Coarse-grained models of polymer chains

The rotational isomeric state model (RIS) is a model chain in which chain conformation is represented by the set of three states, t, g= =. 

This is a Gaussian distribution with a mean square separation (/ ) =aP- between adjacent beads. The bead in a BS chain also indicates a group of monomers as in RF. 

The Gaussian bond (1.4) can easily be stretched to high extension, and allows unphysical mutual passing of bonds. To prevent this unreaUstic mechanical property, the model potential, called the finitely extensible nonlinear elastic potential (FENE), and described by 

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The bond fluctuation model (BFM) [51] has proved to be a very efficient computational method for Monte Carlo simulations of linear polymers during the last decade. This is a coarse-grained model of polymer chains, in which an effective monomer consists of an elementary cube whose eight sites on a hypothetical cubic lattice are blocked for further occupation (see... 

In addressing the coarse-grained models of polymer chains, we first describe the skeletal models without any excluded volume effects, by considering the special case of = 0. We shall address the role of the excluded volume effect in Sections 2.5 and 2.6. 

This conclusion therefore suggests that we could build an equivalent freely jointed chain, without compromising the Afo-dependence of the polymer size given by Equation 2.20, by defining an effective bond length, which is renormalized by incorporating the consequences of Coo. Such a model is the Kuhn chain model and is the basis of all coarse-grained models of polymer chains. 

There is no unique way to construct coarse-grained models of polymer systems. In fact, the choice of model very much depends on the physical problems that one may wish to address, and also many details are fixed from the desire to construct computationally efficient simulation algo-rithms. Thus many variants of models for polymer chains exist, and there may still be the need to invent new ones This section is also not intended to be exhaustive, but rather restricts attention only to the most common models which are widely used in various contexts, as will become apparent in later chapters of this book. 

One of the possible extensions of the HP model is the HA side chain model introduced in [212]. This is a more realistic coarse-grained model of amphiphilic polymers where the dualistic character of each monomeric unit is explicitly represented. 

Finally, we would like to give a short outlook on the stmcture of a single homopolymer chain confined to a very small miniemulsion droplet. Advanced Monte Carlo methods [224] were applied to a simple coarse-grained model of polystyrene in spherical confinement [225]. The polymer chain becomes highly knotted once the confining droplet shrinks (e.g., by evaporation of the solvent) beyond the typical size of the polymer (in good solvent conditions) [225-227]. These simulations may hence lead the way to the synthesis of knotted and unknotted ring polymers in extremely small miniemulsion droplets when the termini of the polymer are chemically linked. 

Fig. 1.2 Typical coarse-grained models of a polymer chain (a) random flight model, (b) bead-spring model, (c) lattice model.

Fig. 5 Illustration of the intemrntation of the coarse-grained models for polymers and solvent In the case of short alkane typically threeC-C bonds are taken together in L effective unit (dotted arcle). The oligomer Q5H34 Cont u,g 50 toms or 16 united atoms, is thus rnduced to an ° XT y 4 4 onng beads along a chain interact with a combination of LJ...

Np —> 00, where the exponent u for good solvents is universal, i.e., independent of the chemical nature of both polymer and solvent, while the prefactor Cp clearly is not. In a coarse-grained model of a polymer chain we can integrate n chemical monomers into one effective subunit (cf. section 1.3) and thus consider an equivalent chain of N = Np/n segments. While we still expect = CN, the explicit... 

Fig. 1.6 Use of the bond fluctuation model on the lattice as a coarse-grained model of a chemically realistic polymer chain (symbolized again by polyethylene). In the example shown n = 3 covalent bonds form one effective bond between effective monomers chemical bonds 1,2,3 correspond to the effective bond I, chemical bonds 4,5,6 to the effective bond II, etc. (From Baschnagel et al. )...

Coarse-grained models have a longstanding history in polymer science. Long-chain molecules share many common mesoscopic characteristics which are independent of the atomistic stmcture of the chemical repeat units [4, 5 and 6]. The self-similar stmcture [7, 8, 9 and 10] on large length scales is only characterized by a single length scale, the chain extension R. 

Fig. 8.8 The bond fluctuation model. In this example three bcmds in the polymer arc incorporated into a singk effecti bond between effective moncmers . (Figure adapted from Baschnagel J, K Binder, W Paul, M Laso, U Sutcr, I Batouli [N ]ilge and T Burger 1991. On the Construction of Coarse-Grained Models for Linear Flexible Polymer-Chains -Distribution-Functions for Groups of Consecutive Monomers. Journal of Chemical Physics 93 6014-6025.)...

A suitable approach to the equilibration of an amorphous polymer system at bulk density becomes much more likely when the fully atomistic model in continuous space is replaced by an equivalent coarse-grained model on a lattice with sufficient conformational flexibility. Different strategies, which seek results at different levels of detail, can be employed to create an appropriate coarse-grained model. Section 4 (Doruker, Mattice) describes an approach which attempts to retain a connection with the covalent bonds in the polymer. The rotational isomeric state (RIS) [35,36] model for the chain is mapped into... 

The mapping procedure provides an explicit connection between an atomistic model of a polymer chain and the corresponding coarse-grained model. For PE, we use an united atom description for the CH2 groups, resulting in a potential of the following type [146,182,183] ... 

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