Concentration Dependence of the Time Lag for Diflusion

Transcription

1 1300 JOURNAL OF POLYMER SCIENCE: PART A-2 VOL. 5 (1967) Concentration Dependence of the Time Lag for Diflusion The permeation technique has been used extensively for measurement of the concentration dependence of vapor diffusion coefficients in polymer membranes. In this a pressure gradient is maintained across the membrane and the pressure on the outgoing side, which is usually effectively zero, obtained as a function of time until a steady state of flow has been established. From the slope of the linear portion of the plot the steady-state permeation rate is obtained and extrapolation back to the time axis gives an intercept L, the time lag for diffusion.1j To determine the concentration dependence of the diffusion coefficient D, the steadystate flux J per unit area and thickness of membrane is obtained as a function of the pressure on the ingoing side and the equilibrium sorption isotherm of the system is also measured. J as a function of the concentration C1 in the ingoing face of the membrane follows and from Fick's law: so that the D versus C relationship may be determined.3 When D is a function of C only, Frisch4 has shown that the dependence of the time lag L on the concentration is given by = 12 l" wd(w) s," D(u)dudw/{ici D(u)du)' (4) where 1 is the membrane thickness, C(z) is the steady-state concentration at the point x; and u and w are dummy variables. When D is constant, eq. (4) reduces to L = 12/60. Thus if the D versus C or equivalent relationship is known from steady-state measurements then the variation of L with C may be predicted and compared with the observed dependence. For several vapor-polymer systems differences have been observed and are usually attributed to physical effects resulting in a timedependent contribution to the diffusion coefficient which vanishes in the steady state.6-8 It is the purpose of this communication to stress that when D varies with C, the relative variation of L with C over a finite concentration range is always considerably smaller than that of D with C. This is suggested by the form of eq. (4) in which D is always r present as part of an integral. More specifically it has been shown9.10 that when log D(u)du is a convex function of LC' y the time lag variation is governed by the inequality 1/6,< L(C1) D(u)du/12Ct,< 1/2. The smaller relative variation of L holds particularly when the diffusion coefficient decreases with increasing concentration and it is possible for the variation of L to be within experimenhl error even although D changes significantly with concentration. To illustrate this point the following calculations were made. Several forms of the functionf(c) in the relation D = DC -0 f( C) were chosen such that, in the range of C 0-10, the diffusion coefficient changed by an order of magnitude. The dependence of D and L on the concentration for the different forms of f(c) are shown in Figures 1 and 2, respectively. The corresponding steady-state concentration distributions are shown in Figure 3. These results clearly illustrate that the relative variation of

2 NOTES 1301 L with C can be very much less than that of D. A comparison of curves E, D, and E, B of Figures 1 and 2 shows that it does not necessarily follow that the larger the departure of D( C) from DC A then the larger is the departure of L( C) from LC -0. Fig. 1. D vs. C CUNM for various forms of f(c): (A)f(C) = Ca; (A ) f(c) = 1 - O.Oo09C8; (B)f(C) = C2; (B )f(c) = c12; (C)f(C) = C; (C )f(c) = C; (D)f(C) = C C2; (D )f(c) = C C2; (E) f(c) = exp (0.2303C); (E ) f(c) = exp - {0.2303C}. Ordinate: D, cm.z/sec.; abscissa: C, arbitrary units. The dotted line represents the abscissa for curves (A)-@). In eq. (3) both the numerator and the denominator increase with concentration. The dominant factor in determining the concentration dependence of the time lag is the term ic DdC. For D increasing with C the percentage change in this term with C compared LC with the term DcdC is greater than that for a corresponding D which decreases with C (Fig. 1). When this term is similar (e.g., curves B and E of Fig. l), then the difference in G(: relations is governed by the difference in the steady-state concentration profiles. r a On the basis of the term DdC (Fig. l), curve B may be expected to have the smaller J O

3 1302 JOURNAL OF POLYMER SCIENCE: PART A-2 VOL. 5 (1967) 1 I 5 10 Fig. 2. L vs. C curves for various forms of f(c). Ordinate: L, min.; abscissa: C units as in Figure 1. Membrane thickness = 0.06 cm. Labels as for Figure 1. time lag, whereas Figure 2 shows the reverse is true. In this case the term 1 x C(x)dz (cf. Fig. 3) is larger for curve B than for curve E, and sufficiently so to counteract the effect of the denominator. For an accurate comparison of steady-state and time-lag data it is necessary to know the D-C or equivalent relationship from zero concentration upwards with a high degree of precision. Usually D is measured as a function of C over a limited range of concentration followed by extrapolation to zero concentration which can introduce a not insignificant source of error unless measurements are made close to zero concentration. Similarly extrapolation of limited J~-C date to give Lc-o( =12/6Dc~) must also be treated with caution (e.g., curves A, C, and D of Fig. 2). More recently Ash et al. expressed eq. (3) in terms of permeation fluxes and solved for the L-C relation graphically." This method has the advantage where accurate curve fitting of the J versus CI data is not possible, but, as before, extrapolation to zero concentration is necessary. Finally, difficulties which may be encountered when comparing steady-state aud timelag data are illustrated with some examples taken from the literature. For several water-polymer systems it waa found that the time lag was independent of concentration, suggesting that D was constant, whereas a steady-state analysis indicated that the d8usion coefficient decreased with concentration.6-8 For example, by fitting a polynomial to the J-CI data of Stannett and WillLms,7 D was observed to decrease by a factor of -5 when the relative humidity increased from 0 to From the Frisch equation it was calculated that the time lag should increase from 45 to 55.6 min. in this range. An arbi-

4 NOTES 1303 trary membrane thickness of 0.04 cm. was adopted in these calculations. The variation in 15, although not large, should certainly be detectable. However, measurements with water vapor are complicated by sorption of the vapor on the glass walls of the apparatus. Although a procedure was adopted to minimize this effect it is questionable whether it was completely eliminated by the technique used.? Within experimental error the same J-CI data can be represented by a straight line up to a relative humidity of Because of the abrupt change in slope of the J-Cl curve with this method of representation it was easier to use the graphical method11 to calculate the L-C dependence. In this case the increase was from 58.2 to 60 min., which in practice would be difecult to detect. The Fig. 3. Steady-state concentration profiles for various forms of f(c). Ordinate: C(z) units as in Figure 1; abscissa: z. Labels as for Figure 1. (F) constant D. form of curve fitting cau therefore be quite critical. However, it is perhaps unlikely that D should suddenly decrease at some nonzero concentration, and the polynomial method of representing the data is probably the better. A similar calculation with the waterrubber hydrochloride system* indicates that, in the relative humidity range of 0-0.8, D decreased by a factor of -10 and the time lag should increase by about 30% (327 to 404 min. for a membrane thickness of cm.). A variation of this magnitude would seem to be outside experimental error even for water vapor time lags and it would appear for this system at least that a real discrepancy exists between steadystate and time-lag date.

5 1304 JOURNAL OF POLYMER SCIENCE: PART A-2 VOL. 5 (1967) In conclusion, for systems in which D decreases with C the variation of the time lag can be very much less than that of the diffusion coefficient. As a result the L-C dependence must be measured with an especially high degree of precision9 if a successful comparison is to be made of steady-state and time-lag results. This work is part of an investigation of water and similar vapors in polymer films. The paper is British Crown Copyright, reproduced with permission of the Controller, Her Britannic Majesty s Stationery Office. References 1. H. Daynes, Proc. Roy. Soc. (London), AW, 286 (1920). 2. R. M. Barrer, Diflwion in and through Solids, Cambridge Univ. Press, P. E. Rouse, J. Am. Chem. Soc., 69, 1068 (1947). 4. H. L. Frisch, J. Phys. Chem., 61,93 (1957). 5. R. M. Barrer and R. R. Ferguson, Trans. Furaday Soc., 54,989 (1958). 6. H. Ymuda and V. Stannett, J. Polymer Sci., 57,907 (1962). 7. V. Stannett and J. L. Williams, in Transport Phenomena in Polymer Film (J. Polymer Sci. C, lo), C. A. Kumins, Ed., Interscience, New York, 1965, p J. D. Wellons and V. Stannett, J. Polymer Sci. A-l,4,593 (1966). 9. H. L. Frisch, J. Phys. Ck., 62,401 (1958). 10. H. 0. Pollak and H. L. Frisch, J. Phys. Ck., 63, 1022 (1959). 11. R. Ash, R. W. Baker, and R. M. Barrer, in press. Physical Chemistry Laboratories Chemistry Department Imperial College London, England Received February 21, 1967 Revised April 10, 1967 J. A. BARRIE D. MACHIN