# Statistical mechanics of “dual‐mode” sorption in polyimides

## Abstract

A statistical mechanical description of the gas takeup or sorption in polyimides below their glass transition temperature is proposed. The model treated is formally similar to Fowler's early model of monolayer adsorption of gases on solid surfaces. It is expected to be appropriate for polymers exhibiting a significant degree of crystallization, e.g., the aromatic polyimides at temperatures well below their glass transition temperature. The model assumes that the penetrant gas molecules bind to a spectrum of states characterized by a distribution of binding energies. If it is assumed that the penetrant molecules bind to the polymer with only two well‐defined interaction energies, namely, zero as well as some particular finite value, a “dual‐mode” expression for sorption is obtained. It is pointed out that this “dual‐mode” or “two‐state” model is not consistent with some recent data obtained on carbon dioxide sorption in a polyimide. Two other simplified models are examined, a “uniform‐density‐of‐states” model and a “three‐state” model. The present treatment yields explicit expressions for the temperature dependence of the sorption parameters. This dependence reflects the thermodynamic redistribution of penetrant molecules among the available fixed density‐of‐states as the temperature is varied. It is pointed out that the temperature dependence of the Langmuir capacity constant obtained from nonlinear least squares fitting the standard “dual‐mode” sorption expression to the polyimide sorption data may lead to an erroneous inference concerning the decrease in “unrelaxed volume” as the temperature is increased from well below the glass transition temperature of the polymer. The “three‐state” model yields sorption isotherms that parallel the data of Sada et al. closely. This suggests the presence of multiple carbon dioxide binding sites in the polyimide. © 1993 John Wiley & Sons, Inc. Copyright © 1993 John Wiley & Sons, Inc.