Stochastic reconstruction of mixed-matrix membranes and evaluation of effective permeability Pavel C ˇ apek a,⇑ , Martin Vesely ´ a , Bohumil Bernauer a , Petr Sysel a , Vladimír Hejtmánek b , Milan Koc ˇir ˇík c , Libor Brabec c , Olga Prokopová c a Institute of Chemical Technology, Prague, Faculty of Chemical Technology, Technická 5, 16628 Prague 6, Czech Republic b Institute of Chemical Process Fundamentals of AS CR, Rozvojová 135, 16502 Prague 6, Czech Republic c J. Heyrovsky ´ Institute of Physical Chemistry of AS CR, Dolejškova 3, 18223 Prague 8, Czech Republic article info Article history: Received 27 November 2013 Received in revised form 21 February 2014 Accepted 1 March 2014 Keywords: Microstructural descriptor Simulated annealing Sample-spanning cluster Random walk simulation Enhanced permeability abstract Microstructures of three mixed-matrix membrane samples made of polyimide and silicalite-1 particles were reconstructed using a stochastic reconstruction procedure. The samples differed in the volume frac- tions of silicalite-1 particles as follows: 0.166, 0.310 and 0.371. The reconstruction revealed the existence of percolation clusters of silicalite-1 particles in the two samples with the volume fraction of silicalite-1 greater than 0.3. In contrast, only the reconstructed microstructure of the first sample contained small clusters of silicalite-1 particles, which did not percolate along any direction. The results of this recon- struction were tested by simulating the random walk of CO 2 molecules in the reconstructed bodies and by predicting the effective permeability of CO 2 . Both original and reconstructed membranes revealed a similar enhanced effective permeability, which exceeded predictions based on the effective medium approximations. Therefore, we suggest that clustering of the silicalite-1 particles was the primary cause of the permeability increase. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction In principle, the separation technology requires membranes with high permeability and selectivity. In this respect, pure dense polymeric membranes have reached their upper limits, repre- sented by a line in the Robeson selectivity–permeability plot for any binary gas mixture [1,2]. For instance, dense membranes made of aromatic polyimides exhibit high separation efficiency for gas- eous mixtures but low permeability for permanent gases and organic vapours [1]. The incorporation of solid fillers, such as microporous sorbents, into polymeric matrices seems to be a promising route towards composite membranes of improved over- all separation efficiency, a concept proven in the mid-1980s [1–5]. Among microporous sorbents, zeolites are of particular interest. Synthesis and application of composite membranes have been concerned with two issues: (i) poor adhesion between the phases associated with the formation of macroscopic voids at the phase interface (sieve-in-cage morphology) [1,2], and (ii) formation of rigidified polymer layers of reduced free volume at the phase interface deteriorating species exchange between the phases. An original way to improve the adhesion between the polyimide matrix and silicalite-1 crystals was introduced by Sysel et al. [6], who used (3-aminopropyl)triethoxysilane to modify the polyamic acid of a controlled mean molar mass. Another situation, possibly eliminating the desired effect of porous fillers, is the clogging of pores with strongly adsorbed species such as solvents. These observations suggest that membrane microphotographs and other types of microstructural information will contribute to the under- standing of functional relationships between the membrane struc- ture and membrane properties [7–13]. In general, a composite material consists of domains of different materials (phases) or of the same material in different states. Pro- vided the properties of all phases are known, the issue is to deter- mine the phase distribution within spatial domains that are much larger than molecules but much smaller than the characteristic length of a macroscopic sample, and the specific interactions on phase interfaces. In this sense, the ideal goal of microstructure analysis is to formulate mathematical models that will quantita- tively account for all macroscopic (effective) properties, such as thermal and electrical conductivity, effective permeability and dif- fusivity, of real composites [14–18]. The great variety of composite materials, along with the spatial distribution and nature of the phases, make the comprehensive mathematical model a challenge. The recent progress of imaging and microscopy has led to the development of novel experimental methods that, with the aid of http://dx.doi.org/10.1016/j.commatsci.2014.03.003 0927-0256/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding author. Tel.: +420 220444407. E-mail address: pavel.capek@vscht.cz (P. C ˇ apek). Computational Materials Science 89 (2014) 142–156 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci