General considerations

Now let us turn to some specific questions related to polymers. Compared to simulations of small molecules, polymers (like many other forms of soft matter) require special at- tention due to the huge number of intra-molecular degrees of freedom. This causes both computational advantages and disadvantages. Polymers are, of course, chain molecules built of repeat units called monomers or (by physicists) ‘beads’. Examples range from the simple, widely used, PE to the technically very important but more complicated BPA-PC, which is used, among other things, for compact discs:

PE (CH2)N polyethylene

PS

P E 0 ((CH2)20), polyethylene oxide

BPA-PC ( C,&C(CH3)2CSH4C03)N bisphenyl A polycarbonate

152 Kurt _Kremer

While three out of the above examples are only soluble in organic solvents, PE0 is also water soluble. Chemistry and biology of course provide many more complex examples.

The molecular weight of a single polymer molecule can easily reach ^N lo5 Daltons (several thousands of monomers). Full scale MD simulations are not suitable (Figure l), but in any case, would be restricted to the study of a very few specific systems. Even then it is very difficult to determine proper intra- and inter-molecular interaction potentials, and one would also need an effective and computationally convenient parametrisation for them. In particular the complicated inter-chain or polymer-solvent interactions are only poorly understood at present.

Despite these seemingly tremendous complications, the situation is not that bad. To what extent do we really need the chemical details? Figure 1 gives a caricature of the various relevant scaling regimes. It also illustrates universality. Properties, which are governed by length scales larger than a few Kuhn lengths are independent of the chemistry.

For example in the limit of long chains of N monomers the mean squared end-to-end distance ( R 2 ( N ) ) = AN’”. The exponent v is universal, and takes values 0.588 ^N 6/5 in a good solvent (for d ^{= 3)} and 0.5 in the melt or in a Theta solvent (see Khokhlov, this volume). The chemical details are hidden in the prefactor A . This suggests a relatively satisfactory situation, namely for most physics questions one can confine the simulation to the simplest and, for computational purposes, fastest models. In many cases we just need the monomer-monomer excluded volume and the chain connectivity. These models are often called coarse-grained models. Later on, we will face the problem of ‘mapping properties back to a given chemical system’ in order to determine amplitudes (like A ) _as well.

Figure 2. Typical models for polymers used in simulations; (a) SAW o n lattice, (b) pearl necklace, (c) bead-spring model.

We now apply the previously described techniques to coarse-grained polymer mod- els. Typically, three classes of models are used for computer simulations of polymers, see Figure 2. The first and historically most widely used model of a single chain is the self- avoiding walk (SAW) on a lattice [3]. In such a walk, each lattice point can be occupied only once; it is trivial to introduce nearest-neighbour energies and other generalisations.

The second model is the direct generalisation of this for continuous space, the pearl neck- lace model. The chain consists of hard spheres of diameter ^CO and a fixed bond length

&.

The third variant is a bead-spring model, mainly used in MD simulations: the monomers are particles which interact with, in most cases, a purely repulsive Lennard Jones inter- action (Equation 1). For the bonded nearest-neighbours along the chain, an additional

Computer simulations in soft matter science 153

spring potential is added which, together with the repulsion, determines the bond length The purpose of a simulation is to generate statistically independent conformations (equilibrium) or to follow the time evolution of a given global conformation (dynamics).

Again, the most natural ansatz would be to simply perform a MD simulation, where Newton’s equations of motion (Equation 2) for a bead spring model are solved numerically.

However, this direct approach cannot be used for an isolated chain. The reason lies in its structural properties: a linear-chain polymer without excluded volume interactions has the structure of a random walk (harmonic chain). Its structure and dynamics can be described by eigenmodes, the Rouse modes, which decouple [23]. Thus solving Equation 2 exactly, without excluded volume or external noise, would never equilibrate the chain.

For a chain with excluded volume, the Rouse modes are no longer eigenmodes. How- ever, the deviations, which affect the large length scales, come mainly from ‘long range’

contacts between monomers which are far apart along the chain but close by in space. For long chains the internal monomer density decays with a power law N1-d”. (The chains are fractal objects with a fractal dimension d f ⁼ l / u ; see Pine, this volume). Consequently, these ‘long range’ collisions, which cause the swelling of the chains, are very infrequent:

equilibration is not guaranteed. The single SAW is an example of the direct relevance of the aforementioned Fermi-Pasta-Ulam problem, which to date is not yet solved [24].

Thus the natural MD approach can only be used for a chain interacting directly with solvent or other chains, or for chains with long range interaction potentials. To avoid this problem one has to couple the motion of chain beads to a stochastic process, for example a heat bath comprising a friction and a random force. In the overdamped limit this results in Brownian dynamics. The other approach is the dynamic Monte Carlo method, where again the conformation changes come from local stochastic jumps. Both approaches follow realistically the (Langevin) dynamics of a chain. Though this leads to information about chain dynamics as well as statics, this should be avoided if the dynamic information is not needed. To illustrate this, we compare in Table 1 a simple liquid (or a lattice gas) and a polymer melt of the same density and total number of particles. The comparison in the

e.

Table ^1. An illustration of the time scales in polymer simulations

Table assumes that we are not close to a critical point of the liquid, where critical slowing down occurs. But in fact, the exponent ^z = 2 . . .3 . 4 in the relaxation time r ^N N z of polymers is analagous to that for critical slowing down [23, 251. (In the language

154 Kurt Kremer

of critical phenomena the inverse chain length corresponds to (2' - T C )/ Tc .) This N - dependent slowing causes significant problems since N values of interest start at around 50 to 100. As a result there is one important rule for polymer simulation: if possible, avoid algorithms with realistically slow physical dynamics. (However, if one wants to study chain dynamics itself, there is no alternative: see Section 4.)