Diffusion coefficients, sorption kinetics

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

Pure PHEMA gel is sufficiently physically cross-linked by entanglements that it swells in water without dissolving, even without covalent cross-links. Its water sorption kinetics are Fickian over a broad temperature range. As the temperature increases, the diffusion coefficient of the sorption process rises from a value of 3.2 X 10 8 cm2/s at 4°C to 5.6 x 10 7 cm2/s at 88°C according to an Arrhenius rate law with an activation energy of 6.1 kcal/mol. At 5°C, the sample becomes completely rubbery at 60% of the equilibrium solvent uptake (q = 1.67). This transition drops steadily as Tg is approached ( 90°C), so that at 88°C the sample becomes entirely rubbery with less than 30% of the equilibrium uptake (q = 1.51) (data cited here are from Ref. 138). 

The moments of the solutions thus obtained are then related to the individual mass transport diffusion mechanisms, dispersion mechanisms and the capacity of the adsorbent. The equation that results from this process is the model widely referred to as the three resistance model. It is written specifically for a gas phase driving force. Haynes and Sarma included axial diffusion, hence they were solving the equivalent of Eq. (9.10) with an axial diffusion term. Their results cast in the consistent nomenclature of Ruthven first for the actual coefficient responsible for sorption kinetics as ... 

The development of mixture sorption kinetics becomes increasingly Important since a number of purification and separation processes involves sorption at the condition of thermodynamic non-equilibrium. For their optimization, the kinetics of multicomponent sorption are to be modelled and the rate parameters have to be identified. Especially, in microporous sorbents, due to the high density of the sorption phase and, therefore, the mutual Influences of sorbing species, a knowledge of the matrix of diffusion coefficients is needed [6]. The complexity of the phenomena demands combined experimental and theoretical research. Actual directions of the development in this field are as follows ... 

Since the fraction of empty sites in a zeolite channel determines the correlation factor (Section 5.2.2), as is well known from single-file diffusion in the pores of a membrane, the strong dependence of the diffusion coefficients on concentration can be understood. This is why a simple Nernst-Planck type coupling of the diffusive fluxes (see, for example, [H, Schmalzried (1981)]) is also not adequate. Therefore, we should not expect that sorption and desorption are symmetric processes having identical kinetics. Surveys on zeolite kinetics can be found in [A. Dyer (1988) J. Karger, D.M. Ruthven (1992)]. 

An intriguing aspect of these measurements is that the values of D determined from NMR and from sorption kinetics differ by several orders of magnitude. For example, for methane on (Ca,Na)-A the value of the diffusion coefficient determined by NMR is 2 x 10 5 cm2 sec-, and the value determined for sorption rates only 5 x 10"10 cm2 sec-1. The values from NMR are always larger and are similar to those measured in bulk liquids. The discrepancy, which is, of course, far greater than the uncertainty of either method, remained unexplained for several years, until careful studies (267,295,296) showed that the actual sorption rates are not determined by intracrystalline diffusion, but by diffusion outside the zeolite particles, by surface barriers, and/or by the rate of dissipation of the heat of sorption. NMR-derived results are therefore vindicated. Large diffusion coefficients (of the order of 10-6 cm2 sec-1) can be reliably measured by sorption kinetics... 

The sorption coefficient (K) in Equation (2.84) is the term linking the concentration of a component in the fluid phase with its concentration in the membrane polymer phase. Because sorption is an equilibrium term, conventional thermodynamics can be used to calculate solubilities of gases in polymers to within a factor of two or three. However, diffusion coefficients (D) are kinetic terms that reflect the effect of the surrounding environment on the molecular motion of permeating components. Calculation of diffusion coefficients in liquids and gases is possible, but calculation of diffusion coefficients in polymers is much more difficult. In the long term, the best hope for accurate predictions of diffusion in polymers is the molecular dynamics calculations described in an earlier section. However, this technique is still under development and is currently limited to calculations of the diffusion of small gas molecules in amorphous polymers the... 

In terms of Eq. (1), the driving force is ApA and the resistance, f2 = L/Pa. Although the effective skin thickness L is often not known, the so-called permeance, PA/L can be determined by simply measuring the pressure normalized flux, viz., Pa/L = [flux of A]/A/j>a, so this resistance is known. Since the permeability normalizes the effect of the thickness of the membrane, it is a fundamental property of the polymeric material. Fundamental comparisons of material properties should be done on the basis of permeability, rather than permeance. Since permeation involves a coupling of sorption and diffusion steps, the permeability is a product of a thermodynamic factor, SA, called the solubility coefficient, and a kinetic parameter, DA, called the diffusion coefficient. 

If the cross coefficients are neglected in equation 5.116 [115] then, for codiffusion, where the species, A and B, are diffusing in parallel, two diffusivities, DAA DA, and, DBB DB, can be defined [114], Thus, to follow the sorption kinetics, during the codiffusion experiment, it is necessary to track both components in the mixture [125], However, in the case of counterdiffusion, where molecule, A, moves in and molecule, B, moves out , one effective diffusivity De = DA DB can be defined to describe the process [90,125], Thus, to track the sorption kinetics, during the counterdiffusion experiment, it is necessary to follow only of one of the components in the mixture. Consequently, to measure these diffusivities equation 5.107 was applied. In the present cases, D, was substituted in equation 5.107 by De, for counterdiffusion. 

It has been demonstrated that the combined application of various NMR techniques for observing molecular rotations and migrations on different time scales can contribute to a deeper understanding of the elementary steps of molecular diffusion in zeolite catalysts. The NMR results (self-diffusion coefficients, anisotropic diffiisivities, jump lengths, and residence times) can be correlated with corresponding neutron scattering data and sorption kinetics as well as molecular dynamics calculations, thus giving a comprehensive picture of molecular motions in porous solids. 

A gravimetric technique to study the sorption, desorption, and diffusion characteristics of uxiter vapor in excised skin under dynamic conditions is described. The technique features a continuously recording microbalance and a humiditygenerating apparatus which provides a stream of air with any given relative humidity. The diffusion coefficient is determined from the kinetics of sorption and desorption. The technique can be used to study other polymeric films, fibers, and powders. 

Calculation of the Diffusion Coefficient from Sorption and Desorption Kinetics in a Plane Sheet... 

Barrow [772] derived a kinetic model for sorption of ions on soils. This model considers two steps adsorption on heterogeneous surface and diffusive penetration. Eight parameters were used to model sorption kinetics at constant temperature and another parameter (activation energy of diffusion) was necessary to model kinetics at variable T. Normal distribution of initial surface potential was used with its mean value and standard deviation as adjustable parameters. This surface potential was assumed to decrease linearly with the amount adsorbed and amount transferred to the interior (diffusion), and the proportionality factors were two other adjustable parameters. The other model parameters were sorption capacity, binding constant and one rate constant of reaction representing the adsorption, and diffusion coefficient of the adsorbate in tire solid. The results used to test the model cover a broad range of T (3-80°C) and reaction times (1-75 days with uptake steadily increasing). The pH was not recorded or controlled. 

---