Statistics of confined self-avoiding walks. Part I. Chain dimensions and concentration profiles

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Abstract
Self-avoiding (SAWs) and random-flight (RFWs) walks of varying number N of steps have been generated inside spheres of varying diameter R, using a random number generator and an ad hoc computer program. The Monte Carlo samples, usually of 100 000 walks, thus obtained, allowed the determination of the following ratios, as a function of N and R: first, the ratio A=(r(N,R))/(r(N, infinity )), where (r(N, R)) stands for the mean (in modulus) end-to-end distance of an N-step confined walk, and (r,(N, infinity )) the same quantity for an N-step non-confined walk; also, the corresponding ratios B, C and D, for the root-mean-square end-to-end distance (r2)12/, the mean radius of gyration (rg), and, finally, the root-mean-square radius of gyration (rg2)12/. If reduced lengths are used, where the reduction length is of the form Nv, v being a scaling exponent, it is found that scaling, i.e. independence of the above ratios with respect to the step number in the walk, is well obeyed. The scaling exponent is equal to 0.592 for SAWs and to 0.500 for RFWs. In order to determine the concentration profiles of end, mid- and overall steps inside the sphere, the last has been divided in a prescribed number of spherical shells, up to 22, of the same thickness, and the number of steps falling inside each shell registered. Again using reduced lengths, it was thus found that all concentration profiles obey scaling, that is, the concentration profile as a function of the reduced distance from the centre of the sphere is defined through a single curve, whatever the value of N. Our results allow a comparison of the parameters for confined SAWs and RFWs.