Short CommunicationNormal diffusion in a medium connected to a subdiffusive medium with absorption
Introduction
Anomalous diffusion occurs in media in which random walk of particles is strongly hindered due to the complex internal structure of the medium. Within the continuous time random walk formalism subdiffusion is a process in which the average time to wait for a particle jump is infinite, while for normal diffusion it is finite; in both cases the average particle jump length is finite. The type of diffusion is usually defined by the relation , where is the mean–square displacement of the diffusing particle; for 0 < α < 1 we have subdiffusion, for α = 1 – normal diffusion, for α > 1 – superdiffusion. Subdiffusion is described by a differential equation with a fractional time derivative (Metzler and Klafter, 2000, Metzler and Klafter, 2004, Klafter and Sokolov, 2011). The situation is more complicated when the diffusion takes place in a system composed of two different media connected together (Kosztołowicz, 2018, Kosztołowicz, 2017a, Kosztołowicz, 2017b, Kosztołowicz et al., 2017). In this paper, we consider a system consisting of a medium A, in which normal diffusion occurs, connected to a subdiffusive medium B in which absorption of diffusing particles may take place. Absorption can be treated as an irreversible reaction P + AC → AC, where P is the diffusing particle and AC represents static absorption centres which are located in the medium B. When the particle P meets AC, then the reaction may occur with some probability controlled by the absorption parameter κ. Such systems are found in biology and medicine. An example of this is the acids diffusion from saliva into the tooth enamel, which is a porous medium, see Lewandowska and Kosztołowicz (2012), Kosztołowicz and Lewandowska (2006) and the references cited therein. This process may cause the development of caries in the enamel. Another example is diffusion of antibiotic into a bacterial biofilm (Steward, 1996). Defending against antibiotic molecules, bacteria produce a thick mucus in the biofilm, which can absorb the molecules. A similar problem, namely diffusion-limited corrosion of static porous copper clusters in thin gap cells containing diffusive cupric chloride, but for the normal diffusion–absorption process in the medium B was considered in Léger et al. (1999). Since subdiffusion may occur in gels or porous media (Metzler and Klafter, 2004, Kosztołowicz et al., 2005), in further considerations we assume that there is subdiffusion in the medium B. However, the results presented later in this paper can be used to describe normal diffusion with absorption assuming α = 1.
The aim of this paper is to present the diffusion model in the system which consists of a diffusive medium A and subdiffusive medium B, in the latter one absorption of diffusing particles may occur, see Fig. 1. Additionally, we assume that the boundary between media is an obstacle for particles diffusing from the medium A to B, but it does not constitute any obstacle when the particles diffuse in the opposite direction. This is the case, among others, when B is a porous medium. Then, the particle that attempts to enter the medium B has ‘to hit’ the channel of the porous medium, with some probability, and when leaving the channel in the medium B and going to the medium A it does so without any obstacles. As far as we know, a model of such system has not been considered yet.
Section snippets
System
We assume that the three-dimensional system is homogeneous in a plane perpendicular to the x axis; thus, we treat the system as one-dimensional. The parameters describing the processes are constant, they depend neither on the concentration of diffusing particles nor on the variables x and t. The system is presented in Fig. 1.
The concentration in the medium A, CA, fulfils the normal diffusion equation, whereas the concentration in part B, CB, fulfils the subdiffusion-absorption equation with
Results
The solutions to Eqs. (1) and (2) are qualitatively different for the κ ≠ 0 and κ = 0 cases. The procedure of solving the equations by means of the Laplace transform method is described in more detail in Kosztołowicz (2018).
How to identify absorption in a subdiffusive medium B observing diffusion in a medium A
Based on the function CA we are going to find a function which is qualitatively different for the cases of κ ≠ 0 and κ = 0. Finding of such function is motivated by practical reasons. Namely, determining it experimentally we can learn if there is absorption in the medium B without observing the process in this medium. Taking into account the fact that both functions (14) and (18) depend on the function (11), we introduce a new function Ξ, defined for both cases as follows
Conclusions
In this paper, we present the model of diffusion in a system in which a medium with normal diffusion is connected to a subdiffusive medium in which the absorption of diffusing particles may take place. We have derived functions ΞA,κ and ΞA,0 that are qualitatively different for the cases of κ ≠ 0 and κ = 0. Knowing the diffusion coefficient DA we can determine the function WAmax. Since WA,κ and WA,0 are measurable experimentally, for example using an interferometric concentration measurement
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