Diffusion in spatially and temporarily inhomogeneous media H. Lehr * and F. Sague ´ s Departament de Quı ´mica Fı ´sica, Universitat de Barcelona, 08028 Barcelona, Spain J. M. Sancho Departament d’Estructura i Constituents de la Materia, Universitat de Barcelona, 08028 Barcelona, Spain Received 30 April 1996 In this paper we consider diffusion of a passive substance C in a temporarily and spatially inhomogeneous two-dimensional medium. As a realization for the latter we choose a phase-separating medium consisting of two substances A and B , whose dynamics is determined by the Cahn-Hilliard equation. Assuming different diffusion coefficients of C in A and B , we find that the variance of the distribution function of the said substance grows less than linearly in time. We derive a simple identity for the variance using a probabilistic ansatz and are then able to identify the interface between A and B as the main cause for this nonlinear dependence. We argue that, finally, for very large times the here temporarily dependent diffusion ‘‘constant’’ goes like t -1/3 to a constant asymptotic value D  . The latter is calculated approximately by employing the effective-medium approximation and by fitting the simulation data to the said time dependence. S1063-651X9602111-3 PACS numbers: 64.60.My, 47.27.Sd I. INTRODUCTION Needless to say, diffusion is a very important physical process and of major practical interest in many fields of sci- ence, ranging from statistical optics to diffusion controlled chemical reactions. Usually, when treating diffusion in liq- uids, one considers the medium to be homogeneous, al- though numerous examples exist where this assumption is not appropriate. In this contribution we would like to study the case where the medium is no longer homogeneous, but has dynamically evolving inhomogeneities. We have opted here to use as a medium a solution of a dynamical equation corresponding to a binary alloy phase separation problem, i.e. in this case the well-known Cahn-Hilliard equation 1,2see, e.g., 3–5 for more recent literature on the theoretical aspects and 6–8 on the applied aspects of this subject. This equation describes the phase separation following a quench of a nonmiscible binary mixture with phases A and B ) inside its coexistence curve. It is known see, e.g., 9 and references therein that the solutions to this equation are very structured, their con- figuration depending on the relative concentration of the phases. Despite the importance of the actual physical situation that leads to the Cahn-Hilliard equation, it will serve here more as a model for a dynamically evolving inhomogeneous medium. Because of the properties mentioned it appears to be an ideal candidate because we are able to study the diffu- sion in a rather rigidly structured medium. The idea for the study is now as follows. We assume that the scalar C has different nonvanishing diffusion coefficients in the phases A and B . If we mix these components the diffusion coefficient at every space and time point will be proportional to the amount of phase A and phase B present at this point. Now, we let these concentrations or rather the difference in molar fraction evolve in time according to the Cahn-Hilliard equation. The question of interest then is the temporal development of the mean-square displacement of the dispersed scalar. Therefore let us call  the variable that according to the Cahn-Hilliard equation describes the temporal development of the difference in molar fraction   t = 2  - + 3 - 2   . 1 with initial conditions being   r ,0 = 0 + .  is a uniform random variable, whose actual range is not of critical importance as long as its average vanishes. Here we have chosen   -0.1,0.1 .  0 is the average difference in mole fraction. This average is an important parameter in- sofar as it determines the configuration of the appearing structures or inhomogeneities. So it is known, e.g., that for  0 =0.4( -0.4) droplets of B ( A ) in A ( B ) appear, while for  0 =0 one finds lamellar structures; see Fig. 1. The equilib- rium stable states of this differential equation lie at  =1. Figure 1 shows these three cases. Figure 1a depicts the choice  0 =-0.4, Fig. 1b  0 =0.0, and Fig. 1c  0 =0.4 at time t =200. The average size of the appearing structures grows according to the Lifshitz-Slyozov time law 2, i.e., with t 1/3 . Due to the conservative nature of Eq. 1, though, the total area occupied by the structures is constant. This means that the number of the structures growing with t 1/3 has to decrease with t -2/3 . Considering now the interface i.e., the borders between zones of positive and negative  ), one * Permanent address: I.N. Stranski Insitut, Sekretariat ER 1, Tech- nische Universita ¨t Berlin, Straße d. 17, Juni 112, D-10 623 Berlin, Germany. PHYSICAL REVIEW E NOVEMBER 1996 VOLUME 54, NUMBER 5 54 1063-651X/96/545/50289/$10.00 5028 © 1996 The American Physical Society