How to Measure Subdiffusion Parameters

Abstract

We propose a method to measure the subdiffusion parameter α and subdiffusion coefficient Dα which are defined by means of the relation x 2 = 2Dα Γ(1+α) t α where x 2 denotes a mean square displacement of a random walker starting from x = 0 at the initial time t = 0. The method exploits a membrane system where a substance of interest is transported in a solvent from one vessel to another across a thin membrane which plays here only an auxiliary role. We experimentally study a diffusion of glucose and sucrose in a gel solvent, and we precisely determine the parameters α and Dα, using a fully analytic solution of the fractional subdiffusion equation.PACS numbers: 05.40.-a, 66.10.-x Subdiffusion occurs in various systems. We mention here a diffusion in porous media or charge carriers transport in amorphous semiconductors [1,2]. The subdiffusion is characterized by a time dependence of the mean square displacement of a Brownian particle. When the particle starts form x = 0 at the initial time t = 0 this dependence in a one-dimension system iswhere D α is the subdiffusion coefficient measured in the units [m 2 /s α ] and α obeys 0 < α ≤ 1. For α = 1 one deals with the normal or Gaussian diffusion characterized by the linear growth of x 2 with t which results from the Central Limit Theorem applied to many independent jumps of a random walker. The anomalous diffusion occurs when the theorem fails to describe the system because the distributions of summed random variables are too broad or the variables are correlated to each other. The subdiffusion is related to infinitely long average time that a random walker waits to make a finite jump. Then, its average displacement squared, which is observed in a finite time interval, is suppressed. The subdiffusion has been recently extensively studied, see e.g. [1][2][3][4]. While the phenomenon is theoretically rather well understood there are very a few experimental investigations. There is no effective method to experimentally measure α and D α . In the pioneering study [4], where D α was determined experimentally for the first time, the interdiffusion of heavy and light water in a porous medium was observed by means of NMR. D α was found, using the special case α = 2/3 solution of the subdiffusion equation. The procedure is neither very accurate nor of general use.Our aim here is to present a method to precisely measure α and D α . The method is described in detail in [5], here we give a brief account of it. For practical reasons, we choose for the experimental study a membrane system containing two vessels with a thin membrane in between which separates the initially homogeneous solute of the substance of interest from the pure solvent. A schematic view of the system is presented in Fig. 1. The membrane does not affect values of investigated parameters. Instead of the mean square displacement (1), our method refers to the temporal evolution of the thickness δ of the so-called near-membrane layer which is defined as a distance from the membrane where the substance concentra...

Random walk model of subdiffusion in a system with a thin membrane

“…We add that the initial condition (65) often occurs when an experimental study of subdiffusion or normal diffusion in a membrane system are conducted, see [2,13,26] and the references cited therein. Let us denote…”

“…In general, the values of the parameters and variables are given in arbitrarily chosen units. However, if we assume that the time unit is second and the space variable unit is a millimeter, then the parameter D α is close to the real value of the subdiffuson coefficients of sugars in a water agarose solution [2] for which α = 0.9. For the normal diffusion of ethanol in a water solvent in a system with artificial nucleopore membrane the values of γ 1 and γ 2 are of the order 10 −1 ÷ 1 [25].…”

“…The process of subdiffusion can occur in media in which the particles' movement is strongly hindered due to the internal structure of the medium, as, for example, in gels [2]. Subdiffusion is usually described by the following subdiffusion equation with the Riemann-Liouville fractional time derivative (here 0 < α < 1) [1] ∂C(x, t) ∂t…”

Time Evolution Of The Reaction Front In A Subdiffusive System

“…Let us note that comparing the right hand sides of Eqs. (A5) and (A6), after simple calculations we get the useful relation (15).…”

“…Subdiffusion occurs in systems where mobility of particles is significantly hindered due to internal structure of the * Electronic address: tkoszt@pu.kielce.pl † Electronic address: kale@amg.gd.pl medium, as in porous media or gels [14,15]. The subdiffusion is characterized by a time dependence of the mean square displacement of transported particle ∆x 2 = 2D α t α /Γ (1 + α), where D α is the subdiffusion coefficient measured in the units m 2 /s α and α is the subdiffusion parameter which obeys 0 < α < 1.…”