An even coarser view on polymers

So far, all models considered allow the correspondence of a bead of a polymer to one or to a few repeat units of a given chemical species. This still means that the number of degrees of freedom that have to be considered is proportional to the number of monomers of a given chain, causing enormous problems if one wants to try to simulate big systems (many chains instead of many monomers). To arrive at a situation where we can simulate many chains we go back to Figure 1. There, three levels of description were illustrated. I have discussed the microscopic and the mesoscopic regime and, in one case, the link between the two. Now we want to consider another step, namely to try to map the chains from the mesoscopic system up to the semi-macroscopic regime where we replace the chains by soft ellipsoidal particles which can strongly overlap in the melt [86, 121. For such a model each chain is represented by a soft ellipsoid which varies its size and shape.

We separate the free energy of a system into an intra-chain part and an inter-chain part. For the total free energy F we make the ansatz:

The first sum runs over all M chains of the system. First let us consider the intra- chain part of the free energy. In a melt, the allowed conformations of a polymer chain are the same as a self-avoiding walk in ‘vacuum’. (The change from self-avoiding to ideal random walk statistics in a melt is a result of the inter-chain interactions causing reweighting of configurations, but not of the intra-chain conformational distribution.) Thus, we characterise the intra-chain contribution to the free energy by the number of states which correspond to a specified moment of inertia tensor of the chain. Denoting this

R

and its eigenvalues R I ,

R2,

R3 (with RI

>

Rz

>

R3), we generate microscopically, in a similar manner to that described in the previous sections, a probability distribution P ( R ) . To each belongs an average intra-chain monomer density distribution p(r,I?) which is sampledas well. Here r is the position vector from the centre-of-mass in the principal frame. The averaging for ^p is carried out over all conformations with a given

B.

Taking into account that the set of allowed conformations of individual chains in the melt and for the isolated chain are identical, the intra-chain contribution to the free energy from chain i is simply given by

Now we assume that the inter-chain interaction is given by the pairwise overlap of the ellipsoids of the different chains. Since each inertia tensor corresponds to a density distribution, we can write for the inter-chain free energy contribution of the pair ij

Here each of the two density distributions is centered on the centre-of-mass of the corre- sponding chain, and E ( N ) is an adjustable parameter accounting for the binary excluded volume as well as the overlapping contribution of the probability distributions. For tech- nical details refer to ^[86].

Computer simulations in soft matter science

20.0

i

15.0

I

J

C=0.67

i

179

melt value

10.0 I ^I , _I

0.0 500.0 1000.0 1500.0

M

CS/pa rt icl e

Figure 21. Evolution of the sample averaged (RL) as a function of Monte Carlo time (4000 particles of N=50). The initial value of E ( N ) = C = 1.0 was changed to values indicated after 600 M C steps. The indicated melt value corresponds to a comparable system with explicit chains with repulsive Lennard Jones interactions and a number density of 0.85 ^{C T - ~} [86].

Here we test this idea for simple coarse-grained polymer models. An extension to a more refined coarse-grained model for e.g. polycarbonate should be straightforward and is an objective of future work. Typical systems consist of 10000 chains of N = 100 monomers.

The simulation procedure is a standard Metropolis Monte Carlo procedure as described in Section 3. The ellipsoidal particles are first randomly distributed in the system, with a distribution of shapes corresponding to isolated SAWS. Then MC simulation is performed such that the ellipsoids can move in space (translation) and can change both the length and the orientation of their principal axes (shape deformation). Figure 21 gives a typical evolution plot of the ensemble averaged-squared radius of gyration of our ellipsoids, as a function of Monte Carlo time steps, for different parameters E ( N ) .

The adjustment of ^E allows a precise mapping of the ellipsoidal model onto the explicit chain models at a given density. To show not only that the end-to-end distance of the ellipsoidal system in the melt agrees with the explicit chain simulation, but also that the chain statistics correspond to Gaussian statistics, we scale the resulting probability distribution function of the radius of gyration for different chain lengths within the random walk scaling scheme. Figure 22 shows this for chain lengths between 25 and 100. Various other control investigations such as the scaling of the correlation hole (the locally reduced density of other chains produced by the self-density of the chain under consideration)

180 Kurt Kremer

3.0

2.0

d-

g

-2 -

c

^1.0

B r

5i

0.0 0.0 0.5 1

.o

1.5 2.0 2.5

R,2/(NoZ/6)

0

Figure 22. Scaled distribution functions of R2,(N) versus the Gaussian normalised value of

I&

for chain lengths as dndicuted [86].

support the conclusion that our chains are now Gaussian. In a very similar way as in earlier studies on phase separations of polymers, one can introduce an E ( N ) which is able to distinguish between two different species. By doing this one is able to investigate phase separation kinetics and morphology development of huge polymer samples. The next step will be the reintroduction of the explicit chains in order to complete the scheme as indicated in Figure 1.

Acknowledgments

The present chapter is a result of longstanding collaborations with many colleagues through the last years. The work on coarse-grained models especially benefitted from a longstanding collaboration with G. ^S. Grest, B. Dunweg, K. Binder, R. Everaers, and M. Putz. CPU time was granted by the German Supercomputer Centre, HLRZ Jiilich.

The multiscale simulations were supported by the BMBF under Grant No. 03N8008EO and the Bayer Corp.. We especially thank the other project partners (group of K. Binder, group of

D.

W. Heermann, group of D. Richter, group of U. Suter and the Bayer polymer physics group) and M. Murat for many helpful disuccions and fruitful collaboration over the years.

Computer simulations in soft matter science ¹⁸¹

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A Molecular Dynamics Study, Langmuir 15 4033

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Equilibrium and flow properties of

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