The spatial evolution of particles diffusing in the presence of randomly placed traps

Fluctuations and self-averaging in random trapping transport: The diffusion coefficient

On the basis of a self-consistent cluster effective-medium approximation for random trapping transport, we study the problem of self-averaging of the diffusion coefficient in a nonstationary formulation. In the long-time domain, we investigate different cases that correspond to the increasing degree of disorder. In the regular and subregular cases the diffusion coefficient is found to be a self-averaging quantity-its relative fluctuations (relative standard deviation) decay in time in a power-law fashion. In the subdispersive case the diffusion coefficient is self-averaging in three dimensions (3D) and weakly self-averaging in two dimensions (2D) and one dimension (1D), when its relative fluctuations decay anomalously slowly logarithmically. In the dispersive case, the diffusion coefficient is self-averaging in 3D, weakly self-averaging in 2D, and non-self-averaging in 1D. When non-self-averaging, its fluctuations remain of the same order as, or larger than, its average value. In the irreversible case, the diffusion coefficient is non-self-averaging in any dimension. In general, with the decreasing dimension and/or increasing disorder, the self-averaging worsens and eventually disappears. In the cases of weak self-averaging and, especially, non-self-averaging, the reliable reproducible experimental measurements are highly problematic. In all the cases under consideration, asymptotics with prefactors are obtained beyond the scaling laws. Transition between all cases is analyzed as the disorder increases.

Non-self-averaging in random trapping transport: The diffusion coefficient in the fluctuation regime

The nonstationary diffusion of particles in a medium with static random traps or sinks is considered. The question of the self-averaging of the diffusion coefficient (or, equivalently, of the mean-square displacement) is addressed for the fluctuation regime in the long-time limit. The property of self-averaging is needed for the result of a single measurement to be representative and reproducible. It is demonstrated that the diffusion coefficient of the surviving particles is a strongly non-self-averaging quantity: In a d-dimensional system its reciprocal standard deviation grows with time exponentially ≈exp[const_{d,1}t^{d/(d+2)}]. The same result is reproduced in the "normalized" formulation "per one survivor on average." The case when all the particles, both the survivors and the trapped ones, are contributing to the diffusion coefficient and its variance is considered also. Non-self-averaging is demonstrated for this case as well, the fluctuations of the diffusion coefficient being of the same order as its average value. The critical dimension, above which the mean-field result becomes exact, is infinite-due to the drastic difference between the classes of trajectories, upon which the corresponding results are being built. In high dimensions the strong non-self-averaging of survivors is preserved. For the case of all the particles taken into account, the nonstrong non-self-averaging is retained for any finite dimension. However, for d→∞ the limiting value of the reciprocal standard deviation, calculated for all the particles, decreases to zero. This signifies restoration of the self-averaging in some sense. In all the cases, the time evolution of the average characteristics and of their variances is governed by the decaying concentration of the survivors in fluctuational cavities.

Structure and dynamics of short chain molecules in disordered porous materials: A molecular dynamics simulation study

The static and dynamic properties of short polymer chains in disordered materials are studied using discontinuous molecular dynamics simulations. The polymers are modeled as chains of hard spheres and the matrix is a collection of fixed hard spheres. The simulations show that the chain size is a nonmonotonic function of the matrix concentration for all polymer concentrations. The dependence of polymer diffusion D on the degree of polymerization N becomes stronger as the matrix concentration is increased. At high matrix concentrations we observe a decoupling between translational and rotational diffusion, i.e., the rotational relaxation time becomes very large but the translational diffusion is not affected significantly. We attribute this to the trapping of a small number of polymers. Under these conditions the polymer chains diffuse via a hopping mechanism.

Monte Carlo simulation and self-consistent integral equation theory for polymers in quenched random media

The conformational properties and static structure of freely jointed hard-sphere chains in matrices composed of stationary hard spheres are studied using Monte Carlo simulations and integral equation theory. The simulations show that the chain size is a nonmonotonic function of the matrix density when the matrix spheres are the same size as the monomers. When the matrix spheres are of the order of the chain size the chain size decreases monotonically with increasing matrix volume fraction. The simulations are used to test the replica-symmetric polymer reference interaction site model (RSP) integral equation theory. When the simulation results for the intramolecular correlation functions are input into the theory, the agreement between theoretical predictions and simulation results for the pair-correlation functions is quantitative only at the highest fluid volume fractions and for small matrix sphere sizes. The RSP theory is also implemented in a self-consistent fashion, i.e., the intramolecular and intermolecular correlation functions are calculated self-consistently by combining a field theory with the integral equations. The theory captures qualitative trends observed in the simulations, such as the nonmonotonic dependence of the chain size on media fraction.