Dual sorption theory

While the permeability of most rubbery polymers to most gases is pressure independent, the permeability of glassy polymers usually is observed to decline with increasing pressure. This is accounted for by the dual sorption theory if one assumes that both the Henry and Langmuir type sites contribute to per-... 

Both Equations [18.1] and [18.5] describe ideal cases. In many experimental situations, sorption isotherms are non-linear, showing a negative behavior compared to Henry s law. Some cases can be described using a dual sorption theory (Paul and Koros, 1976) in which the global sorption capacity is the result of two different contributions, one that follows Henry s law and a second that follows Langmuir behavior ... 

The solubility can be obtained from equilibrium measurements in which the volume of gas taken up is determined when the polymer sample is brought into contact with a gas at a known applied pressure. For glassy polymers where the solubility of a gas often deviates in the manner shown in figure V - 10b, such deviadon can be described by the dual sorption theory [13 -15], in which it is assumed that two sorpdon mechanisms occur simultaneously, i.e. sorpdon according to Henry s law and via a Langmuir type soiption. This is shown in figure V -11. 

Figure V - 11. The two contributions in the dual sorption theory Henry s law and Langmuir type sorption.

Vieth W, Howell J, Hsieh J (1976) Dual sorption theory. J Memb Sci 1 177-220 Weitsman YJ (1987a) Stress assisted diffusion in elastic and viscoelastic materials. J Mech Phys Solids 35(l) 73-94... 

Figure 3.3.4. Solubility of methane in oriented polystyrene. Reprinted from Journal of Membrane Science, Vol. 1, W. R. Vieth, J. M. Howell and J. H. Hsieh, Dual sorption theory," pp. 177-220. Copyright (1976), with permission from Elsevier.

The basic premise of the dual-sorption theory is that a glassy polymer contains microvoids, or holes created during the process of cooling the polymer below its glass transition temperature. Therefore, gas molecules can either dissolve in the bulk polymer or go into these holes. Consequently, glassy polymers have a high gas solubility due to the availability of dual-sorption modes. It is found that gas dissolution in the matrix follows Henry s law, whereas that in the holes displays a Langmuir-type dependence. Thus, at equilibrium, the total gas concentration in the polymer can be separated into a normal sorption term c, and a hole contribution [65] ... 

In closing this section, it should be noted that the dual-sorption theory has... 

The form of the sorption isotherm predicted from the dual sorption model is shown in Figure 2.28. Because the expressions for sorption contain three adjustable parameters, good agreement between theory and experiment is obtained. 

The basic assumptions of the dual mode sorption theory as it applies to the transport model of Vieth and Sladek, have been stated by Vieth et al. in their excellent review of the subject The sorption isotherm was described by the combination of a Henry s law dissolved component, Cp, and a Langmuir hole filling term, Ch, i.e.. 

In some cases deviations from Fickian behavior in glassy epoxy polymers (19-25) have been adequately described using dual mode sorption theory (22-24). This theory is based upon the premise that the sorbed penetrant exists in two thermodynamically distinct populations. These populations consist of molecules adsorbed in "holes", and species simply dissolved in the polymer matrix. 

Figure 20 Schematic representation of the dependence of concentration on partial pressure illustrating predications of Henry s law and dual mode sorption theory.

It appears that moisture transport in conditions of less than 100% relative humidity is largely dominated by diffusion. The diffusion phenomenon is best described using dual mode sorption theory. 

The three parameters of the dual mode sorption theory, ko, Ch. and b, are determined by analyzing an isothermal plot of C versus p. For gas sorption at low pressure, where bp 1, the sorption isotherm in Equation 1 reduces to the following linear expression. 

Plots of permeability coefficients versus (1 + bpY should be linear. The intercept and slope of the line correspond to koDp and CnbDn, respectively. Once the equilibrium parameters from the dual mode sorption theory are known, the two diffusion coefficients. Do and D//, can be determined. 

Dual-sorption models are based on the premise that whilst some penetrant molecules diffuse normally in the polymer matrix, others will be affected by polymer-penetrant interactions or micro-void filling. Carter and Kibler (1978) addressed this problem in terms of the probability that a water molecule may react with a polymer molecule. Their model is based on the theory that moisture in a polymer network can be either bound or free. The probability that a free water molecule becomes bound is y and the probability that a bound water molecule is... 

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