Model self-avoiding walk

C. Domb and F. T. Hioe, to be published J. Chem. Phys. (in press) F. T. Hioe Self-Avoiding Walk Model of a Polymer Chain, Thesis, University of London,... 

In the random walk and self-avoiding walk models, described above, all energies are either zero (no interactions) or infinite (complete exclusion for overlapping monomers). In a more general case, finite, but non-zero interaction energies could be considered. In this case, different states... 

Fig. 16. Moves used to equilibrate coil configurations for the self-avoiding walk model of polymer chains on the simple cubic lattice (upper party end rotations, kinkjump motions and crankshaft rotations f 107]. From time to time these local moves alternate with a move (lower pan) where one attempts to replace an A-chain by a B-chain in an identical coil configuration, or vice versa. In the transition probability of this move, the chemical potential difference Ap as well as the energy change SjF enter. From Binder [2S8]...

Medvedevskikh Yu. G. Statistics of Linear Polymer Chains in the Self-Avoiding Walks Model/Ih. G. Medvedevskikh, Condensed Matter Physics. 2001, vol. 2. Ra 26,209-218. 

Monte Carlo Study of the Interacting Self-avoiding Walk Model in Three Dimensions. 

Big. 7.23 Ratio between the Flory-Huggins critical temperature, Jf = N 1 — y)z(/ 2ks) and the actual critical temperature Tc for the self-avoiding walk model of polymer mixtures on the simple cubic lattice (Fig. 7.3) plotted versus concentration 1 - of sites taken by monomers (upper part) and versus the inverse square root of the chain length (lower part). Upper part refers to Af = 16 (for energy parameters coincide this is marked by a sohd dot). Curves are only drawn to guide the eye. Both the Flory approximation, eq. (7.34), which implies = 1, i.e., a horizontal straight line, and the Guggenheim... 

In Section 7.2.4 it was shown that via a finite size scaling analysis a meaningful extrapolation of simulation data to the thermodynamic limit is possible, and in this way one can extract estimates for both critical exponents (/ , 7) and amplitudes C+(1V), C- N) and B N) of the collective scattering function above Tc (eq. [7.16]) and below Tc -ScoiK = 0) =C- N)r, t = I — T/Tc — 0 or the order parameter (/>" )= B lf)t, respectively. While for short enough chains N < 32), data both for the simple self-avoiding walk model of Fig. 7.3 and for the bond fluctuation model are nicely consistent with the expected critical exponents for the three-dimensional Ising models /3 0.325,7 Pi 1.241), for N>64 one rather finds effective exponents ... 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

The simplest model of polymers comprises random and self-avoiding walks on lattices [11,45,46]. These models are used in analytical studies [2,4], in particular in the numerical implementation of the self-consistent field theory [4] and in studies of adsorption of polymers [35,47-50] and melts confined between walls [24,51,52]. 

A self-avoiding walk on a lattice is a random walk subject to the condition that no lattice site may be visited more than once in the walk. Self-avoiding walks were first introduced as models of polymer chains which took into account in a realistic manner the excluded volume effect1 (i.e., the fact that no element of space can be occupied more than once by the polymer chain). Although the mathematical problem of... 

To obtain a clearer insight into the relationship between the Ising and self-avoiding walk enumerations, it is better to pass to the logarithm of the partition function, In Z, since this eliminates the need to consider disjoint graphs. We can then find a property of the Ising model which corresponds to each of C , u , (R 2), f (x) dx. The Ising enumerations are much more... 

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.)...

In his paper Domb presents a detailed analysis of the statistical properties of self-avoiding walks on lattices.1 These walks serve as models for linear polymer chains with hard-core intramolecular interactions associated with the exclusion of multiple occupancies of the lattice sites by the chain so-called chains with excluded volume. 

This subsection has to be very brief, since only rather preliminary qualitative studies are available [34,35]. The model used are self-avoiding walks with N=16... 

It should also be remarked that, in attempts to find the actual configurations of macromolecules, lattice models have played important roles (73). The main interest here is the investigation of self-avoiding walks on a given lattice as a model of a real chain. One tries to find, for example, the mean square length of the random walk as a... 

The bond fluctuation model [72] is used to simulate the motion of the polymer chains on the lattice. In this model, each segment occupies eight lattice sites of a unit cell, and each site can be a part of only one segment (self-avoiding walk condition). This condition is necessary to account for the excluded volume of the polymer chains. For a given chain, the bond length between two successive seg-... 

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