Self-avoiding limit

Since u e/8 for chains close to 4 dimensions, R ) of such a chain in the self-avoiding limit is given by replacing u in eq 2.15 with e/8, i.e.. 

For e 0 this ratio in the self-avoiding limit is equal to (1/6)[1 — (e/96)], which yields 0.165 for e = 1. As mentioned in Chapter 2, the corresponding value estimated from computer simulation experiments is 0.155. The difference cannot be neglected simply. 

In a veiy good solvent, i.e., in the self-avoiding limit, u tends to u, so u —> 1, which gives jy oo and hence cj —> 1. Thus we find with eq 3.31 and 3.32 that V for 3-dimensional chains in the self-avoiding limit is... 

Thus the Douglas-Freed RG theory allows us to determine the self-avoiding limit of q r as a function of 2, regarding not only the exponent to but also the prefactor. Probably the prefactor value 1.63 is approximate, but its proximity to the computer simulation value 1.64 (see Section 1.4 of Chapter 2) is worthy of note. 

The ratio as /aR at the self-avoiding limit is found to be 0.982 from eq 3.53 and 3.56, which is considerably different from the simulation experimental value 0.93 (see Section 1.6 of Chapter 2). 

Fixed points A and B are version. of free draining of the chain while C and D relate to a nondraining chain. The fixed points with u = 0 correspond to the 0 state, while u = 7T e/2 correspond to the self-avoiding limit (the maximum good solvent). 

In the case of a nondraioing coil in the self-avoiding limit (point P) = 27r e, so... 

Accx)rding to Equations 34 and 36, expressions for D for a nondrainiiig coil in the maximum good solvent (the self-avoiding limit)... 

Thereby, F represents by itself a free energy of random walks independent on the conformational state of a chain F(x) brings a positive contribution into F and the sense of this consists in a fact that the terms F(x) and S(x) represent the limitations imposed on the trajectories of random walk by request of the self-avoiding absence. These limitations form the self-organization effect of the polymeric chain the conformation of polymeric chain is the statistical form of its self-organization. 

A self-avoiding walk corresponds to the limit of a restricted walk as r- oo. Such a walk is no longer Markovian, and no general analytical methods have been found for determining its properties. However, evidence has been accumulating to indicate that the basic characteristics differ in an essential manner from those of restricted walks. We now proceed to discuss this evidence in further detail. 

The majority of attention in earlier discussions of self-avoiding walks was focused on the behavior of mean square length with increasing n, and on the question whether converges to a limit, as for restricted... 

Figures 1 a, 2a to compare with djd l and mj(n — I)a , in Figures 1 b, 2b. The limit and slopes in Figure 1 b are exact but the general pattern of behavior of the other plots is sufficiently similar to give us confidence in the conclusions. (The convergence in three dimensions is more rapid since excluded volume plays a smaller part. Similarly the self-avoiding walk approximation provides a closer fit to the correct behavior of the Ising model.)...

The consideration of the sets of mutually self-avoiding walkers has a wider import in revealing a fundamental invariant (the number of walkers) which turns out to be important in transfer-matrix solutions, as applied to polymer graphs extending even to the 2-dimensional limit [141,142]. Also this invariant or "order" has physical implications [71,72,84,85,152]. 

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. ... 

Lax et a/. (1981) have applied scaling law theory to a free polymer behaving as a self-avoiding walk in a thin slab of thickness h. This limits the span of the polymer chains leading to distortions in the direction parallel to the slab plane. The rms end-to-end distance of a self avoiding walk of isolated chains is given by... 

This limiting value of u corresponds to uq = oo, and is called the fixed point of the system. Thus, the L dependence of Q of the self-avoiding chain is given... 

Computer simulations of self-avoiding ring chains have given information on the ratio (5 (r))/(5 (/)) at the limit of large chain length. By suitable extrapolation of data from Monte Carlo simulations of off-lattice chains of N (the number of bonds) up to 99, Bmns and Naghizadeh [78] estimated this ratio to be 0.559. Chen [79] found 0.568 on the basis of Monte Carlo calculations on... 

The various sequences were analysed using standard methods of asymptotic analysis of power series expansions as described in [38]. For self-avoiding wedks and polygons, it is easy to prove that the limit Km oo(cn) " exists by use of concatenation arguments [4], We assume the usual asymptotic growth of the sequence coefficients c, viz ... 

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