Monte Carlo method attractive chains

Mazur and McCrackin used the Monte Carlo method to study attractive chains drawn on simple cubic or face-centred cubic lattices. 

A self-avoiding chain constructed on a lattice by applying the Monte Carlo method, is produced with a probability which is proportional to the factor IF(12) corresponding to the chain configuration 12 (see Section 2.2). To calculate mean values, we must simultaneously correct this effect and take the attractive interactions into account. With each chain with p(12) contacts, is associated a weight [ 1F(12)] -1 w n where w = exp[ - fiJ (with J 0). 

The mean value of any function 0(12) of the chain configurations is given by 

Mazur and McCrackin calculated a few quantities in terms of In w. The link numbers N of the chains which they considered belong to the range 0 N 100 and differ by factors of 10. For each N, the number of constructed chains is of the order of 20000. 

Mazur and McCrackin calculated the mean square distance between end points and for the largest values of N, they represented it in the form 

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The best-known physically robust method for calculating the conformational properties of polymer chains is Rory s rotational isomeric state (RIS) theory. RIS has been applied to many polymers over several decades. See Honeycutt [12] for a concise recent review. However, there are technical difficulties preventing the routine and easy application of RIS in a reliable manner to polymers with complex repeat unit structures, and especially to polymers containing rings along the chain backbone. As techniques for the atomistic simulation of polymers have evolved, the calculation of conformational properties by atomistic simulations has become an attractive and increasingly feasible alternative. The RIS Metropolis Monte Carlo method of Honeycutt [13] (see Bicerano et al [14,15] for some applications) enables the direct estimation of Coo, lp and Rg via atomistic simulations. It also calculates a value for [r ] indirectly, as a "derived" property, in terms of the properties which it estimates directly. These calculated values are useful as semi-quantitative predictors of the actual [rj] of a polymer, subject to the limitation that they only take the effects of intrinsic chain stiffness into account but neglect the possible (and often relatively secondary) effects of the polymer-solvent interactions. 

It is evident that this approach is impractical for this simple problem because the Hory distribution provides the same information in a much more efficient way. The Monte Carlo methods become more attractive when modeling complex microstructures for which no analytical solutions are possible, such as for terpolymers, branched or crosslinked chains, and chains with branching resulting from chain walking with late transition-metal catalysts. The Monte Carlo techniques have been used to model a variety of polyolefin microstructures effectively and are the most powerful, albeit the most computational time consuming, of all modeling techniques [91-96]. 

In the layer with decreased relative permittivity surrounding the ions, the free energy of the solvent is lower than in the absence of the electric field of the ions. The approach of the ions towards one another requires the mutual inter-penetration of the solvate spheres, i.e., the release of a certain amount of solvent from ihc solvate sphere of the ions. This process needs work, and this work appears as a repulsive force between the ions, (This effect lends stability to electrolyte solutions, for in the absence of such repulsive forces, attraction between the charges would favour the precipitation of the solid salts.) By taking into account such repulsive forces, it was possible to interpret the positive deviation of the average activity coefficients of the ions from the Debye-Hiickel limiting law (hypernetted chain equations, HNC, calculation by the Monte Carlo method [Ra 68, Ra 70],... 

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