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, it is clear that the number of interface segments per chain is expected to show a corresponding decrease (see Table 3). The average length of the bridge segments shows a marked decrease with increasing
while they are as short as 2.4 units in system Ms,5oThis behavior is rather obvious, since the length of the bridge segments is expected to depend on the average distance between the surfaces of neighboring filler particles. On the other hand, bridge segments are approximately equal in length (12-13 units) in systems Ms,20 and Mi6,36The same is true for systems Mg^o and Mio,36, where the length is nearly 7 units. Therefore, the length of the bridge segments (a fundamental
122 parameter for the elastic behavior of filled polymers) appears to be a function of the ratio af/(p, a n d decreases with decreasing this ratio. Figures 5 and 6 also show that loop segments are always shorter than bridge segments, and that their average length decreases with increasing (p and increases with increasing crj. In practice, with the exception of extremely crowded systems, the length of the loop segments is roughly half that of the bridge segments. The data listed in Tables 2 and 3 can be utilized to extract a general picture of the molecular arrangements in polymer/nanofiller systems, best described in the form of simplified two-dimensional schemes of the kind shown in Figure 7. The distribution of filler particles in these schemes is arbitrary, but consistent with the results of the simulations, and the chains are ideal average chains, in the sense that number and length of the various kinds of segments and all other features correspond approximately to the average values found in each case. The scheme in Figure 7 corresponds to system Ms,3o- The interface shells of adjacent particles are very close in this system, and partly superimposed in most cases (the overall additive volume of Nf spheres of diameter aj + 4cr centered on the particles is as high as 71% of the total volume). In fact, nearly 15% of all interface units belong to the interface shells of two different particles. The average chain is in contact with 5.2 different particles, and contains as many as 12.2 interface segments. These are connected by 4.4 bridge segments, 2.9 loop segments and 4.9 direct connections. The lengths of interface, bridge and loop segments are 3.8, 7.4 and 4.2 units, respectively. The average chain also contains terminal segments of total length 9 units. Accordingly, Figure 7 shows 5fillerparticles arbitrarily distributed in two dimensions in such a way that their interface shells are very close and partly superimposed. The schematic chain is in contact with all these particles and contains 13 interface segments of length 4 units, 4 bridge segments of length 7 units, 3 loop segments of length 4 units, 5 direct connections between different interface shells and two terminal segments of total length 8 units. Similar schemes can be obviously obtained from the data in Tables 2 and 3 for all systems that have been simulated and, by interpolation, for systems with intermediate compositions.
1.5
Predicting the molecular arrangements
The results obtained from the simulations of dense systems have also enabled to establish a set of simple approximate rules allowing to predict the molecular arrangements in polymer/nanofiller systems, provided that thefillerparticles can be considered nearly spherical and distributed
123
Figure 7. Schematic two-dimensional picture of the mutual arrangement of filler particles and chains in system Ms,30- The dotted circles delimit the interface shells of the various particles; the interface units are shown in white.
at random. These rules, roughly valid for all systems listed in Table 1, with the exception of extreme cases (systems very crowded with small particles or systems containing a large proportion of free chains), are based on a number of considerations that can be briefly summarized as follows. The density of units in a spherical shell between 0.8cr and 2a from the surface of a filler particle is found for all dense systems to be approximately l.l<j~3; therefore, the fraction of interface units is given by h « l.lVsNf/NpLp-(fa + f3 + ...) « lMVs/Vf)ip/(l-)/V; once Lj is known, the number of interface segments per chain can be evaluated as by Ni « f\Lp/Li. Pc is expected to be equal to the number of filler particles having their center in a sphere of diameter aRg + af + 4a, where Rg is the rms radius of gyration of the chains and a is a constant. In fact, for the dense systems studied, Pc is quite well approximated by Pc « rc
where 0ext is the imposed direction direction of alignment. We have chosen eext = 0.5. In the first part of Fig. (7) we show the value of (P2), defined with respect to the external axis (full line) and with respect to an average axis specified by the ensemble of units (dashed line), in the
144
V...-''"'*'"
0.5-
CM
V
0.0 10
-0.5 J
Figure 7. Plot of (P2) as a function of time defined for a system of 200 chains of semi-rigid units of 10 beads with additional orienting potential: the full line represents the order parameter calculated with respect to a fixed axis, while the dashed line represents the order parameter defined with respect to an averaged axis.
presence of an external field parallel to the stable initial director. The two order parameters are relatively close (0.8 vs. 0.7). The central region of Fig. (7) shows the development of the systems under the pull of an external potential, which is instantaneously rotated perpendicular to the initial director: the chains spread, assuming a disordered transient configuration. Finally, in the last part of the simulated switching experiment the system reaches a new equilibrium state, almost completely aligned to the external field. A more pictorial view of this behavior is given in Fig. (8), which shows snapshots of the time evolution of the system at selected times.
5.
Summary
The characterization and analysis of model DPD systems for liquid crystals polymers requires certainly a more thorough investigation than the few examples presented in this communication. Orienting units can be modified by changing the ratio between internal and external spring k/ke, the equilibrium distance between beads, the repulsive factor a and so on. Once a given orienting unit has been defined, it can be used as building block for analyzing different LCP model systems. A potentially
145
Figure 8. Snapshots of a system of 200 chains of semi-rigid units of 10 beads with internal orienting potential, in the presence of a switching external potential (only a fraction of units is shown).
146 relevant field of application is the simulation of rheological observables, which are hardly treated in the framework of standard molecular dynamics studies. Finally, more versatile models can be built by including in the semi-rigid units couples of adjacent beads interacting via an additional orienting potential.
Acknowledgments This work was financed by the Italian Ministry for Universities and Scientific and Technological Research, project PRIN ex-40% and by the Pundago para a Ciencia e a Tecnologia, Portugal, through the research grant SFRH/BD/2982/2000 to A.E. Gomes.
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[9] J.H.R. Clarke, Molecular dynamics of amorphous polymers. In The physics of glassy polymers, Edited by R. J. Young, R. N. Haward, London: Chapman and Hall, pag. 33, 1997. [10] M.J. Cook, M.R. Wilson, Mol. Cryst. Liq. Cryst, 363:181, 2001. [11] P. Ziherl, M. Vilfan, S. Zumer, Phys. Rev. E, 52(l):690, 1995. [12] P. Ziherl, S. Zumer, Phys. Rev. E, 54(2):1592, 1996. [13] A. Polimeno, L. Orian, A.E. Gomes, A.F. Martins, Phys. Rev. E, Part A, 62(2):2288, 2000. [14] M. Lukaschek, A. . Gomes, A. Polimeno, C. Schmidt, G. Kothe, J. Chem. 117(9):455, 2002.
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[15] K. Binder, W. Paul, J. Polym Sci. Part B - Polym. Phys., 35(1):1, 1997. [16] P. Espafiol, Europhys Lett, 39(6):605, 1997. [17] P.J. Hoogerbrugge, J.M.V.A. Koelman, Europhys. Lett, 19(3):155, 1992; J.M.V.A. Koelman, P.J. Hoogerbrugge, Europhys. Lett, 21(3):363, 1993. [18] R.D. Groot, P.B. Warren, J. Chem. Phys., 107(ll):4423, 1997. [19] Y. Kong, C.W. Manke, W.G. Madden, A.G. Schlijper, J. Chem. 107(2):592, 1997.
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SOME THINGS WE CAN LEARN FROM CHEMICALLY REALISTIC POLYMER MELT SIMULATIONS Wolfgang Paul, Stephan Krushev Institut fur Physik, Johannes Gutenberg- University, 55099 Mainz, Germany Wolfgang. [email protected]
Grant D. Smith, Oleg Borodin, Dmitry Bedrov Department of Materials Science and Engineering and Department of Chemical and Fuels Engineering, University of Utah, Salt Lake City, Utah 84112, USA
Abstract
We present in this contribution results from Molecular Dynamics (MD) simulations of a chemically realistic model of 1,4-polybutadiene (PB). The work we will discuss exemplifies the physical questions one can address with these types of simulations. We will specifically compare the results of the computer simulations with nuclear magnetic resonance (NMR) experiments, neutron scattering experiments and dielectric data. These comparisons will show how important it is to understand the torsional dynamics of polymers in the melt to be able to explain the experimental findings. We will then introduce a freely rotating chain (FRC) model where all torsion potentials have been switched off and show the influence of this procedure on the qualitative properties of local dynamics through comparison with the chemically realistic (CRC) model.
Introduction Polymers are complex objects displaying non-trivial structure from the scale of a (typically) carbon-carbon bond (1 A) to the radius of gyration of the coil (Rg « 10-100 A[l]). Here we will be concerned with melts of simple linear polymers where the polymer coils behave as if they were random walks (RG OC N where N is the degree of polymerization of the chains). Connected with the spread in length scales is an even wider spread in time scales: from local bond-length and bond-angle vibrations (10~15 —10~13 s) over conformational transitions between isomeric states 149 P. Pasiniet al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 149-170. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.
150 in the dihedral potential (10~12 -10~ 10 s) to the self-diffusion and overall configurational relaxation of the whole chain which for short chains scales as N2 times the time-scale for conformational transitions and for long chains as N3 times that time-scale. With N ranging from 100 (short chains) up to 10000 and more (long chains) these time scales cover 10~8 s up to 10 s. The local dynamics is naturally strongly dependent on the exact chemical nature and structure of the polymer one studies. The large scale dynamics, however, is largely universal and is described with the Rouse model whereas for longer chains the tube model and reptation concept is believed to describe the chain dynamics [2]. It is easy to see that no single simulation method can capture the physics of polymer dynamics on all these length and time scales [3]. For situations where we can ignore quantum effects (which can, however, be important in polymer crystals [4]) MD simulations with chemically realistic force fields are the method of choice to study local relaxation. Besides the necessity for carefully optimized chemically realistic force fields there is always the question to address whether the simulation is able to equilibrate the model system at a given thermodynamic state point of temperature, density and chain length. In general this means that the simulations are limited to polymer melts well above the glass transition temperature (or density) and to chains not exceeding the entanglement molecular weight, although some newly developed Monte Carlo techniques [5, 6] help to overcome the latter limitation. In the next section we will present quantitative comparisons of MD simulations for 1,4-polybutadiene with data from NMR experiments, neutron scattering and dielectric relaxation. We will also discuss the question of increasing dynamic heterogeneity upon cooling down of the simulated systems as can be observed in the torsional dynamics. In the ensuing section we will then present a comparison of the dynamics in the simulations of the chemically realistic model with that of a freely rotating chain model. This will allow us to address the question of applicability of mode coupling ideas to describe the glass transition in polymer melts.
1.
Quantitative Comparison to Experiment
We will present MD simulations of a chemically realistic united atom model for PB employing a carefully validated quantum chemistry based force-field [7]. We will study in this section a random copolymer of 50% trans 1,4-butadiene, 45% cis 1,4-butadiene and 10% vinyl groups. This polymer was synthesized with an average molecular weight correspond-
151 ing to about 30 repeat units and characterized to possess the above chemical microstructure [8]. We have 40 chains of 30 repeat units in our simulation box and we will be using a united atom model for the CH, CH2 and CH3 groups. The simulations are performed in the NVT ensemble using the Nose-Hoover thermostat [9, 10] after determining the correct density at atmospheric pressure for each temperature. With a total of about 5000 united atoms, these systems can be simulated today over a time range of more than 100 ns (about 2 ns of real time trajectory per day). The ability of the simulation to quantitatively reproduce experimental data on relaxation processes in polymer melts strongly rests on the implementation of the correct force fields for the dihedral angles. These typically possess barriers separating the isomeric states and it is by correlated jumps over these barriers [11-13] that the relaxation processes come about. The frequency of these jumps is exponentially sensitive to the height of the barriers. In order to be able to make a parameter free quantitative comparison with experiment one therefore needs to carefully determine this part of the force field [7], which is done through parameterization against high-level quantum chemistry calculations. The validation of this force field then proceeds through comparison with experiment.
1.1
NMR Experiments
An experimental technique which is very sensitive to the local dihedral barriers is C13 NMR spin lattice relaxation time measurements. For polymers like PB this technique observes the reorientational motion of the CH bonds as quantified through the second Legendre polynomial of the CH bond orientational autocorrelation function. It is sensitive to the local chemical environment giving rise to 12 different resonances which can be identified [14]. A CH bond at the sp2 carbon of the double bond in a cis group next to a trans group relaxes differently from one next to another cis group and differently from the ones in a trans group and differently from the CH bonds at a sp3 carbon. The vinyl group alone gives rise to six different resonances. In a computer simulation one measures the CH vector autocorrelation function and determines the second Legendre polynomial according to
:>CH/
(1)
152 Prom this function one obtains the spectral density by Fourier transformation 1 roo 1 r°
(2)
=5/
The spin-lattice relaxation time finally is given by evaluating the spectral density at the Larmor frequencies of carbon and hydrogen atoms — = K [J(UJU - o/c) + 3J(uc) + 6J(uu + uc)}
(3)
where u;#, uc are Larmor frequencies^ is a constant depending on the hybridization of the nucleus and n is the number of hydrogen atoms attached to the carbon atom under study. To be able to evaluate this function from a stored trajectory of a simulation of a united atom model, we have to reinsert hydrogen atoms into the system. This is done using the information on equilibrium (T = 0) bond lengths and angles which allows the determination of the hydrogen positions from the positions of the carbon backbone atoms (united atoms) alone. In Fig 1 we show a comparison between experiment and simulation for the 12 resolvable resonances for two different temperatures. As one can see, there is a very good agreement (better 273 K 353 K
Q experiment | simulation
1.0
0.1
i
trans-cis trans-tra.ns cis-cis
ris-trira
trtns
cis
v
trtre-QL
(X
vinjl-1 vinyl-^ cis-vinyl
resonance Figure 1. Comparison of the spin lattice relaxation times determined from MD simulations and experiment for two different temperatures. The figure shows data for 12 different resonances (cis-cis for example indicates an sp2 carbon in a cis group next to another cis group, trans an sp3 carbon of a trans group)
than within 20%) between simulation and experiment for all but one
153 resonance. The one that is not reproduced is a rigid body rotation of the side group which does not lead to a conformation relaxation and which we therefore did not try to match more accurately. The temperature dependence of the spin lattice relaxation time for two resonances can be seen in Fig. 2 The spin lattice relaxation time decreases when 3.0 -
H i f f f"
2.5
O cn
• 1.0 0.8 0.6
- 2.0 m
m*
0.4
H • ffi 4$
cis-cis (si lirulation) trans (si mul ation) cis-ci s (experi ment) trans (experiment)
m M
#
• • i
rv i
2.8
_
3.0
3
3.2 10^
3.4 10"
3.6 10^
3.8
1/T Figure 2. Comparison of the spin lattice relaxation times determined from MD simulations and experiment for the cis-cis (sp2) and the trans (sp3) resonance as a function of inverse temperature. Also shown are results for the nuclear Overhauser enhancement which is a measure of the non-exponentiality of the observed relaxation
the autocorrelation time for the observed relaxation increases. For high temperatures, where the nuclear Overhauser enhancement is still close to three, the spin lattice relaxation time traces the temperature dependence of the autocorrelation time of the torsional transitions and the agreement between simulation and experiment is a quantitative validation of the torsional force field [14]. From comparing to the observed torsional autocorrelation function in the simulation one can learn that this identification of the physical motion seen in the NMR experiment breaks down below about 300 K. In Fig. 3 we compare different measures of the local orientational mobility of the chains. The most basic one is the mean time between torsional transitions which we determined with a time resolution of 1 ps along the simulated trajectory. This time scale shows an Arrhenius temperature dependence. It is different for the three relevant torsion potentials along the chain, the allyl bonds next to a cis or trans double bond and the /3 bond at the connection between two butadiene monomers. All these time scales, however, simply follow
154 •trans ally] -cisallyl
io3
IO4 ,.
10*
10
10°
IO1 2.4 10"* 2.6 L0-* 2.8 10"4 3.0 LO"5 3.2 10* 3.4 10* 3.6 1 0 *
3.8
Figure 3. Temperature dependence of several measures of the local orientational mobility of the chain. The lower set of curves pertaining to the right abscissa shows the mean time between torsional transitions for the three relevant torsional angles along the chain. The upper two sets of curves give the integrated autocorrelation time for the second Legendre polynomial of the CH vector orientation, TCH, and the integrated autocorrelation time for the torsion angle autocorrelation function, Trcm-
an Arrhenius law with their energetic dihedral barrier as activation energy in the observed temperature range, so that there are no packing effects observable on the dihedral dynamics in this range. Contrary to the mean time between torsional transitions, the autocorrelation time for the torsions (thick lines in the Fig. 3, defined as the time integral of the following correlation function /(*)
=
(4)
increases in a Vogel-Pulcher like manner. This shows the growing importance of back-jump correlations for the torsional transitions: a torsional transition which occurs with its Arrhenius rate is immediately reversed with an increasing probability upon reducing the temperature thus leading to no decorrelation of the torsional angle. The final set of curves denoted by their resonance name in the figure legend gives the autocorrelation time of the second Legendre polynomial for the CH vectors, i.e., the time integral of (1). These time scales pick up contributions from different torsions adjacent to the carbon atom under study and lie intermediate to the autocorrelation functions of these torsions.
155
1.2
Neutron Scattering Experiments
The comparison between neutron scattering experiments and MD simulations can be done on several length and time scales. Comparing to incoherent scattering as for example measured in time of flight experiments [15-17] one focuses on small length scales (momentum transfers of about one inverse Angstrom) and time scales below 100 ps and the motion of individual atoms. This yields complementary information to the one obtained from the comparison with NMR experiments. On larger length and time scales it is a challenge to reproduce in the simulations the dynamics of a single chain in the melt as observed in neutron spin echo (NSE) experiments [18]. These experiments measure the configurational relaxation of a polymer chain on varying length scales in form of the intermediate coherent scattering function of the chain (5)
Here the sums run over all atoms of the same chain, so one imagines an experiment where one has a few deuterated chains in a protonated matrix, giving rise to strong coherent scattering between the atoms of one single chain.
3 CO
10
15
20
scaled time (us) Figure 4- Single chain coherent intermediate scattering function for PB at 353 K compared between experiment and simulation. Simulation times are scaled by a factor of 0.8 to account for a difference in center of mass diffusion coefficient.
156 This quantity is easily calculated from the computer simulation and Fig. 4 shows a comparison [8] of the results from the simulation (lines) to the experimental data (symbols) for a momentum transfer range of q = 0.05 A" 1 to q = 0.3 A" 1 . It turned out that there is an overall difference in center of mass diffusivity of about 20% between simulation and experiment similar to earlier experience [19], so for the figure the experimental time points are rescaled by a factor 0.8. This makes the whole set of scattering curves, measuring the configurational relaxation of the polymer chain on different length scales superimpose. Polymer melt relaxation on these length and time scales and for chains below the entanglement molecular weight, as in our case, is typically analyzed within the Rouse model. The equation of motion for this chain of phantom beads
(dWna(t)) (dWna(t)dWm/3(t'))
= 0
(6)
= 6nm5ap8(t - t')2(#) = P"( e ~ x + ^ 2 — 1) where x = qRg, which describes the scattering of a Gaussian coil. The obtained value for the radius of gyration agrees well with the directly measured value. The single chain structure factor shows no temperature dependence in the depicted temperature range and agrees perfectly between the FRC and CRC models. This is a peculiarity of PB since for this polymer all minima in the different dihedral potentials are iso-energetic, which also explains the lack of temperature dependence of the single chain structure factor of the CRC model.
3-
O CRC 273 K FRC 273 K
q [A"1] Figure 12. Melt structure factor for PB as obtained from simulations of the CRC model and the FRC model at 273 K.
The other quantity characterizing the melt structure is the liquid structure factor. This is shown over a wide momentum transfer range in Fig. 12 at 273 K. For the calculation we have used the united atoms as scattering centers of equal scattering strength, calculating in this way the structure of the actual simulated systems. A quantitative comparison to the structure factor of PB would be improved by reinserting the hydrogen atoms into their mechanical equilibrium positions [26] and explicitly using the scattering lengths of the carbon and hydrogen atoms in the system. It is gratifying, that even so the position of the amorphous halo at q = 1.47 A" 1 agrees nicely with the experimental results [27]
165 and also the behavior at higher momentum transfers is comparable (to as large q as there are experimental data available). We have performed both sets of simulations at the equilibrium density of the CRC model and this result shows that under these conditions the liquid structure is the same in both models. The relevance of these findings becomes obvious when we think about the mode-coupling theory of the glass transition. In this theory one assumes that the relevant slow variables for describing the glass transition are density fluctuations and one tries to describe the arrest of the structural relaxation. Starting from the Liouville equation and using the Mori-Zwanzig projection operator formalism one arrives at the following formally exact equation [28]
4>q(t) + n2q(j>q(t) - uj>q(t) + n2q f
dt'mq{t - t')$q(t') = o ,
where cf>q(t) = (6pq{t)6p*(0))
(11)
is the correlator for density fluctuations or intermediate scattering function. ttq is a microscopic frequency scale and mq(t) is a memory kernel containing the essential physics of the problem. In the idealized version of MCT this kernel is again expressed in terms of coupled density fluctuations and the coupling constants are completely determined by the static structure of the melt, and here again mainly by the two-point correlation function of the liquid structure factor. Upon lowering the temperature towards the glass transition or increasing the density this coupling induces a qualitative change in the dynamics. In the supercooled liquid regime a two-step relaxation develops consisting of the final a or structural relaxation and a plateau or MCT-/3 relaxation regime intervening between the microscopic dynamics and the structural relaxation. This /3-regime is the time regime of caging. Upon lowering the temperature, the life-time of the plateau (cage) increases until it is infinite at Tc and all correlation functions only decay onto their plateau value. For the incoherent density correlations this plateau value is the Debye-Waller factor of the glass. For the glass transition in the bead spring model of Bennemann et al. [29, 30] this picture was essentially confirmed. In the preceding paragraphs we have shown that we have two models at hand which show the same static structure on the level of the twobody correlation functions. Do they have the same dynamics? The high temperature behavior of the CRC model (curve at T = 353 K) and the behavior of the FRC model agree. One observes a crossover from short time ballistic and vibrational motion to a subdiffusive Rouse-like regime determined by the connectivity of the chains. For the CRC model at
166
10°
t[ps] Figure 13. Mean square displacements of the sp3 carbons along the chain back-bone as a function of time for several temperatures for the CRC model and for T = 273 K for the FRC model.
273 K, however, one observes a plateau intervening between the short time motion and the Rouse-like regime and this plateau becomes more pronounced upon lowering the temperature to 240 K. It starts at around t = 1 ps and extends almost to 100 ps for 240 K. Within a mode coupling picture treating only the density fluctuations as slow variables this finding is unexpected. Also, the simulation temperatures are still in the high temperature liquid-like regime of PB (the experimental estimate for the temperature Tc where mode-coupling effects should be observable is 220 K). This slowing down is not due to packing effects which mode-coupling theory tries to capture but obviously due to the presence of intramolecular barriers against dihedral rotation. On the time scale of 1 ps the fast vibrational dynamics of the bond angles and torsion angles is damped out and this time scale is not strongly dependent on temperature. The mean time between torsional transitions, however, as we have seen increases in an Arrhenius-like fashion with decreasing temperature. Consequently we are observing a separation of time scales between the vibrational dynamics and the relaxational dynamics governed by the torsional transitions. At 240 K the mean waiting time between torsional transitions has reached about 100 ps and this is exactly the time scale of the break-up of the plateau. For shorter times the
167 mean-squared displacement curves only pick up contributions from the fast-moving torsions in the waiting time distribution and upon lowering the temperature these become fewer and fewer (at a fixed time). In principle the memory kernel in the mode-coupling equation contains contributions from three particle correlations, however, for all systems studied so far in computer simulations, these only slightly modified the predictions of the theory and helped improve agreement between simulation and theory [31]. Also, there has been an extension of the theory taking chain connectivity into account [32] which improved agreement with the simulations of the bead-spring model, but it remains to be seen whether an application of this theory to the two models presented here can account for their strongly different dynamic behavior.
0.0
Dt [A2] Figure 14- Comparison of the single chain coherent intermediate scattering function for the CRC (full lines) and FRC (dashed lines) models at 353 K. The difference in segmental friction is absorbed into a rescaling of the time axes by the chain center of mass diffusion coefficient.
A different conclusion, however, has to be drawn when we look at the dynamics on larger length and time scales [34], i.e., at the regime which is typically described by the Rouse model. These are length scales larger than the statistical segment length of the chain and time scales where connectivity dominates which are several times the typical time scale between torsional transitions. On these time scales the difference in local dynamics observed in Fig. 13 gets absorbed into just one effective rate constant, the segmental friction £. One would therefore conclude
168 that on these time and length scales the dynamics is the same after rescaling time scales for the difference in segmental friction and this is exactly what we find in Fig. 14. Note that here we really have the same chemical polymer and the same local packing for the CRC and FRC models which might explain the difference in behavior compared to the experimental results for polyisobutylene and polydimethylsiloxane [33] which have about the same statistical segment length but different local packing.
3.
Summary
In this contribution we have discussed chemically realistic MD simulations of 1,4-polybutadiene melts to exemplify which physical questions can be addressed by this technique. These simulations are targeted at understanding molecular structure and relaxations on small to intermediate length and time scales. They rely on carefully validated force fields which are able to quantitatively reproduce experimental data on well defined model systems. These force fields become available today thanks to a combination of experiments, high-level quantum chemistry calculations and simulations of simple model systems. Having established the ability of the simulation to reproduce available experimental data without any adjustable parameters one can then proceed to exploit the strong points of the simulation approach. Where the range of thermodynamic parameters coverable in the simulation is often strongly limited, for those thermodynamic state points where a simulation in full equilibrium is possible one gets the complete information on the system under study down to every coordinate and momentum of every particle. This allows for measuring properties and correlations not available to experimental techniques, like for instance the distribution of waiting times between torsional transition, which are then instrumental in understanding and explaining effects of dynamic heterogeneity in polymer melts. Another strong point of the simulation approach is its ability to selectively change parts of the model Hamiltonian. In this way one can compare a chemically realistic model of PB with a freely rotating chain version of the same polymer and does not have to switch to a completely different polymer with some of the same properties like is unavoidable in experiments [33]. With this approach we could establish that identical structure on the two-body correlation function level (single chain and liquid structure factors) does not imply identical dynamics which raises questions on the applicability of the mode-coupling theory of the glass transition to polymer melts.
169
Acknowledgments This is a paper on results from Molecular Dynamics simulations, but obviously much of this work would not have been possible without the close collaboration with the following colleagues from the experiment side: M. D. Ediger, M. Monkenbusch, X.H. Qiu, D. Richter, L. Willner. The authors acknowledge funding from the German Science Foundation under grant PA473/3-l,2, BMBF under grant 03N6015 and the American Chemical Society under grant ACS-PRF-3321AC7.
References [I] K. Binder editor, Monte Carlo and Molecular Dynamics Simulations in Polymer Science, Oxford University Press, 1995. [2] M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, 1988. [3] S.C. Glotzer and W. Paul, Annu. Rev. Mater. Res. 32: 401, 2002. [4] R. Martonak, W. Paul and K. Binder. Phys. Rev. E, 57:2425, 1998. [5] V.G. Mavrantzas, T.D. Boone, E. Zervopoulou and D.N. Theodorou, Macromolecules, 32:5072, 1999. [6] N.C. Karayiannis, V. G. Mavrantzas and D. N. Theodorou. Phys. Rev. Lett., 88:105503, 2002; N.C. Karayiannis, A.E. Giannousaki, V.G. Mavrantzas and D.N. Theodorou, J. Chem. Phys., 117:5465, 2002. [7] G. D. Smith and W. Paul, J. Phys. Chem. A, 102:1200, 1998. [8] G.D. Smith, W. Paul, M. Monkenbusch, L. Willner, D. Richter, X.H. Qiu and M.D. Ediger, Macromolecules, 32:8857, 1999. [9] S. Nose, Progr. Theor. Phys. SuppL 103:1, 1991. [10] W.G. Hoover, Phys Rev A, 31:1695, 1986. [II] R.H. Boyd, R.H. Gee, J. Han and Y. Jin, J. Chem. Phys., 101:788, 1994. [12] G.D. Smith, Do Y. Yoon, W. Zhu, M.D. Ediger, Macromolecules, 27:5563, 1994. [13] W. Paul, G. D. Smith and Do Y. Yoon. Macromolecules, 30:7772, 1997. [14] G.D. Smith, O. Borodin, D. Bedrov, W. Paul, X. Qiu and M.D. Ediger, Macromolecules, 34:5192, 2001. [15] G.D. Smith, W. Paul, D.Y. Yoon, A. Zirkel, J. Hendricks, D. Richter and H. Schober, J. Chem. Phys., 107:4751, 1997. [16] K. Karatasos, F. Saija, J.-P. Ryckaert, Physica B, 301:119, 2001. [17] O. Ahumada, D.N. Theodorou, A. Triolo, V. Arrighi, C. Karatasos and J.-P. Ryckaert, Macromolecules, 35:7110, 2002. [18] G.D. Smith, W. Paul and D. Richter, Chem. Phys., 261:61, 2000. [19] W. Paul, G.D. Smith, Do Y. Yoon, B. Farago, S. Rathgeber, A. Zirkel, L. Willner and D. Richter, Phys. Rev. Lett, 80:2346, 1998. [20] G.D. Smith, W. Paul, M. Monkenbusch and D. Richter, J. Chem. Phys., 114:4285, 2001.
170 [21] H. Frolich, Theory of Dielectrics, Oxford University Press, 1958. [22] G.D. Smith, O. Borodin and W. Paul, J. Chem. Phys., 117:10350, 2002. [23] A. Arbe, D. Richter, J. Colmenero and B. Farago, Phys. Rev. E, 54:3853, 1996. [24] A. Aouadi, M.J. Lebon, C. Dreyfus, B. Strube, W. Steffen, A. Patkowski and M.R. Pick, J. Phys. Condens. Matter, 9: 3803, 1997. [25] S. Krushev and W. Paul, Phys. Rev. E, 67:021806, 2003. [26] W. Paul, Do Y. Yoon and G.D. Smith, J. Chem. Phys., 103:1702, 1995. [27] D. Richter, B. Prick and B. Farago, Phys. Rev. Lett. 61:2465, 1988. [28] W. Gotze and L. Sjogren, In Transport Theory and Statistical Physics, S. Yip and P. Nelson, eds., Marcel Decker, 1995, pp 801. [29] C. Bennemann, J. Baschnagel and W. Paul, Eur. Phys. J. B, 10:323, 1999. [30] M. Aichele and J. Baschnagel, Eur. Phys. J. E, 5:229, 2001; 5:245, 2001. [31] F. Sciortino and W. Kob, Phys. Rev. Lett, 86:648, 2001. [32] S.-H. Chong and M. Fuchs, Phys. Rev. Lett, 88:185702, 2002. [33] A. Arbe, M. Monkenbusch, J. Stellbrink, D. Richter, B. Farago, K. Almdal, and R. Faust, Macromolecules, 34: 1281, 2001. [34] S. Krushev, W. Paul and G. D. Smith, Macromolecules, 35: 4198, 2002.
MONTE CARLO SIMULATIONS OF SEMI-FLEXIBLE POLYMERS Wolfgang Paul, Marcus Muller, Kurt Binder Institut fur Physik, Johannes Gutenberg-University, 55099 Mainz, Germany Wolfgang. [email protected]
Mikhail R. Stukan, Viktor A. Ivanov Physics Department, Moscow State University, Moscow 119992, Russia Abstract
We present Monte Carlo simulations on the phase behavior of semiflexible macromolecules. For a single chain this question is of biophysical interest given the fact that long and stiff DNA chains are typically folded up into very tight compartments. So one can ask the question how the state diagram of a semiflexible chain differs from the coil-globule behavior of aflexiblemacromolecule. Another effect connected with rigidity of the chains is their tendency to aggregate and form nematically ordered structures. As a consequence one has two competing phase transitions: a gas-liquid and an isotropic-nematic transition potentially giving rise to a complicated phase diagram.
Introduction The physics of semi-flexible macromolecules has received increasing attention over the last two decades. Imagine a polymer with local bonded interactions giving rise to a substantial stiffness of the chain, i.e., the persistence length p or Kuhn length (statistical segment length) is much larger than the length of a chemical bond. When one looks at very long chains of this nature L 3> p, where L is the contour length of the chains one recovers normal Gaussian chain behavior and all the scaling properties [1] theoretical physics has been emphasizing for a long time after early work on worm-like chain molecules. In the biophysical area and also for many applications of stiff macromolecules it is important not to study the limit L —» oo but to look at the properties of chains with L ~ p. A most significant problem from biology in this context is the foldability of long and stiff DNA chains 171 P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 171-190. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.
172 into very tight compartments in the cells [2]. This folding is often in the form of tightly wound up toroidal structures. What is the region of stability of these structures in the state diagram of a semi-flexible chain of length L (or equivalently degree of polymerization N)l Note that we will refer to a state diagram as opposed to a phase diagram because no thermodynamic limit is taken here. The existence of equilibrium toroidal structures [3-9] as well as the kinetics of the folding process [10-13] have found much interest in recent years and have been studied in 2 as well as 3 dimensions using off lattice models as well as lattice models. We will look at this state diagram and its dependence on chain length in the next section. When one studies solutions of very stiff chains one always has to deal with aggregation and ordering effects, even at very low concentrations. This has to do with the fact, that these chains have little conformational entropy and very little translational entropy to lose upon aggregation, but much interaction energy to gain by forming ordered aggregates. So the important physics of these solutions is a competition and interaction between a liquid-gas and an isotropic-nematic transition. Both transitions have been studied separately in great detail by computer simulations. The liquid crystalline behavior in solutions of semiflexible polymers has been addressed in many simulation studies over the years [14-22] the interplay between both transitions, however, has received relatively little attention [23, 24]. Which transition will prevail in the thermodynamic phase space spanned by density and temperature for which choices of stiffness of the chains? We will address this question in section 1.2.
1.
State Diagram of a Semi-flexible Chain
In all of the work we will present in the following we will employ the bond-fluctuation lattice model [25-28] a rendering of which is shown in Fig. 1. The repeat units of the polymer chain (monomers) are represented by unit cubes on a three dimensional simple cubic lattice. They are connected by a set of allowed bonds (108 in total) comprising 5 different bond lengths (2, \/5, \/6, 3 and \/l0 ) and allowing for 87 different bond angles between consecutive bonds. The properties of this model, for single chains as well as for solutions or melts, are intermediate between classical lattice random walks and continuous space polymer models like the bead-spring model. In the wide range of relevant length and time scales in polymer physics [29, 30], Monte Carlo simulations of such coarse-grained lattice modelshave proven to be particularly efficient for
173
Figure 1. model.
Rendering of the three-dimensional version of the bond-fluctuation lattice
the examination of the generic phase behavior as well as generic relaxation processes of polymer systems. For the simulation of the state diagram of a semi-flexible chain we will employ the following Hamiltonian H
-1
N
(1)
where di is the angle between two consecutive bonds k and Zj+i, b is the stiffness energy in units of T, N is the chain length and ?7LJ is the nonbonded attractive interaction which we take in the form of a discretized Lennard-Jones potential =
f - e / T [2(r - 2)3 - 3(r - 2)2 + 1] r = 2, >/5, I 0 other (2)
174 We will vary the parameters b (stiffness) and (3 = e/T (inverse temperature in units of the interaction strength). For the simulation we will employ a mixture of local random hopping moves, where one tries to move a randomly chosen monomer into a randomly chosen lattice direction subject to bond and excluded volume constraints and slithering snake moves, where one tries to attach a randomly chosen bond at one end of the chain and cut off the first bond at the opposite end. The first of these types of moves is efficient in equilibrating local structure and the second one in equilibrating large scale structure because it displaces the center of mass of a chain with each successful slithering snake move by a distance 0(1), whereas the local hopping moves only lead to a displacement O(l/N). To characterize the shape of the globules occurring in the simulation upon decreasing the temperature we measured the three principle moments of inertia M^M^M^ (eigenvalues of the gyration tensor) and constructed the following shape parameters from them M3
M
M1 + M3 and K
^W
,. (3)
A sphere is characterized by K\ = l,i^2 = 1, an ideal rod by K\ = 0, K2 = 1 and an ideal disk by K\ — 0.5, K2 = 0.5. We can furthermore characterize a tendency of the chain to wind up, like we would expect for a toroidal structure, by measuring the following quantity
a=
X^ ^ x k
h ISIIS h
(4)
where /$, i = 1 , . . . , iV — 1 denote the bond vectors along a chain. For a random coil, disordered globule or nematic globule with hairpins into different directions this quantity will be small, whereas it will be large for a chain winding around (an object or an empty volume) in a consistent direction.
1.1
Mean Field Scaling Theory
Looking at the coil-globule transition for chains of finite length we have to consider several ways to define the transition point which all agree in the thermodynamic limit but show different finite size behavior. The true transition temperature would be the ^-temperature where the second virial coefficient of the chains vanish. In a simulation one would use a finite size scaling plot to determine this temperature and look for the temperature, where R%/N (neglecting logarithmic corrections) is
175 independent of N (see Fig. 2). One can also define a transition temperature for instance as the temperature of the maximum offluctuationsin the measurements of R2, which would correspond to the temperature of vanishing free energy difference between the coil state and the globule state (see Fig. 3). For the finite size effects on this temperature theory [31, 32] predicts
Ttr/e - I ~ - p 3 / 4 i v 1 / 2 .
Rearranging for the stiffness at the transition as function of temperature (and ignoring constants) we obtain a prediction for the coil-globule transition line in our state diagram: p ~ (8* — #~~1)4/3JV2/3
(5)
To predict the coil-torus and torus-globule transition lines we have to make an ansatz for the excess free energy of the collapsed states with respect to the coil state: ** = -^elastic ~r -^attraction • -^surface •
The elastic bending free energy describes the loss of conformational entropy of a stiff macromolecule inside a cavity of size R ^elastic ^ NTp ( ^ '
where a is the segment size. The polymer volume fraction inside the collapsed state (torus or sphere) can be written as ~ Na3/Rr2 where R = r for a sphere and r < R for a torus. The attraction based part of the free energy is then Na3 ^attraction =
N j
N
where nmax is the number of neighbors in the densely packed collapsed state. In analogy, the surface contribution to the free energy is given by the fact that a monomer on the surface misses some neighbors
_ _
RrNo?
^surface — - e
where n m a x is the number of neighbors on the surface of a densely packed collapsed globule or torus. Assuming now 0 = 1 (which relates R and r) and neglecting the numbers of neighbors as irrelevant prefactors, we obtain the following ansatz for the scaling behavior of the excess free energy
^ - e N
+ e^.
(6)
176
Minimizing with respect to R gives the following scaling relations for big and small radii and the excess free energy T?
R ^ CLN^^TP1^
B~~^^
'
T rsj n N2 ^ T)~^~ ^ & ^
•
— ^"Nf^l^r^-I^R~^L/^>— (7)
Prom t h e condition F = 0 we obtain t h e coil-torus transition line
p ~ f3N2
(8)
and from the condition r = R we obtain the torus-sphere transition line p ~ pN1/s .
1.2
(9)
State Diagram
Let us begin this section on results by providing the determination of the ^-temperature of our model [7] Prom Fig. 2 we can read off a 2.0 G B O
ON = 120 QN = 160 ON = 200
1.5
0.59
v
0.63
0.67
0.71
G—ON = 20 Q
0.5
0.0 0.0
E3N = 4O
O ON = 60 A—AN = 80 <JN = 100
150, respectively, for chain length N = 40. We expect the toroidal structures to be contained in the subensemble with a > 150. All the structures in this sub-ensemble are characterized by a shape parameter K\ around 0.5. Looking now at the density profiles obtained for the structures in this sub-ensemble in Fig. 7 we can identify one set of structures with a hole in the middle (toroids) and another one which is densely packed in the middle. Since these structures also have K\ ~ 0.5 they have to be disks, which can be verified
179
Figure 5. Snapshot figure of a toroidal globule for a chain of length N = 240 at stiffness 6 = 15 and inverse temperature j3 = 1.
by inspection of selected configurations. There is a rich state diagram of these semi-flexible chains and whether there are other possible shapes of the globules occurring at larger stiffnesses and for longer chains is an open question.
2*
Solutions of Semi-flexible Chains
When we now go from the regime of dilute solutions and basically isolated chains to the regime of semidilute and dense solutions of semiflexible chains we can expect two main effects. Due to the stiffness of the chains and at sufficiently large aspect ratio the chains will prefer a nematic alignment and we expect to see an isotropic to nematic transition. In the limit of almost rigid rods this transition happens at very dilute concentrations. When the chains become moreflexiblethe transition density increases. This behavior was studied for a lyotropic system in [22]. The Hamiltonian employed in this simulation of the bond-fluctuation model was given by H{b, $) = J2tb(bb
b0)2 + Y,e# ti
cos
#(! + co c o s #o)
180
2500 -I
(a)
20001500-
500-
0
50
100 150 200 250 300 350 400 450 500
a Figure 6. Histogram of the values for the winding measure a obtained in the simulation for chain length N — 40. One can partition the measurements into subensembles with a < 150 and a > 150, respectively. The line is a fit using two Gaussians.
(a)
Figure 7. Density profiles for the structures found in the sub-ensemble with a > 150 for chain length N = 40. The left panel shows the toroidal structure and the right panel is the profile for disk-like structures.
where the sums run over all bonds and all bond angles in the system, respectively. The parameters were chosen as €& = 1, €# = 0.67, 6o = 0.86 and Co = 0.03. Performing a simulation at constant density, (j) = 0.5, which is a melt density in this model, one observed an isotropic-nematic transition at a temperature of about T = 0.25 for chains of length N =
181 20 (the temperature and chain length together determine the aspect ratio of the molecules and this in turn determines the location of the isotropic-nematic transition). When one now introduces an attraction between the monomers leading to a coil-globule transition for the isolated chain, on the one hand the tendency for nematic alignment is increased and on the other hand a liquid-gas critical point is introduced into the model. For flexible chains a Hamiltonian of the form H = -en where n is the number of contacts, i.e., the number of monomer pairs within the neighbor shell 2 < Ar < \/6, introduces a liquid-gas critical point at Tc/e = 1.79 [34]. We combine stiffness and attraction into the model Hamiltonian H = -en + Y, t'biP - bo)2 + J2 e'# c o s tf(! + co cos
tf0)
(10)
where e'b = e6/0.25 = 4 and e'o = e^/0.25 = 2.68. When we set T = 1 the intramolecular part keeps the chain stiffness constant at the value it had at the isotropic-nematic transition for e = 0. Increasing e from zero we expect a shift and a widening of the coexistence region between isotropic dilute or semidilute liquid and nematic dense melt. Within mean field theory the possible shapes of the phase diagrams for such a model system have been studied theoretically in reference [35]. Depending on the chain stiffness the liquid-gas critical point can be either observable or buried in the two-phase region of the isotropicnematic transition (see Fig. 8). One can have a coexistence between dilute/isotropic and dense/nematic in the simplest case, but also three phase coexistence regions are possible between isotropic gas, isotropic liquid and nematic liquid or isotropic gas, nematic semidilute liquid and nematic dense liquid. We therefore expect to have to look for phase transitions in two coupled order parameters, density and orientation. To vary the density in the simulation we will employ a grand-canonical simulation technique using the configurational bias scheme for chain insertion and deletion [36]. We performed simulations for chain length N = 20 in a box of linear size L = 90. The linear dimension of the box is therefore 2.5 times larger than the ground state length of the polymer chains (all bond lengths equal to 2 and all bond angles equal to 0). We can therefore fit 2 — 3 nematic domains into the linear dimension of the box (some test runs also used L = 50 to look for finite size effects). In addition to the grand-canonical configurational bias moves we will also perform slithering snake moves of the chains (one suggests a new bond at one
182
— stiff chains, p=20 stiffer chains, p=50 rod-like chains, p=500
3-
». VI
r i\V___
\
1-
0- : i . •' 1 • i • 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
•
1
1
1
1
1
1
1
1
1
Figure 8. Theoretical phase diagram for the solutions of semi-flexible chains of different stiffnesses, parameterized by the persistence length p.
chain end and, if successful, cuts off the first bond at the opposite end) and local random hopping moves [23]. By increasing the chemical potential we will start to fill the box and at some point observe a gas-liquid transition either with or without isotropic-nematic transition accompanying it. The nematic order can be monitored with the help of the Mayer-Saupe tensor x
Qa/? =
iVp(JV-l) j
NP(N - 1) §
2
(ii)
where the sum runs over all bond vectors in the system, Np is the number of polymer chains, N the degree of polymerization and a and (3 are Cartesian coordinates. The largest eigenvalue of this tensor is the orientational order parameter, S. When one limits the sum in this definition to only run over the bonds in one single chain and then averages the largest eigenvalue of the so-defined tensor over all chains, one can define a chain nematic order parameter, 5'chainThe transition from a high-density nematic state to an isotropic or nematically ordered (semi-)dilute state can be observed when we start from a columnar crystal completelyfillingthe lattice and then reduce the chemical potential to decrease the density. A typical curve for the density
183
O-O 0.00 D-D 0.08 AAQ.15 OO0.20 v v 0.30 >>0.40
15 20 (a)
25
30
35 40 |Ll
45
50
55 60
Figure 9. Density as a function of chemical potential for two paths: increasing the chemical potential (filled symbols) and decreasing the chemical potential (open symbols). The legend gives the values of the attractive interaction strength in the simulations.
as a function of chemical potential for both paths is shown in Fig. 9. For e = 0 their seems to be a continuous curve from the dilute to dense regime with a kink around /J, ~ 52. With increasing e, however, a stronger and stronger hysteresis loop starts to develop indicating an increasingly strong first order transition. The first order nature of this jump in the density goes along with a jump in the nematic order parameter shown in Fig. 10. Again the curve seems to be continuous for e = 0 indicative of the weakly first order nature of the isotropic-nematic transition in this case. Increasing the attractive interaction a jump in the order parameter develops which is especially pronounced for stronger attraction and the path of decreasing chemical potential. Let us indicate two points which are worrisome in these last two figures. Firstly we can observe that the density of the nematic phase is around 0.9 which means that we have to work with an almost filled lattice with extremely long relaxation times for all chemical potentials larger than the one for which the jumped occurred. Nevertheless, both for increasing and decreasing chemical potential path the densities of the
184
1.0
jQ
0.8 0.6 0.4 o o o.oo
0.2 0.0
20
D-D 0.08 A-A0.15 O-O0.20 VV 0.30
25
30
35
40
45
50
55
60
Figure 10. Nematic order parameter as a function of chemical potential for two paths: increasing the chemical potential (filled symbols) and decreasing the chemical potential (open symbols). The legend gives the values of the attractive interaction strength for the simulations.
nematic dense phase nicely match up with each other. This is not the case for the order parameter leading us to the second worrisome point. Upon increasing the chemical potential we do not jump to the almost perfectly ordered nematic state (5 < 0.9), to which the columnar phase quickly relaxes, but to a state with a much reduced order parameter of S ~ 0.6. The reason for this reduction in order can be seen in Fig. 11. The model systems form nematic domains separated by domain walls running across the simulation box. These domain walls are extremely stable (there is little enthalpic driving force for their dissolution) and they may also be stabilized by finite size effects of the simulation box, due to a size mismatch between the simulation box and the preferred equilibrium domain size. For a small simulation box (for instance for linear size L — 50) one can avoid the multidomain state, however, at the cost of artificially enhancing the ordering tendency due to this finite size effect which leads to a larger error in the determination of the transition chemical potential. Ideally, the bimodal line in density and nematic order as a functionof temperature and chemical potential would be determined by applying the equal weight rule to the probability distributions of density and ne-
185
Figure 11. Snapshot of a multidomain state of our model system which is reached by increasing the chemical potential (density) from the dilute side.
matic order observed in the simulations. At the first order phase transition one should observe a bimodal structure of (semi-)dilute and dense in the density distribution and isotropic and nematic in the order parameter distribution. This bimodal structure can be observed for the density, however, we were not able to observe it in the order parameter. For the small simulation volume L = 50 we could observe two peaks and also observe a single system tunneling between the two states in the course of the simulation. For the large system L = 90 no bimodal structure could be obtained because the multidomain states fill up the order parameter histogram obtained in the simulation between the isotropic and nematic values. Employing expanded ensemble techniques to allow for a frequent tunneling between dilute and dense states in the density histograms was possible, however, due to the weak coupling between density and nematic order parameter this also did not lead to an improvement in the order parameter distribution. Having only the chemical potential at our disposal as the control variable for the simulation we can directly manipulate density but we were not able to improve the sampling for the order parameter. An alternative way to determine the location of the bimodal in the phase diagram consists of finding the loci of equal pressure between the isotropic and nematic states. For a lattice model pressure determina-
186 tion is more difficult than for a continuum model and the best working method for this task is the repulsive wall method of Dickman et al. [37] which has been successfully applied to the bond fluctuation model before [38]. It turned out, however, that this method contained finite size effects which had not been taken into account before. After quantifying these effects and establishing a method how to avoid them [39] we were able to calculate the pressure in our simulation volume for the isotropic state over a large range of densities to high accuracy. The unsolved problem here is the question what is actually meant by the osmotic pressure of the system and whether one should not rather consider a pressure tensor (the repulsive wall method anyhow measures only the normal component of the pressure tensor in the system and assumes that the bulk of the system is homogeneous and isotropic to relate that measurement to the osmotic pressure.
°'°00.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Figure 12. Phase diagram for the isotropic-nematic transition. The determination of the coexistence densities and error bars is described in the text.
Having excluded the applicability of more precise methods of locating the binodal lines in the phase diagram we resorted to a very simple minded determination. From the simulation results shown in Fig. 10 we can determine the limits of stability coming from the disordered as well as from the ordered side, i.e., we are getting the loci of the spinodal lines. We then assume that the binodal value for the chemical potential
187 lies in the middle between these two spinodal values (which would be true only for a symmetric phase diagram) and with this value determine the coexisting densities using the results shown in Fig. 9. We then use a conservative error estimate by giving the error through the values for the spinodal densities. The resulting phase diagram is shown in Fig. 12. We can identify a type of phase diagram predicted from the theory for intermediate stiffnesses. We do not see a coexistence between two nematic states as was predicted for very large stiffness and also no liquid-gas coexistence a lower densities. The liquid-gas coexistence point seems to be buried within the two-phase region of the isotropic-nematic transition and we have indications [23] that it may become observable when we slightly reduce the intrinsic chain stiffness.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 13. Change of the persistence length of the chain upon going from the isotropic to the nematic state.
We also note that the stiffness of our chains is lower by a factor of 15 compared to the theoretically predicted region. Some numerical discrepancy was expected due to the mean-field nature of the theory. May be, however, the theory also underestimates the stiffening the chains undergo upon ordering. When we measure the persistence length in the simulation defined by the decay of orientational correlation between
188 bonds along the chain
N-l-l
C(l) =
we observe a very strong stiffening effect shown in Fig. 13. Upon approaching the isotropic-nematic coexistence around 0 = 0.5 there is a small increase in chain stiffness and upon transition into the nematic phase the chains practically stretch out completely.
3,
Summary
We have discussed in this contribution some results on the phase behavior of semi-flexible polymers. The results we presented were taken exemplarily from simulations of the bond-fluctuation lattice model. For single chains the introduction of chain stiffness imparts much complexity onto the state diagram of a semi-flexible chain of finite degree of polymerization. It is this additional complexity which, for instance, manifests itself in the existence of a region in the stiffness-temperature plane, where a toroidal globule is the thermodynamically most stable structure which makes these chains interesting model systems for biopolymers like DNA and their packing properties in the cells. With increasing chain length at constant stiffness (i.e., the ratio of contour length over persistence length increases) the region of stability of the toroidal globule becomes smaller and it vanishes in the thermodynamic limit. In that limit, however, there is a region of stiffnesses where a nematically ordered dense globule is stable [40]. When one leaves the single chain limit and goes to semidilute solutions of semi-flexible chains one immediately has to face their tendency for aggregation. Stiff chains lose practically no conformational entropy and very little translational entropy upon aggregation but gain much in interaction energy. For semi-flexible chains this introduces a liquid-gas coexistence with a very low density in the gas phase into the phase diagram. At the same time, the chains stiffen upon aggregation and one observes an isotropic-nematic transition. For the range of stiffnesses we introduced into our model Hamiltonian there was just one simultaneous liquid-gas and isotropic-nematic coexistence. We did neither find two coexisting isotropic phases of different densities nor two coexisting nematic phases of different densities. This situation may, however, change upon reducing the stiffness or changing the length of our chains.
189
Acknowledgments The authors profited from helpful discussions with A.R. Khokhlov, A. Yu Grosberg, V. V. Vasilevskaya, P. G. Khalatur, A. N. Semenov, I. A. Nyrkova and J. Baschnagel. We also acknowledge funding by the Deutsche Forschungsgemeinschaft, grant no. 436 RUS 113/223, INTAS grants 01-607 and YSF 2001/1-174 and the Russian foundation for Fundamental Research, grant 03-03-32773.
References [I] P.G. De Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, New York, 1979. [2] A. Yu. Grosberg, T.T. Nguyen, and B.I. Shklovskii, Rev. Mod. Phys., 74:329, 2002; V.A. Bloomfield, Biopolymers, 44:269, 1998. Chap. 7 at http://biosci.umn.edu/biophys/BTOL/supramol.html. [3] H. Noguchi, S. Saito, S. Kioaki and K. Yoshikawa, Chem. Phys. Lett, 261:527, 1996. [4] K. Yoshikawa, M. Takahashi, V.V. Vasilevskaya and A. R. Khokhlov, Phys. Rev. Lett, 76:3029, 1996. [5] H. Noguchi and K. Yoshikawa, Chem. Phys. Lett, 278:184, 1997. [6] H. Noguchi and K. Yoshikawa, J. Chem. Phys., 109:5070, 1998. [7] V.A. Ivanov, W. Paul and K. Binder, J. Chem. Phys., 109:5659, 1998; V.A. Ivanov, M.R. Stukan, V.V. Vasilevskaya, W. Paul and K. Binder, Macromol. Theory Simul., 9:488, 2000. [8] M.R. Stukan, V.A. Ivanov, A. Yu Grosberg, W. Paul and K. Binder, J. Chem. Phys., 118:3392, 2003. [9] Yu. A. Kuznetsov and E.G. Timoshenko, J. Chem. Phys., I l l : 3744, 1999. [10] H. Noguchi and K. Yoshikawa, J. Chem. Phys., 113:854, 2000. [II] T. Sakue and K. Yoshikawa, J. Chem. Phys., 117:6323, 2002. [12] B. Schnurr, F.C. MacKintosh and D.R.M. Williams, Europhys. Lett, 51:279, 2000. [13] B. Schnurr, F. Gittes and F.C. MacKintosh, Phys. Rev. E, 65: 061904, 2002. [14] T.M. Birshtein, A.A. Sariban and A.M. Skvortsov, Polymer, 23: 1481, 1982. [15] P.G. Khalatur, Y.G. Papulov and S.G. Pletnava, Mol. Cryst Liq. Cryst, 130:195, 1985. [16] A. Baumgartner, J. Chem. Phys., 84:1905, 1986. [17] A. Kolinski, J. Skolnik and R. Yaris, Macromolecules, 19:2560, 1986. [18] M.R. Wilson and M.P. Allen, Mol. Phys., 80:277, 1993. [19] M. Dijkstra and D. Frenkel, Phys. Rev. E, 51:5891, 1995. [20] F.A. Escobedo and J.J. de Pablo, J. Chem. Phys., 106:9858, 1997. [21] A. Yethiraj and H. Fynewever, Mol. Phys., 93:693, 1998. [22] H. Weber, W. Paul and K. Binder, Phys. Rev. E, 59:2168, 1999.
190 [23] V.A. Ivanov, M.R. Stukan, M. Muller, W. Paul and K. Binder, J. Chem. Phys., 118:10333, 2003. [24] W. Hu, D. Prenkel and V.B.F. Mathot, J. Chem. Phys., 118: 10343, 2003. [25] I. Carmesin and K. Kremer, Macromolecules, 21:2819, 1988. [26] H. P. Wittmann and K. Kremer, Comp. Phys. Commun., 61:309, 1990. [27] H.-P. Deutsch and K. Binder, J. Chem. Phys., 94:2294, 1991. [28] W. Paul, K. Binder, D.W. Heermann and K. Kremer. J. Phys. (France), 1:37, 1991. [29] K. Binder editor, Monte Carlo and Molecular Dynamics Simulations in Polymer Science, Oxford University Press, 1995. [30] S. C. Glotzer and W. Paul. Annu. Rev. Mater. Res., 32:401,2002. [31] A. Yu. Grosberg and A. R. Khokhlov, Adv. Poylm. Sci., 41:53, 1981; A. Yu. Grosberg and A. R. Khokhlov, "Statistical Physics of Macromolecules", Americal Institute of Physics, 1994. [32] A.Yu. Grosberg and D. V. Kuznetsov, Macromolecules, 25: 1970, 1992; 25:1980, 1992; 25:1991, 1992; 25:1996, 1992. [33] B.T. Stocke, presentation at the Europolymer Congress, Stockholm, 2003. [34] N. Wilding, M. Muller and K. Binder, J. Chem. Phys., 105:802, 1996. [35] A.R. Khokhlov and A.N. Semenov, J. Stat Phys., 38:161, 1985. [36] B. Smit, Moi Phys., 85:153, 1995. [37] R. Dickman, J. Chem. Phys., 87:2246, 1987. [38] H.-P. Deutsch and R. Dickman, J. Chem. Phys., 93:8983, 1990. [39] M.R. Stukan, V.A. Ivanov, M. Muller, W. Paul and K. Binder, J. Chem. Phys., 117:9934, 2002. [40] U. Bastolla and P. Grassberger, J. Stat. Phys., 89:1061, 1997.
MACROMOLECULAR MOBILITY AND INTERNAL VISCOSITY. THE ROLE OF STEREOREGULARITY Giuseppe Allegra Dipartimento di Chimica, Materiali e Ingegneria Chimica "G. Natta" Via L. Mancinelli 7, 20131 Milano, Italy giuseppe.allegraOpolimi.it
Sergio Bruckner Dip. di Scienze e Tecnologie Chirniche, Universita Via Cotonificio 108, 33100 Udine (Italy).
Abstract
Macromolecular dynamics at the scale of a few chain bonds is largely controlled by the " internal viscosity" effect if the energy barriers hindering the skeletal rotations are sufficiently large. In an extensive spin-echo neutron scattering analysis, Richter and co-workers (Macromolecules, (2001), 34, 1281) investigated by spin-echo neutron scattering the dynamic properties of polyisobutylene (PIB) and polydimethylsulfoxide (PDMS) in toluene solution, the latter polymer being currently assumed to have very small rotational barriers. Analysis of the data according to a theory proposed by one of us (G.A.) enabled them to obtain realistic values both of the rotational barrier around C-C bonds (w 3kcal/mol) and of the natural frequency of the rotational jumps for PIB. - A problem related to chain internal viscosity concerns the iso- and syndiotactic forms of polystyrene (respectively i-PS and s-PS). After a careful conformational analysis it is shown that i-PS has very large effective energy barriers due to interactions between phenyl rings. This effect is compounded with that of the intrinsic rotational barrier and helps explaining the kinetic difficulty to crystallise of i-PS as compared with s-PS.
Introduction The motion of a long polymer chain is bound to take place via rotations around chain bonds, see Figure 1. In turn, each rotation implies that a conformational energy barrier must be surmounted, see Figure 2; 191 P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 191-201. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.
192 consequently the process entails energy dissipation for reasons analogous to those discussed by Eyring in his viscosity theory for a system of heavy spheres [1]. The dynamical-statistical description of this phenomenon has been the subject of a large body of investigations, and in the last few years several experimental results have been made available [2-4]. In the present paper we shall discuss the results in the light of a theoretical approach due to the present author and to Fabio Ganazzoli [5-8]. The main ideas will be concisely presented in this paper.
\ isKw^.,
Figure 1.
J
/
.2k
Any deformation of a polymer chain implies chain rotations.
CH,,CH,
0° Figure 2.
60°
120°
180°
240°
300°
6
The internal energy for the rotation around the central bond in n-butane.
193
1.
Internal viscosity
It is useful to decompose ideally the chain conformation into Fourier sinusoidal components. Each component consists of the alternation of a compressed chain strand comprising a large amount of gauche skeletal rotations, and of an extended strand with a larger proportion of trans rotations, thus creating a wave-like distribution of compressed/extended regions. Each wave moves towards either chain direction, and the superposition of these travelling waves produces the overall chain conformational dynamics. The propagation of each travelling wave along the chain is controlled by internal viscosity, i.e., the dissipation effect produced by the energy barriers hindering the rotations. The resulting dynamics may be described through the Langevin dynamical equation, where each chain atom experiences four forces, namely: 1) the elastic force
K[R(k + 1, t) - 2R(*, t) + R(fc - 1, t)] « kd^t];
(k =
where Coo is the polymer characteristic ratio, / is the C-C bond length, ks Boltzmann's constant, T the temperature and the position vector of the fc-th chain atom (k = 1,2...AT) at time t (see Fig.3);
K=3 K=5 ^
^
\
K=4
^
K=2
K=l LJ
/
R(K,t)
Figure 3. Scheme showing the vector representation of the chain atoms from an arbitrary origin (point 0).
2) the friction force
194 dR(k,t) ^ dt 3) the Brownian force X(fc, t) (purely stochastic). 4) the polymer internal viscosity, to be identified with an additional force proportional to *R(k,t)
dkdt
{k
3kBT
' -~cJ*}
The sign ambiguity is related with the direction of propagation of the deformation. The rotational characteristic time To is AE where AE is the average rotational barrier, see Fig.2, whereas A~l is proportional to the rotational frequency.
2,
Recent experimental investigations
A.Arbe, D. Richter and coworkers [4] carried out careful neutron spinecho investigations on the dynamics of monodisperse chains of PDMS (a polymer with a small value of the rotational barrier, therefore of TQ) and of PIB with a similar length, in toluene solution (see Figure 4). They had previously studied PIB melts with the same technique [3]. The classical Rouse-Zimm theory, with no internal viscosity, accounts satisfactorily for the data from PDMS. Conversely, the results from PIB could only be interpreted after adding the internal-viscosity, according to the approach proposed by Allegra and Ganazzoli [5-8] (see Figures 5,6). In the case of PIB, the numerical results for TO were (30 < T < 100°C, dilute toluene solutions) r0 = 1.27 x H r 1 2 e x p ( - | ^ )
[s]
{RT in kcal/mol)
Analogous calculations carried out for type-I inversions yield a much lower energy barrier, i.e. around 4kcal/mol. The result for AE (= 3.1kcal/mol) is in a reasonable agreement with the current value of the rotational barrier for alkyl polymers. The prefactor is consistent with the rotational frequency around C-C bonds, in one of the limiting cases considered by Kramers [9]; this is currently referred to as the natural oscillation frequency, whereby the medium viscosity produces a very small friction, however sufficient to provide the Brownian energy at constant temperature. In the case of PIB in
195
\ CH3
CH 3
CH 3
CH3
CH3
Figure 4-
Scheme showing the structure of PIB (above) and PDMS (below).
the melt the theoretical interpretation is qualitatively similar but the value of AE is much larger (« lOkcal/mol instead of « 3kcal/mol, in the temperature range 100 - 200° C). It is reasonable to assume that in the molten state the skeletal rotations of adjacent chains are strongly coupled, the degree of coupling becoming larger and larger as the glass transition temperature Tg is approached.
3. 3.1
Steric hindrance to rotational propagation Isotactic Polystyrene (i-PS)
In addition to internal viscosity, arising from intrinsic rotational potentials, essentially due to interactions among electron pairs on adjacent atoms, the chain conformational motion may be hindered by steric interactions among side groups belonging to the same chain. A significant
196
Figure 5. Chain dynamic structu factor of PDMS (empty symbols) and PIB (full symbols) in toluene solution at 300K (a) and 378K (b). The corresponding <J(= 4TT sin 6/X ) values are indicated. Lines through the points are guides to eye.
example is provided by isotactic polystyrene (i-PS). With isotactic polymers, left- and right-handed chain strands with a threefold helical conformation follow one another with alternating types of conformational inversions. The resulting model is qualitatively shown in the following Scheme, where T and G respectively stand for a trans and a gauche rotation around a skeletal bond (see Fig.2).
197
0.2
2.8
3.2 3.6 1000/T(K)
Figure 6. T-dependence of the solvent viscosity (dashed line) and the characteristic time TO deduced for the conformational transitions in PIB (diamonds). The solid line through the points is the best fit to an Arrhenius law.
Scheme:
While in the conformational inversion of type I side groups of adjacent monomer units point away from the inversion site, in type II they point towards the inversion site. We carried out a computational project to evaluate the internal energy profile involved in the displacement of both inversion points from one monomer unit to the next [10]. Our conformational calculations were performed with the CERIUS computer program. By far the most critical conformational path is the one involved with type II-inversion. In this case a significant steric hindrance arises from interactions between 4-th and 5-th neighbouring phenyl rings, see Fig.7. As it may be seen in Fig.8, skeletal rotations around 5 consecutive chain bonds were driven together in the search for the lowest-energy path that shifts transition point II by one monomer unit, see Scheme.
198
Figure 7. Molecular models of i-PS involved in a conformational transition shifting the inversion point (II in Scheme 2) by one monomer unit. The starting point is (a), the final model is (c), the intermediate model (b) is recorded near the energy maximum.
As shown in Fig.8, the resulting energy barrier turns out to be about 15kcal/mol; it effectively increases the natural barrier producing TQ. The resulting energy plot is given in Figure 10. The energy barrier is in the vicinity of 6kcal/mol.
3.2
Syndiotactic Polystyrene (s-PS)
In this case the transition shown in Fig.9 was investigated. As in the isotactic case, it implies the displacement of the inversion point by one monomeric unit. The transition does not involve strong conflicts among side groups, and it was performed by driving four chain bonds one at a time without co-operative changes of other bonds.
199
Figure 8. Energy (black circles) as a function of the conformational path in transition II for i-PS (see SCHEME). Open circles represent the torsional contribution to the total energy. The reaction coordinate along the abscissa corresponds to changing by less than 10° one of five consecutive rotational states at a time, along the minimum-energy path (see ref.[10] for details).
4,
Some concluding remarks on internal viscosity and steric rotational hindrance
Generally speaking, within a polymer chain in a low-viscosity solvent strain propagation takes place via rotational rearrangements and its velocity cannot exceed the limiting value r~h (bonds/s along the chain contour), where reff is the characteristic time of the effective rotational barriers. In a polymer like PIB, with a relatively small steric hindrance to skeletal rotations, we have refj = TO, or the characteristic time due to the intrinsic energy barrier (mainly due to electron-pair interactions). The resulting linear rate is around lcra/s, for a coiled chain with N = 100 skeletal atoms at 300K. The same limiting rate of strain propagation should hold in a highly swollen polymer network. In the case of i-PS, a polymer with a strong steric hindrance to skeletal rotations, chain dynamics in general - and crystallisation rate in particular may be strongly depressed in comparison with other polymers having a chemically similar but stereochemically different structure. In this case the effect of the intrinsic energy barrier is compounded with the steric effect, and we may write as a first approximation r
e//
= T
0{intrinsic)
200
Figure 9. Molecular models of s-PS: the starting model (a), the final model (c) and an intermediate model recorded at the relative minimum corresponding to eight consecutive trans states.
It is interesting that isotactic polystyrene (i-PS) crystallises more slowly and to a lower degree than the syndiotactic polymer (s-PS), in qualitative agreement with the results reported. These considerations suggest new possible experimental verifications of internal viscosity in highly swollen polymer networks. We point out that in an unswollen (bulk) polymer sample, strain propagation takes place at a much larger rate via direct interatomic contacts (the sound velocity). At lower temperatures, in the bulk polymer the effective rotational energy barrier (E increases, as an increasing number of adjacent chains tend to couple their skeletal rotations. At the glass temperature Tg the effective barrier AEeff is expected to go to very large values. As a last remark, we point out that friction phenomena occurring within a bulk, amorphous polymer sample sliding on a hard surface [11] are basically controlled
201
E 12 (kcal/mof) 8 6 4 2 -
Figure 10. Energy as a function of the conformational path for s-PS. See also the caption to Fig. 8.
by large-frequency dynamic effects; a quantitative investigation on this issue is under way in our laboratory [12].
References [I] H. Eyring, J. Chem. Phys., 4:283, 1936. [2] D. Richter, M. Monkenbusch, J. Allgeier, A. Arbe, J. Colmenero, B. Farago, Y. Cheol Bae and R. Faust, J. Chem. Phys., 111:6107, 1999. [3] D. Richter, M. Monkenbusch, W. Pykhout-Hintzen, A. Arbe and J. Colmenero, J. Chem. Phys., 113:11398, 2000. [4] A. Arbe, M. Monkenbusch, J. Stellbrink, D. Richter, B. Farago, K. Almdal and R. Faust, Macromolecules, 34:1281, 2001. [5] G. Allegra , J. Chem. Phys., 61:4910, 1974. [6] G. Allegra and F. Ganazzoli, Macromolecules, 14:1110, 1981. [7] G. Allegra, J. Chem. Phys., 84:5881, 1986. [8] G. Allegra and F. Ganazzoli, Advances in Chemical Physics, I. Prigogine and S.A. Rice (Eds.), 75:265, 1989. [9] H.A. Kramers, Physica, VII:4, 1940. [10] S. Bruckner, G. Allegra and P. Corradini, Macromolecules, 35:3928, 2002. [II] N. Maeda, N. Chen, M, Tirrell and J.N. Israelachvili, Science, 297:379, 2002. [12] G. Allegra, paper in preparation.
PROTEIN ADSORPTION ON A HYDROPHOBIC GRAPHITE SURFACE Giuseppina Raffaini and Fabio Ganazzoli Dipartimento di Chimica, Materiali e Ingegneria Chimica "Giulio Natta", Politecnico di Milano Via L. Mancinelli 7, 20131 Milano, Italy [email protected]
Abstract
We review here recent atomistic simulations of the adsorption of some protein fragments on a hydrophobic graphite surface. Fragments of unlike secondary structure containing either Q-helices or /3-sheets and of unlike hydropathy were taken into account. The simulations were carried out with simple energy minimizations to describe the initial adsorption on a bare surface, and with molecular dynamics runs to study the final and most stable adsorption geometry in a dielectric medium. Large conformational changes were found, involving complete denaturation of the fragments and a large spreading on the surface, with some degree of bidimensional ordering. The kinetics of surface spreading is also briefly reported. Finally, the statistical hydration of the isolated and adsorbed fragments was also investigated by explicitly accounting for the solvent.
Introduction Biomaterials are synthetic or natural materials intended to interact with biological systems. The success of an implant in human body is conditioned by the interactions between the biological system and the biomaterials [1], usually mediated by proteins [2, 3]. There are many experimental techniques for measuring the amount of adsorbed proteins on solid surfaces and for studying their conformational changes [4]. Examples of these techniques are depletion methods, quartz crystal microbalance measurements, ellipsometry, total internal reflection fluorescence, Fourier transform infrared reflection, neutron and X-ray reflectivity, Atomic and Scanning Force Microscopy, and circular dichroism spectroscopy. In spite of a large number of experimental investigations, the mechanism of adsorption and binding of proteins on solid surfaces is still only partially understood. In recent years, computer simula203
P. Pasini et al. (eds.), Computer Simulations of Liquid Crystals and Polymers, 203-219. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.
204
tions have added significant new insights to our understanding of these phenomena, complementing both theory and experiments. In this paper, we discuss some recent advances in this field, focusing in particular on our simulation results. At first, we briefly review both theoretical approaches and coarse-grained simulations with an emphasis on the relevance of atomistic simulations. Afterwards, we discuss our simulation methodology and the protein fragments we took into account, and then we report our results for the initial and the final adsorption stage on a hydrophobic surface in a dielectric medium. Later on, we give a preliminary overview of the kinetics of spreading of the adsorbed protein fragments, and then we discuss the simulations in the explicit presence of water. Finally, we give an outlook to further issues we are currently tackling.
1.
Short background of theoretical and simulation methods
Early theoretical approaches to study protein adsorption on a solid surface adopted semimacroscopic models derived from colloid science, providing encouraging results when electrostatic interactions with a charged surface are dominant. For instance, Roth and Lenhoff [5, 6] modelled the protein as a charged sphere having the same volume as the protein under consideration, and including also the van der Waals interactions through the Hamaker approach calculated the equilibrium constant for the protein- surface interaction. Later, the full protein structure with the actual charge distribution at its envelope was accounted for within the same methodology [7], or using Brownian Dynamics simulations [8], but still assuming throughout a rigid molecular structure equal to the crystallographic one. On the other hand, simulations with coarse-grained models on a lattice or in continuous space are often used in polymer science to describe the general, often universal, large-scale behaviour. Accordingly, similar models have been used to investigate certain aspects of protein folding [9], or the kinetics of absorption and denaturation [10], considering an appropriate copolymer simply formed by hydrophilic and hydrophobic units (an amphiphilic copolymer) on a lattice [9, 10]. In addition to oversimplify the molecular structure of proteins, these approaches can be plagued by lattice artefacts, which favour locally ordered structures and strongly constrain the conformational degrees of freedom, which is a crucial limitation in particular for compact structures. The lattice models were also improved upon by considering the possibility of directional interactions among the units that mimick the formation of
205 antiparallel /3-sheets [11], or by adding further terms that penalise the formation of kinks, thus effectively enhancing the backbone stiffness [12]. The latter model was employed to investigate adsorption on attractive surfaces, providing a rough description of the large molecular rearrangements involved in the process. In this way, it was claimed that bundles of a-helices, coarsely modelled on a cubic lattice, may undergo denaturation and subsequent surface refolding to a /3-sheet structure [12]. We feel such claims to be largely unsubstantiated, being obviously affected, and most likely dictated, by the lattice structure. It must be stressed therefore that the above-mentioned approaches, though fairly satisfactory from certain viewpoints, cannot describe the specific rearrangements and possible denaturation of real proteins maximizing the interaction energy, in particular on neutral or on hydrophobic surfaces [13-16]. Therefore, progress in the understanding of these important biological phenomena must account for the specific protein structure, including also the atomistic details. Only in this way one can achieve a realistic description of molecular rearrangements, conformational changes and possible ordering after adsorption on a solid surface. On the other hand, atomistic molecular dynamics simulations are lacking, to the best of our knowledge. Therefore, we recently tackled this problem using molecular mechanics (MM) and molecular dynamics (MD) methods, studying both the intramolecular rearrangements on a surface and the energetics and kinetics of the adsorption process for a possible comparison with the experimental data. Our methodological approach and some results were already reported in Refs. [17], and [18], indicated in the following as paper I and paper II for brevity.
2.
Simulations details
In this paper, we discuss our recent atomistic simulations of the protein adsorption on a hydrophobic surface, considering a flat graphite surface chosen because of its simplicity and of its rigidity, so that it can be treated as a fully rigid body. This surface may be viewed as a zero-order approximation of pyrolytic carbon, widely used in certain body implants such as cardiac valves. As proteins, we considered human serum albumin, the most abundant blood protein, and fibronectin, a protein present in the extracellular matrix and in body fluids that is involved in the early stages of blood clotting. Due to the very large size of these proteins, we selected two albumin fragments, the A and E subdomains, and the fibronectin type I module shown in Figure 1. Thus, we chose protein fragments with unlike secondary structures, a-helices or antiparallel /3-sheets. As starting geometries we used the experimen-
206
tal structures deposited with the Protein Data Bank [19] (human serum albumin, 1AO6, and fibronectin type I module, 1FBR, Ref. [19]), while the graphite planes were prepared from scratch.
subdomain A
subdomain E
X
0 is the energy required to desorb the molecule and bring it back to the free, optimized state. We also calculated the where Efrozen strain energy, Estrain, defined as Estrain == Efrozen-Efree, is the energy of the molecule in the frozen geometry it adopts upon adsorption. A larger interaction energy and a greater strain energy take place when a larger number of amino acids are close to the hydrophobic surface. Therefore, we determined the number of amino acids in contact with the surface, n °, taking conventionally 5A as the upper limit for the contact distance, and found a significant positive correlation between or Estrain and ?\AO &s shown in Figure 4. Considering separately 5A
209
the data points for the protein fragments, the best-fit lines through the origin are given by [17, 18] 71(3) • n « kJ mol l for the albumin subdomains 54(1) • n ° kJ mol"1 for the fibronectin module
{
^'
17(1) • n ° kJ mol x1 for the albumin subdomains 13(1) - n o kJ mol" for the fibronectin module
(2)
5A
with the figure in parentheses giving the standard error on the last significant digit.
• fibronectin • albumin o 10
20
30
40
n5A
Figure 4- Interaction energy Eint (left) and strain energy E'strain (right) plotted as a function of nsoA- The results obtained for the two albumin subdomains in different orientations are stained for the two albumin subdomains in different orientations are shown with empty symbol, and for the fibronectin module with full symbols. The solid lines are the best-fit lines through the origin given by eqs. (1) and (2).
For all protein fragments, the driving force for adsorption, locally modifying the secondary structure, consists of the favourable van der Waals interactions with the surface, mainly due to the hydrophobic or at least the less hydrophilic residues. We remark that according to eqs. (1) and (2) Eint increases with the number of residues in contact with the surface faster than Estrain- By extrapolation, we infer that the fragments may undergo much larger deformations so as to maximize their interaction with the surface. We also point out that the initial optimizations in the dielectric medium show many widely different energy minima. Therefore, the configurational phase space of adsorbed proteins
210 displays a rugged energy landscape, reminiscent of glassy states. However, it turns out that the energy barriers separating the local minima are often not prohibitively large, and can be easily surmounted through a suitable kinetic energy input in the MD runs.
4.
Final adsorption stage by molecular dynamics in the dielectric medium
Selected geometries, in particular the lowest- and the highest-energy states obtained in the previous section, were subjected to MD runs in search of the best adsorption geometry after optimization of many instantaneous snapshots. Interestingly, when equilibrium was achieved in the MD runs, further energy minimizations produced only relatively minor readjustments, mainly involving local features with modest energy changes. We report in Figure 5 the optimized geometries obtained in the final adsorption stage. We found a very similar behaviour for both albumin subdomains and therefore we report just one case.
Figure 5. Side and top view of two best adsorption geometry after the MD runs and subsequent energy minimizations. At left we show the albumin A sub domain, and at right the fibronectin module. Note the lack of any secondary structure in both cases.
Both albumin subdomains show an extensive denaturation with the formation of a monolayer of amino acids. Such a large molecular spreading on the surface is consistent with experimental data obtained for the whole protein on a hydrophobic surface [15]. On the other hand, for the fibronectin module it is more difficult to form a monolayer on the graphite surface because it contains four disulfide bridges acting as intramolecular crosslinks. We studied how such crosslinks affect or even hinder the conformational changes during the MD runs by per-
211 forming similar MD simulations after replacing the disulfide groups with thiol moieties. Preliminary results indicate that indeed eliminating the crosslinks allows the molecule to achieve a much larger flattening on the surface, suggesting that eventually a monolayer can also be formed by the modified (i.e., crosslink free) fibronectin module in the long run. It can also be observed in Figure 5 that the most stable geometries found for the final adsorption stage display a bidimensional order, whereby the backbone trajectory roughly tends to give an antiparallel arrangement. This feature can be seen more clearly in the albumin subdomain (see Figure 5 at left), but it is also largely present in the fibronectin module. Such ordering is due to strong dipolar interactions that involve the side groups of the facing amino acids, and therefore it does not lead to the formation of true /3-sheets. From the energetic viewpoint, we note that the interaction energy per amino acid in contact with the surface is equal to 56 mol"1 for the albumin subdomains and to 57 kj mol"1 for the fibronectin module, solely due to the dispersion (or van der Waals) forces. Not surprisingly, these values are very similar, since the protein fragments are made up of the same 20 natural amino acids. Note that for the albumin subdomains the value of this interaction energy is lower than that extrapolated from eq.(l), namely 71 kJ mol"1, because the initial adsorption is dominated by the hydrophobic aminoacids, whereas in the most stable state all residues interact with the hydrophobic surface, including also the hydrophilic ones. The strain energy per residue in contact with the surface in the most stable state amounts to 17 kJ mol"1 for the albumin subdomains and to 15 kJ mol"1 for the fibronectin module. These values are surprisingly close, but we believe such result to be accidental, in view of the different secondary structure of the two protein fragments. The large interaction energies just discussed strongly suggest that protein adsorption on hydrophobic surfaces is usually irreversible. Such conclusion is consistent with experimental results, which show that the slow spreading of albumin over a hydrophobic surface eventually leads to a smaller coverage in terms of the adsorbed mass of protein per unit surface due to the lower bare surface that remains available for other molecules [15, 16]. Unfortunately, analogous data for the fibronectin module are lacking. The conformational changes can also be investigated using the radius of gyration Rg and its components, or the principal axes of the molecules as geometric descriptors. During the adsorption process, we found a pronounced increase of Rg compared to the isolated fragments [17, 18] For the isolated albumin A subdomain, Rg measures 12.1 A , increasing to 21.5 A in the final state, while Rg is equal to 16.2 A for the isolated
212 fibronectin module, and to 28.5 A in the final state. Upon spreading, we also found that the systems become also much more anisotropic. In fact, the principal axes of the isolated albumin A subdomain are equal to 9.6 - 5.6 - 4.9 A, and become equal to 19.3 - 9.3 - 1.0 A in the final state, while for the isolated fibronectin module they are equal to 14.4 - 5.9- 4.4 A, and in the final state to 23.2 - 16.1 - 3.5 A. Needless to say, in all cases the shortest axis is orthogonal to the surface. For all the protein fragments, the very large interaction energy above reported and the great anisotropy, in particular the very small molecular thickness, suggest that in the final adsorbed state the protein fragments cannot be easily removed under a shear stress, such as due for instance to the flow of the surnatant solution.
5.
Kinetics of surface spreading
We are currently analysing the kinetic data for the molecular spreading on the surface, and therefore here we only report a few preliminary and qualitative results. The MD runs show that the final adsorption state is attained through a liquid-like spreading of the protein fragments, often accompanied by tilting of the whole molecule, as reported in Figure 6, or by multi-stage changes with sequential disruption of the secondary structure, as reported in Figure 7. In Figure 6 we report the kinetics of the fibronectin module from the least stable initial adsorption geometry to the final best adsorption stage, shown through the time evolution of the potential energy Epot and of the distance d of the molecular centre of mass from the surface. The process is best described in terms of an initial backbone tilting towards the surface, which leads to a faster decrease of d compared to Epot. Similar plots are shown in Figure 7 for the albumin E subdomain, where the multi-stage changes involve the sequential, and quite abrupt, adsorption of denatured strands. For the fibronectin module in different orientations, we often found also an interesting equilibrium state with a continuous breaking and reforming of /3-sheets through minor displacements of the backbone trajectory. Note that in some cases the overall conformation remains quite similar to the isolated molecule. On the contrary, starting from the lessstable initial adsorption geometry, we found larger rearrangements and a complete denaturation with a stronger adsorption after the backbone tilting above discussed in connection with Figure 6. In the most stable state, found after optimization of many selected snapshots, the fibronectin module and the albumin subdomains optimize the surface interaction by spreading as much as possible, thus maximiz-
213 2000
i 100
200
'(ps)
Figure 6. The less stable initial adsorption geometry (upper picture) leading to the most stable final adsorbed state (lower picture), at left, and plots of the potential energy and of the distance of the molecular centre of mass from the surface as a function of time at right.
ing their footprints. Quantitative estimates of the footprints and of the exposed surface can be found in paper I and paper II. Here we limit ourselves to briefly mention that the changes in molecular size and in the overall anisotropy that take place upon surface spreading can also be easily monitored as a function of time. We used as geometrical descriptors the radius of gyration, Rg, and in particular its components parallel and perpendicular to the surface, and the principal axes to characterize the molecular anisotropy. The overall shape of the isolated protein fragments in a globular conformation is essentially isotropic, and it does not show major changes upon the initial adsorption. On the contrary, during the surface spreading the molecular shape becomes strongly anisotropic, as already anticipated in the previous section for the final adsorption state. We report in Figure 8 an example of the time change of Rg and of its components during the kinetic process already shown in Figure 6. From the right-hand- side plot of Figure 8, it can be seen that the decrease of the perpendicular component yielding the molecular thickness is somewhat faster than the full spreading that maximizes the molecular footprint, monitored in turn through the parallel components.
214 1600
1 0
200
400
600
800
'(PS)
Figure 7. The same as in Figure 6 for the albumin E sub domain. Note the different time scale.
0
250
500
750
1000
'(ps)
Figure 8. Radius of gyration and its squared components calculated during the MD runs of the fibronectin module already shown in Figure 6.
A more detailed analysis of the kinetics of spreading on the graphite surface shall be reported in a future paper [23].
215
6,
Hydration of the adsorbed protein fragments
The simplest way to investigate the hydration of the adsorbed fragments is through the accessible surface area exposed to the solvent. However, although this area can be taken as proportional to the dispersive interaction energy with the solvent, it is not indicative of the full solvation energy because it does not account for the electrostatic contribution, in particular for the hydrogen bonds. Nevertheless, we note that the fragment spreading on the surface enhances both the molecular footprint and the accessible area to the solvent. In this way, it is possible to maximize both the hydrophobic interactions with the graphite surface and the interactions with the solvent, hence the molecular hydration, at least on the exposed side. Quantitative results for our protein fragments can be found in papers I and II. A more lengthy, but also a more powerful tool for studying the hydration of the protein fragments consists on the explicit inclusion of a very large number of water molecules in the simulations. Therefore, we considered the protein fragments in different adsorption geometries introducing them in a box containing a few thousand of water molecules with periodic boundary conditions, and saved many different frames during the MD runs to calculate the pair distribution function (PDF) of the water molecules around the protein backbone. This function yields the probability density of finding the water molecules (or its oxygen atoms in the present case) at a distance r from the backbone. In Figure 9 we report such PDF for all the fragments we studied.
0.3
final 0.2
i initial adsorption
isolated 4 K# while w is a set of positive parameters that are used as weighting factors. In the expanded ensemble, the probability of observing the system in the state m is ^
(10)
(9)
227
By setting wm = a/Zm
(11)
with a an arbitrary constant, Eq. (10) becomes P(m\w) = ^
(12)
The converse is also true, that is, if we have a uniform distribution P(m\w) = 1/M, then wm oc Zml. The PMF AT is the free energy of the system in state ra, relative to that of state fc, Af = F(m) - F(k) = -kBT log (Zm/Zk)
(13)
^From Eq. (10), we obtain
When one has achieved uniform sampling, Eq.(12), the PMF reduces to = F(m) - F(k) = -kBT(\ogwm
- logwk)
(15)
The PMF can be computed in a simple way from a set of weights w that results in uniform sampling in the expanded ensemble. We now describe a practical way to determine such a set. The reader is referred to previous publications for a more detailed description [6, 11]. An initial set w° is proposed, and gradually updated on the fly on the basis of the number of visits to each state. In analogy to the original implementation of the DOS method, a histogram of the visits to each state is monitored. Every time the system visits a state m, the weight for this state is changed to Wm -> Wm/f
(16)
and the corresponding entry of the histogram is incremented. Number / is a convergence factor. Monte Carlo moves from a state o to state n are accepted with probability P(o -* n) = min(l,exp(-/3(C/n - Uo) + \ogwn - \ogwo)) Once the histogram is deemed flat (e.g., when the entry is greater or equal than 80% of the average), tor / is updated (e.g., / —> y/J). After this, the but the set of current weights is retained for the
(17)
minimum histogram the convergence fachistogram is cleared, next iteration. The
228
simulation proceeds until log/ falls below a given threshold. In ordinary DOS simulations, this threshold is typically set to 10~7 or 10~8 in order to achieve sufficient accuracy in the calculation of the density of energy states, g(E). However, in this application, the set of weights (that corresponds to a "density of separation states") can be accurately estimated with much less stringent convergence factors. Typically, it is sufficient to set the threshold to log fthr ~ 3 x 10~3, corresponding to five successive updates of the convergence factor. One of the benefits of the EXEDOS method is that it provides an additional test for convergence: in addition to the estimate of the PMF obtained from the weights, one can also integrate the average force on the colloidal particles measured at each state. The average force F(£m) is a simple average of the sum offerees acting on a given colloidal sphere, along the direction of increasing reaction coordinate. Thus, for the sphere-wall interaction, it is the average total force (between the colloid and all the mesogens, and also with both walls) along the z-axis. For the sphere-sphere interaction, F((m) is the total force (between the given colloid and all mesogens, the walls and the second sphere) acting along the line connecting the centers of the spheres. In both cases, a second estimate of the PMF may be obtained as F(C)dC
(18)
Both estimates, Eq. (15) and Eq.(18), must agree if the method has truly achieved uniform sampling. Therefore, in addition to the condition / < /thr? the agreement of both estimates can be viewed as another criterion of convergence. After both methods agree, either one can be used. In this work we have chosen to present the estimates obtained from integration of the forces as these curves have a smoother appearance. To extract local static properties of the system, such as the density p and the tensor order parameter Q, a separate series of NVT runs were performed with spheres fixed at a few selected values of the reaction coordinate. The corresponding simulation configurations were analyzed by binning the LC molecules in a rectangular grid and computing the quantities of interest for each bin. In particular, for each bin with a volume VB the density was found as p = NB/VB, where NB is the number of molecules within this bin; the tensor order parameter was computed as 1
NB
229 where I is the identity tensor. Prom the tensor order parameter one can compute the scalar order paramete S (sometimes denoted also as P2) and the director, n. If we denote the largest eigenvalue of Q as Ai, then (20) and n is the unit eigenvector associated with Ai.
1.2
Dynamic Field Theory
In some instances, it is possible to describe the configuration of the LC just by specifying the director field n(r); this is commonly referred to as the "continuum theory". The free energy of the system is a functional of the spatial derivatives of n : T = I / Ki(V • n) 2 + K2(n • (V x n))2 + K3(n x (V x n))2dr
(21)
However, since this approach assumes a constant, homogeneous value of the scalar order parameter, it fails to describe the core of topological defects, where P2 deviates from its bulk value on length scales of the order of £. The continuum theory for n also fails to describe multidomain systems because it cannot handle the variation of P2 across domain boundaries. As an alternative to the continuum theory, the mesoscopic approach can be based in a dynamic field theory for the tensor order parameter, Q. This tensor can be viewed as a coarse-graining of the microscopic probability distribution function ^(u,r,t) for the molecular orientation u. In this sense, Q corresponds to the symmetric, traceless part of the tensor of second moments of ip at the point r and time t:
Q(r, t) = I (uu - i l ) V(u, r, *)du.
(22)
The local state of the liquid crystal is therefore described by five degrees of freedom. In this model, the properties of the liquid crystal are described by a Landau-de Gennes free energy that contains two contributions: first, a contribution of the form
Ts
11
= 11 ii ~ ff ))
t tr (( QQ 22) )
t r {t rQ{ Q3 3) ) + " ir T^Q 2 ) 2 d r -
(23)
which describes the excluded volume effects that drive the first order transition from the isotropic to the nematic phase. The coefficients A
230
and U are phenomenological parameters that depend on the liquid crystal of interest. These (and the other phenomenological parameters of this theory) can be selected by comparison with empirical data. They can be assigned a microscopic interpretation [5]: A corresponds to pfc#T, while U is proportional to <JI<JQP. In this model, that transition occurs at U = 2.7. The second contribution to the Landau-de Gennes free energy describes the long range elastic forces dominant in the nematic phase:
(24) The coefficients Li, L2, L3 are related to the splay, twist and bend elastic constants K\^K2^K^ through the equations [5]
L2 = ^ = r ^ ,
(25b)
^ = ^ ,
(25c)
In the one constant approximation [K\ = K2 = K$ = JFC), L\ = K/(2S2) while L2 and L3 become zero; the long range elastic contribution simplifies to
)dr
(26)
J '
and the total free energy is jr = jr5
+
jPe.
(27)
The relative strength of the contributions from Eq. (23) and Eq.(26) depends on the type of liquid crystal that one studies: for polymeric liquid crystals, short-range interactions are expected to be dominant, while long-range interactions are dominant for low-molecular weight liquid crystals. The evolution equation for Q that we use in this work corresponds to a particular case of the Beris-Edwards dynamical formalism [5] and is given by
f(MS))
The coefficient T = 6£>*/(l - 3tr(Q 2 )/2), where D* is the rotational diffusivity coefficient for the mesogens.
231 In Eq. (28) it is understood that | £ represents the symmetric part of the functional derivative of T with respect to Q: 2
Inspection of Eq. (28) shows that Q will remain symmetric and traceless as it evolves. When the functional derivatives in Eq. (28) are evaluated, one obtains a system of partial differential equations for the components of the tensor Q. We solve this system with a finite difference method over a rectangular grid. One can construct characteristic length and time scales from the parameters of the theory. As an instance, it is convenient to introduce the quantity £ = ^18Li/AU as a characteristic length for changes of the order parameter. This length corresponds to a few times the molecular length. For the time scale, we choose the quantity T =
(6D*A(1-U/3))~1.
2.
Clusters of particles
2.1
Mapping of simulation and field theory length scales
We first consider the insertion of one spherical particle into a nematic liquid crystal. The particle is fixed in the middle of the cell (both in simulations and in the field theory). Prom past theoretical studies, it is known that defect structures that arise in the proximity of an interface [16], as well as the nature of their interactions [17], depend on the anchoring properties of the liquid crystal. Experimentally, the orientation and strength of anchoring at various interfaces depend on the system under study. As an instance, it is possible to control the anchoring properties at the surface of water droplets immersed in a liquid crystal through the use of amphiphilic molecules adsorbed at the droplet interface [15]. For the case of solid surfaces, the anchoring properties can be controlled using self-assembled monolayers (SAMs) of alkanethiols or other compounds [1]. In molecular simulations, the anchoring properties are determined directly through the choice of interaction potentials. For the Gay-Berne fluid, Andrienko et al. have shown that a colloidal particle immersed in the bulk nematic phase exhibits strong homeotropic anchoring (i.e. the 2
It turns out that, for this model, the indicated functional derivative is already symmetric.
232
Figure 4- Two of the possible defect structures associated with a strong hometropic sphere: (a) Hedgehog point defect (b) Saturn ring disclination line.
orientation of the molecules is perpendicular to the surface) [18]. In our simulations, we observe strong homeotropic anchoring at the surface of the colloids and also at the confining walls; hence, in the theory we only consider this type of anchoring by imposing the corresponding boundary condition to the field Q. Two possible defect structures that can occur for a homeotropic colloidal particle are the Saturn ring disclination line and the Hedgehog point defect. (See Fig. 1.2.1) In this work we consider particles with a radius R of the same order of £ (which corresponds to the characteristic length for changes of the order parameter). For such small particles, the Saturn ring configuration is expected to be more stable than the Hedgehog configuration [12, 13]. Our Monte Carlo simulations and field theory agree when they indicate that the particle is surrounded by a Saturn ring. To map the characteristic length scale of the theory (£) into the units of the Monte Carlo simulations (CTQ), we can estimate the ratio (30)
by analyzing the dependence of the Saturn ring radius (a) as a function of the particle size (R). One can start by plotting the Monte Carlo simulation values for the ratio a/R as a function of R/GQ. In Fig. 4 one can observe that a/R —» 1.2 for large particles. This limit compares well with the value obtained using continuum theory (a/R ~ 1.236) [14]. We can overlay the field theory results by plotting a/R as a function of /?/(£/&£,); the best match is obtained by setting kL = 2£, and this provides us with the desired mapping of scales.
233
Figure 5. Size of the Saturn ring (in units of the particle radius) as a function of the particle size.
2.2
Sphere/substrate interactions
In this section we consider the interaction between a particle and a wall. We first present the results from the theory. Figure 1.2.2 shows the director profile and scalar order parameter maps for a particle of radius R = 0.36£ located at the surface of a wall. Comparing these maps with those for a particle further away, (Fig. 1.2.2 ), we observe that the disclination line is no longer at the equatorial plane, but is shifted towards the wall. Similar deviations of the position of the ring are observed in the Monte Carlo simulations. We define the potential of mean force AF(s) as the free energy difference when the center of the particle is located at s + R and when it is at the center of the cell. (Hence, s is the separation between the wall and the surface of the sphere.) We present AF(s) for three different radii in Fig. 6. The theory predicts a positive, repulsive potential between the particle and the wall. The intensity of this repulsion increases with the radius of the particle; the simulations predict a global repulsive force, with the occurrence of oscillations and local extrema. The presence of these minima and maxima is due to the formation of smectic-like modulation of the LC density by the confining wall. (See Fig. 7) A comparison of the density profile and the PMF reveals that configurations with the particle's center between two density peaks are more stable. These features, originated in the molecular structure of the LC, cannot be captured by the coarse-grained approach of the field theory because it assumes a constant density for the liquid crystal.
234
5
10
1$
Figure 6. Contour map of the scalar order parameter for a particle of radius R = 6 in the proximity of the wall
1
Li
250 Vi
{-,
200
3
150
[
'
—
R-3o 0 (MO)
_
R-6a*0^)
„
•A • RHoJciiiooiy) 4, R-5a0Crbeacy) A
1
100
1
1
5\D Vi • A
50 0-
-
A
1
Figure 7. Potential of mean force as a function of the radius of the sphere, from simulations and field theory. See the text for an explanation of the mapping of length and energy scales between these two approaches.
235
Figure 8. Contour map of the density of mesogens, for a particle of radius R = 6 in the vicinity of the wall
It is interesting to compare quantitatively both approaches. In order to do that, we need to find a scaling factor so that = A/e.
(31)
This factor is obtained by matching the maxima of the PMF obtained from the theory and the simulations for R = 4 0)
(27)
and we have a slow, logarithmic dependence of S% on a (-> 0). The analogous result for the three-dimensional case is: A2H ~ 5^ = n x 0.1218..., as found long ago by Ronca and Allegra[26]. We see the interesting feature that the three-dimensional network has an unperturbed m.s size which, in the absence of any squeezing force, is comparable to that of a single chain strand. However, the energy vs. network size curve is infinitely steep in this case, so that further compression of this network requires the application of infinitely strong forces.
2. 2.1
Polymer-mediated adhesion The model
We are interested in modelling adhesion phenomena produced by a thin layer of polymer chains confined between two rigid parallel walls.
258 Our aim is to derive the small-strain tangential and longitudinal moduli of the adhesive layer. We will not address its large-strain behaviour and ultimate failure properties, which are also very important for applicative purposes. The chains are assumed to be very long, so that their unperturbed radii of gyration are much larger than the wall-to-wall distance L. As a matter of fact, we consider a single infinitely long chain, compressed by the walls to melt-like densities (see Figure 2). We assume that, at a given instant in time, all the chain monomers in contact with the walls react and form permanent chemical bonds with it. The original infinite chain is thus broken up into shorter subchains, with their two end monomers either on the same or on opposite surfaces ("loops" and "bridges", respectively). The distribution of lengths for the loops and bridges, as well as their relative probabilities, is not given a priori, but will be derived from statistical principles. Thus, formation of irreversible polymer-surface bonds corresponds to taking an "instantaneous snapshot" of an equilibrium configuration the system — an idea pioneered some time ago by Deam and Edwards in the context of polymer networks [27]. Unlike the previous section, now there are no cross-links between the chain(s). Note that the present system corresponds to the "Model B" of ref. [13]. In the same paper, we also investigated the properties of a so-called "Model A", consisting of monodisperse polymer chains bridging the two surfaces. We adopt a lattice model, following earlier treatments by Di Marzio [31], Scheutjens and Fleer[32], Silberberg [33] and one of us[34]. The statistical segments of the polymer lie on the edges of cubic cells. The lattice model is in many ways artificial, but it has also some distinct advantages. In particular, it allows the enumeration of the chain conformations and we implement it in a way which accounts exactly for finite chain extensibility (unlike Gaussian models). The more realistic representation of the interaction with a solid wall, compared with the harmonic potential of the previous section, is an additional bonus. The polymer is assumed to be in a rubbery amorphous state above its glass transition temperature, with a perfectly uniform density across the slab. The latter constraint is implemented by application of a suitable shortrange polymer-wall attractive potential.
2.2
The transfer matrix
We consider two walls at a distance L, in cubic lattice units (equal to the length of the chain segments). Thus, L is also the number of lattice "layers" between the bottom and the top wall. We denote by P^ the
259
Figure 2. Representation of a very long chain filling the space between two parallel walls, forming loops, bridges and trains of variable length. The slab width is in this case L = 12. All the beads in contact with a wall are irreversibly bonded to it, after reaching configurational equilibrium. L x L transfer matrix: 4 1 0 0 0
1 4 1
0 1 4
0 0 1
0 0
1 0
0 0 0 4 1
(28)
1 4
Its (i, j) element represents the number of ways of placing one bond on the lattice, so as to connect a bead on layer i with a bead on layer j . Its eigenvalues A^ and eigenvectors a^ = [ai^,a2fc,... ,a>Lk] a r e : =
4 + 2 cos
(29) (30)
We illustrate the usage of the transfer matrix by writing the partition functions — equal to the total number of allowed conformations, since there are no energetic terms associated with chain bending or bead-bead
260
interactions — of a chain of n segments, whose ends are constrained to lie either on the same ("loop", LP) or on opposite walls ("bridge", BR). In the first case we have: 1 0 0
Z LP (n,L) = [ 1 0 0
L k=l
whereas in the second case:
0
0
0 0 0
L
(32) jfe=l
2.3
Statistical population of loops and bridges
As mentioned above, we consider a infinitely long chain, which is initially unconstrained and free to reach statistical equilibrium. The chain meanders between the walls, forming loops and bridges of all possible lengths n. We label their partition functions by ZLP and ZBR. These differ from the partition functions of the previous section since: a. In the present context, a bridge/loop presesents a chain travelling from one to the other/same surface, without ever touching either wall in the intermediate steps (otherwise, the chain would be further broken down into shorter bridges and/or loops). b. We wish to model a situation with a uniform polymer density across the wall-to-wall slab. However, a phantom chain interacting with purely repulsive walls would have a density maximum at the middle, since it suffers from "entropic repulsion" [15, 34]. This is understood by recognizing that, on a simple cubic lattice, a segment within the bulk has six possible orientations. Instead, when one of its end beads is in contact with a wall, the segment has only five possible orientations. This can be remedied by placing a suitable "premium" on the monomers which are directly in contact with a wall. In practice, the above requirements are implemented in the following way:
261 i. The two terminal bonds are constrained to be orthogonal to the surface, so that the effective number of free bonds in a subchain of n segments is n — 2. ii. We use a matrix P of order L - 2 instead of L as in Eq.(28), to fordid the internal beads from visiting the upper and lower layer. Hi. The premium for the atoms at either wall is obtained by multiplying the partition function of each subchain by a factor of | . There is only one such factor, even though each subchain has two terminal beads, to avoid double counting of the chain-wall contacts. These points may be summarized as: (33) Z x (n, L) = \zx{n - 2, L - 2) (n, L > 2; X = LP or BR). o Finally, one should not forget that the chain may also form "trains" of segments in contact with the walls (see again Figure 2). These will be dealt with implicitly, since a train may be viewed as a sequence of "one-segment loops". All that is required is the additional rule: iv. A one-segment loop (a chain bond lying flat on a wall) has the partition funtion: 24
ZLP(1,L) = —,
(VL)
(34)
given by the | premium times the number of possible orientations of the segment (four). The relative populations of loops and bridges are expressed by their probabilities: % ^ ,
(35)
where 6n is the partition function of a generic unconstrained n-segment section of an infinitely long chain. It ensures the fulfilment of the normalization condition: f > L p ( n , L ) + p B R ( n , £ ) ] = l.
(36)
71=1
These quantities are given graphically in Figures 3 and 4, for some selected strand lengths or slab widths. We observe that, for a given strand length, the loop and bridge probabilities are virtually identical up to wall-to-wall distances of the order of the unperburbed radius of gyration, i.e. up to L ~ y/n. Afterwards, the loop probability tends to
262
\n=25
10
15
20
25
30
35
Figure 3. Probabilities of loops (solid lines) and bridges (dashed lines) of selected lengths n, as a function of the wall-to-wall distance L.
a constant value, whereas the bridge probability drops to zero. Short loops are always more more likely to occur than long ones. Instead we observe a crossover for bridges, since a short bridge is more probable than a long one for narrow slits, and less probable for wider slits. For a given wall-to-wall distance L, the most probable bridges have a strand length n~ L2. The average strand length is defined as oo
— y n
»,£)].
(37)
n-\
It has a simple expression, which may be obtained by the following argument. We know from ref.[34] that, if the density on the internal layers is equal to p, the density on the terminal ones is | p . The total number of beads within the slab is thus: (38)
where S is the slab area (->• oo). Since the infinite superchain is broken into two separate strands whenever one of its beads touches one of the
263
10
50
100
150
200
250
300 350
Figure 4- Probabilities of loops (solid lines) and bridges (dashed lines) for some selected wall-to-wall distances L, as a function of their length n.
walls, we may obtain (n) from the ratio of the total number of beads and the number of beads on the terminal layers: 3. 1 N (39) = With
L
(WG>
,
(8) G
This relation has also been tested in NEMD simulations [1] where both the polymer and the solvent molecules where treated microscopically, i.e.
273
no friction coefficient was used explicitely. In the NEMD simulations the relation (8) is well obeyed at all shear rates studied. This is surprising since no friction is used in the underlying dynamics. To understand the basic physical processes, we introduce a simple model with 3/2 "degrees of freedom" rather the few thousand involved in the NEMD simulations.
2.
A Simple Model
In the spirit of dumbbell models [9], we mimic the dynamics and the shape of polymer molecule by that of a particle at position r where r = 0 corresponds to its center of mass. A force F acting on this particle can be chosen such that the time average of r 2 coincides with the mean square radius of gyration of the polymer coil in equilibrium. Furthermore, the effect of the flow of the background fluid is taken into account via a "friction" force proportional to the difference between the velocity of the particle and the flow velocity v(r). For this model, reduced (dimensionless) variables are used, but they are denoted by the same symbols as the corresponding physical quantities. The reduced mass is put equal to 1. Then the equation of motion reads
£='-)•
Qq
(13)
O
where aeq is recalled as the equilibrium value of the alignment in the nematic phase. Thus Aeq is equal to AK at the transition temperature provided that « = 0. In the limit of small shear rates 7, the tumbling parameter is related to the Jeffrey tumbling period [18], see also [25]. Within the Ericksen-Leslie description, theflowalignment angle x in the nematic phase is determined by cos(2X) = -7i/72 = 1/Aeq.
(14)
304
A stable flow alignment, at small shear rates, exists for |Aeq|l only. For |Aeq| < 1 tumbling and an even more complex time dependent behavior of the orientation occur. The quantity |Aeq| - 1 can change sign as function of the variable #, cf. Fig. 4. For |Aeq| < 1 and in the limit of small shear rates 7, the Jeffrey tumbling period [18] is related to the Ericksen-Leslie tumbling parameter Aeq by (15) for a full rotation of the director.
-1
0.5
-0.5 temperature
Figure 4- The tumbling parameter as function of the temperature or concentration variable $ for AK = 145,1.25 and n = 0 (upper and lower thin curves) as well as AK = 1.25,1.05 and K = 0.4 (upper and lower thick curves). The dashed horizontal line marks the limit between the flow aligned (Aeql) and the tumbling (Aeq < 1) states.
In the following, both AK and K are considered as model parameters. The first one is essential for the coupling between the alignment and the viscous flow. The second one influences the orientational behavior quantitatively but does not seem to affect it in a qualitative way. If one wants to correlate the present theory with the flow behavior of the alignment in the isotropic phase, on the one hand, and in the nematic phase, on the other hand, for small shear rates where the magnitude of
305 the order parameters is practically not altered, it suffices to study the case AK 7^ 0, K = 0 in order to match an experimental value of A by the expression (13). Mesoscopic theories [12, 14, 23] indicate that n ~ AKThus we also study the case K ^ 0, in particular K = 0.4.
1.4
Scaled variables: stress tensor
The stress tensor (5) associated with the alignment is related to the relevant quantities expressed in terms of scaled variables by
m
B
K
r e f
^r I
K
,
$ = #* + ^V6 OAK
statt3a* • **,
(16) where a* = CL/CLK and 3>* = $/$ r ef i n (16). The dimensionless shear stress S a l associated with the alignment is defined by
Then, Eq. (16) is equivalent to
\ ^
W
h
(18)
where Gai is a shear modulus associated with the alignment contribution to the stress tensor, and the product Ao a^ is essentially one parameter entering the theoretical expressions. The quantity 7?ref = GaiTai serves as a reference value for the viscosity. With the scaling used here, the dimensionless (first) Newtonian viscosity, in the isotropic phase, is ^New = * + 77fso w i t t l ^fso = ^iso/^ref- For high shear rates the dimensionless viscosity rf approaches the second Newtonian viscosity rj*so. The total deviatoric (symmetric traceless) part of the stress tensor, in units of G a i, is denoted by a. In terms of the quantities introduced here it is given by , cf. (4), a = -p = 2 77iS0r - £d = 2r,[soT + V2G al S a l .
(19)
In the following, we will denote quantities in reduced units by the same symbols as the original ones, unless ambiguities could arise.
1.5
Basis tensors and component notation
The symmetric traceless alignment tensor has five independent components. It can be expressed in a standard [34] ortho-normalized tensor
306
basis as follows: 4
a = £ aKTk,
T° = fi/2 ezez,
T1 = 0 7 2 (exex - eyey),
k-0
(20) where Tl with i = 1,..,5 are the basis tensors by which a is uniquely expressed. The orthogonality relation and the expression for the coefficients ax are given by T% : Tk = ^ and ax = a : Tfc. Using these basis tensors, from (2) we obtain a system of five ordinary differential equations 1 0 3 \/3 AK7 2 1 2
1 VSn'yao , 3
(21)
m
'
where $0
=
(^ ~ 3ao + 2a )ao + 3(ax + a2) — - ( a 3 + a 4 ) ,
$i
=
($ + 6ao + 2a )ai — - v 3 ( o 3 — a 4 ) , (22)
and a2 = ag + a2 + a2 + a2 + a4. The parameters ft, 7?, AK were introduced in the foregoing section. Prom their definition we see, that the order parameters satisfy |a*| < (15/4) 1 / 2 for i = 1,2,3,4, and - 5 1 / 2 / 2 < a0 < V2 The corresponding expansion with respect to the basis tensors and the component notation can be used for the other second rank irreducible tensors. Prom equations (17) and (16) one deduces expressions for the (dimensionless) shear stress axy, and the normal stress differences N\ —