Predicting particle transport through an aging polymer using vacancy diffusion Aaron W. Thornton a,b, * , Anita J. Hill b , Kate M. Nairn b , James M. Hill a a Nanomechanics Group, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia b CSIRO MMT, Locked bag 33, Clayton Sth MDC, Vic 3169, Australia Available online 26 October 2007 Abstract Understanding the process of particle transport is important for various applications such as separation, storage and blockage of selected particles within a polymer. The diffusivity of particles has been related to the fractional free volume f within a sample by the expression D(f) =A exp(B/f) for certain constants A and B. Polymers are known to undergo physical aging such that the free volume distribution changes over time towards an equilibrium. This phenomenon has been well explained by the vacancy diffusion model estab- lished by Curro et al. [J.G. Curro, R.R. Lagasse, R. Simha, Macromolecules 15 (1982) 1621]. Using both the diffusion expression and the vacancy diffusion model, this paper models particle transport in aging, unaged and aged polymer samples. Ó 2007 Elsevier B.V. All rights reserved. PACS: 63.30.h; 82.32.Lr Keywords: Polymer; Transport; Diffusion; Vacancy; Fractional free volume 1. Introduction Glassy polymers are in a non-equilibrium state below their glass transition temperatures (T g ). When annealed below T g , the molecules within the polymer move towards equilibrium, with a consequent increase in density, and decrease in both free volume and molecular mobility. This process is known as physical aging. Reviews of non-equi- librium behavior and physical aging can be found in Tant and Wilkes [2] and Struik [3]. One important property which is altered by aging in a wide range of polymers is permeability [4,5]. In this paper we combine a model for aging with a model for particle dif- fusion, and hence can predict the effect of aging on particle transport. Curro et al. [1] proposed a vacancy diffusion model for the physical aging process in polymers. Vacancy diffusion is the idea of vacancies diffusing to a place where they can- not contribute any longer to any mobility dependent pro- cess. In our case this place is the external surface. In Curro’s model, the diffusion mechanism is described a form of the Doolittle [6] equation D(f) =D r exp(B(1/f  1/f r )) where f is the fractional free volume and B, D r and f r are material constants. Fractional free volume f is commonly expressed as f= (v  v 0 )/v where v is total specific volume and v 0 is occupied volume usually calculated by Bondi’s group contribution method [7] which is based on the pack- ing densities of molecular crystals at absolute zero. McCaig et al. [8] suggested that vacancy diffusion is not the only process responsible for aging and that lattice contraction must also occur. Lattice contraction implies an overall decrease in size of all the free volume elements within the polymer. However, following the aging process using positron annihilation lifetime spectroscopy (which yields separate parameters related to the size and concentration of free volume elements) shows that the 1567-1739/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2007.10.051 * Corresponding author. Address: Nanomechanics Group, School of Mathematics and Applied Statistics, University of Wollongong, Wollon- gong, NSW 2522, Australia. Tel.: +61 2 4221 3531. E-mail address: aaron.thornton@csiro.au (A.W. Thornton). www.elsevier.com/locate/cap www.kps.or.kr Available online at www.sciencedirect.com Current Applied Physics 8 (2008) 501–503