PHYSICAL REVIEW E 95, 012155 (2017) Structure-diffusion relationship of polymer membranes with different texture Monika Krasowska, * Anna Strzelewicz, and Gabriela Dudek Department of Physical Chemistry and Technology of Polymers, Silesian University of Technology, Ks. M. Strzody 9, 44-100 Gliwice, Poland Michal Cie´ sla M. Smoluchowski Institute of Physics, Jagiellonian University, Lojasiewicza 11, 30-059 Krak´ ow, Poland (Received 30 September 2016; revised manuscript received 30 November 2016; published 27 January 2017) Two-dimensional diffusion in heterogenic composite membranes, i.e., materials comprising polymer with dispersed inorganic fillers, composed of ethylcellulose and magnetic powder is studied. In the experimental part, the morphology of membranes is described by the following characteristics: the amount of polymer matrix, the fractal dimension of polymer matrix, the average size of polymer matrix domains, the average number of obstacles in the proximity of each polymer matrix pixel. The simulation work concentrates on the motion of a particle in the membrane environment. The focus is set on the relationship between membranes morphology characterized by polymer matrix density, its fractal dimension, the average size of domains, and the average number of near obstacles and the characteristics of diffusive transport in them. The comparison of diffusion driven by Gaussian random walk and L´ evy flights shows that the effective diffusion exponent at long time limits is subdiffusive and it does not depend on the details of the underlying random process causing diffusion. The analysis of the parameters describing the membrane structure shows that the most important factor for the diffusion character is the average size of a domain penetrated by diffusing particles. The presented results may be used in the design and preparation of membrane structures with specific diffusion properties. DOI: 10.1103/PhysRevE.95.012155 I. INTRODUCTION A lot of transport processes encountered in nature in everyday experience can be described using subdiffusion. Scientists constantly develop methods which improve both an analytical and numerical description of this phenomenon. The aim of this research is to draw attention to the role of the structure on which the transport takes place. Because the structure is the element which can be influenced, i.e., created or modified, presented results bring us closer to designing membranes with specific transport properties. Most of the materials encountered in nature are mixtures of compounds and they are investigated in a twofold ap- proach. The first one is the structure characterization and the second is the exploration of dynamic properties [1–7]. The structure of a complex system is usually a “structure with variations” and is characterized through a large diversity of composition, strong interactions between individuals, their activity, movement, and anomalous evolution in time. The problem of diffusion in complex systems usually does not follow the classical laws which describe transport in ordered crystalline media, i.e., the diffusion process usually does not follow Gaussian statistics and the second Fick’s law [8–12]. Such anomalous transport can be found in a wide diversity of complex materials. The list of systems displaying anomalous diffusion is quite extensive, e.g., porous systems, biological cells, fractal geometries, and polymeric networks [1,11,12]. Research of anomalous diffusion in crowded two- dimensional environments, especially in biological cells and model membranes, has been a focus recently in a number of simulations, theoretical and experimental studies [11,13–21]. Nowadays, many researchers try to answer the question how * Monika.Krasowska@polsl.pl the highly crowded environment, like in biological cells, affects the dynamic properties of passively diffusing particles. They demonstrate and discuss in the papers how anomalous diffusion with strongly non-Gaussian features arise in different model systems [13,14,22]. Some of the observations described in [14,16] can be also relevant for the diffusion of particles in other membrane systems. The basis of our further considerations comes from the assumption that the total displacement of a diffusing particle is a sum of a large number of atomic moves  x (t ) = x t 1 +  x t 2 +···+ x t n , where 0 <t 1  t 2 ···  t n  t , and  x i denotes a random move at the moment t = i . Thus, the particle position at the time t equals  r (t ) =  t i t  x t i . (1) For such a particle one can define its square displacement: x 2 (t ) = | r (t ) − r (t = 0)| 2 , where  r (t = 0) is the initial posi- tion of particle. The mean square displacement (MSD) can be defined as 〈x 2 (t )〉 = 〈| r (t ) − r (t = 0)| 2 〉, (2) where 〈·〉 denotes the ensemble average, i.e., average over a set of different diffusive particles. Typically, the MSD increases linearly in time. The emergence of anomalous transport causes a non-linear increase of the MSD: 〈x 2 (t )〉∼ D α t α , (3) where α = 1 and D α is the generalized diffusion coefficient and has the dimension cm 2 s −α . The anomalous diffusion behavior is intimately connected with the breakdown of the central limit theorem, caused by either broad probability distributions of  x t i or non-vanishing correlations between  x t i and  x t j for i = j . Taking into 2470-0045/2017/95(1)/012155(10) 012155-1 ©2017 American Physical Society