The self-avoiding walk SAW

For previous reviews of Monte Carlo methods for the self-avoiding walk, see Kremer and Binder and Madras and Slade (Ref. 11, Chapter 9). 

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All the theoretical work described so far has assumed conformational ideality. That is, the intramolecular pair correlations are presumed to be independent of fluid density (and composition in an alloy) and can be computed based on a chain model that only accounts for short-range interactions between monomers close in chemical sequence. This assumption can fail spectacularly in dilute good solution where the effective intrachain monomer-monomer interaction is repulsive in a second virial coefficient sense. " For such good solvent conditions, the polymer mass/ size relationship no longer obeys the ideal random walk scaling law R but follows the self-avoiding walk (SAW) law i /V" with v = ... 

Let us consider a simple self-avoiding walk (SAW) on a lattice. The net interaction of solvent-solvent, chain-solvent and chain-chain is summarized in the excluded volume between the monomers. The empty lattice sites then represent the solvent. In order to fulfill the excluded volume requirement each lattice site can be occupied only once. Since this is the only requirement, each available conformation of an A-step walk has the same probability. If we fix the first step, then each new step is taken with probability q— 1), where q is the coordination number of the lattice ( = 4 for a square lattice, = 6 for a simple cubic lattice, etc.). 

For a self-avoiding walk (SAW) the coil end-to-end distance, R, scales with,... 

A more realistic model is that of chains with excluded volume, i.e., selfintersections of the chain are not allowed because two monomers occupying the same lattice site have infinite interaction energy. These chains, which constitute a subgroup of the ideal chains, are also called self-avoiding walks (SAWs) (see Figure 4). Because of the lack of finite interaction energy in this model, the SAWs are equally probable, and the partition function is the total number of different SAWs, which, in contrast to ideal chains, is unknown exactly... 

Figure 4 Chains of N = 11 steps (bonds), that is, 12 monomers (solid circles) on a square lattice (open circles). Immediate chain reversals are not allowed therefore the maximum directions v available are 4 for the first step and 3 for the later steps, (a) An ideal chain (random walk) starting from the origin (1) the chain intersects itself and the last step (dashed line) goes on the third one. (b) A self-avoiding walk (SAW) is not allowed to self-intersect. The SAWs are a subgroup of the ideal chains, (c) A SAW with a finite interaction e between nonbonded nearest-neighbor monomers. The total energy of the chain is 6e.

The authors [4, 5] considered random walks without self-intersections or self-avoiding walks (SAW) as the critical phenomenon. Monte-Carlo s method within the framework of computer simulation confirmed SAW or polymer chain self-similarity, which is an obligatory condition of its fac-tuality. Besides, in Ref [4], the main relationship for polymeric fractals at their treatment within the framework of Flory conception was obtained ... 

We introduce first the (lattice) Self-Avoiding Walk (SAW) model of polymer chains, their critical statistics and the criteria indicating effects of lattice disoder on the critical behaviour. Prominent indications for the effect of disorder on the SAW statistics are then discussed. Next, some mean field and scaling arguments are discussed for the SAW statistics in disordered medium percolating lattice in particular. 

The generator of the self-avoiding walk statistics or the distribution function Gjv(r), which represents the number of A-stepped SAW configurations with end-to-end distcince r, is not Gaussian as in the case of random walk (for random walk GN(r) exp[—r /A]). From the distribution function GAr(r), one can obtain the asymptotic behaviour of various moments. A brief summary is as follows The statistics of SAWs are characterised by the connectivity constant fi (Gn = M A" ), which is nonuniversal and depends... 

Self-avoiding walks (SAWs) constitute the simplest, yet non-trivial model for studying the static behavior of a linear polymer embedded in a good solvent. Such a polymer is a chain-like array of TV -t-1 monomers rigidly connected to each other, in which the only residual interaction between non-consecutive monomers is a (short-ranged) monomer core repulsion [1-6]. 

It has been noticed several times [3, 4] that the configuration of soft polymer chains differs from the RW trajectory in one important aspect it must not intersect. This limitation, known as long-range ordering effect or excluded volume effect, requires new statistics, i.e., statistics of self-avoiding walks (SAW). The attempts made so far [3] have not succeeded in solving this problem completely. 

We see that the direct data on coils are not quite conclusive. It is then helpful to return to theoretical calculations. There do exist rather accurate numerical studies on real chains on a lattice. The chain is still represented by a random walk as in Fig. (1.1), but the main difference is that now this walk can never intersect itself. We call it a self-avoiding walk (SAW). 

Equation [2] is known as a simple sampling MC method, and such techniques were already in use in the 1950s in a polymer context and stiU find some applications more recendy, " but only for the study of self-avoiding walks (SAWs) on lattices, for rather short chain lengths N. Although SAW is one of the most primitive models for a macromolecule in dilute solution trader good solvent conditions, this model is still widely in use, as its simplicity allows the use of both elaborate mathematical methods and very efficient simulation algorithms. " ... 

A polymer chain with an excluded volume can be modeled by a self-avoiding walk (SAW) on the lattice. Unlike the random walker we have seen in Section 1.2 for the ideal chain, this walker is not allowed to visit the sites it has already visited. Its trajectory is close to the conformation of a real chain with excluded volume on the lattice. For the SAW to represent a real chain, the SAW must be equilibrated by moving the chain around on the lattice. Figure 1.39 illustrates a difference between the random walk and the SAW on a square lattice. Apparently, the dimension of the latter is greater The excluded volume swells the chain. SAW is widely used in computer simulations to calculate properties of the polymer chain that are difficult to obtain in experiments. 

Problem 1.11 A self-avoiding walk (SAW) generated on a lattice is slightly different from the trace of a real chain on the same lattice. Explain why. 

Lattice sites are labeled in the order in which they are visited, starting out from the origin (0). Each step consists in adding at random an elementary lattice vector [( l,0)fl, (0, l)a, where a is the lattice spacing] to the walk, as denoted by the arrows, (b) Same as (a) but for a non-reversal random walk (NRRW), where immediate reversals are forbidden, (c) Two examples of self-avoiding walks (SAWs), where visiting any lattice site more than once is not allowed trials where this happens in a simple random sampling construction of the walk are discarded. From Kremer and Binder [2]. 

Figure 2.13 Normalized probability distribution of the end-toend distance. The top curve is for the Gaussian chain (random walk, RW) and the bottom curve is for a fully swollen coil (self-avoiding-walk, SAW).

For long flexible polymer chains it has been customary for a Imig time [1, 2] to reduce the theoretical description to the basic aspects such as chain connectivity and to excluded volume interactions between monomers, features that are already present when a macromolecule is described by a self-avoiding walk (SAW) on a lattice [3]. The first MC algorithms for SAW on cubic lattices were proposed in 1955 [164], and the further development of algorithms for the simulation of this simple model has continued to be an active area of research [77, 96, 165 169]. Dynamic MC algorithms for multichain systems on the lattice have also been extended to the simulation of symmetric binary blends [15, 16] comprehensive reviews of this work can be found in the literature [6, 81, 82]. It mms out, however, that for the simulation both of polymer blends [6, 9, 21, 82, 170, 171] and of solutions of semiflexible polymers [121 123], the bond fluctuation model [76, 79, 80] has a number of advantages, and hence we shall focus attention only on this lattice model. 

Flexible lattice polymers are typically modeled by self-avoiding walks (SAW). The total number of conformations for a chain with N monomers is not known exactly. For N oo... 

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