKNOWLEDGEMENTS This work was supported in part by the National Science Foundation under Grant No CTS 01-00854. We are grateful to Alex Nikolov and Yurko Duda in collaboration with whom some of the results reported here have been obtained.
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J. Texter and M. Tirrell, AIChE J., 47 (2001) 1706. H. A. Stone and S. Kim, AIChE J., 47 (2001) 1250. D. T. Wasan and A. Nikolov, Nature, 423 (2003) 156. B. V. Derjaguin and L. Landau, Acta Physicochim., 14 (1941) 633. E. J. W. Verwey and J. Th. G. Overbeek, The Theory of Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. [6] R. G. Horn and J. N. Israelachvili, J. Chem. Phys., 75 (1981) 1400. [7] J. N. Israelachvili, Intermolecular and Surfaces Forces, 2nd ed., Academic Press, London, 1992. [8] P. Richetti and P. Kekicheff, Phys. Rev. Letter, 68 (1992) 1951. [9] J. L. Parker, P. Richetti, P. Kekicheff, and S. Sarman, Phys. Rev. Letter, 68 (1992) 1955. [10] P. Kekicheff, and P. Richetti, Prog. In Colloid and Polym. Sci., 88 (1992) 8. II1] A. D. Nikolov and D. T. Wasan, J. Colloid Interface Sci., 133 (1989) 1.
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[12] T. Biben, J.-P. Hansen, H. Loven, in Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution, (S.-H. Chen, J. S. Huang, P. Tartaglia, Eds.) Kluwer Academic Publishers, Dordrecht, Netherlands, 1992, 23. [13] D. Henderson and M. Lozada-Cassou, J. Colloid Interface Sci., 114 (1986) 180. [14] D. Henderson, J. Colloid Interface Sci., 121 (1988) 486. [15] A. Trokhymchuk, D. Henderson, and D. T. Wasan, J. Colloid Interface Sci., 210 (1999) 320. [16] L. S. Ornstein and F. Zernike, Proc. Acad. Sci. Amsterdam, 17 (1914) 793. [17] J. K. Percus and G. L. Yevick, Phys. Rev., 110 (1958) 1. [18] J. A. Barker and D. Henderson, Rev. Mod. Phys., 48 (1976) 587. [19] J. L. Lebowitz and J. K. Percus, Phys. Rev., 144 (1966) 251. [20] D. Henderson, F. F. Abraham, and J. A, Barker, Mol. Phys., 31 (1976) 1291. Reprinted as Mol. Phys., 100 (2002) 129. [21] P. Attard, D. Bernard, C. Ursenbach, and G. N. Patey, Phys. Rev. A., 44 (1991) 8224. [22] D. Henderson, 11th Symposium on Thermophysical Properties, Boulder CO, June 23-27, 1991; Fluid Phase EquiL, 76 (1992) 1; J. Chem. Phys., 97 (1992) 1266. [23] B. V. Derjaguin, Kolloid Z., 69 (1934) 155. [24] M. S. Wertheim, Phys. Rev. Letters, 10 (1963) 321. [25] E. Thiele, J. Chem. Phys., 39 (1963) 474. [26] M. S. Wertheim, J. Math. Phys., 5 (1964) 643. [27] R. J. Baxter, Aust. J. Phys., 21 (1968) 563. [28] J. Perram, Mol. Phys., 30 (1975) 1505. [29] W. R. Smith and D. Henderson, Mol. Phys., 19 (1970) 411. [30] D. Henderson and W. R. Smith, J. Stat. Phys., 19 (1978) 191. [31] J. L. Lebowitz, Phys. Rev., 133 (1964) A895. [32] M. Toney, J. N. Howard, J. Richer, G. L. Borges, J. G. Gordon, O. Melroy, D. G. Wiesler, D. Yee, and L. B. Sorensen, Surface Sci., 335 (1995) 326. [33] R. Roth, R. Evans and S. Dietrich, Phys. Rev. E, 62 (2000) 5362. [34] A. Trokhymchuk, D. Henderson, A. Nikolov and D. T. Wasan, Langmuir, 17 (2001) 4940. [35] S. Asakura and F. Oosawa, J. Chem. Phys., 22 (1954) 1255. [36] D. Henderson, D. T. Wasan, and A. Trokhymchuk, J. Chem. Phys., 119 (2003) 11989. [37] D. Henderson, D. T.Wasan and A. Trokhymchuk, Condens. Matter Phys., 4 (2001) 779. [38] M. S. Wertheim, L. Blum and D. Bratko, in Micellar Solutions and Microemulsions (S.H. Chen, R. Rajagopalan, Eds.) Springer: New York, 1990; Chapter 6, p.99. [39] A. Trokhymchuk, D. Henderson, A. Nikolov and D. T. Wasan, J. Colloid Interface Sci., 243(2001) 116. [40] J. J. Salacuse and G. Stell, J. Chem. Phys., 77 (1982) 3714. [41] L. Blum and G. Stell, J. Chem. Phys., 71 (1979) 1300. [42] J. Y. Walz and A. Sharma, J. Colloid Interface Sci., 168 (1994) 4851. [43] M. S. Wertheim, J. Stat. Phys., 42 (1986) 459; 42 (986) 477. [44] M. S. Wertheim, J. Chem. Phys., 85 (1986) 2929; 87 (1987) 7323. [45] J. Chang and S. Sandier, J. Chem. Phys., 102 (1995) 437; 103 (1995) 3196. [46] E. Vakarin, Yu. Duda and M. F. Holovko, Mol. Phys., 90 (1997) 611. [47] M. F. Holovko and E. V. Vakarin, Mol. Phys., 84 (1995) 1057. [48] M. F. Holovko and E. V. Vakarin, Mol. Phys., 87 (1996) 1375. [49] E. Vakarin, M. F. Holovko and Yu. Duda, Mol. Phys., 91 (1997) 203. [50] D. Duda, D. Henderson, A. Trokhymchuk and D. T. Wasan, J. Phys. Chem. B, 103 (1999)7495.
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[51] H. K. Christenson, D. W. R. Gruen, R. G. Horn and J. N. Israelachvili, J. Chem. Phys., 87 (1987) 1834. [52] M. L. Gee and J. N. Israelachvili, J. Chem. Soc, Faraday Trans., 86 (1990) 4049. [53] E. Waisman and J. L. Lebowitz, J. Chem. Phys., 52 (1970) 430; 56 (1972) 3086, 3093. [54] L. Blum in Theoretical Chemistry: Advances and Perspectives, 5 (H. Eyring and D. Henderson, Eds) Academic Press, New York, 1980, 1. [55] P. Debye and E. Httckel, PhysikZ., 24 (1923) 185. [56] M. Wertheim, J. Chem. Phys., 55 (1971) 4281. [57] G. Gouy, J. de Physique, 9 (1910) 457; D. L. Chapman, Phil. Mag., 25 (1913) 475. [58] R. Parsons and F. G. R. Zobel, J. Electroanal. Chem., 9 (1965) 333. [59] D. Henderson, L. Blum, and J. L. Lebowitz, J. Electroanal. Chem., 102 (1979) 315. [60] S. L. Carnie and D. Y. C. Chan, J. Chem. Phys., 73 (1980) 2949. [61] D. Y. C. Chan, D. J. Mitchell, B. W. Ninham and B. A. Pailthrope, J. Chem. Phys., 69 (1978)691. [62] L. Blum and D. Henderson, J. Chem. Phys., 74 (1981) 1902. [63] W. Schmickler and D. Henderson, J. Chem. Phys., 80 (1984) 3381. [64] L. Blum, D. Henderson, and R. Parsons, J. Electroanal. Chem., 101 (1984) 389. [65] W. Schmickler and D. Henderson, Progr. Surface Sci, 22 (1986) 323. [66] A. Trokhymchuk, D. Henderson, A. Nikolov, and D. T. Wasan, Phys. Rev. E, 64 (2001) 012401. [67] W. G. McMillan and J. E. Mayer, J. Chem. Phys., 13 (1945) 276. [68] L. Beloni, J. Phys.: Condens. Matter, 12 (2000) R549. [69] A. A. Louis, Philos. Trans. R. Soc. London, Ser. A359 (2001) 939. [70] C. N. Likos, Phys. Rep., 348 (2001) 267. [71] A. Trokhymchuk, D. Henderson, A. Nikolov, and D. T. Wasan, J. Phys. Chem. B, 107 (2003) 3927. [72] J. Gao, W. D. Luedtke, and U. Landman, J. Phys. Chem. B, 101 (1997) 4013.
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Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.
Chapter 8
Theory of emulsion flocculation D. N. Petsev Department of Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, New Mexico 87131 1. INTRODUCTION Emulsions are colloidal dispersions of liquid in liquid, for example, a mixture of oil and water. As a result of the mixing, one may obtain oil droplets in water (O/W) or water droplets in oil (W/O). The sizes of the droplets could be in the micrometer and even submicrometer range. A problem of both fundamental and practical importance is that of the emulsion stability against flocculation. Colloid stability is often analyzed in the framework of the Derjaguin-LandauVerwey-Overbeek (DLVO) theory [1-4]. DLVO theory suggests that the stability against aggregation in a colloidal dispersion (e.g., emulsion) depends on the balance between van der Waals attraction and electrostatic repulsion. However, these two forces do not represent a full account of the whole variety of interactions that may occur in colloidal systems. These include steric repulsion [5,6], depletion attraction [7-12], hydration and hydrophobic interactions, oscillatory surface forces, etc. [13]. In the case of emulsions, the colloidal particles are fluid. The droplet fluidity and interfacial mobility may have a strong impact on the emulsion behavior and particularly on their stability against flocculation [14-26]. Emulsion stability also always requires the presence of surface-active molecules or fine solid particles that adsorb at the droplet interface and prevent the droplets from rapid coalescence [4]. Even in the presence of a stabilizing additive, emulsions are thermodynamically unstable. The contribution of the interfacial free energy is proportional to the total area of contact between the two phases and is usually positive. Destroying the droplets and separating the phases macroscopically allows for considerable reduction of this unfavorable term, and therefore of the overall free energy. The time scales on which such event occurs may vary from seconds to years. Hence, emulsions are usually kinetically stable. The analysis of emulsion stability provides not only fundamental challenges of great scientific interest, but is also very important for various
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practical aspects. In some cases (as in food industry, paint production), the main concern is to obtain stable emulsions while in others (e.g. oil recovery) destabilization and oil separation is desired. Knowledge of the basic principles and mechanisms governing emulsion stability presents both academic and applied interest. Stability loss in emulsions may occur in four different ways [27,28]: • Creaming (or sedimentation) is due to density difference between the two immiscible liquids, the lighter phase will tend to go up while the heavier will move downwards. In the case of an O/W emulsion the oil droplets will accumulate at the top, forming a cream layer. • Flocculation occurs when the droplets stick to each other and form aggregates due to the presence of a minimum in the interaction energy. The aggregates could be compact or may form an expanded gel-like structure. However, the individual droplets remain separated by a layer of the continuous phase - thin liquid film (see Fig. 1). Flocculation may be weak and reversible or strong and irreversible. Flocculation enhances creaming since it forms large floes. It is also often a prerequisite step to coalescence.
Fig. 1. Sketch of two deformed droplets in the presence of long range repulsion (e.g., electrostatic). If no long range forces are present, the droplets deform upon contact of their surfaces. Otherwise a liquid film with thickness h is formed.
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• Coalescence is the process of the fusion of two or more droplets to form a larger droplet. In this case the droplets are no longer separated by the continuous phase and have become a single entity. The thin liquid film that may have separated them before in the flocculation stage has ruptured under the action of the attractive forces, or due to hydrodynamic instabilities [20,29-31]. • Ostwald ripening is a process of molecular diffusion transfer of oil from the smaller droplet to the larger droplet and is driven by chemical potential differences. The present chapter discusses only emulsion flocculation. Coalescence is outlined in Chapter 9 of this book. As mentioned above flocculation is a process during which the droplets aggregate, but remain separated by thin films of the continuous phase as shown schematically in Fig. 1. The formation of such configuration is due to an attractive interaction but the stability of the film between the interfaces proves the presence of repulsive force at shorter distances. Attractive and repulsive energy contributions have different dependence on the separation of the droplets and their superposition may exhibit a complex distance behavior.
Fig. 2. A sketch of DLVO interaction energy as a function of the separation (liquid film thickness) between two infinite flat surfaces. This picture is more relevant to solid surfaces. The droplets shown are to indicate qualitatively the effect of the attractive forces on the film formation (more detailed discussion is given in the text).
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In DLVO interaction, it is described by the combination of van der Waals attraction and electrostatic repulsion [1-4]. Van der Waals energy diverges as \lh at small separations, h, and decays as \/h6 at large distances. The electrostatic repulsion is finite at small distances and decays exponentially. Because of this, the overall interaction energy is dominated at small and large separations by attraction, while the intermediate region could be repulsive because of the electrostatic contribution, see Fig. 2. Fig. 2 suggests the presence of a strong and short-ranged repulsion also at very small separations. This repulsion is not included into the original version of DLVO theory [1-4]. Its possible origin is discussed briefly further in this chapter. The particular shape of the energy curve in Fig. 2, however implies that flocculation may take place in the far (secondary) minimum II or in the near (primary) minimum I. The attraction in the secondary minimum is much weaker than that in the primary and so is the flocculation. Droplets, flocculated in the secondary minimum would separate more easily than those in the primary. Flocculation in the primary minimum could have been preceded by such in the secondary minimum. Emulsions with an energy barrier for coalescence could still be amenable to flocculation. The kinetics of flocculation, as well as that for transition between secondary to primary minimum and film rupture depends on both direct (e.g., vane der Waals, electrostatic, etc.) and hydrodynamic interactions [15,20,26,29,31-35]. In the present chapter we discuss the droplet interaction energy and its relation to the kinetics of flocculation. A brief overview of the typical interactions that might be encountered in emulsion systems is given in Section 2. Section 3 outlines the mechanism and kinetics of droplet flocculation and film thickness transitions. Section 4 presents briefly some of the experimental methods for studying droplet interactions and flocculation and Section 5 contains the conclusions.
2. INTERACTION ENERGY BETWEEN TWO EMULSION DROPLETS 2.1. Energy density per unit area Let us consider the interaction between two infinite plane-parallel surfaces (representing the interacting droplets) separated by a layer of the different medium (representing the disperse phase, e.g., water for aqueous suspensions). This is a one-dimensional problem and its theoretical treatment is relatively simple. It also provides a foundation for further more elaborate treatment relevant to more realistic particle shapes (see below). The possible interactions between emulsion droplets may have a different physical origin [1-5,11-13]. Some of them are briefly outlined below.
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2.1.1. Electrical double layer (electrostatic) interactions Electrostatic interactions are due to the charge that many colloidal particles (or emulsion droplets) acquire when immersed in water. The reason for this charge is dissociation of surface ionic groups or adsorption of ions from the solution. In the particular case of emulsions, the surfaces charges are usually due to the use of ionic surfactants as a stabilizing additive. Electrostatic interactions are repulsive in general and play an important role in explaining colloid stability in DLVO theory [1-4]. In some special cases like interaction between surfaces with different charge (or potential) attraction at very short distances could be observed [1,2], see also [14]. Even the simple case of two plane parallel surfaces has no exact and general solution for the electrostatic interactions. There are, however, two reasonable approximations, briefly outlined below. Low surface potentials (weakly charged droplets). This approximation refers to the case (e$!olkT} 1. This is a widely used approximation, known as the nonlinear superposition [12] and reads (see also[l-3,36]) fel{h)-
64CelkTK-1 tanh2 ^
exp(-^).
(4)
There is a number of other and more elaborate expressions for the electrostatic interactions available [1-3,36], and there are always the alternatives of numerical solutions, but the expressions shown above cover a very wide range of reasonable experimental conditions relevant to emulsions and offer rather accurate results. The surface charge (or potential), of emulsion droplets, could be conveniently controlled by using a mixture of ionic and nonionic surfactants [37]. 2.1.2. Van der Waals (dispersion) interactions Van der Waals interactions are due to the forces acting between the individual molecules in the macroscopic colloidal particles (e.g., droplets). These forces are between dipoles and induced dipoles and are typically short ranged. Integration of all these molecular interactions over the volumes of the interacting macroscopic bodies gives an overall energy that is considerably longer ranged. This was first done by de Boer [38] and Hamaker [39], assuming pairwise additivity of the interactions between the individual molecules in the colloidal particles. Such integration, sometimes referred to as Hamaker approach, could often be performed analytically for a number of particle shapes and geometries. Van der Waals interactions are always present between particles immersed in a liquid of a different refractive index and dielectric constant. Their magnitude may be negligible, especially when compared to other interactions that are present. However, van der Waals interactions diverge at contact as \lh and for sufficiently small particle separations they may easily become the dominant force.
Theory of Emulsion Flocculation
319
The van der Waals energy of interaction per unit area for two infinite parallel slabs is [1-3,12,13,36]
AH is the Hamaker constant and depends on the material properties of the droplets and the surrounding fluid. It was shown later [40,41] that due to phase shift between the interacting dipoles, AH might actually decay with the separation and hence, is not truly a constant. This effect is called electromagnetic retardation and may become substantial for h> C/LJ, where c is the speed of light and co is the frequency of dipole oscillation. Further developments of the theory of dispersion interactions [42-44] (see also [12]) lead to elaborate approaches for calculation of the Hamaker constant. These approaches usually require a substantial numerical effort but useful approximate analytical results are available [12] for the non-retarded
and fully retarded
AH =±kT 4
£L^2+iV e, - e 2
47T«2 nf + n{
«±ZA:'l
(7)
h
cases, where S\ and si are the dielectric permittivities, while n\ and n2 are the refractive indices for the droplets and the surrounding medium respectively. hp is Planck's constant, a> is the fluctuation frequency of the interacting molecular dipoles and c is the speed of light. Obviously, retardation affects only the second (frequency dependent) term of Hamaker constant [see Eqs. (6) and (7)]. The important difference between the two expressions above is that the nonretarded AH is a true constant with respect to the separation, h. The second term in the right hand side of the expression (7) for the fully retarded AH, however, decays as \lh [compare to (6)]. Hence, retardation leads to a faster decay of the van der Waals energy. A very useful expression that interpolates between (6) and (7) has been suggested {see [12]}
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D.N. Petsev
where H = «2 njf + rq\ v '
— c
and
.
(8)
i~2/ 3
3 2
F(H) = ±H /•"C + ^)°iK-^) < f c M 1+ M '
Equation (8) covers the whole range from nonretarded to fully retarded van der Waals interactions. The presence of electrolytes in the solution containing the interacting colloids decreases the value of the Hamaker constant [45]. The first term in (6), (7) and (8) is the only one affected by electrolyte screening and decays exponentially with the separation h. While the electrolyte effect is substantial for high concentrations (e.g., about 1 M NaCl), it has virtually no effect when the electrolyte amount is lower [12]. Typical numerical values of Hamaker constants for oil/water/oil or water/oil/water systems (like most emulsions) are between 3 and 4xlO"21 J (see e.g., [13]). Dispersion interactions may also induce repulsion along with the attraction. It is due to the interactions between atoms and molecules at very small separations. Analysis of this repulsive contribution for colloidal particles has been done recently [46] and showed that the respective free energy of repulsion has the form
fZP{h) = Const
x^.
h
(9)
2.1.3. Steric interactions Steric interactions occur when there is an overlap of the adsorbed polymer layers that may cover the colloidal particles. Emulsion droplets covered by a nonionic surfactant are stable because of the steric repulsion between the hydrophilic headgroups of the amphiphile. Since the adsorbed polymers induce steric interactions, their interactions with the solvent are extremely important [5,12]. For good [47,48] and ©-solvents [49], the interaction is usually repulsive, see also [50], but for poor solvents it could be attractive [12,51]. The first two (repulsive) cases could be modeled theoretically relatively simply, the last one allows only for numerical treatment. Nonionic surfactants usually have
321
Theory of Emulsion Flocculation
polyoxyethylene headgroups, which are known to become more hydrophobic with increasing temperature. Hence, emulsion stabilized by such a surfactant may switch from a stable to an unstable (flocculated) state by varying the temperature. The free energy of interaction per unit area due to polymer steric repulsion in good solvent is approximately given by [47,48]
fsl(h) = 2kTI3/2Lg
h-5l4+HlIA-^
for hK
12 \{L + h)2
' +
h(2L + h)
(L
+
hf
^ . _ _ ^ 2i?2 2 h h(L + h) (L + h)[2(L-a) + h] (h2+4R2)yh2+4R2-h)
2R2a(2L2+Lh + 2ah)
2h[2(L-a) + h]
h(L + h)2[2(L-a) + h\
+
where L = a + va 2 — R2 . Eq. (29) could be significantly simplified if one assumes that the deformations between the droplets is small, or (Vla) < 1. In this case, the van der Waals energy contribution becomes
rr /, N Uvw(h,r) =
A
4 2 H Attractive interactions favor the deformation (film formation). Hence, strong van der Waals forces or depletion interactions, induced by smaller colloids, may lead to droplet deformation. > Repulsive interactions suppress film formation. A typical example is electrostatic repulsion: high surface potentials and/or low electrolyte concentrations do not favor the formation of plane a parallel film between the droplets in a doublet. Addition of electrolyte decreases the repulsion and therefore facilitates deformation. > The lower the interfacial tension the higher the droplet deformability. Interfacial tension resists any extensions of the droplet interface and thus the formation of film in the doublet. Similar is the role of high bending energy which increases the interfacial rigidity. ^ Larger droplets deform more readily than small ones if all the remaining parameters are the same. This could be attributed to the higher capillary pressure of small droplets that make them behave more like rigid spheres.
Fig. 5. Countour diagram of the interaction energy of two charge stabilized droplets. a= 1 urn, AH = l x l O " 2 0 J , y = l mN/m, tf0 = 100 mV, Ce, = 0.1 M.
Theory of Emulsion Flocculation
331
This is illustrated in Fig. 5, which presents a contour diagram of the interaction energy between two droplets as a function of the film thickness, h, and film radius, r. The droplet size chosen for this calculation is 1 \im, the electrolyte concentration corresponds to 0.1 M monovalent salt and the Hamaker constant is AH = l x l ( T 2 0 J, ^ 0 = 5 0 mV and y = 1 mN/m. A minimum in the energy surface is formed at r/a = 0.055 and h/a = 0.0069. The attraction energy is -29.1 kT. If the deformability is ignored, the energy vs. distance dependence runs along the ordinate axis and the energy minimum is -27.3 kT at h/a = 0.0058. From the ratio r/a (at the energy minimum), one may calculate the equilibrium contact angle, a using [18] a = sin—-. a
(34)
Increasing the electrolyte concentration or lowering the droplet surface potential leads to greater deformation. More details on the structure (values of the film radius and thickness) are given elsewhere [14,17,18,21,24,25]. These also include the effect of other types of colloidal forces like steric, depletion and structural. While steric repulsion acts similarly to the electrostatics, depletion may turn into oscillatory structural force at a high volume fraction of the small colloids [21,25]. These forces lead to metastable states of macroscopic emulsion films containing one or more layers of trapped particles [55,56,81,82] and their role for miniemulsion droplets (submicrometer sized and below) is usually destabilizing, leading to a deep depletion minimum without any small particles in it [21,25]. 3. KINETICS OF DROPLET FLOCCULATION The kinetics of flocculation of solid colloidal dispersions was first analyzed by Smoluchowski [83-85]. The theoretical model he developed was based on the notion that the time determining step is the approach of two colloidal particles due to diffusion. It has subsequently undergone numerous refinements and generalizations [2,86-96]. These include more realistic direct and hydrodynamic interactions and account for higher order particle correlations but mostly for solid colloidal particles. Still, the major differences between solid and fluid particles are the fluidity and flexibility of the droplet interfaces. 3.1. Droplet motion and hydrodynamic interactions The motion of a viscous droplet in unbounded viscous fluid without surfactants presents a relatively simple hydrodynamic problem and was solved
332
D.N. Petsev
by Rybczynski [97] and Hadamard [98]. However, in most real cases there are always surface-active substances present and they change the droplet movement considerably. Analyses of single droplet motion in the presence of surfactants were given by Levich [99]. The relative motion of two droplets presents a more difficult problem. The case of absence of surfactant and interfacial tension gradients has been considered by Davis et ah, who suggested numerical solutions for the mutual mobilities of non-deformable droplets [100] as well as accounting for the deformation in lubrication approximation [101,102] or using a boundary integral algorithm [103]. More detailed analysis of the hydrodynamic interactions between droplets is also given in Chapters 10 and 11 of this book. Emulsion droplets in absence of surfactants are usually unstable against coalescence and are unable to form doublets or higher order aggregates of individual droplets separated by thin films of the continuous phase. The presence of surfactant stabilizes the emulsion droplets and strongly affects the hydrodynamic interactions. When two droplets are close to each other, the main contribution to the hydrodynamic resistance (energy dissipation) comes from the thin gap between them, or the thin liquid film. The hydrodynamics of displacing solvent from the film during the approach of two surfactant stabilized emulsion droplets has been studied extensively [31,33,34,104-106]. These studies considered millimeter and sub millimeter sized droplets where the main force, bringing the two droplets together, is usually buoyancy. Knowing the magnitude of the buoyancy force and the hydrodynamic resistance, one may calculate the velocity, V{h,R) and hence the time of film thinning, r, viz. r{hm,hf,R) = ^ ^ - ^
V(h,R) = FBC(h,R)
(35)
where FB is the buoyancy force and
, , R2 , R4 ' ah
l l ah
where rj is the solvent viscosity and ss is a parameter that takes into account the interfacial mobility and droplet bulk viscosity - see [31,104]. For tangentially immobile droplet interfaces, ss = 1. This is the case when the interfaces are
334
D.N. Petsev
Fig. 7. Components of the hydrodynamic resistance tensor as defined by Eq. (37). These are valid for small separations between the droplets.
Theory of Emulsion Flocculation
335
completely saturated with adsorbed surfactants that are soluble in the continuous phase only. If there are no surfactants, or if they are soluble only in the droplet phase then ss decreases and may become about 0.001 [31,104,105]. In absence of deformation (R = 0) and tangentially immobile droplet surfaces, ChR =CRR = 0 a n d (hh =3-7rrya2/2/z, which corresponds to the result for two solid spheres of radius a [12]. Another limiting case is a very thin film with a large radius, Rlh^>\, when the contribution of the curved surfaces around the film could be ignored. In this case the Reynolds formula for the approach of two solid parallel disks [108], (hh = 3nrjR 12}?, is applicable. The component QhR increases as R3 while (RR becomes infinite for the case of tangentially immobile droplet surfaces and finite radius since es — 1, see Eq. (37). Fig. 7 represents the dependence of the different tensor components on the film thickness, h, and radius, R. 3.2. Brownian Flocculation 3.2.1. Effect of the droplet interactions on the flocculation kinetics If the direct and hydrodynamic interactions are known one may calculate the steady flux of droplets toward a given central droplet, J, and hence - the rate constant of pair flocculation
j
n^
WF
where nx is the droplet number concentration at infinite distance from the central droplet, Do is the Stokes-Einstein diffusion coefficient [109] and WF is the Fuchs factor defined by [2,12,86]
WF=4D0af dr Jo
[
\\ D{r)
2 2 r
J
-
(39)
The integral in (39) is taken over the distance between the flocculating droplets. U(r) is the interaction energy and D{r) = kTI(,(r) is the diffusion coefficient at small separations [15,26], also see below. The rate constant, kf, defined by (38), can be introduced in the kinetic equation for the formation rate of doublets of droplets [15]
336
D.N. Petsev
£ =*/-*,
CO)
«2 is the number of doublets formed and nx is the single droplet number at infinity. Detailed analysis of the above equation for the particular cases of droplet flocculation and coalescence is performed in Ref. [110]. The flocculation and coalescence kinetics of Brownian emulsions is not affected by deformability for diluted systems, in the presence of only attractive interactions [15]. In such systems the time determining step for a flocculation or coalescence of two droplets is the approach from large distances. On the other hand, the presence of strong repulsive interactions, e.g. high surface potentials, may prevent the droplet deformation for low electrolyte concentrations ^ 0.1 M of monovalent electrolyte. Therefore, the kinetics of flocculation and coalescence of very diluted or charged emulsions with low concentration of electrolyte present will be identical to solid dispersions with the same particle size, Hamaker constant, surface potential and other interaction determining parameters [15], see also [111,112]. In the case of charge-stabilized emulsions at high ionic strength (above 0.1 M monovalent electrolyte), the situation could be different. For such high salt concentrations droplet flocculation often occurs and is accompanied by droplet deformation [32,113-116]. The droplets may flocculate in the secondary minimum of the DLVO energy [117] and eventually jump into the primary similarly to macroscopic foam and emulsion films [29,118]. The primary doublets may further coalesce. This case was recently analyzed [26] and was shown that the mean time for transition between the secondary into the primary energy minimum, r depends on the droplet deformation and film formation. This mean transition time could be obtained by solving the following equation [26] (see also Fig. 6) —
^
V\Peq (R,h)D(R,h).Vr(R,h)\ = - 1 , where
611
Peg (/?,/,) = e x p - ^ P
X
'
(41)
and D = kT^(R,hf
Since the inverse friction matrix is [26] >•-! _
1 ( (>RR ~ C M | _
1
|
(M
the components of the diffusion tensor, D, become
~ChR\
,,2\
337
Theory of Emulsion Flocculation
kT
Chh
D K R =
CRRCM
kT
=
~ChR
+
,
ChR CRRCHH )
n
~kT i ChR
kTQm = CRRChh ~ChR
CRRCM-I
The dependence of the diffusion tensor components on the film thickness, h, and radius, 7?, is given in Fig. 8. Note that DRR is infinite at R = 0 and becomes zero for es — 1. The transition (or coalescence) time equals twice the mean time necessary for a system (doublet of droplets) to move from the secondary minimum (Rmin,hmm) to the saddle point (RsaddAadd) (see Fig. 5). Hence, the boundary conditions for Eq. (41) are T R
{ min. Kin ) =T>
T R
(44)
{ sadd, Kadd ) = °
The multiplication by two is needed to account for the fact that a system at the saddle point can either cross it or return with equal probability [119,120]. In the case of tangentially immobile droplet surfaces DhR = DRh = DRR = 0 , Eq. (41) becomes [26]
"^^.^.ijiJlML-l. Peq{R,h)dh
H
,45)
dh
Equation (45) can be integrated directly and c Km,
dz
r°°
^ • ^ ^ ) = J L f ( > M ^ M J , dyPeq{R,y).
(46)
If hmin is at the secondary minimum and hsadd corresponds to the top of the barrier that has to be overtaken, Eq. (46) gives the mean transition time between the secondary and the saddle point (Fig. 6) [26]. It is interesting to compare (46)
338
D.N. Petsev
Fig. 8. Components of the diffusion tensor as defined by Eq. (43). These are valid for small separations between the droplets.
Theory of Emulsion Flocculation
339
to (35). The latter applies to film thinning and thickness transitions due to the action of a well-defined deterministic force (e.g., buoyancy), while the former is valid for processes that are driven by random Brownian forces. That is why the mean times given by (46) and (41) are statistical averages and depend on the probability distribution in the configuration space defined by Peq(i?,/?) = exp —U(R,h)I kT , as well as on the diffusion coefficient Dhh(R,h) = kT/(hh(R,h) and/or the diffusion tensor D(R,h) = kT[C)(R,h)\'1. Analysis of the dynamics of secondary-primary film transitions, based on (44) demonstrated that the deformation of the droplets may increase the mean transition time with orders of magnitudes when compared to referent nondeformable particles, for more details and examples see Ref. [26]. Eq. (46) and the underlying model do not take into account wave disturbances of the droplet (and more importantly, film) surfaces. Such disturbances are known to trigger film thickness transitions and breakups in macroscopic films (e.g. large millimeter sized droplets) [29-31]. According to experimental evidence [22,118,121] and some theoretical estimates [122], film surface corrugations and wavelike deformations do not occur for small films, like those that form between Brownian micrometer and submicrometer droplets. Still the analysis of some model systems based on Eq. (44) showed that the effect of the droplet deformability could be very substantial at high electrolyte concentrations [26]. 3.2.2. Flocculation and coalescence kinetics of dilute emulsions Calculating the flocculation and/or coalescence rate constant for diluted emulsions is relatively simple if only the Dhh(R,h) = kT /(,hh(R,h) term is taken into account and the other tensor terms are assumed to be zero, DhR = DRh = DRR = 0 [15]. In this case the problem could be solved by calculating the steady flux of droplets toward a single one as suggested by Smoluchowski for solid colloidal particles [83-85]. The important difference in case of droplets is the possibility for interfacial deformation due to hydrodynamic and direct forces. The droplet deformation in this case may start at distances that are substantially larger that those corresponding to the distant secondary minimum of the direct droplet interactions. The reason is the addition of an effective Brownian force to the action of the surface colloidal forces. The droplet flux is
J =4 ^
D W
* G +£ & * 4 2 L coml [ dr
kT
(47)
dr
where r is the distance from the central droplet, nx is the droplet number concentration at infinity, D(r) is the diffusion coefficient, P(r) is the probability
340
D.N. Petsev
to find a droplet at a distance r from the central droplet, and U(r) is the pair interaction energy. Eq. (47) can be solved for the probability function using the boundary conditions P -> 1, U -> 0 for r -»oo P^O, forr^O
(48)
and the result is [15,123]
pooexp[^(r)/^r] , . P(r) w = exp
U[r)\ —-1 itr
*r
D(r)r2 ^yi— . r ooexp U(r)lkT\ I ^—df Jo D(r)r 2
(49)
To determine the total force (Brownian and direct), it is convenient to rewrite Eq. (47) in the form [15] J R(\-e.\ -*- = - i *-!-. Vh 2h
(55)
Then Eq. (52) becomes Fr=Chh+ChR^^Vh
(56)
where Qhh + C,hRR{\ — es)/2h is the effective friction coefficient. Knowing the total force, one could easily obtain the deformation separation distance h, (where the surface curvature inverses its sign, and therefore the subscript /) from the relationship [31,34], see also [15] h -
FT
J
^rd)
p - p l
r
\
( oo even without droplet deformation. As regards coalescence, even though it depends on droplet deformation, the chapter is aimed at determining the time of coalescence irrespectively of the details of this process and therefore the mentioned issue will not be considered here. Investigation of small droplets also allows to consider them as solid particles, i.e., to neglect the effect of the surface and internal mobility of droplets on their sedimentation and on interparticle interaction. Since usually droplets of emulsion are covered with a layer of adsorbed maeromolecules, their mobility, especially for micron-sized droplets, diminishes to zero.
Fig.4. Scheme of aggregation, disintegration and coalescence. Characteristic times of different stages: TS SC{- collision of two singlets (formation of a secondary doublet), TSC]^Sdisintegration of a secondary doublet, rsd^pd ~ transformation of a secondary doublet into a primary one, rpd sd~ transformation of a primary doublet into a secondary one, TC coalescence time.
360
N.O.Mishchuk
2.3. Method of calculation of characteristic times The most consecutive model of aggregation and disaggregation in dilute suspensions, including calculation of characteristic times of subprocesses, is the model of Muller [41, 42]. This model could be directly used for description of non-coalescing droplets and expanded with account of coalescence [26-29]. If emulsion is rather dilute, multiplets practically do not exist and therefore it is enough to take into account the fluxes of single particles (singlets) Jj directed to or from a certain arbitrarily chosen particle (fig.3). Let us suppose that a system has reached a quasi-equilibrium, in other words, the aggregation and disaggregation are mutually equilibrated and the concentrations of singlets «1, primary rip and secondary ns doublets are constant. Naturally, at such quasi-stationary regime, diffusion fluxes are constant and can be described by the equation [41, 42]
L dr
kT dr J
y }
where v(r) is the concentration of singlets at the distance r from a chosen particle, U{r) is the energy of interaction of two particles, D = 2DQ is the coefficient of reciprocal diffusion of two singlets, defined through the diffusion coefficient of single sphere Do = kT 167rr/a0. At short distances D(r) = 2DQ /p(r), where p(r) is the factor taking into account hydrodynamic resistance between the particles approaching each other (for example,
p{r) = l +
ao{r-2ao)[44]).
To define fluxes J\ -J4, presented in Fig.3, Muller proposed to solve Eq.(2) with the following boundary conditions. The flux Jx leads to appearance of secondary doublets and satisfies the boundary conditions of Smoluchowski [45] - the absence of particles in the secondary pit and constant concentration of particles at the infinity v(r e S) - 0
and v(r = <x>) = NQ = const
(3)
The flux J2 leads to disintegration of a secondary doublet and satisfies the conditions |V(r)4;zr2dr = 1
and v(r = <x>) = 0
(4)
i.e., at the moment of doublet disintegration there is only one particle in the
Coalescence Kinetics of Brownian Emulsions
361
secondary pit and, during the process of disintegration, the particle that moves away interacts only with the second particle from the doublet. The flux J 3 defines the probability of transformation of secondary doublets into primary ones and satisfies the condition v(r e P) = 0
(5)
and the first condition in (4), that is, the absence of a particle in the primary pit and the presence of a single particle in the secondary pit. Finally, the flux J 4 defines the probability of transformation of primary doublets into secondary ones and satisfies the conditions
^v(r)4nr2dr = \
H
v(r€S) = 0
(6)
(rsP)
i.e., at the moment of transition there is a single particle in the primary pit and no particle in the secondary pit. According to Eq.(2) and conditions (3-6), the expressions for fluxes can be presented in the following way [42]:
R2TS
R2WTS
R2WTP
where 1^,1^,1^, are the "powers" of primary and secondary minimums, Wis the retardation factor [46, 47] which can be presented as
^ = \r2
exp(- U{r)) \P}Q txV{U(r'))dr' dr « r^
(8)
(r'Y
W=
f (reB)
P(r
'hxp{O(r'))dr'
fcxp(-O(r))r2dr
Fs= (reS)
(9)
(r'Y >
v
I> =
Jexp(-£/(r)/ kT)r2dr (reP)
(10)
362
N.O.Mishchuk
Here the distance r is normalized on diameter of particle 2a 0 U(r) = U(r)/kT. Taking into account the achievement of a local equilibrium J] n\ = 2 J2 %
H J^fip-J^ng
and
(11)
and existence only of singlets and doublets, the characteristic times of subprocesses have been expressed [42] as
r
s sd
'
-
l
1
-
J,
sd s
T
J3
-J_-^l
> ~J2~2D0'
R2WT^
1 l
sd,pd
~
SnD0RN0'
—
„ _. ^-^0
i '
l
pd,sd~
j
R2WTP
~~ ~ r. *M ^^O T
m\
(
v1J/
Although Eqs.(7-13) are received for a quasi-equilibrium state, which can be reached only at t » Ts sd'^sd,s'^sd pd>^pd,sd> they c a n t>e a l s o u s e c l when analyzing transition to an equilibrium state [42]. It is interesting to note that the time of transition from a secondary to a primary pit depends not only on the barrier height but also on the depth of the secondary pit, and the characteristic time for inverse transition depends on the depth of the primary pit. The larger the depth of the primary and secondary pits and, correspondingly, the larger the values of FS,TS and Tp, the less "readily" the particle leaves its place and the larger is the time of transition from the secondary pit into unconstrained state TS(J}S or into primary pit fsdnd • As a consequence, the flux J\ is divided into two fluxes J2 and J 3 , the intensities of which, according to Eq.(7), correlate as J2 UT, = W^s l^s ^W . This means that, due to the barrier, the probability of formation of a primary doublet is W times smaller than the probability of disintegration of an aggregate. It should be stressed that, according to the model presented above, even if a secondary pit is very shallow, i.e. the characteristic times Tsj s and/or ^sd,pdare small the transition from two singlets to a primary doublet formally has two stages: transition from singlets to a secondary doublet and transition through the barrier from a secondary doublet to a primary one. Although coalescence was not investigated in [42], formally it is possible to introduce the notion of the flux J 5 and the characteristic time of rupture of
Coalescence Kinetics ofBrownian Emulsions
363
thin film between two droplets (or the time of coalescence) Tc = 1/ J 5 , as it was done, inter alia, in [19, 26]. Some problems of correct calculation of interaction energy and characteristic times, necessary to ensure the proper description and understanding of existent processes, were discussed in the review [29]. 2.4. System of differential equations for description of coagulation and coalescence According to papers [42] and [19, 26], the change of concentrations of singlets (n s ), primary ( n p d ) and secondary ( n s d ) doublets can be described by interconnected differential equations
dns^_nj T
d?
+2nsd
+npd
r
s,sd
(i4)
f
sd,s
c
^ = -3L-»J^- + - U+ -^L 2r
d?
s,sd
dn
pd
{Fsd,.!
nsd
=
npd
r
&
r
pd,sd
npd
r
sd,pd
sd,pd)
(15)
r
pd,sd
r
c
or, with introduction of dimensionless values, by the equations djh^_n2+2^L dt
T
+^ L
(1?)
T
sd,s
c
d
-^=i-J—+-^1
dt dn
pd dt
2
T
y sd,s nsd
= T
sd,pd
as)
T
sd,pd)
npd T
c
where the concentrations of singlets and doublets are normalized on the initial droplet concentration NQ: ns=nsl NQ, npd =rtpdl NQ and nsd =nsd INQ and all characteristic times are normalized on the time of formation of secondary doublets Tj = T; I Ts sd .
364
N.O.Mishchuk
Since the size of the chapter is limited, the possibility of transformation of a primary doublet into a secondary one and correspondent terms npd I Tpd scj at transition from Eqs. (14-16) to Eqs. (17-19) are neglected. Eqs.(17-19) take into account the process of formation of secondary doublets, their possible disintegration into two singlets or transformation into primary doublets. Coefficient 2 in Eq. (17) shows that disintegration of a doublet results in two singlets and coefficient 1/2 in Eq.(18) shows that two droplets form a doublet. Eqs. (17-19) assume that formation of triplets is negligible. This is possible only in the case when the number of doublets in the system is not too high. The condition necessary for this will be discussed in Section 2.5. The terms npci I rc describe coalescence. The number of coalesced droplets is directly proportional to the number of formed thin films, i.e., the number of primary doublets, and inversely proportional to the characteristic time of their rupture r c . Since existent experimental methods do not allow to measure relatively small changes of droplet size, the coalesced droplets are included into Eqs. (17) and (19) as initial singlets. Taking into account that it is practically impossible to create an absolutely monodisperse emulsion, the 26% difference between the radiuses of initial ( |
(34)
T
sd,pd)
the solution of which could be received according to the scheme proposed in [19, 26], where the linearization of a similar equation was performed by replacing function ns (t) with its initial value ns (t = 0)« 1. The exactness of solution at t > tst (29) could be higher when ns is replaced with its quasiequilibrium value ns « ns (t -> oo) - c\ (26), which was obtained for simpler emulsion without transition into a primary minimum. After presentation of Eq.(34) as
^V^^L dt2
dt
\
]-2c,-^+ T- ^ + ^— T T sd,s
sd,pd)
sd,pd
(35)
369
Coalescence Kinetics ofBrownian Emulsions
its solution is expressed very simply: a f
e-
_e~a2'
\
2(af2-a,) where
1 1 2ci ^,=20,+-!-+—!-;?,=—*T
sd,s
T
(38)
T
sd,pd
sd,pd
Concentrations of primary doublets and singlets are obtained from Eqs.(33, 36, 21) as
a2(l-e-^)~a,(l-e-a^) "/«/= o ^"7°
H;
. . . 2 «5=l- n^-2«pd
..„ (39)
Emulsion will remain singlet-doublet if, in addition to condition (27), it satisfies the condition of slow transition into the primary minimum, which is possible if hd,Pd»hd,s
and ^d,Pd>1
( 4 °)
In the opposite case, the number of primary doublets increases considerably, i.e., the total number of doublets rises and the possibility of triplet formation appears. Under conditions (40) q\ pci » TS^SCJ » TSCI^ ), at first, the faster processes equilibrate and that leads to coincidence of the beginnings of curves for singlets (1-3 and 4-6) and doublets (l'-3' and 4'-6'). A difference between the curves at the same value of TS(Jp(j but different values of TSCI s is clearly seen in Fig.6. Therefore, both the barrier that should be overcome by droplets and their location and interaction energy are of importance. The lower the interaction energy in the secondary minimum and, correspondingly, the smaller the time of secondary doublet disintegration, the lower is the probability of transition into the primary minimum at the same height of the barrier. Since secondary doublets transform into primary ones slower than they disintegrate, the concentration of primary doublets in the beginning of the process is low and does not affect the concentration of singlets and secondary doublets. However, as the number of primary doublets increases, the number of droplets participating in aggregation and disaggregation in the secondary minimum decreases and that leads to a decrease of singlet concentration. As a result, maximums of the curves 4'-6' and bends in the curves 4-6, 4*-6* appear demonstrating the qualitative change of the investigated process. Similarly to aggregation in the secondary minimum (Section 2.5), in the present case, not only the time of transition to a quasi-stationary state but also the values of corresponding concentrations depend on the characteristic times of each subprocess. Both in case of insurmountable and surmountable barriers, the concentration of secondary doublets grows and the concentration of singlets decreases as the time of doublet disintegration TS(JS increases (see transition from curves 1, 1' to 3, 3' and from curves 4, 4',4" to 6, 6', 6"). At very low TSCJs, the probability of secondary doublets disintegration is to such extent higher than the probability of their transformation into primary doublets that the concentration of secondary doublets becomes independent of the latter process. This conclusion is proved by small difference between curves 1' and 4', which coincide at the given scale of the figure. It is useful to compare the kinetic curves at fixed TSJS and different T
sd pd ( s e e Fig-7)- The decrease of the barrier leads to acceleration of
transitions into the primary minimum and affects the number of secondary doublets and residuary singlets. However, even at the transition times that are commensurable with the time of rapid coagulation (TSJ nd=^> curves 5, 5',5") or even smaller ( r 5 j n ( i < l, see curves 6,6",6"), the general rate of coagulation is considerably lower than the rate of rapid coagulation. This means that at a very
374
N.O.Mishchuk
Fig.7. Dependence of concentrations of singlets ns (curves 1-6), secondary nscj (V6') and primary rip^ (l'-6') doublets and the sum of doublets and singlets (l*-6*) on time / . The calculations are performed at TSCJ s =0.1 and
TSCJ n ^ = 1 0
(curves 1,1',1",1*); 5
(2,2',2",2*); 3 (3,3',3",3*); 2 (4,4',4",4*); 1 (5,5',5",5*) and 0.3 (6,6',6",6*), correspondingly. Curve 0* is calculated at zsc[ s =1000 and TSCJ pj =1000. Figs.c and d present the same results on different scales.
shallow secondary minimum and, consequently, low xs£s, droplets scatter sooner than they receive an opportunity to overcome the barrier. The analyzed peculiarities of primary doublet formation could be used for
Coalescence Kinetics of Brownian Emulsions
375
theoretical explanation of the well-known experimental fact - at reduction of electrolyte concentration, the deceleration of coagulation in comparison with rapid coagulation is smoother than according to the calculation of the retardation factor (see, for example, [46, 47]). Let us compare Figs.7d and 5d. Curve 1 * in Fig.7d coincides with curve 6* in Fig. 5d and the fan of curves 2*-6* in Fig.7d is the branching of the same curve 1 * that appears owing to the change of transition time TSCJpcj. Since in Fig.7 all values of x^
pcj
are larger than TscjfS , at short time t < TSCJs curves
l*-6* practically coincide. This means that if the measurement is performed within the time interval 0 < t < TS^^S , there will be a difference between curve 0* for rapid coagulation and curves l*-6*, caused not by transition into the primary minimum but by disintegration of secondary doublets. In other words, this will be "slow" coagulation, although not in its classical meaning. It will be slow due to reversibility of aggregation in the secondary minimum. The qualitative illustration of two different types of "retardation" is shown in Fig.8, where the factor of retardation at transition into the primary minimum without disintegration of doublets (curve 1) and the factor of the "retardation" caused by the change of the slope of kinetic curves related with the reversibility of aggregation in the secondary minimum (curves 2,2',2") are presented.
Fig.8. Factor of retardation W as a function of electrolyte concentration. Curve 1- traditional factor of retardation (see Eq.(9)), curves 2,2',2' - "retardation" factor caused by reversibility —21 of aggregation in the secondary minimum. Hamaker constant is ^ = 2 1 0 J, surface potential y/ = 25mV, radius OQ =0.5/jm. Curve 2" is calculated at the linear change of surface potential (// from 25mV at C=\molll to 50mV at C = 10~ moll I. Volume fractions of droplets are 1% (curve 2) and 10 % (curves 2',2").
376
N.O.Mishchuk
The numerical calculations were performed for arbitrary chosen parameters with account of retardation and screening of van der Waals interaction according to the models [30, 31] that particularly were used in [28, 29]. It was also assumed that experimental measurements of the number of droplets were performed at t - 0.2, which corresponds to the practically linear section of the kinetic curves in Fig.7d. As seen from the curves in Fig.8, at reversible aggregation, theoretical "retardation" of the process at reduction of electrolyte concentration could be even slower than that obtained experimentally, for example, in [48, 49]. The degree of "retardation" depends both on the change of volume fraction of droplets (2, 2') and on possible change of surface forces related with the change of surface potential (curves 2', 2"). The dependence of the position of curve 2 on volume fraction could be used to prove the "non-traditional" mechanism of coagulation retardation and to separate it from other mechanisms, for example, those proposed in [48] and other papers. Choice of the time of measurement is also important. For example, in Fig. 7d, the kinetic curves at /S in formation of primary doublets is especially important for "water in oil" emulsion, where the thickness of the double layer is very large and, consequently, TSCJ^ diminishes to zero. 2.7. Single energy minimum. Coalescing droplets The simplest version of coalescence takes place after aggregation directly in the primary minimum, i.e., without complicative transition from the
311
Coalescence Kinetics ofBrownian Emulsions
secondary minimum to the primary one. This is possible at low surface charge when two minimums merge in one. The system of three differential equations (17-19) in this case is reduced to two: ^
= -,,2 + 2 - ^ + ^
dt
T
(45)
x
pd,s
c
dn
pd n2s f 1 l) -jr = ^-npd\ +— dt 2 \rpd,s Tc)
where zpds
( 46 )
has the same meaning as rsds
in Section 2.5. The system of
equations can be solved using the scheme of linearization presented in Section 2.6. The solution for the number of doublets is e-ft'-e-/ht nPd rnd=—-. 2{/32-px)
(47)
where
A , 2 = " f [-1± P f ]; P2=2cXp + ^ ^ ; 9 2 = ^ 2
1
i
T d s
p \ \
P>
Tc
Tc
(48)
and quasi-equilibrium concentration of doublets c\p is expressed by Eq.(25) with the replace of ts^^s by rp^s. Since the total number of droplets at coalescence is not preserved, the concentration of singlets can be found from Eqs.(45, 46) as
(ns + 2npd)=-
y
or
ns=\-2npd
-nc
where
is the decrease of the total number of droplets caused by coalescence.
(49)
378
N.O.Mishchuk
The location of the maximum for doublet concentration can be found similarly to Eq.(44) as
_ln(/V/?2) 'max
KJ1J
n n
P2 ~ P\
However, contrary to the previous case, where the similar expressions defined the location of maximum number of secondary doublets and the bend of kinetic curve for the sum of primary and secondary doublets, in the given case, this is the location of the maximum concentration of one type of doublets. In Fig.9 presented below, calculations are performed specially for shallow minimums and, correspondingly, quick disintegration of doublets, in order to make the role of doublets disintegration more illustrative. As seen from the figures, the rate of coalescence depends on aggregation reversibility. The larger the doublet lifetime r ^ , the higher is the probability of coalescence. The increase of
TSC[S
(compare Figs. 9a,b and 9 c,d) leads not only to
stronger decrease of ns and ns +np(^ but also to more pronounced maximums of npci that demonstrate faster coalescence. It should be underlined that the intensification of coalescence is caused not by the decrease of the time of film rupture r c , but by the increase of the number of droplets that are in the energy minimum and wait for the rupture of the film. Attention should be also paid to the fact that although the total number of droplets decreases, according to the curves in Fig.9, the emulsion remains to be singlet-doublet, since npci lns < 0.15 when the time of investigation is not too long. To show the role of doublet disintegration for a wider range of parameters, the decrease of the total number of droplets is calculated for a few sets of characteristic times of disaggregation and coalescence (see Fig. 10). The influence of rp^s depends on its correlation with the time of coalescence r c . If these values are of the same order (see curves 1, 1' or 2, 2'), the number of coalesced droplets changes weakly. However, when they differ to the great extent (compare curves 1 and 1" or 2 and 2"), the number of coalesced droplets decreases considerably. It is interesting to note that, excluding the beginning of the curves, i.e. the time interval t s = 0.3 (Figs, c, d) and
TC = 1000 (curves 1,1', 1"); 10 (2, 2', 2"); 3 (3, 3', 3"); 1 (4, 4',4"). Curve 0* is calculated for T
pd
5
= 1000 and rc = 1000.
380
N.O.Mishchuk
Fig. 10. Dependence of the decrease of total number of droplets nc on time at tp^s = 100 (1 3); Tpd,s=™ (1'- 2') and
tpdjS=\
(l"-2") for r c =30 (1,1',1"); r c =100 (2,2',2");
r c =1000(3).
Taking into account that, at numerical calculations for Fig. 10, rather large values of coalescence time r c were used, the coefficient fi\ for this limiting case can be written in a simpler view:
A=^ Tc
!
(52)
2c]p+l/TpdfS
Thus, it is inversely proportional to the coalescence time. This is not surprising since it is coalescence that is the slowest process limiting the rate of emulsion destabilization in the given system. 2.8. Two energy minimums. Coalescing droplets Let us analyze a more general case of coalescence with the possibility of transition from the secondary minimum to the primary one that can be described by three equations (17-19). Since the solution of these equations is more complicated than in the previous cases, fist of all the numerical solution will be analyzed (see Figs. 11 and 12). The comparison of Figs. 11 and 12 clearly points to the influence of characteristic time isd,pd o n m e formation of primary doublets and, consequently, on the rate of coalescence at a fixed time of rupture of thin films rc. The lower energy barrier leads to quicker appearance of primary doublets
381
Coalescence Kinetics ofBrownian Emulsions
and rupture of thin layers and that, in its turn, causes quicker decrease of numbers of singlets and secondary doublets. Similarly to the case of noncoalescing droplets, at the fixed value of rsd S , the slope of curves in the
Fig.ll. Dependence of concentrations of singlets ns (curves 1-5), secondary « 5 ^(l'-5') and primary tip^ (2"-5") doublets, sum of doublets n^ (l*-5*) and sum of singlets and doublets n +n
s d
Tsd,pd
(curves ]**-5**)
On
time t.
Curves 0,0** are received at r ^ ^ =1000,
=1000, r c = 1 0 0 0 ; curves l,l',l", 1*,1** at 7 ^ = 0 . 3 ,
TC =1000; the others at r ^ ^ =0.3, rsd,pd
=1°
and 7
r^^=1000
and
c =1000 (2, 2', 2", 2*, 2**); 10(3,
3', 3", 3*, 3**); 3 (4, 4', 4", 4*, 4**); 0.1 (5, 5', 5", 5*, 5**), correspondingly.
N.O.Mishchuk
382
beginning of aggregation is independent of TSC[p j , which was taken considerably larger than r ^ s. The initial slope of curves is also independent of rc,
the value of which is comparable with TSCJS and is considerably smaller
than TSC{pc[. The latter result is caused by the need to overcome a barrier before coalescence starts. This means that instead of comparing TSCI s and r c , it is necessary to compare the values of TS(jtS and T
T
T
sd sp + c >> sd s'
me
T
sd,sp
+T
c • Since
initial slopes of curves are defined by a faster
process- the disintegration of secondary doublets.
Fig. 12. The same as Fig. 11, but rsd^s = 0.3 is replaced with tgd^pd =
Coalescence Kinetics of Brownian Emulsions
383
As seen from numerical analysis, slow rupture of thin film in case of quite high barrier (large TSCJpj) leads to a slow change of the total number of droplets. This means that the system should be examined for a very long time and therefore the exactness of the investigation can be low owing to different accompanying processes, the influence of which increases with time. The decrease of the total number of droplets calculated for a few sets of parameters is shown in Figs. 13 a,b. The curves in Fig. 13a are similar to curve 2" in Fig. 10, which is also shown in Fig. 13a to make comparison easier. Indeed, the calculations for curve 2" were performed at xc =100, r ^ =1 (since in Fig.10 a single minimum was investigated, rpci s is the analogue of ts^^s), therefore the curves in Fig. 13a can be considered as the complication of the process shown in Fig. 10. The droplets represented by curve 2" participate in three processes (appearance of a doublet, its disintegration or coalescence). In the case shown in Fig. 13a, the additional process appears, that is the overcoming of a barrier. It might seem that, when a barrier exists, coalescence should be slower. However this is true only for a short period of time when the number of primary doublets is very small (see beginning of curves 2,3 and 2" in Fig. 13a). Later, at a low barrier (curves 1-3), coalescence occurs quicker than in a single minimum case (curve 2"). This non-trivial result has a very simple explanation. The disintegration of a doublet from a single shallow minimum occurs quicker than its possible coalescence. In case of two minimums, after formation of secondary doublets, some of them disintegrate into singlets, while others transform into primary doublets, which wait until thin films rupture. For a single energy pit, the number of coalesced droplets is proportional to the ratio of times iS(j s lxc . For two pits, the TSC[ S/TSCJ sp part of secondary doublets transforms into primary ones, and the l / r c part of transformed doublets coalesces. Thus, the rate of coalescence is proportional to Tsd,s/[rsd,spTc)- This very rough qualitative evaluation of the role of two energy pits is reflected by kinetic curves for the number of coalesced droplets. At a high barrier (TSCJ^P = 10, curve 4), when the probability of formation of primary doublets is low, at fixed tc, the rate of coalescence for two minimums is lower than for a single minimum. In contrast, at low barrier (and small TSCJsp, see curves 1,2), the probability of formation of primary doublets is high and therefore the rate of coalescence for two minimums is higher than for a single one. It is also worthwhile to analyze results of numerical calculations at fixed r sd,pd' a s presented in Fig. 13b, in which curve 3 coincides with the same curve in Fig.l3a and curve 2" from Fig. 10 is also shown repeatedly. The shorter the
384
N.O.Mishchuk
Fig.13. Dependence of the decrease of the total number of droplets nc on time a) TC = 100, TSCJ^S = 1 and Tsd,pd = °-' (curve 1), 1 (curve 2), 3 (curve 3), 10 (curve 4); 6) TSCI pd =0.1 and r c = 1 0 0 (curves 1-3) and r c = 1 0 (curves l'-3'); xscjs
equals O.I
(curves 1,1'), 0.3 (curves 2,2') and 1 (curves 3,3'). Curve 2" is copied from Fig. 10.
time TSCI s, the lower is the number of coalesced droplets. Droplets from secondary doublets, instead of transformation into primary doublets, "prefer" to leave aggregates and therefore the probability of coalescence decreases. The decrease of r c , when other parameters are fixed, leads to quicker coalescence, as shown by curves l'-3' inFig.l3b. Unfortunately, the analytical solution of differential equations for a low barrier and slow rupture of thin layer is rather difficult. Therefore we will analyze analytically only the opposite limiting case of a relatively high barrier and quick rupture of thin films. This case is more interesting from the experimental point of view than the quick rupture of film and a low barrier, since the latter process would look as rapid coagulation and on its background it would be difficult to define the rate of coalescence. When condition T
r
sd,pd »
(53)
c
is fulfilled, it may be assumed that each primary doublet merges into a single droplet immediately after it is formed, i.e., as may be described by the condition dn
pd dt
nsd
=
T
sd,pd
_npd=Q T
c
( 5 4 )
385
Coalescence Kinetics ofBrownian Emulsions
In the framework of the above scheme, the solution of differential equations (17-19) with account of (54) takes the following view: e-nt
n
_e-r2l
e-r]t_e-nt
T
sd =
"pd = —
(55)
T
n-n
sd,Pd
n-n
and d /
n
\
—\ns+2nsd) at
pd
=
or
n
=1 -2nsd
TC
- In
-nc
(56)
ft=-S_
(58)
d F
where
*Tsd,Pd
nnTsd,pd
ri,2=^f-l± P f 1 P3=2c 1+ -U^- ; 2
I
\|
P3 I
T
sd,s
T
sd,pd
T
sd,pd
The maximum of the kinetic curves for secondary doublets is reached after the time _ Hn 172)
(59)
\jy;
' m a x ~~
72-71 In the limiting case when xc I tsdnd ~^> 0, the same time corresponds to the maximum of primary doublets (curve 5"in Fig. 12b) and, consequently, to the maximum of the total number of doublets (curve 5* in Fig.l2d). However, when ratio ?clTsd,pd'v& n o t t 0 ° l ° w ' condition (53) is violated and the maximum of curves for primary doublets and the total number of doublets shift to larger times (see, for instance, curves 3", 4" in Fig.12b and curves 3*, 4* in Fig.l2d). 3. DESIGN OF EXPERIMENTAL INVESTIGATIONS Comparison of kinetic curves for total concentrations of singlets and doublets (see Figs. 7c, 9c, l i e and 12c) shows that trends of curves seem to be very similar at considerably different subproccesses of emulsion destabilization.
386
N.O.Mishchuk
This means that there may be different interpretation of the experimental data obtained by absorption or scattering of light. In particular, absorption of light does not allow discrimination between two aggregated droplets and two droplets that have already coalesced. Moreover, taking into account that absorption of light depends on the length of the used wave, droplet size, size and shape of the aggregate and total concentration of droplets, the applicability of the used method and equipment for the specific investigated system should be thoroughly controlled. It is clear that for this aim it is necessary to develop a special technique that allows to count not only the general number of droplets and aggregates but also singlets, doublets and multiplets separately. Such apparatus based on the combination of streaming ultramicroscopy and single particle light scattering is used for a long time to investigate aggregation in suspensions [53, 54]. As distinct from streaming ultramicroscopy that owing to hydrodynamic flow could affect the stability of secondary doublets, transition into primary minimum and the process of coalescence, videomicroscopy [15-20] is a method that, in principle, does not influence the state of emulsion. Moreover, videomicroscopy allows not only to count the number of doublets and multiplets but also to observe the behavior of chosen doublets (see, for instance, Fig.3 in paper [17]) and follow general changes in emulsion. Therefore it seems to be the only reliable way to define the rate of rupture of thin films against a background of all subprocesses that lead to drop coalescence . Owing to the fact that the analysis of the videopictures [15-19] was not automated, the information about the rupture or coalescence of doublets was not sufficiently complete. Development of special computer programs for videoimage processing could allow to analyze large data files. Only in this case, careful study would allow separation of primary (irreversible) and secondary (reversible) doublets and direct definition of the mechanism of slow coagulation. Beside the development of special videotechnique and corresponding software tools, details of experiment arrangement also play an important part. As it was repeatedly written above, at a given ratio between the density of droplets and medium, droplet size should be limited to avoid the influence of gravitation. The scattering of emulsion size should be minimal, as that could allow not only to weaken the gravitational component of coagulation but also to avoid the change of interaction energy and, correspondingly, characteristic times for different pairs of droplets. One can easily see that the strict monodispersity is less important for rapid coagulation and more important for all other subprocesses of aggregation. Indeed, the most important factor for rapid coagulation is the efficiency of collision between droplets. The analysis of collision efficiencies [55] for identical 4DQQQ and different (Z)o + Z)jX^o +a\) droplets with radiuses QQ a n ^
Coalescence Kinetics of Brownian Emulsions
387
a\ shows that, owing to inverse proportion between diffusion coefficients and radiuses Dq\~\l a§\, t n e efficiencies differ weaker than the radiuses. For example, the efficiencies of collision between initial (a 0 ) droplets only and between initial ( CIQ ) and coalesced (a\ = iflciQ = 1.26ao) droplets differ for 1.3%. When two types of quite different droplets (ag ar>d a\ - 2«o) a r e present, the efficiencies differ for 12.5% only. Emulsion with such scattering of droplet sizes, which does not considerably affect Smoluchowski time, can be easily obtained using even very old methods (see, for instance, [56]). Unfortunately, all other characteristic times depend on radiuses considerably stronger and therefore the aforementioned radius scattering is too large to provide a small difference between the values of TS^JS, tsci,pd a n d Tc for different pairs of droplets. This is caused both by the dependence of characteristic times r,- on droplet radiuses f,- ~ R l2D§~a^ (12-13) and by their exponential relation (see Eqs. (12, 13) and (8-10)) with the energy of interaction U{r), which is a function of particle radiuses 2a o «i /(ao + a\) [2]Indeed, calculation of the characteristic time of doublet disintegration [28, 29] showed that a relatively small change of droplets radius leads to a sharp change of its value. It is clear that the larger the scattering of radiuses and of characteristic times, the more inaccurate is the definition of the time of thin film rupture. Thus, the design of the experiment should provide for the preparation of emulsion with maximum monodispersity. The experiment also requires to select such electrolyte concentration and volume fraction of droplets that would allow forming of singlet-doublet emulsion with prevalence of singlets, relatively low number of doublets and almost total lack of triplets and other multiplets. It is this type of emulsion that makes it possible to observe kinetics of singlet and doublet concentrations, define the characteristic time of disaggregation and calculate the time of thin film rupture. For simplification of theoretical analysis all times were normalized on the time of formation of secondary doublets Ts>scj. However the choice of the optimal value of this time is very important. It should be about several minutes; otherwise bends in kinetic curves, which are of significance for interpretation of investigated processes, could be lost in the time interval between preparation and investigation of emulsion. Since, at the given difference of droplets and medium densities, the choice of droplet size is limited because of gravitation influence, it is necessary to change both the size and volume fraction of droplets. Thus, for example, for one-percent emulsion (a = 0.01) and radius a0 = 0.5jam, Ts sci is approximately equal to 10 sec, but for OQ =\/MI it is already 1.5 min.
388
N.O.Mishchuk
Decreasing the volume fraction in 10 times results increases tgm to 15min. With account of condition (26), the dimensionless disintegration time should be T sd s K< 0-25, therefore the bend in kinetic curves should appear within a few seconds, a few dozens of seconds or a few minutes, correspondingly. It is also necessary to draw attention to the fact that while at rapid coagulation of emulsion the main developments take place in the bulk of emulsion, at slow coagulation the key role could be played by wall effects. First of all, this refers to the abovementioned emulsion concentration or dilution near the upper or bottom wall of the cuvette caused by gravitation. This means that behavior of droplets should be investigated not near the wall but in the bulk of emulsion. For this aim the cuvette should satisfy a few conditions. It should be vertical with the height about 5-10 cm so that the state of emulsion in the central part of cuvette was independent of creaming. With account of the droplet sedimentation velocity Usecj =2aoApg/9r/,
for example at Ap =
O.Olg/cm , such height of the cuvette allows to investigate the properties of emulsion with one micron droplets during a week, with 3-micron droplets during two days and so on. The thickness of the cuvette should be about several hundreds of microns to avoid natural convection. It is also necessary to stress that investigations may not be carried out just near the vertical wall of the cuvette since, owing to hydrodynamic resistance between a droplet and the wall, the latter hampers diffusion [57] and, correspondingly, slows down the rate of droplet collision. The requirement of studying emulsion at a certain distance from the wall does not cause any considerable complications, since investigations should be performed at strongly diluted emulsions, which are sufficiently transparent to allow video filming in the depth of the cuvette. Another important fact to be considered is the attraction of droplets to walls, in result of which the droplets are either attached to the walls or spread on their surface. Therefore the walls should have a strong charge of the same sign as the droplets to create a high energy barrier between them. They can also be covered with non-charged macromolecules, overflowing primary pits in the gap between walls and droplets and in this way preventing aggregation in the primary minimum. For example, in [28] it was proposed to use for this aim agarose gel, which immobilizes electric double layer hydrodynamically [58] and, hence, is supposed to overflow the primary minimum. In addition, the adsorption of macromolecules should be irreversible, since even their small admixture in emulsion changes its behavior. In conclusion, it should be said that the theoretical investigation presented in [26-29] and, for a more general case, in the given chapter shows the role of different factors in destabilization of emulsions and observable kinetic of coalescence. The comprehensive experimental study and theoretical analysis of
Coalescence Kinetics ofBrownian Emulsions
389
kinetic behavior of singlets and doublets allow to discriminate the role of reversibility of aggregation, transformation of secondary doublets into primary ones and coalescence in the general process of emulsion destabilization. The characteristic time of rupture of thin film between two droplets regardless of the nature of this rupture could be determined. Although the experience of application of videomicroscopy in such research is not quite wide, the accomplished experimental investigations [15-20] confirm the possibility of designing a device to measure coalescence and disintegration time of doublets, which, in combination with emulsion dynamics modeling, could be a useful tool in analyzing emulsion properties. Development of theoretical modeling and experimental investigation that takes into account overcoming of energy barrier and provides a thorough analysis of all existing subprocesses would refine the insight on emulsion properties and allow to optimize emulsion technologies with respect to stabilization and destabilization. REFERENCES [I] J.A. Kitchener and P.R. Musselwhite, in: "Emulsion Science", I.P. Sherman (Ed.), Academic Press, New York, 1968. [2] B.V. Derjaguin, Theory of Stability of Colloid and Thin Films, Plenum, New York, 1989. [3] I.B. Ivanov and P.A. Kralchewski, Colloids Surf. A, 128 (1997) 155. [4] I.B.Ivanov. (ed.), Thin Liquid Films. Marcel Dekker, New York, 1988. [5] K.D. Danov, I.B.Ivanov, T.D.Gurkov and R. Borwankar, J. Colloid Interface Sci., 167,
(1994)8. [6] [7] [8] [9]
I.B.Ivanov, K.D. Danov and P.A. Kralchewski, Colloids Surf. A, 152 (1999) 161. D.N.Petsev, Langmuir 2000, 16, 2093-2100. R.G.P. Borwankar, L.A. Lobo and D.T. Wasan, Colloids Surf. A, 69 (1992) 135. J. Sjoblom, in: J. Sjoblom (ed.), Emulsion and Emulsion Stability, Marcel Decker, 1996, p. 393. [10] V.Mishra, S.M.Kresta and J.Masliyah, J. Colloid Interface Sci., 197, (1998) 57. [II] K. Khristov, S.E. Taylor, J. Czarnecki and J. Masliyah, Colloids Surf. A, 174 (2000) 183. [12] T.Gilespie and E.K.Rideal, Trans.Faraday Soc, 52 (1956) 173. [13] E.G.Cockbain and T.S.McRoberts J. Colloid Sci. 8 (1953) 440 [14] H.Sonntag and H.Klare. Z.Phys. Chem. 223 (1963) 8. [15] S.S. Dukhin, O. Saether and J. Sjoblom, in: K.L. Mittal and P. Kumar (eds.), Emulsions, Foams, and Thin Films, N.Y., Basel, 2000. [16] S.S. Dukhin, O. Saether and J. Sjoblom, in: J. Sjoblom (ed.), Encyclopedic Book of Emulsion Technology, Marcel Decker, 2001, 71. [17] O. Holt, O. Saether, J. Sjoblom, S.S. Dukhin and N.A. Mishchuk, Colloids Surf. A, 123124(1997) 195. [18] O. Saether, J. Sjoblom, S.V. Verbich, N.A. Mishchuk and S.S. Dukhin, Colloids Surf. A, 142(1998) 189. [19] O. Holt, O. Saether, J. Sjoblom, S.S. Dukhin and N.A. Mishchuk, Colloids Surf. A, 141 (1998)269. [20] O. Saether, J. SjQblom, S.V.Verbich and S.S.Dukhin, J. Disp. Sci. Technol., 20 (1999) 295.
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[21] W.B.Russel, D.A.Saville and W.R.Schowalter, Colloidal Dispersions, Cambridge University Press, New York, 1989. [22] J.N. Israelachvili, Intermolecular and Surface Forces, 2 nd Edition, Academic Press, London, 1991. [23] D.N.Petsev, V.M.Starov and I.B.Ivanov, Colloids and Surf. A, 1 (1993) 65-81. [24] W.Albers and J.Th.G. Overbeek, J. Colloid Interface Sci., 14 (1959) 518. [25] K.D. Danov, N.D. Denkov, D.N. Petsev and R. Borwankar, Langmuir, 9 (1993) 1731. [26] S.S. Dukhin and J. Sjoblom, J. Dispers. Sci. Technol., 19 (1998) 311. [27] S.S. Dukhin, J. Sjoblom, D.T. Wasan and O. Saether, Colloids Surf. A, 180 (2001) 223. [28] N.A.Mishchuk, R.Miller, A.Steinchen and A.Sanfeld, J.Colloid Interface Sci., 256 (2002) 435. [29] S.S.Dukhin, N.A.Mishchuk, G.Loglio, L.Liggieri and Miller R., Adv. Colloid Interface Sci., 100-102(2003)47. [30] Ya.I. Rabinovich and N.V. Churaev, Kolloid Zh., 52 (1990) 309. [31] V.N. Gorelkin and V.P. Smilga, Kolloid Zh., 34 (1972) 685. [32] N.A. Mishchuk, J. Sjoblom and S.S. Dukhin, Kolloid Zh., 57 (1995) 785. [33] E.D. Shchukin, E.A. Amelina and A.M. Parfenova, Colloids Surf. A, 176 (2001) 35. [34] N.A. Mishchuk, S.V. Verbich, S.S. Dukhin, O. Holt and J. Sjoblom, J. Disp. Sci. Technol., 18(5) (1997) 517. [35] A.S.Dukhin, Kolloid Zh., 49 (1987) 630. [36] V.M.Voloschuk and Y.S. Sedunov, Coagulation Processes in Disperse Systems. Hydrometeoizdat: Leningrad, 1975 (in Russian) [37] M.Smoluchowsi, Z.Phys.Chem. 92 (1918) 129. [38] V.G.Levich, Physico-Chemical Hydrodynamics, Prentice Hall, New York, 1962. [39] A.S.Kabalnov, A.V.Pertsov, Yu.D. Aprosin and E.D.Shchukin, Kolloidn.Zh. 47 (1985) 1048. [40] S.S. Dukhin and J. Sjoblom, in: Emulsion and Emulsion Stability, J. Sjoblom (Ed.), Marcel Decker, 1996,41. [41] V.M. Muller, Kolloid. Zh., 40 (1978) 885. [42] V.M. Muller, in: 'Surface Forces in Thin Films", B.V. Derjaguin (Ed.), Nauka, Moscow, 1979,30. [43] S.Ljunggren, J.C.Eriksson and P.A.Kralchevsky, J.Colloid Interface Sci. 191 (1997) 424. [44] V.M. Muller, Colloid J., 58 (1996) 598. [45] M.Smoluchowski, Phys.Z., 17 (1916) 557, 585. [46] N.A. Fucks, Z.Phys., 89 (1934) 736. [47] B.V. Derjagin and V.M. Muller, Doklady Acad./Nauk SSSR, 176 (1967) 869. [48] H.Kihira, N.Ryde and E. Matijevic, J.Chem.Soc.Faraday Trans., 88 (16) (1992) 2379. [49] D.Snoswell, J.Duan, D.Fornasiero and J.Ralston, J.Phys.Chem., B, 107 (2003) 2986. [50] RJ.Pugh, Adv.Colloid Interface Sci., 64(1996)67. [51] N.Mishchuk, D.Fornasiero and J.Ralston, J. Phys. Chem., A. 106 (2002) 689-696. [52] S.A.K. Jeelani and S.Hartland, J.Colloid Interface Sci., 206 (1998) 83. [53] N.Buske, H.Hedan, H.Lichtenfeld, W.Katz and H.Sonntag. CoI.Polym.Sci. 258 (1980) 1303.
[54] [55] [56] [57] [58]
H.Sonntag, V.Shilov, H.Gedan, H.Lichtenfeld and C.Durr. Colloids Surf., 20 (1986) 303. N.A.Fuchs, The Mechanics of Aerosols, Pergamon, Oxford, 1964. M.A. Nawab and S.G. Mason, J.Colloid Sci.13 , 179, 1958. J.Goldman, R.G. Cox and H.Brennet, Chem.Eng. Sci., 22 (1967) 637. C.J. van Oss, R.M. Fike, R.J. Good and J.M. Reinig, Analyt. Biochem., 60 (1974) 242.
Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.
Chapter 10
Hydrodynamical interaction of deformable drops A.Z. Zinchenko and R.H. Davis Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309-0424 1. INTRODUCTION For small emulsion drops with diameters in the range of 1-100 /im, the surface deformation is typically small, and much theoretical work has been done to date to investigate drop interactions with negligible shape distortion. Another relevant assumption for small emulsion drops is the neglect of fluid inertia on the microscale, making it possible to use simplified (Stokes) equations for the fluid motion. Exact semi-analytical solutions have been constructed to describe hydrodynamical interaction of two surfactant-free spherical drops at arbitrary surface separations and orientations, both in a quiescent liquid and shear flows [1-8]. These results allowed us to study the effects of drop interactions on the collision efficiency of Brownian and non-Brownian drops in gravity-induced and shear-induced motion, and on the non-Newtonian rheology of semi-dilute emulsions [4,8,911]. For 'well-mixed' concentrated emulsions of spherical drops, numerical multipole techniques were used to evaluate the effective properties (viscosity, sedimentation rate) [12]. These studies have been complemented by asymptotic analyses for two-drop interactions at small surface separations [4-5,13-15]. In particular, it was found that the lubrication resistance of the liquid film between two non-deformed drops with arbitrary viscosities is inversely proportional to the square root of the gap, and therefore this resistance does not preclude drops from coming into contact (in a finite time) even without van der Waals attraction. This finding is in stark contrast to solid-particle interaction; it is well known (e.g., [16]) for the latter case that the lubrication resistance of the thin film is inversely proportional to the gap, and so collisions of small solid particles are not possible
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without attractive molecular forces, non-continuum effects, or microscale surface roughness. The present review focuses on the effects of surface deformation on drop interactions. While deformation is relatively unimportant for describing the behaviour of solitary and well-separated drops of small size (1-100 lira) in dilute emulsions, it may have a crucial effect, as it turns out, on the efficiency of drop coalescence and on the collective behaviour of highlyconcentrated emulsions. The reason is that the deformation of the thin film between two surfaces approaching each other is known to preclude drops from coming into contact, unless van der Waals forces become important [17-18], and so the coalescence efficiency of slightly deformable drops is a result of subtle interplay between the resistance of the thin film and short-range molecular attractions. The degree of deformation on the drop lengthscale is measured by the capillary number Ca ~ /ief//cr
,
where /j,e is the dynamical fluid viscosity outside the drops, U is the characteristic velocity of the flow relative to the drops, and a is the constant surface tension (the effects of surface tension gradients due to surfactants or temperature variations are not included here). When Ca 0. Most importantly, the contact force (6) or (9) from the outer solution enters the integral balance (16) for the film as the lubrication force. The thin-film system of equations (12)-(17) (or its simplifications for immobile or non-deformable films, constant driving force F, or no van der Waals attractions) have been derived and used by a number of authors [33,15,17,18,34] to study head-on collisions. In Refs. [20,21], film drainage equations (12)-(13), (15)-(17) were combined with the outer solution equations (6)-(7) (or (9), (11)) to describe three-dimensional coalescence in gravity- or shear-induced motions, but without the pressure-gradient term in Eq. (15), which has limited the theory in Refs. [20,21] to moderate viscosity ratios A tcou, the time-dependence of the contact force through Eq. (7) or (11) is taken into account. The exact choice of the initial separation h°min does not affect the overall solution in the range of interest t > tcou (since, for most of the time during t < tcou, deformation is unimportant). This simplest matching strategy based on Eq. (21) is asymptotically correct for Ca -> 0 (except for the special case of sliding collisions, when F(fic) m 0 or F((3C,8C) & 0, which have a negligible probability in random emulsions), but it is limited to moderate viscosity ratios A 1, is used instead of Eq. (20): w -67r/i e - — $(p) — — , p = AJ ——, (23) h-min at N where $ ( p ) has an exact expression [14]:
H
^ ^ f ^ n + i ^ i + i ^ + e]
(24)
'
The lubrication form (20) is the limiting case of (24) for p -C 1. Equating (23) to the contact force (6) or (9), and integrating backwards from Kiin = 0 to hmin = h°min, yields
*""*'=£ IT* • where
^
*(p)
-
n(n + 1)
F
(2n + l)Po]
and po = X(2h^nin/R)1^2\ the contact force F and d^/di in Eq. (25) are taken from equations (6)-(7) or (9)-(ll). Equation (25) relates the collision angles (3C (or f3c and 0c) to the initial angles (30 (or (30 and 90) at t = 0. In particular, for gravity-induced motion,
sin/30 = s i n / 3 c e x p [ - ^ ^ l [ aApgR J
.
(27)
For relative centre-to-centre trajectories in the plane 9 = n/2 in shearinduced motion, Eq. (25) yields ! - ( ! - £ ' ) cos 2/?e = 8fcI(l-g') 1 - (1-Bt)co82/30 (l + k)3D*
, , '
{
(for general trajectories, 9 ^ TT/2, it is also possible to relate 9O, (30 to 9C, (3C through the second equation (11), but the algebra becomes more involved). Knowing /3O (or (30,90) and the initial film profile (22) allows us to start integrating the thin-film equations together with the contact-motion equations. Again, the choice of initial separation h°min has no effect on the solution in the range of substantial film deformations (for /i/a8- < O(Ca)).
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For AGz1/2 < 1, the matching strategy (25) and that of Refs. [20, 21] yield practically identical results, and both do not contain any adjustable parameters; for \Call2 > 0(1), however, the matching rule (25) is the one to use. The numerical solution of thin-film equations (12)-(17) is greatly complicated by numerical stiffness, so that the simplest, explicit algorithms require extremely small time steps (especially at the initial stage of film deformation), and are very impractical for systematic coalescence efficiency calculations (when these equations need to be solved many thousand times). Far more efficient, semi-implicit, absolutely stable algorithms have been developed recently; see the original works [19, 20] for details. To demonstrate how the theory is used to calculate the coalescence efficiency, consider the case of two drops in gravity-induced motion with size ratio k — a\/a,2 = 0.5, viscosity ratio A = 1, Ca = 2.2 x 10~3, and the non-dimensional Hamaker parameter 5 = 1.1 x 10~3, where
Fig. 3. Dynamics of the minimum surface separation for two trajectories close to critical.
Fig. 3 presents the minimum surface separation, hmin/a2, scaled with the larger radius 02, versus the angle (3, for two different collision angles /3C = 1.035 rad (solid line) and 1.040 rad (dashed line). The minimum separation is located along the line of centres, or at the rim when a dimple forms. When (3C — 1.035 rad, the van der Waals attraction has enough time to pull the surfaces together, leading to coalescence. In contrast, for a slightly
Hydrodynamical Interaction of Deformable Drops
403
higher value of j3c = 1.040 rad, the strength of the van der Waals attraction is insufficient for coalescence, the drops reach minimum separation of hmin/o-2 — 7.6 x 10~6 at (3 « TT/2 and eventually separate. The critical value (3fu in the range 1.035 - 1.040 is slightly different from f3?it m 0.99 found in Ref. [20] for the same conditions, simply because we used in the present calculations much larger cutoff radii r max in the numerical solution of thin-film equations (12)-(16) for full numerical convergence. Fig. 4 shows the film profile evolution for a slightly supercritical value of pc = 1.040. The profiles are plotted in non-dimensional variables [20]: h = hR/b2
,
r = r/b
,
(30)
where b is defined by irb2a/R = ApgR3 and is a measure of the film radial extent. At /3 — 1.101, the film just starts to dimple. Subsequently, the dimple radius decreases with increasing (3. At (3 = 1.712, when the surface clearance hmin has reached its last local minimum (Fig. 3), the dimple disappears altogether, causing very rapid drop separation thereafter.
Fig. 4. Dynamics of the film profile for f3c = 1.04 rad.
Once the critical collision angle f3^u is found by trial-and-error (which typically requires extensive calculations and an efficient thin-film algorithm), Eq. (19) may be used to determine the corresponding critical offset parameter d^\ separating coalescence (c?oo < d^11) and noncoalescence (doo > d^f). The coalescence efficiency is defined as / dcrit •^12 =
\2 •
(<Jl)
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Most importantly, knowing E\i allows us to solve the population dynamics equation (e.g. [36-38]) and predict the evolution of the drop-size distribution in dilute emulsions due to coalescence. It was found [20] that there is a narrow 'second coalescence zone' for collision angles j3c close to n/2, but its contribution to the coalescence efficiency £12 is typically very small, so Eq. (31), assuming a single coalescence zone d^ < d^lt, is quite adequate for practical purposes. Figure 5 shows the coalescence efficiency for slightly deformable (solid curves) and spherical (dashed lines) drops with Ca — 0.001 and A = 1 as a function of the non-dimensional Hamaker parameter S. Figure 6 presents E12 as a function of the capillary number at k = 0.5 and 8 = 4 x 10~4. It can be predicted from Fig. 6, for example, that deformation decreases the coalescence efficiency of two drops with a\ = 262 /xm, a-i = 525 /im, a = 10 dyn cm"1, A = 0.5, Ap = 0.1 g cm"3, and A = 10~14 erg (5 = 4 x 10"4, Ca — 5.8 x 10~3) by about five fold. The sharp dependence of S on the drop size (see Eq. (29)) is the reason for steep behaviour of Eyi in the transition range from spherical to deformed drops. Similar calculations can be performed for shear flow [21], although they are technically more involved, since the upstream interception area for deformable drops is no longer a circle, and two collision angles, j3^%t and #£"*, have to be found by trial-and-error. The collision efficiency, E\%, is defined as En = Jn/J°12
,
(32)
Fig. 5. Coalescence efficiency for gravity-induced motion at Ca = 0.001 and A = 1. Reproduced by permission from Ref. [20].
Hydrodynamical Interaction of Deformable Drops
405
Fig. 6. Coalescence efficiency for gravity-induced motion at k = 0.5 and S = 4 x 10 4 . Reproduced by permission from Ref. [20].
where J\2 is the flux of pairs through the upstream interception area, and Jf2 is the corresponding Smoluchowski's value [40] assuming the neglect of drop interactions, shape distortions and molecular attraction. Figure 10 from Ref. [21] shows, for example, that the deformation effect on the coalescence efficiency of two drops of ethyl salicylate in diethylene glycol for shear flow with k = 0.5, 7 = 1 s"1, a = 1.9 dyn cm"1, A = 0.1, /j,e = 0.35 g cm"1 s"1, A = 5 x 10~14 erg becomes significant when the average radius a = (ai + 02)/2 exceeds 300 /zm. At a = 300 /xm, the capillary number (2) is 0.005, which is expected to be within the range of validity of the asymptotic theory, even for this small A = 0.1 (see Sec. 4). For A = 0(1), it is also possible to perform scaling for the range of drop sizes where deformations have a significant effect on shear-induced coalescence. For two drops with A = 0(1), coming into 'apparent contact' without van der Waals attraction in a general 3D collision, the minimum surface separation h along the trajectory scales like h ~ RCa4^ (Sec. 4). Roughly, if the molecular force Fw ~ AR/h2 becomes competitive with hydrodynamic forces FH ~ HeiR2 at this separation, the drops will coalesce. Substituting the definition (2) of the capillary number Ca into the criterion Fw ~ FH yields r A5rr8 1 1/17
*~&]
(33)
In deriving this scaling, we neglected all non-dimensional numerical factors that depend only on k and A.
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A.Z. Zinchenko and R.H. Davis
3. COMPUTATIONAL METHODS FOR INTERACTING DROPS WITH ARBITRARY DEFORMATIONS 3.1 Boundary-integral Method For drops with finite deformations (Ca = 0(1)), the asymptotic method of Sec. 2 is obviously not valid, and so theoretical analyses in this case rely more heavily upon numerical calculations. If the Reynolds number is still small, typically requiring a highly viscous matrix fluid, the boundaryintegral method is most appropriate. This method was pioneered by Rallison and Acrivos [22] for one-drop calculations, and since then it has received considerable development, including binary interactions [19, 26-28, 41-42], and multiple drops in a periodic box [23-25, 43-44] to simulate locally homogeneous emulsions. The essence of this method is to reduce the Stokes equations (1) in the fluid domains inside and outside the drops to a system of integral equations for the fluid velocity u on drop surfaces only, using the reciprocal theorem for Stokes flows [30] and the Green function (which corresponds to the Stokes flow generated in an unbounded fluid by a point force). In particular, for two drops moving in an unbounded liquid, this system of boundary-integral equations takes the form u(y) = 2K £ / ti(a:) • T(x - y) • n(x)dSx + F(y) 0=1
,
(34)
•/6 1. For drops in close approach, however, the spectrum of K-values for Eq. (34) becomes nearly continuous, and so elimination of just marginal values n = ±1 from the spectrum by deflation does not alleviate all difficulties for extreme A; it is better to use in this case more sophisticated, biconjugate-gradient iterations of Lanczos [42], after deflation. For numerical solution of Eq. (34) (or its deflated version), the drop surfaces must be discretized. There are two general approaches to surface discretization: (i) global parametrization (e.g., [46-47]) and (ii) unstructured triangulation. The first approach (with a clear analogy to globe parametrization by a system of longitude and latitude lines) is losing popularity and is now considered generally inferior to the second method, primarily due to the presence of coordinate singularities (poles) in the global parametrization approach, with a negative effect on numerical stability. In dynamical simulations, drops often start from spherical shapes. Unstructured, almost uniform triangulations of a sphere can be constructed by simple techniques. In the first scheme [45], each face of a regular icosaedron inscribed into the sphere is divided into four triangles, the new vertices are projected radially on the sphere, and the process is repeated as many times as necessary, giving discretizations with N& = 20 x Ak (k = 0,1,2,...) triangles;
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A.Z. Zinchenko and R.H. Davis
Fig. 7 shows an example for N& = 1280. The second scheme [42] differs only in that we start from a regular dodecaedron, project each pentagon face centre radially on the circumscribed sphere, connect the projection with the face vertices, and then proceed as in the first method; this scheme produces triangulations with N& — 60 x 4k [k = 0,1,2,...). Subdivision of each triangle face into m2 smaller triangles [43] may also be practiced; when the latter procedure for small m < 5 is combined with the first two schemes, it gives additional possibilities NA = 720, 1500, 2160, 2880, 6000, 6480, etc., but still with highly uniform unstructured triangulations (the maximum-to-minimum mesh edge ratio being within 1.19-1.22).
Fig. 7. An example of surface triangulation into 1280 elements starting from a regular icosaedron.
When drops move (and deform), one can advect the mesh nodes either with the interfacial fluid velocity u or with the normal velocity (u • n)n (both found from the solution of Eq. (34)), to update the drop shapes. A familiar difficulty with both strategies, making them unsuitable, is that the internode distances become highly irregular, invalidating the mesh after a short simulation time. One remedy [43] is to add an artificial tangential velocity to the node motion (which has no effect on shape evolution, for sufficiently small time steps); this additional velocity is constructed by some local rules to prevent mesh degradation for a long simulation time, based on the idea of internode 'springs'. Unfortunately, in the original form, this idea of 'grid tension' leads to numerical stiffness, with tight stability limitations on the time step. A more advanced, adaptive restructuring method [48] combines a dynamical spring-like mesh relaxation (performed iteratively) with topological mesh transformations (node addition/subtraction/reconnection). The latter approach is very flexible and was used, in particular, in drop pinch-off simulations [48]. With isotropic mesh restructuring into compact elements, however, it is not easy to reach convergence (i.e., independence of the global results from triangulation) for elongated shapes, because adequate azimuthal resolution requires a
Hydrodynamical Interaction of Deformable Drops
409
very large total number of boundary elements as a drop stretches. A very different approach to mesh control called 'passive mesh stabilization' [42, 26, 23-25] uses fixed topology triangulations (i.e., with a fixed number of nodes and fixed connections) and seeks to prevent mesh degradation by constructing an additional global tangential field on each surface Sa separately from the solution of a variational problem. These additional node velocities are required to minimize, in some sense, the 'kinetic energy' of disordered mesh motion (as opposed to 'potential energy' in the simplest grid tension method), thus avoiding excessive numerical stiffness. Suitable forms of the kinetic energy function were found both for compact [42] and highly stretched [26] shapes. This method provides less control over the mesh than does adaptive restructuring [48]. However, it was found to be surprisingly flexible in a variety of problems, including dimpled shapes and highly stretched drops closely approaching breakup [26]; most importantly, it was possible to systematically demonstrate, with passive mesh stabilization, the independence of the results (global shapes, dynamics of thin neck thinning, etc.) from the degree of triangulation even for a modest number of boundary elements [26]. The results for finite deformations discussed below are all obtained using passive mesh stabilization. Local surface fitting by a paraboloid has become a popular method to calculate the curvatures k(x) and normals n(x) in the nodes of an unstructured method. The first parabolic fitting algorithm is, probably, due to Rallison [49]; a much simpler, more general and efficient version was subsequently developed [42]. Numerical discretization of the singular integrals (34) and (37) by a trapezoidal rule [49] (see [42] for details) is preceded by singularity and 'near-singularity' subtractions [43] (with u(x*) and f(x*) subtracted from u(x) and /(as), where a;* is the mesh node on Sp nearest to y); the latter greatly improves the quality of the numerical solution. For gas bubbles or low-viscosity drops (A • 0, and requires special desingularization techniques [26]. Using the curvatureless form is also crucial for successful drop breakup simulations, when a fixed-topology mesh is used (allowing boundary elements to stretch); examples are given in Sec. 5. Besides binary interactions in an unbounded medium (relevant to dilute emulsions), as described by Eq. (34), it is of great interest to simulate multidrop interactions with periodic boundaries. Namely, a random system of TV ^> 1 drops with centres in a box V is assumed to be continued triply-periodically into the whole space. The idea of periodic continuation is borrowed from statistical physics (e.g. [51]) and allows us to accurately simulate the effective properties of locally-homogeneous concentrated emulsions away from the boundaries with a limited number of drops N. For emulsion sedimentation (Sec. 6.2), the periodic box V is stationary, and can be taken as a cube (Fig. 8). For emulsion shear flow (Sec. 6.1), the box V is initially a cube, but it is then skewed by the flow, until the whole system repeats itself in a cyclic manner [43] (Fig. 9). The main modification to the boundary-integral equation (34) is in using Hasimoto's [52] periodic Green function. The details are involved, however, and the reader is referred to the original papers [23-25]. 3.2 Multipole Acceleration of Boundary-integral Calculations Direct point-to-point calculation of all boundary integrals (34) and (37) has 0(N2N^) computational cost per time step, which severely limits the number of drops N and/or the number of triangular elements N& per drop possible in dynamical simulations, especially for contrast viscosities (A 7) to the boundary integrals for y, y' and y" are evaluated, respectively, by (i) reexpansion of Lamb's singular series from x° to x°s, (ii) pointwise calculation of Lamb's singular series, and (iii) direct point-to-point summation. Reproduced by permission from Ref. [23].
The periodic Green function is partitioned into the free-space and 'farfield' part. Each block contribution to the free-space part of the boundary integrals is expanded as Lamb's singular series [30] (which may be viewed as an expansion in inverse powers of the distance from the block centre). Interactions between well-separated blocks (e.g., B1 and Bs, with shells P 7 and V$, respectively) are handled by Lamb's series reexpansions from singular to a regular form (i.e., in positive powers of the distance from the centre x°6). If the shells V^ and V1 overlap, but mesh node y' of block B% is 'well outside' shell X>7, Lamb's singular series for block B1 is used directly to calculate this block's contribution to the boundary integrals for node y' (Fig. 10). Only if node y" of block B( is inside shell £>7 (or is outside, but
Hydrodynamical Interaction of Deformable Drops
413
too close to X>7), should we use direct summations to calculate the contribution of block B7 to the boundary integrals for node y" (Fig. 10). The far-field contributions are handled by Taylor expansions about block centres. Thus, the essence of the algorithm is to use efficient multipole/Taylor expansions as much as possible, and employ direct summations only as the last resort. Technical details are quite involved, though, and the interested reader is referred to the original papers [23-25]. For N — O(102 — 103) and NA ~ 103, the computational gain of this algorithm over the standard summation is 2-3 orders of magnitude at each time step, which has made such large long-time simulations feasible [23-25]. Perhaps contrary to expectations, 3D boundary-integral simulations for only two drops in close approach is a very difficult problem, when the capillary number is small and the drops are nearly non-deformed. One reason is a very tight stability limitation on the time step stemming from the Courant condition. Another difficulty is localization of stress, requiring very high resolution in the narrow gap between two drops. Less obviously, the outer region (away from the gap) also needs unlimited resolution as Ca —> 0; the reason is that, in the non-dimensional form of (37), the curvature deviation (of order O(Ca)) from a uniform value is divided by Ca, to produce an 0(1) effect. As a result, for Ca a and < K >a are the average values of |g| and K over Sa. In addition, an appropriate small constant is added to (43) to make zero the total flux through Sa (to conserve mass). On the main, smooth part of the drop or bubble, the modification to the normal velocity u, • rii is very small, of order ea < q >a (or even smaller, if the surface is nearly spherical, as is often the case in bubble cusping). In the cusp region, however, where K may attain high values, (43) gives a stronger negative correction to the normal velocity, and the exponential dependence keeps the cusp curvature under control. The principal curvatures k\ and ki for Eq. (44) do not have to be accurate, and so the method remains effectively curvatureless. Repeating the simulation in Fig. 18 with a small cusp-smoothing (ei = 62 = e = 6.67 x 10~3) allows us to proceed much farther and predict the further motion (Fig. 19). Namely, a larger bubble is being sucked into the dimple formed on a smaller one, resulting in bubble capture. This interesting phenomenon of deformation-induced capture was first discovered experimentally by Manga and Stone [27-28] for two bubbles in corn syrup, although adequate computer simulations were not available at that time. Manga and Stone [27-28], however, offered a far-field asymptotic analysis for well-separated drops with small deformations by the 'method
424
A.Z. Zinchenko andR.H. Davis
Fig. 19. The same simulation as in Fig. 18 in the (x, z)-plane, repeated with smoothing (e = 6.67 x 10~3) and leading to bubble capture. Reproduced by permission from Ref. [26].
of reflections', to qualitatively explain the onset of bubble capture. Most strikingly, simulations in Figs. 18-19 show that the critical offset Ax% for bubble capture can be much larger than the geometrical Smoluchowski's value 1 + k, which is in accord with experiments [27-28]. In practice, this capture phenomenon is followed by bubble coalescence when the van der Waals attraction drains the thin film, but the actual coalescence time (which may be quite large [27-28]) depends on the details of near-contact physics. Additional calculations [26] confirm global numerical convergence of the results in Fig. 19 in the limit N& —> oo, e —»• 0, and their independence of the relation between e and N& in this limit, thus making the choice of smoothing (43) purely technical. The cusp-smoothing, in combination with the curvatureless algorithm, is therefore a correct method to predict the global shape evolution both with point and line 'singularities' (limitations are discussed in Ref. [26]). The local structure of cusps,
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however interesting, is irrelevant to predicting global dynamics of cusped shapes. A different mode of bubble capture is shown in Fig. 20. The parameters k = 0.7, A = 10"3, -6 = 7, Axo = 2.3 and Az0 - 10 are the same as in Fig. 9 of Ref. [26], but we used here much finer triangulations (N& = 15360 triangles per bubble, instead of N& = 3840 in Ref. [26]), and, accordingly, a four times smaller smoothing parameter ei = 8.2 x 10~4 fa — 0). Only a point singularity forms on the smaller bubble, while the other bubble remains smooth. Unlike in Fig. 19, the smaller bubble is swept around the larger one and eventually becomes sucked into the dimple formed at the rear of the larger bubble. This later mode of bubble capture is known experimentally [27-28] as 'entrainment.' The present results are close to those in Fig. 9 of Ref. [26] and show just a slightly faster capture; again, a good global accuracy is achieved without fully resolving the cusp. Calculations similar to those in Figs. 19-20 (although, for more limited resolutions) were used systematically [26] to find the critical offsets (Axo)cr for capture by trial-and-error (Fig. 21). The critical offset increases with B due to increased deformation-induced alignment but becomes only a weak function of B at large Bond numbers. Also, the critical offset is very sensitive to the size ratio and is the largest for bubbles of nearly the same size (although the mutual approach is slow in this case). Additionally, a comparison of the dark squares and the crosses in Fig. 21 at a\/a2 = 0.7 shows that the capture efficiency is slightly reduced when the bubbles (A = 10~3, which is representative of A < 0(1O~2)) are replaced with drops having A = 0.1. Further increase in A to 0(1) values leads to interaction-induced breakup, instead of capture. For ai/02 = 0.7, A = 1, B = 5.31, Ax0 = 1, and Azo = 5.09 (Fig. 22), the smaller drop is swept around the larger one, stretches due to hydrodynamical interaction, and starts necking. The reason for stretching is that the hydrodynamical field in the wake of the larger drop is mainly an extensional flow. The simulation in Fig. 22 was repeated for four different numbers of triangles iVAl = 2160, 3840, 6000 and 8640 on the smaller drop [26], to assess accuracy. After t = 6, the drops continue to separate, and the smaller drop experiences neck pinchoff, while the other drop remains compact. In Fig. 23, only the evolution of the breaking drop is shown. Despite some local mesh imperfections inherent in our fixed-mesh topology breakup simulations, the global convergence is quite good (Fig. 24), especially in the top bulbous area; in particular, the radius 03 = 0.920ai of the main (top) fragment after breakup is accurately predicted [26]. Moreover, these simulations accurately describe the temporal
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Fig. 20. Bubble capture through 'entrapment' at ai/a2 = 0.7, A = 10~"3, B = 7, NA = 15360, ei = 8.2 x 10" 4 .
Fig. 21. The non-dimensional critical capture offset far upstream for bubbles and low-viscosity drops. Dark squares are for A = 10~3, crosses are for A = 0.1 (at a.i/a.2 = 0.7 only). Reproduced by permission from Ref. [26].
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Fig. 22. Relative buoyancy-driven motion of two drops with 0,1/0,2 = 0.7, A = 1, B = 5.31; 3840 elements on the smaller and 1280 elements on the larger drop. Reproduced by permission from Ref. [26].
dynamics of the neck thinning. As breakup is approached, the local neck shape becomes axisymmetrical (even though the whole problem is 3D), thus allowing us to introduce an equivalent neck radius r = (<STOm/7r)1/'2, where Smin is the minimum area of cross-sections orthogonal to the line of maximum elongation. This observation supports, incidentally, the body of local axisymmetrical studies on viscous pinchoff (e.g., [59-62]). In particular, for A = 1, a local self-similar axisymmetrical solution [62] predicts a linear dependence r(t) at pinchoff, with the dimensional slope of —0.034cr/^e- For our simulations in Fig. 24, the dynamics of the equivalent neck radius is given in Fig. 25. The small lack of accuracy at the bottom of the drop at large times (Fig. 24) does not seem to appreciably affect the dynamics of the neck thinning, and excellent numerical convergence with respect to triangulations is observed in Fig. 25; this agreement may be due to the universality of self-similar thinning at pinchoff. Our dimensional slope dr/dt at the end of the simulation is about —0.029cr//xe, which is close to the theoretical result [62] —0.034a//ie; a plausible reason for small discrepancy is that the global drop shape at the end of our simulation
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Fig. 23. The stretching and breakup of the smaller drop. The parameters are the same as in Fig. 22, except that 8640 triangular elements are used for the smaller drop.
Fig. 24. Comparison of the absolute positions and shapes of the smaller drop in the (x, z)-plane for the simulation of Fig. 23 using JVAl = 3840 (dashed lines), 6000 (solid lines), and 8640 (dotted lines). Reproduced by permission from Ref. [26].
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Fig. 25. The non-dimensional neck radius vs. time for the simulations in Figs. 23-24 with different triangulations of the smaller drop. The results for N^i = 8640 and 6000 are practically indistinguishable. Reproduced by permission from Ref. [26].
is not quite axisymmetrical, causing a small error in the direction of the minimal cross-section. Another possible reason is that the ultimate slope dr/dt is approached only for very thin necks, which was seen previously in axisymmetrical simulations [62-63]. In any case, using dr/dt between —0.029cr//ie and — 0.034a/fie to extrapolate the neck radius in Fig. 25 to zero yields tight bounds on the non-dimensional breakup time: t^ = 7.95 — 8.02. Breakup simulations similar to those in Figs. 22-24 were also made for contrast viscosities (A ^ 1), and critical horizontal offsets (Axo)^. for breakup were also calculated [26]. An additional mode of 3D two-drop interaction in gravity-induced motion predicted theoretically [26] is 'combined capture and breakup.' Namely, after the smaller drop is swept around the large one, it becomes sucked into the dimple formed on the larger drop, like in the pure capture phenomenon (Fig. 20), but simultaneously undergoes considerable elongation and starts necking. This behaviour strongly suggests that the smaller drop will break without being released from the dimple, as has also been observed in axisymmetric simulations [63]. Another, and very interesting phenomenon, discovered experimentally [64] for moderate A and moderate-to-large B is 'pass-through', in which the leading drop forms a torus and the smaller drop passes through its centre, sometimes in multiple cycles reminiscent of leap-frogging. It can be noted that a gravity-induced breakup simulation for a set of parameters close to ours in Figs. 22-23 was also performed by Cristini
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et al. [65] using isotropic mesh restructuring [48] (with a growing number of boundary elements as a drop stretches). Their mesh algorithm is very versatile, with an excellent local control over the mesh, and even allows the calculations to proceed after primary breakup by using mesh splicing [48]. With isotropic restructuring, however, it is often more difficult to obtain convergent results (in particular, an accurate temporal dynamics of drop elongation in primary breakup), and a very large total number of elements may be needed for this purpose, compared to 'passive' mesh stabilization [26]6. MULTIDROP SIMULATIONS 6.1 Shear Flow of Concentrated Emulsions One of the useful applications [24] of multipole-accelerated, boundaryintegral simulations in a periodic box is to study non-Newtonian rheology of a concentrated emulsion of deformable drops in a steady shear (7x2, 0, 0) (where 7 > 0, without a loss of generality). For simplicity, monodispersity is assumed. A large number N of spherical drops of radius a with centres in a periodic box V (initially, a cube) starts from a random non-overlapping configuration (prepared by a standard Monte-Carlo method, e.g. [66]), and the system is subject to shear, which causes the drops to deform and the periodic cell V to skew (excessive cell skew is simply avoided [43, 24] by restructuring periodic cells V at half-integer strains jt). At every time moment t > 0, the quantity of interest is the space-average stress tensor Eij, which can be expressed in terms of surface integrals (e.g., Ref. [67]) over interfaces only. For N ^> 1, the only essential components are [68] the shear stress £12, the first £11 — £22 and the second £22 — £33 normal stress differences. It is convenient to introduce non-dimensional quantities M* = E12 /(/vy), M = (En - £22)/(/ie|7l) and N2 = (£ 22 - E M ) / ^ ! ) Fig. 26 presents a typical snapshot [24] of the simulations, at drop volume fraction c = 0.5, N = 200 drops in a periodic cell, N& = 1280 triangular elements per drop, and capillary number Ca — fieja/a = 0.1; the whole simulation was continued to strains jt ss 28 with about 3000 time steps. For the number of drops N attainable in present-day dynamical simulations (roughly, N = O(102 — 103) with multipole-accelerated codes [23-25]), a single configuration is not representative, and time averaging of the stress Ey is essential. Fig. 27(a-c) presents the trajectories of the dimensionless effective viscosity //* and normal stress differences JVi, N2 vs. strain 7^ at c = 0.5, A = 1, N = 100 and capillary numbers Ca = 0.025, 0.1 and 0.2. These results were produced by the algorithm
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Fig. 26. A snapshot of an emulsion shear flow simulation at c = 0.5, A = 1, Ca = 0.1, and strain -yt = 4.45. The centres of 200 independent drops have been mapped into the unit cube, an initial periodic cell. Reproduced by permission from Ref. [24].
[24], but we used here much higher resolution N& = 8640 at Ca = 0.025 essential for Ca • 0 is approached, however, averaging becomes a formidable task and requires strains jt of order 30-40, or more, especially for the normal stress differences. Additionally, time-step stability limitations and required surface resolution are much tighter, making, again, Ca -C 1 a far more difficult case. One physical reason why large (but still subcritical) deformations are easier to study is that such drops tend to orient and stretch along the shear flow, thus greatly reducing drop interactions and statistical fluctuations. Although small systems (TV ~ 10) can be simulated by the much simpler, direct boundary-integral method (without multipole acceleration), it is quite important to use large systems N > 0(1O2) in emulsion-rheology simulations at high drop volume fractions. First [24], the statistical fluctuations for N ~ 10 were found to be much larger, necessitating much larger strain intervals for averaging; however, dynamical calculations for N ~ 10 do not proceed much faster than for N ~ 102 when multipole acceleration is used [24]. Secondly, and most importantly, when statistical errors for N ~ 10 are eliminated by adequate time averaging, the systematic errors can still be quite large, especially for the normal stress differences. For example, at Ca = 0.05, A = 1, and c=0.5, the N = 10 approximation
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Fig. 27. The trajectories of the effective viscosity (a), and first (b) and second (c) normal stress differences for c = 0.5, A = 1, NA = 1280 - 8640 and N = 100, with Ca = 0.025 (thick lines), 0.1 (dashed lines) and 0.2 (thin solid lines). The initial configuration for Ca = 0.1 calculations (shown in a,b only) was taken from a steady state with Ca = 0.2
was shown [24] to overestimate the average of N\ almost 1.5-fold (note that the first normal stress difference in this case is essential, comparable to the shear stress). In contrast, the average results for iV = 100 and 200 showed very good convergence [24]. The time-averaged results for the effective shear viscosity (< fi* >) and normal stress differences (< Ni >, < iV2 >) at A = 1, N = 100 and different concentrations c (from 0.3 to 0.5) are shown in Fig. 28(a-c); at
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c = 0.55, the behaviour is complicated by "phase transition" [24] and will not be discussed here. For moderate capillary numbers {Ca > 0.1), the results are identical to those in Ref. [24], but for Ca = 0.025 and 0.05, we used much higher resolutions (N& = 8640 and 6000, respectively) in the present calculations to improve the accuracy. Several trends in Fig. 28(a-c) are interesting to discuss. The first is a sharp dependence of the emulsion viscosity on Ca at high concentrations (Fig. 28a), so that most of the shear-thinning occurs for drops with only small deformation. This shear-thinning occurs because the drops slide more easily past each other when the shear rate and, hence, capillary number are increased, due to drop deformation.
Fig. 28. Time-averaged (a) effective viscosity and (b,c) normal stress differences at A = 1.
Unlike for the viscosity < / / >, there are no general mechanical principles to predict the signs of < N\ > and < N2 >, but our calculations always yield positive < N\ > and negative < N2 > (which is in accord with
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more limited simulations [43-44J for small systems (N = 12) at 30% concentration, and with single-drop calculations [69]). However, our results for c = 0.3, 0.4 and 0.45 indicate that the first normal stress difference, < N\ >, extrapolates to small but negative values at Ca = 0 (Fig. 28b). The second normal stress difference, < N2 >, although inevitably subject to some statistical errors, is seen to remain negative as Ca —>• 0 (Fig. 28c). Negative signs of both < N\ > and < N2 > at Ca = 0 are also in agreement with the asymptotic rheological calculations [9] for semi-dilute emulsions of spherical drops on the level of pairwise interactions. The physical reason for predicted sign change of < N\ > at small Ca (Fig. 28b) is that, for moderate-to-large Ca, drop deformation has the primary effect on N\ while, at Ca and < A^ > at Ca —> 0 indicate that, in physical units, the normal stress differences En — E22 and E22 — £33 are linear, not quadratic functions of the shear rate |-y|, as 7 —> 0. On the other hand, in the phenomenological theory of simple nonNewtonian liquids (e.g. [68]), assuming certain 'smoothness hypotheses', the constitutive equation for slow flows is shown to be of Rivlin-Ericksen [70] type. This general form is, indeed, confirmed by a small-deformation analysis of a single drop in a linear flow [71-72]; in particular, the normal stress differences are expandable in even powers of the shear rate 7, as 7 -> 0. Our rigorous simulations (Fig. 28) reveal that the Rivlin-Ericksen phenomenological theory becomes inapplicable to concentrated emulsions due to drop interactions. It should be noted, however, that, for Ca to the capillary number in the range Ca -C 1 at high concentrations, so that most of the shear thinning is observed again for nearly non-deformed drops.
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Fig. 29. A snapshot of the emulsion shear flow simulation for c = 0.55, A = 3, Ca = 0.1 and jt = 6.38. The centres of 100 independent drops have been mapped into the unit cube, an initial periodic cell. Reproduced by permission from Ref. [24].
Fig. 30. The trajectories of the effective viscosity for c = 0.55, A = 3, and N = 100. Reproduced by permission from Ref. [24].
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6.2 Sedimentation of Concentrated Emulsions of Deformable Drops In this section, we discuss sedimentation (or, equivalently, creaming under buoyancy) of concentrated emulsions of deformable drops starting from a homogeneous initial state with no deformation. Although this problem did not attract attention in the literature until recently [23, 25], sedimentation of interacting drops with moderate-to-large deformations may have interesting applications in phase separations. Under normal gravity, the Bond number for emulsion drops is typically small, so the sedimentation would occur only with small drop deformation. If such an emulsion, however, is placed in a centrifuge (thereby increasing the acceleration due to gravity by an order of magnitude, or more), the Bond number may become 0(1), causing strong deformation (and drastically changing the sedimentation regime!). Another example is emulsion sedimentation in the miscibility gap, with low surface tension. The behaviour of interacting deformable drops settling under gravity is strikingly different from that for freely suspended drops in shear flow. Figure 31 presents snapshots of the simulation at Bond number B = Apga2/a = 1.75, drop volume fraction c = 0.35, viscosity ratio A = 1, N = 600 drops in a periodic cell and N& = 960 triangular elements per drop, for different non-dimensional times t (scaled with fj,e/(Apga)). The initial 'well-mixed' state of non-overlapping spherical drops of radius a (not shown) was prepared by the standard Monte-Carlo method. The main quantity of interest is the sedimentation rate, i.e., the volume-averaged fluid velocity over the drop phase, in the reference frame where the emulsion, on average, is at rest (only the vertical component, U, is essential). The non-dimensional rate U/Uo (where Uo is the settling velocity of an isolated spherical drop) for this simulation is presented in Fig. 32a by the thick line. An apparent non-existence of the steady state is due to drop clustering, and is akin to the instability of a dilute suspension of sedimenting solid rod-like particles predicted analytically by Koch and Shaqfeh [29] by far-field analysis of the pair distribution function (for more recent studies on rod-like particles, see Ref. [73-74]). Indeed, most drops in our simulation (Fig. 31) acquire prolate shapes and become partly oriented along the vertical, and there is some qualitative analogy between the two systems. The main difficulty of the emulsion sedimentation study lies in a strong dependence of the instability growth on the initial random configuration; thin lines in Fig. 32a are for nine other random initial conditions showing considerable dispersion, as time proceeds. Probably, to make one realization representative in a wide time range, systems with N > O(105)
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Fig. 31. Snapshots of the emulsion sedimentation simulation from a homogeneous initial state of spherical drops for B = 1.75, c = 0.35, A = 1 and TV = 600. The drops sediment downward. Reproduced by permission from Ref. [25].
Fig. 32. The dependence of the sedimentation rate on random initial configurations at B = 1.75, c = 0.35, A = 1 and NA = 960: (a) 10 realizations with N = 600; (b) 10 realizations with N = 300. Reproduced with permission from Ref. [25].
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would need to be considered (since statistical fluctuations decay slowly, ~ N~ll2), which is far beyond the present-day capabilities; ensemble averaging over many realizations must be performed instead at each time moment, with assumptions made about the probability density of the initial states. In what follows, we assume [25] that all initial configurations of non-overlapping spheres have equal probability; a standard Monte Carlo method for 'hard spheres' (e.g. [66]) is known to generate realizations satisfying this condition. To study the effect of N, similar calculations were done for 25 uncorrelated initial realizations with N = 300; the first 10 are given in Fig. 32b showing even larger dispersion. Remarkably, however, ensemble-averaged results for N = 300 and 600 are in excellent agreement over a wide time range (Fig. 33) and are thus representative of the macroscopic behaviour; in contrast, a smaller system N = 100 greatly underestimates the average sedimentation rate (Fig. 33), except for very small times. The physical reason why very large systems are imperative in this problem is due to clustering; as time grows, so does the cluster size, and more drops are required to make the results box-size independent. In addition to the effect of N, the numerical effect of drop triangulation on the results in Fig. 33 was also studied and found to be small in a wide time range [25]. Calculations were also performed at the same A and B to study the effect of drop volume fraction (from 0.15 to 0.4) on the ensemble-average
Fig. 33. The ensemble-averaged sedimentation rate for B = 1.75, c = 0.35, A = 1 and N& = 960, with 14, 25 and 20 realizations for TV = 600, 300 and 100, respectively. Vertical bars show statistical errors (with 67% confidence level) for N = 100 and 600; statistical errors for N = 300 are similar to those for N = 600. Reproduced by permission from Ref. [25].
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sedimentation rate (Fig. 34); at c = 0.15, quite large systems N — 1200 were used (Fig. 35). At c — 0.4, the instability and clustering grow slowly (Fig. 34). This slower growth is likely due to geometrical constraints at high concentrations hampering drop deformation. The instability growth rate is higher at c = 0.35. Emulsions with 25% drop volume fraction show the strongest instability, with the sedimentation rate increasing 2.15 times from its minimum by t = 150 (by this time, a single drop would have fallen a distance equivalent to 40 radii). At c = 0.15, the initial sedimentation rate is, of course, higher than at c = 0.25 (due to less backfiow and hindrance at lower concentration), but it grows somewhat more slowly as time proceeds. The average drop length and deformation at c = 0.15 are also somewhat smaller than at c = 0.25, thus helping to explain why the emulsion instability at c = 0.15 is not as strong as at c = 0.25. The existence of the optimum volume fraction for emulsion instability and clustering (about 0.15—0.25 for A = 1 and B = 1.75) simply follows from the fact that, as c —>• 0, there are no interactions, and, hence, no deformation of sedimenting drops - a primary cause of clustering and instability. It remains an open question, however, if the graphs of the average sedimentation rate for different drop volume fractions can intersect (so that a more concentrated system would start sedimenting faster than a less concentrated one). We
Fig. 34. The ensemble-averaged sedimentation rate for B = 1.75, A = 1, N& = 960 and different drop volume fractions. Twelve realizations with TV = 400, 14 realizations with N = 600, 14 realizations with N = 800 and 10 realizations with N = 1200 were used for c = 0.4, 0.35, 0.25, and 0.15, respectively. Reproduced by permission from Ref. [25].
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Fig. 35. A typical snapshot of a clustered configuration with B = 1.75, c = 0.15, A = 1, N = 1200 and N& = 960. Drops sediment downward, t = 140. Reproduced by permission from Ref. [25].
cannot rule out that further evolution would lead to massive, statistically significant drop break-up changing the trends in Fig. 34. Surprisingly, however, in none of the calculations for Fig. 34 have we seen individual drop break-up; the time-scales where drop break-up might have a significant effect on the sedimentation rate are much larger than those in Fig. 34. The phenomenon of instability of sedimenting emulsions is strongly sensitive to the Bond number: When B is decreased, the instability growth slows down dramatically, due to less deformation and clustering, which is demonstrated in Fig. 36 for two random realizations. It is expected, however, in the spirit of the Koch-Shaqfeh [29] theory for a dilute suspension of rod-like particles, that there is no critical Bond number for the instability. Similar calculations for A ^ 1 are much more expensive and could be done only for individual realizations (e.g., Fig. 37), without ensemble averaging. Even under this limitation, it was possible to study the effect of A on the sedimentation rate starting from the same random initial configuration (Fig. 38). The most interesting observation from Fig. 38 is that the instability growth accelerates dramatically, when A is decreased from 1 to 0.25, and the results for A = 0.1 show the same trend; in the latter case, however, we could not proceed to large times due to numerical difficulties.
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Fig. 36. The effect of the Bond number on the instability growth for two random realizations (1 and 2) with c = 0.35, A = 1, N = 300 and JVA = 960. Solid lines, B = 1.75; dashed lines, B = 0.9. Reproduced by permission from Ref. [25].
Fig. 37. Snapshots of the emulsion sedimentation simulation from a homogeneous initial state of spherical drops for B = 2.5, c = 0.4, A = 0.25, N = 125 and 7VA = 1500. Drops sediment downward. Reproduced by permission from Ref. [25].
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Fig. 38. The effect of the viscosity ratio A on the instability growth for an individual realization with B = 2.5, c = 0.4, N = 125. Reproduced by permission from Ref. [25].
6.3 Flow of a concentrated emulsion past a deformable drop or bubble Another problem [25] which can be studied by large-scale numerical simulations is the steady gravity-induced motion of a deformable bubble (or drop still called a 'bubble' in what follows) through a concentrated emulsion at low Reynolds numbers. When the bubble is much larger than the emulsion drops, the emulsion can be treated, in principle, as an effective medium with the macroscopic boundary conditions (no flux, continuity of velocity and stress) on the bubble surface. For a Newtonian form of the effective stress tensor and a spherical bubble, the classical solution of Hadamard and Rybchinski [75] shows that no deformation will occur, irrespective of the nonzero surface tension value; surface tension, however, is required for the stability of the drop. Unfortunately, a Newtonian form of the effective stress tensor is not an accurate approximation for concentrated emulsions of deformable drops (Sec. 6.1), and a more complicated constitutive equation valid for arbitrary kinematics would be required, which is problematic and could be done, at present, only in an ad hoc manner. For a bubble comparable in size with the emulsion drops, the problem can be studied instead by rigorous, first-principle numerical simulations, without a constitutive equation. A large bubble of non-deformed radius a& and surface tension G\, is placed in a cubic box, together with N 2> 1 drops of non-deformed radius a^ forming a random emulsion. The whole system is continued triply periodically into the whole space. The emulsion drops are deformable (with surface tension aa) and neutrally buoyant, so that the
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motion in the system is entirely due to density difference pb — pe between the bubble and the surrounding medium. The bubble and the emulsion drops are assumed to start with spherical shapes. The main quantity of interest is the settling bubble velocity (in the reference frame where the whole system is at rest) as a function of emulsion concentration c, bubble Bond number Bb = \pb- Pe\alg/ab
,
(46)
bubble-to-emulsion-drop size ratio ab/ad and surface tension ratio o^/orf, bubble-to-medium-viscosity ratio A& = w,//i e , and emulsion-drop viscosity ratio Xd = p-d/p-e- The limit N —> oo must be taken to approach the solution for a single bubble in an unbounded medium. The solid line in Fig. 39a shows the non-dimensional settling bubble velocity Ub/U0 (where
Fig. 39. The non-dimensional settling velocity of a bubble through a concentrated emulsion at c = 0.45, ab/ad = 2, \b = 0.05, ab/ad = 2, and \d = 1, with (a) N = 800, (b) N = 200. The bubble and each emulsion drop are discretized by 2160 and 1280 elements, respectively. The horizontal lines are the stationary levels for Bb = 4. Reproduced by permission from Ref. [25].
Uo is the isolated bubble velocity, in the absence of emulsion drops) in a 45% concentrated emulsion for N = 800 and Bb — 4; time is scaled with Me/(|/°6 — Pe\ga>b)- The initial decline in Ub/U0 is due to emulsion-drop interactions increasing the effective viscosity around the bubble; a statistical steady state is reached at t ~ 50. Despite a relatively large Bb = 4, the bubble deformation was found to be small (although an emulsion at c = 0.45 and Ad = 1 is noticeably non-Newtonian, see Sec. 6.1). In contrast, the emulsion drops experience large deformations in the vicinity of the bubble, with typical oblate and prolate shapes upstream and downstream, respectively (Fig. 40); the other drops (away from the bubble) are only slightly
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Fig. 40. A typical snapshot of emulsion drops in the vicinity of the bubble, from the simulation in Fig. 39 with N = 800 and -B& = 4 at t = 31. The bubble rises upward. Reproduced by permission from Ref. [25].
deformed, but they create a necessary background for the simulation. A similar simulation with N = 200 (solid curve in Fig. 39b) shows strong sensitivity of the initial velocity Ub/Uo (at t = 0) to N (0.68 for iV = 200 vs. 0.80 for N = 800; the limit must be 1 at N —>• oo). This dependence on N merely reflects strong artifact interactions between single bubbles in different periodic cells at t = 0, since strictly spherical emulsion drops with \d — 1 have no effect at all. Fortunately, at finite 5&, the necessary steady-state results for Ub/U0 are less sensitive to N (0.578 at N = 200 vs. 0.619 at N — 800), and thus the limit N —>• oo can be practically achieved in this problem. Similar calculations at Bb = 2 (dashed lines in Fig. 39a,b) give slightly worse convergence of the steady-state results (0.566 for N = 800 vs. 0.521 for N = 200). It appears that the screening effect of emulsion-drop deformation weakens as Bb is decreased, and convergent steady-state simulations for smaller Bond number may require N > O(103); the same is obviously true for larger size ratios ab/ad- Nevertheless, our simulations demonstrate the possibility of solving the problem without an ad hoc constitutive equation for an emulsion.
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7. SOME PROSPECTS FOR FUTURE RESEARCH In this chapter, we have demonstrated some contemporary progress made in the area of deformable drop interactions and coalescence, and concentrated emulsion flows by rigorous theories and first-principle numerical simulations. Still, there remains a large number of issues to be resolved, and a large room for further development. The hydrodynamical theory of coalescence (Sec. 2), based on matching the thin-film 'inner' solution with the 'outer' solution was shown to be a correct approach for arbitrary 3D interaction of slightly deformable drops. Applications to coalescence efficiency calculations, however, have been made so far [20, 21] only for classic, unretarded van der Waals attractions. To make this theory more realistic, additional account for electromagnetic retardation and other colloidal forces may be necessary. Besides, real emulsions are rarely uncontaminated, and it would be practically important to include surfactant effects on interaction and coalescence, which opens a wide area of research; some initial studies have already been made [76]. For multidrop interactions, it appears possible to further improve multipole-accelerated boundary-integral algorithms and make them suitable for accurate manythousand drop simulations with large deformations, with more powerful computer resources available in the near future. For solid particle simulations, the lattice-Boltzmann (LB) method was found to be a very promising tool [77-79]. It would be useful to explore if LB can become a competitive method for deformable drops, in terms of accuracy and speed. When fluid inertia cannot be neglected (which is often the case for large drops), the problems become much more involved; a front-tracking method [80] appears to be a suitable tool for deformable drops, although much remains to be done to improve the accuracy of this method. In the area of computational rheology of concentrated emulsions (Sec. 6.1), generalization for time-dependent shear flows would be straightforward; a major challenge, however, is to develop a simulation method suitable for an arbitrary history of macroscopic deformation (different from shear flow).
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Emulsions: Structure Stability and Interactions D.N. Petsev (editor) © 2004 Elsevier Ltd. All rights reserved.
Chapter 11
The role of inertial effects and conical flows in breakup of liquid threads Vakhtang Putkaradze Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131-1141, USA email: [email protected]
We study the appearance and relevance of conical flows in the problem of droplet breakup. First, we give the theory of break-up of a slender jet of fluid in air [1]. We then proceed to the two-fluid break-up when a thread of one fluid is rupturing up while being surrounded by another fluid. The two-fluid break-up shows the appearance of conical shapes [2]. Inspired by these experiments, we develop a theory of exact solutions of Navier-Stokes equations which describe the flow of two fluids fluid separated by an interface in the shape of a single or double cone. We also describe the extension of these solutions to two dimensions. 1. INTRODUCTION When observing a dripping faucet in the kitchen, one can notice that the process of forming an individual droplet can be separated into two parts. The first part is the slow accumulation of water in the end of the faucet so that the gravitational force pulling the droplet down can overcome the surface tension holding the droplet inside the faucet. Once enough mass is accumulated, the droplet falls from the faucet with ever-increasing speed. The "slow" phase finishes when substantial volume of the droplet appears from the faucet; from that time on, the surface tension's role becomes destructive, snapping the liquid thread extending from the faucet to the droplet.
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The process of snapping the liquid thread and forming the droplet is very fast, so inertial effects are important for understanding this process. Formation of droplets and the role of inertial effects when air is the surrounding substance is understood quite well. Less well understood is the role of inertial effects when a thread of one fluid snaps while being surrounded by another fluid. The major difficulty of describing the fluid motion at the very moment of droplet formation lies in the fact that at this point in time and space Navier-Stokes equations exhibit a singularity. This singularity is physical, as at the moment when the droplet is born, the thread becomes infinitely thin in finite time. An amazingly beautiful and complex dynamic during droplet break-up has been revealed in great detail in [3]. We shall start by giving the reader an introduction into the theory of droplet formation in air, where a rigorous theory balancing surface tension, viscosity and inertia can be derived. 2. 2.1.
DROPLET FORMATION IN AIR Derivation of equations of motion
In this section, we follow the derivation of equations for fluid break-up in [1] without getting into too many technical details. The analysis of this problem has been mainly completed, and we will concentrate on giving the reader enough information to see the difference between this case and the thread break-up in case the ambient fluid is present. The reader is advised to refer to the original article in order to fill in the gaps in the derivation. Let us start with the derivation of the mass conservation at the point of breakup. We assume that the fluid is incompressible and the motion has radial symmetry with the axis of symmetry being z. By assumption, the velocity has only two components: vr, at the direction of the polar radius r (perpendicular to the thread's axis) and vz at the direction of z (along the thread's axis). To make further progress, let us identify the small parameter e inherent for this problem as the ratio of the typical scales in r and z direction. Thus, we re-scale r as r —> er, whereas the z coordinate remains unchanged: z —> z. Let us therefore assume the following expansion for the z component of the velocity: vz = v0(z,t)
+ e2r2v2{z,t)
+ ...
(1)
The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads
451
The incompressibility requires that WV
Id ~ rdr ^Vr'
+
dvz ~dz ~
'
^
which in turn yields an expansion for the r component of the velocity: Id
loot?
Suppose the equation for the thread is F(r, z, t) = h(z, t) — r = 0. The kinematic condition on the free surface requires that molecules which are on the surface remain on the surface, or DF/Dt = 0, where D/Dt is a full (material) derivative. This leads to
dh dt
dh dz
^7 + vz-
vr \z=h = 0
(4)
Integrating (2) with respect to rdr from r — 0 to r = h(z, t) and using (4) gives the equation of mass conservation -7T- + w (voh2) = 0.
(5)
Equation (5) is easy to understand, as the area of the radially symmetric thread at point z is given by S(z,t) = nh(z,t)2, so (5) describes simply the advection of volume of a radially-symmetric fluid element. The velocity of fluid VQ is still unknown. To complete the system, we must consider the z-component of Navier-Stokes equation, which connects VQ and V2- The full Navier-Stokes for three-dimensional velocity v in vector form reads: dv 1 — + (v • V)v = — V p + i/Av,
(6)
with p being pressure and v being kinematic viscosity. Let us consider the z-component of this equation and do appropriate rescaling of spatial coordinates (r —> er , z —> z), and velocities (1,3). Keeping only the terms of the lowest order in e we obtain dvo_ dt
dvo__ _ldp dz pdz
(. \
dhjo\ dz2 J
where p is pressure and v is kinematic viscosity. To connect pressure with the velocity components, we utilize the boundary conditions at the free
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surface. The normal dynamic boundary condition (vanishing of the stress normal to the surface) gives p = 7/c, where 7 is the surface tension coefficient and K is local curvature of the surface. Finally, the tangential dynamic boundary condition (vanishing of the tangential stress) yields the connection between VQ and v(r]), ip{rj) satisfy
I»
+
#') + W = |
+
3 < ^
i ( * 2 + ^(*2)') + ( « 2 ) ' = o
(13) (I")
To specify the boundary conditions on the solutions of (13,14), let us notice that far away from the singularity in non-rescaled variables, both height and velocity should be finite. Non-rescaled variables z',t', which are close to singularity, correspond to very large values of 77. This is referred to as matching of inner solution (12) and outer solution, describing the evolution of fluid far from the singularity. The inner and outer solution must match,
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Vakhtang Putkaradze
in particular, both the height and velocity of the inner solution should remain finite. Technically speaking, it is an assumption that both height and velocity should tend to a constant value (as this constant value can be zero). However, this assumption is well-justified by extensive comparison between the theory, numerical simulation and experiment. Thus, we posit (v) -> A±V2,
77->±oo,
n
rx
The boundary conditions (47) close the system and allow one to find a solution of (13,14). Since the conditions on the solution are asymptotical, the solution is not unique and additional solutions satisfying (13,14) and boundary conditions (15) were found [8]. The universal law of scaling was later studied in more details in [9]. Notice that the inner solution satisfying (15) grows quadratically away from the singularity. Let us now investigate how the inner solution described by the self-similar ansatz merges with the outer solution. Close to the singularity, when the length scale of the self-similar solution becomes small, we can see that when 77 —> ±00, | ^ | ^ . /
W
±
(
z
_
2
o
)
,
(16)
so the outer solution must be quadratic close to the singularity. More precisely, Viter-^-^o)2-
(17)
Thus, the outer solution for the neck's width h(z, t) consists of two parabolas with (very) different curvatures, joined smoothly at the singularity. It is interesting that in this particular case the microscopic processes driving the inner solution specify the asymptotics of the outer solution (17), whereas in most standard cases, it is the large-scale processes at the outer solution which are driving the asymptotic of the inner solution. The process we have described is universal and is observed in any fluid rupturing in air, provided that the process of rupture does not develop an instability. However, for very small viscosities, this universal self-similar process of rupture can be hard to observe, as the viscous length scale lv can become extremely small. On the scales much larger that lu, the outer solution forms conical shapes. To describe the flow of fluid in conical geometry, we shall later introduce a self-similar ansatz separating radial and
The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads
455
angular variables. While that description lacks the power of Cosserat equations describing the dynamics of rupture, it will introduce the contribution of nonlinear terms in a consistent way. However, before we do that, we will describe another set of experiments showing rupture of a thread of one fluid inside another fluid, when the appearance of conical interfaces is more pronounced.
3. 3.1.
PHYSICAL EXPERIMENTS SHOWING CONICAL INTERFACES Lava lamps: break-up of a thread of one fluid in another fluid
We see that in the case of a liquid bridge rupturing in air close to the singularity, the slender-body approximation works excellently. We shall now turn our attention to the case when a liquid bridge of one fluid is rupturing while being surrounded by another fluid, which is immiscible with the first. A very beautiful demonstration of this process, familiar to everybody, is the lava lamp. In the experiment (or the lava lamp), a lighter fluid is slowly rising through heavier highly viscous fluid. At some point, the rising bubble of lighter fluid separates so far away from the bulk of it that a neck is formed, which is then ruptured by the surface tension. Watching a lava lamp, we notice that the process of forming a neck is extremely slow, yet the process of rupturing is extremely fast, even when the viscosity of surrounding fluid is high. This makes recording the exact moment of break-up challenging; but these difficulties were successfully resolved in [2]. The photographs taken at the very moment of break-up show the formation of two cones with a common tip and axis of symmetry. Let us suppose that fluid i, i = 1,2 has kinematic viscosity ?].;. The angle of the cones depends on the ratio of viscosities A = v\jvi) but is practically independent of other parameters of the experiment. However, a recent study [10] showed that the break-down of a droplet is not self-similar for certain fluids (water and silicone oil) and remembers the initial as well as boundary conditions. Careful experiments and numerical simulations showed formation of exceptionally thin and slender threads of fluid (with the diameter of the order of several nano-meters with length up to a few millimeters) bridging the gap between the two conical sections drifting apart.
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Vakhtang Putkaradze
The theory of the phenomenon of two-cone formation was developed by [2] and [11] (see also [12] for the numerical analysis of the process of break-up of a thin liquid thread). The theory and simulations are based on assuming Stokes' flow (low Reynolds number) on both inner and outer fluid. Based on the integral re-formulation of the motion of the interface, the theory provides a dynamic way of predicting the interface position, along with velocity field in space. The formation of two cones at the instant of the drop pinch-off is confirmed by the numerical simulation. The cones' angles also compare favorably with experimental data. In what follows, we shall develop an exact solution of two fluids separated by an interface in the shape of a cone (or two cones) without assuming that Reynolds number is small. We are motivated by the following argument: if the shape of the interface is exactly conical very close to the origin, there is no typical length associated with it, and, as a consequence, no Reynolds number can be defined. We show that it is possible to find exactly self-similar (in space) stationary solutions respecting the conical geometry and taking into account both linear and nonlinear terms. However, in searching for the stationary solutions of this type we sacrifice the dynamic description of the interface afforded by the Stokes flow. Thus, our solutions should be considered as supplementary to those found in [2] and [11]. In our development of exact conical solution, we shall first start with the case when only a single conical interface was present. While we show that mathematically, no exact solution of this problem satisfying the requirements of regularity at the cone's axis and all boundary conditions at the interface can be found, such flows are also of interest as the interface in a shape of a single cone is observed in the experiment of two-fluid selective withdrawal [13, 14]. In the selective withdrawal experiment, a lighter fluid (oil) is withdrawn through a tube which is positioned at a certain elevation from the interface between the lighter fluid and heavier fluid (water). The interface between two fluids is deformed by the flow. For small values of withdrawal rate, only the lighter fluid enters the tube; whereas for high enough values of withdrawal rate, both fluids are withdrawn. At some critical value of the flux, stationary flows are observed for which the interface assumes the shape of an almost perfect cone. The rounding of the cone tip is much smaller than the capillary length, and the appearance of such singular conical shapes makes the phenomenon interesting for our studies. It has also been demonstrated that it is possible to use the thin slender
The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads
457
threads observed close to the bifurcation in two-fluid withdrawal regime to coat microscopical particles used in medical applications [15]. The theory of this phenomenon in the regime of two-fluid withdrawal based on the assumption of slender thread entering the tube is also in preparation [16]. 4. 4.1.
FORMAL DESCRIPTION OF CONICAL FLOWS: SELFSIMILAR SOLUTIONS General considerations
Motivated by appearance of conical interfaces in the two-fluid flow experiments, we shall derive exact solutions of Navier-Stokes equations describing the flow of two immiscible fluids, with the interface separating the fluids being a cone. Solutions of this type have velocities growing without bound at the conical tip as 1/r, where r is the spherical radius from the tip of the cone. Therefore, stress tensor components diverge as 1/r3 at the tip of the cone. Since the curvature of the cone is growing as 1/r at the tip, forces induced by surface tension diverge as 1/r2, i.e., slower than viscous forces. Thus, this solution persists close to the tip of the cone even if the surface tension is introduced. Note that far way from the tip the conical interface will be deformed by surface tension (and gravity in the case of density difference). If r = 0 were the only singularity of the flow, such two-fluid conical solution would be very interesting, not only as a model of what happens in selective withdrawal close to the tip of the cone, but also as an example of a physically realizable solution with finite energy exhibiting singularity in space. Indeed, if the singularity would be present only at r = 0, one could in principle set up an experimental apparatus with prescribed flow at the boundaries, say r = R leading to the singular solution. Unfortunately, as we show below, our exact solution must exhibit infinite velocity not only at the tip of the cone, but also at the all of the cone's axis. We show that the solution without singularities at the axis can only be achieved with unphysical fluids having negative viscosity or density ratio. The singularity at the axis is typical of the three-dimensional solutions of single fluid of this type. A swirling motion can be successfully incorporated into this ansatz as well; however, we shall not use it here, since the formulas become extremely complex and analytical treatment of two-fluid flow is almost impossible. The situation is similar for the case of two-fluid two-cone flows. We
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Vakhtang Putkaradze
assume that the interface is located at two positions 9 = B\ and 9 = 02, which describes two coaxial conical surfaces. In this case, it is still impossible to achieve a solution with no singularity at the axis 9 = O,vr. Nevertheless, the two-cone flows are still of interest and we shall outline their structure as. The flows described in this section are natural extensions of the singlefluid flows bounded by conical interfaces, which are a class of analytical solutions of Navier-Stokes equations [17] - [21]. One of the important features of the conical flows of this type is that a singularity on the axis of the cone is ubiquitous. Much work has been devoted to the explanation of this singularity. In particular, it was recently proved rigorously [22] that specifying either tangential stress or pressure on the interface makes the regular solution unique. However, it seems to be impossible to satisfy all boundary conditions and still obtain a regular solution. No such result was available for the case of two-fluid flow until now. We will also show that it is possible to derive an exact two-fluid conical solution of Navier-Stokes equation which does not have a singularity on the axis if one of the boundary conditions on the interface is violated. For example, a solution can be achieved if a slip boundary condition on the interface between two fluids is assumed, and we will describe such a solution below for the case of two cones. A solution for a single cone is possible as well, but will not presented here, as it is a simple technical extension of the two-cone case. Instead, we shall choose a more pedagogical approach. First, we discuss the impossibility of a regular solution for the single-cone case satisfying all the regularity requirements. Then, we show that if a slip condition on the interface is assumed, a two-cone solution is possible. We shall just mention that a single-cone regular solution with a slip on the interface is possible (the proof of this fact is relatively easy), but a twofluid conical solution satisfying all the regularity requirements is impossible (which is rather tedious to show) [23]. We shall begin with a derivation of equations, common for both single - and two-cone solutions. 4.2.
Derivation of equations and boundary conditions
We begin by introducing the spherical system of coordinates (r, 9, 0), with the origin being at the cone's tip, and 9 = 0, IT axis aligned with the axis of the cone. Two fluids denoted by a subscript j — 1,2 are separated by
The Role ofInertial Effects and Conical Flows in Breakup of Liquid Threads
459
the interface at 6 — #*. Let us assume that fluid with subscript 1 occupies the area above the interface, i.e., fi < /z*, and fluid 2 is in the area /z > /z*. We assume the velocities of each fluid using the ansatz vr, = VjFjW/r
vej = VjfjW/r
v^ = T^/r,
(18)
where Uj are kinematic viscosities of each fluid and Fj, fj are dimensionless. In this section, we shall put Tj = 0 (no swirling), greatly simplifying the formulas. This ansatz leads to a first order equation of Riccati type, first obtained by Slezkin [24]. We shall repeat the derivation of the Slezkin equation briefly, since the presence of the interface modifies the final result and requires the expressions for stress tensor components in order to match them at the boundary. Incompressibility condition 1 9 , , 1 d . . . x r2Qr
\
>J> rsmQdO
J
leads to a relationship between F.j{9) and fj(6):
which can be simplified introducing the new coordinate /J, = cos 9 and
The incompressibility condition is then simply
If we now define pressure in each fluid as 2
pj(r,n) = ?i£-Pi(li),
(20)
from the Navier-Stokes equations (^-component) we obtain the pressure in the form
Pl
= f-\JLd[i
2 1 - [il
+ Aj,
(21)
where Aj is an integration constant. Substituting (20, 21) in the r-component of Navier-Stokes equations and integrating twice, we obtain the full version of the Slezkin equation for g(fi): (1 - /z 2 )^ + 1
m
+l-g]= -AjU2 + 2BjH + Cr
(22)
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Vakhtang Putkaradze
Two new integration constants Bj,Cj are introduced in addition to Aj from (21). To solve (22) in each fluid, we have to specify g.j at the interface for each fluid, as well as three unknown constants Aj, Bj, Cj, which gives a total of eight unknowns. In addition, the position of the interface /z* is unknown. In order to complete the system, we have to find the corresponding number of boundary conditions. The kinematic boundary conditions are the continuity of the r and 9 components of the velocity. Moreover, since the flow is stationary, vgj must vanish at the interface 9 — 6>*, or ji — /z*. Thus, the kinematic boundary conditions on the 9 components of velocity are
