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Fig. 4.-Filtratiori of a salt solution through an ion-exchange membrane. “Permaplcx C-10” membrane-0.099 N NaCl maximum pressure about 1500 p.s.i.: (1) normality of filtrate us. volume filtered; (2) normality of NaCl solution in cell vs. volume in cell (calculated except initial and final points). Dotted line shows normality of original solution.
The observed salt filtering effect of the ion-exchange menihranes is believed to be due t o the electrical properties of the membrane, but we have no proof yet that it is not a size or shape effect. If it is indeed an electrical effect, then it is related to other ion transport phenomena across the membrane such as eiectricA ion transfer, electro-osmosis and diffusion. These relationships
are expressed by the general flux equations of the thermodynamics of the steady state.l6,l6 Additional studies on the relationship between these processes as well as further experiments are in progress. ( 1 5 ) A. J . Staverman,
Trans. Faraday SOC., 48, 176 (1952). (113) S. W. Lorimer, E. I. Boterenbrood and J. J. Hermans, Discussion of the Faraday Society on “Membrane Pllenomena,” Netting. ham, 1956 (in preas).
SOME PROPERTIES OF DIFFUSION COEFFICIENTS I N POLYMERS BY R. M. BARREK Contribution ,from the Cheinislry Deparlment, Imperial College of Science and Technology, London, S. W . 7, England Received July 90.1866
An account has been given of several recent developments in sorption and diffusion in some polymers. The five diffusion coefficients characteristic of any binary mixture can all be obtained in penetrant-polymer mixtures, but past measurements in a variety of systems have probably given erroneous diffusion coefficients D because it has only recently been realized that in them the diffusion coefficients are a function of time as well as of concentration. Present methods of interpretation cannot allow for simultaneous time and concentration dependence of D. The most reliable results for this reason refer to elastomer-penetrant systems, where time effects are a t a minimum owing to short relaxation times of polymer molecules. For the same reason in olymers exhibiting less chain mobility and stronger inter-chain bonds the steady-state methods of measuring diffusion coeikients are preferable to transient state procedures. A classification of penetrant-polymer systems has been given, with examples of each category. Factors influencing the concentration de endence of diffusion coefficients D are discussed, and also the experimental observations leading to the zone theory of tge diffusion mechanism and the quantitative formulation of this theory. Relations between viscous resistance and diffusion are indicated, and generalized functional relationships between Doand E J T in the Arrhenius equation D = Doexp ( -E,/RT) are considered. Selectivity in the transmission of molecules through organic membranes also has been considered.
Investigation of transport processes of simple molecules within polymers usually involves measurements of sorption, permeability and diffusion of penetrant in and through the polymer. Interpretation of these basic phenomena 1)rings contact with a considerable region of the physical chemistry Of high polymers’ Sorptiorl and equilibrium properties characteristic of polymer and penetrant amenable to thermodynamic and
statistical thermodynamic analysis1-6; diffusion coefficients are rate constants, amenable to “irre( 1 ) E . u . . G. Gee, Quart. R e , ) . ,1, 265 (1947). , 4.10 (1941); d n n . N . Y. Aead. ( 2 ) hf. Hiiggins, J . Chen. P h ~ s .9,
~l(l~,y~a~ns,
sc~;3)4~;, Faradall sot,, 40, 320 ( 1 g 4 ~ ) , ( 4 ) P.6. Flory, Proc. Roll. SOC.(London), 8 2 3 4 , GO, 73 (195G). ( 5 ) R . M. Barrer, Trans. Faiadzy sot., 4 3 , (1947). ~ ( 6 ) H L. Frischand c. v. Stannett. J. T’olimrr SCZ., is, 181 (1954).
SOMEPROPERTIES OF DIFFUSIONCOEFFICIENTS IN POLYMERS
Feb., 1957
versible thermodynamic” and kinetic treatment.’#* Diffusion may Le connected with other rate processes such as dielectric relaxation or viscous flow for which a spectrum of relaxation times is exp e ~ t e d . ~ * ~ ~In’ Othis paper some recent progress is considered. 1. The Five Diffusion Coefficients in a Binary Mixture.-Meyer,I1 Darken, Hartley and Crank,13aand Carman and SteinI3b have discussed the physical significance of diffusion coefficients measured under different conditions. Five diffusion coefficients characterize any binary mixture of species A and B (whether gaseous, liquid or solid). First we may think of pure diffusion streams of A and B along the x-coordinate uncomplicated by any kind of mass flow. That is, the origin of the x-coordinate is in a plane normal to it and moving so that there is no mass flow across it. Per unit cross-section normal to the x-direction, in which diffusion occurs, the streams are where DA,Dn, CA and CB are, respectively, intrinsic diffusion coefficients and concentrations of A and B. However, in a binary mixture which undergoes no volume change on mixing a t constant pressure and which fills a constant volume vessel, a complication arises when we consider transport through a unit cross-section fixed with respect to the walls of the vessel and normal to x. Since in general, J A # J B , a compensating mass flow must occur to maintain uniform pressure throughout the diffusion volume. If VA and VB are the specific volumes of A and B, and CA and CB,are their concentrations in g. per unit volume, then for unit volume of solution
+ VBCB= 1
VACA
(1.3)
Moreover the total volume flow of A through the fixed unit cross-section (due to mass flow and pure diffusion) must equal the corresponding total volume flow of B through unit cross-section, since uniform pressure is maintained. These two equal and opposite flows, JIA, J’Bare J’A = -DABVA a-;c A dX
bCB J’B = -DBAVB(1.4) ax
where DABand DBAare two interdiffusion coeficients, and where J ‘ A = J’g
only one mutual interdiffusion coefficient describes the combination of mass flow and pure diffusion. It can now be shown easily that14
+ and that when VA = 1 7 ~ DAB = NADB + NBDA
DAB = VACADB VBCBDA
(1.5)
Since V Aand VB are not in general equal to zero the only way in which 1.3, 1.4 and 1.5 can simultaneously be satisfied requires that DAB = DBA. Thus (7) R . RI. Barrer, Trans. Faradau Soc., 86, 644 (1939); 88, 322 (1942). 39, 237 (1043). (8) H. Eyring, J . Chem. Phys., 4, 283 (1930). See S. Glasstone, I
