Number of self-avoiding chains on a lattice

Though the static properties of polymers in solution are now quite well-known, only a small fraction of them has been proved in a perfectly rigorous manner. We owe the main results, concerning numbers of chains to the British mathematician Hammers ley1 4 who studied the problem around 1960, and some refinements are due to Kesten (1963).5 These are summarized below in a modified form. 

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In a general way, for a self-avoiding chain on a lattice, the average number of contacts per link can be defined by... 

If a polymer solution is modeled by an assembly of self-avoiding walks on a lattice, a basic physical quantity is the osmotic pressure II. Carrying out a simulation with a fixed number JT of chains of length at a lattice of volume V with one of the dynamic algorithms described in Section 1.2.2, the osmotic pressure is not straightforward to sample. If one had methods that yielded the excess chemical potential and the Helmholtz free energy Fjr, one would find II from the thermodynamic relation... 

Let ZN be the number of self-avoiding chains with N links, drawn on a d-dimensional square lattice, from an origin O- The quantity A(N) = N llnZN tends to a limit A when N- oo. ... 

There have been a number of computer simulations of block copolymers by Binder and co-workers (Fried and Binder 1991a,ft), and this work was reviewed in Binder (1994). Although computer simulations are limited due to the restriction on short chain lengths that can be studied, finite size effects and equilibration problems at low temperatures, the advantages are that the models are perfectly well characterized and ideal (monodisperse, etc.) and microscopic details of the system can be computed (Binder 1994). In the simulations by Binder and co-workers, diblocks were modelled as self- and mutually-avoiding chains on a simple cubic lattice, with chain lengths N = 14 to 60 for/ = 1.A purely repulsive pairwise interaction between A and B segments on adjacent sites was assumed. A finite volume fraction of vacancies was included to speed the thermal equilibration process (Binder 1994). 

Then we turn out attention to the self-avoiding walk for a lattice chain. Let, after j steps, one end enter the volume u around the other end more exactly on anyone of the neighboring/- 1 sites. Because of excluded volume effects, immediate reversals are forbidden for the end, so there exist (/- l)ps possible paths (FUs) for another step. The probability of these paths being vacant (unreacted) is 1 - D. The total number of paths available for the end is therefore (/- l)ps (1 -D). Of these only one path can lead to the other end. Thus the probability of the two ends closing a ring can be written as... 

On the other hand, the number of self-avoiding walks is larger than the number of stair-chains on a lattice (Fig. 2.6). On a translation lattice, the stair... 

A polymer liquid can be represented by a set of self-avoiding chains, drawn on a lattice so that each lattice site is on one of the chains but only one. Again, we may try to evaluate the number Z of configurations of the system or its entropy S = In Z. 

Figure 2.26. End-to-end distance of self-avoiding walks on the cubic lattice, plotted as a function of the number of bonds, N, of the chain. The solid line and the dashed line represent the theta chains and athermal chains, respectively. The dash-dotted line has a slope of 1/2. (From Ref. 18.)...

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.). 

A one-dimensional lattice is made of regularly-spaced points on a straight line. Drawing a self-avoiding chain by starting from an origin O is a trivial operation (see Fig. 3.15). For any value N of the number of links, it is possible to draw two chains of length... 

For a self-avoiding chain with N + 1 points (N links) on a square lattice or a simple cubic lattice, a contact can be defined as two points that are non-adjacent on the chain, and nearest neighbours on the lattice. Now let us consider the selfavoiding chains which have a given origin, N links, p contacts, and the same end-to-end vector r let ZN p r ) be the number of configurations of these chains. This number has been counted by W. Orr39 for N < 8 on the square lattice and for N < 6 on the simple cubic lattice. 

The condition of self-avoidance of a random walk trajectory on //-dimensional lattice demands the step not to fall twice into the same cell. From the point of view of chain link distribution over cells it means that every cell cannot contain more than one chain link. Chain links are inseparable. They cannot be tom off one from another and placed to cells in random order. Consequently, the numbering of chain links corresponding to wandering steps is their significant distinction. That is why the quantity of different variants of iV distinctive chain links placement in Z identical cells under the condition that one cell cannot contain more than one chain link is equal to Z I Z-N) ... 

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. 

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