Jan., 1957
93
THETIMELAGIN DIFFUSION
THE TIME LAG I N DIFFUSION* BY H. L. FRISCH’ Contribution from the Department of Chemistry, University of Southern California, Los Angeles, Cal. Received August 9 , 1066
Explicit expressions for the time lag in linear diffusion of a gas or vapor through a membrane with concentration dependent diffusion coefficient are obtained without explicitly solving the diffusion equations. Generalizations of the method used to obtain the time lag- are indicated. The use of the derived relations is sketched by applying them to several cases of physital interest.
Introduction If one is measuring the rate of flow of a gas (or any other solute) through a membrane (or thin slab of material) in which the gas dissolves there will be an interval from the moment the gas comes into contact with the membrane until it emerges a t a constant rate on the other side. By analyzing stationary and non-stationary states of flow it is possible to measure the diffusion constant, the permeability constant and the solubility of the gas in the membrane. By solving an appropriate diffusion equation in the case that the diffusion coefficient of the gas is a constant, Do,a simple algebraic expression is obtained by direct calculation2 which determines the intercept L, the so-called time lag, in terms of Do,the thickness of the membrane I and the concentrations a t the boundaries of the membranes, which the transmitted gas pressure-time curve makes on the axis of time. This experimental technique has proven in many instances to be a very valuable one in elucidating the diffusion constants of gases in both metals and non-metals.2 Unfortunately in many physical systems of current interest the diffusion coefficient is not a constant but the (differential) diffusion coefficient a t a given place and time in the membrane, D, is a function of the concentration, c, a t the given place and time, ie., B = D(c). The form of this function is often known to a sufficient approximation. Among such systems can be found those involving organic vapors diffusing through various polymer memb r a n e ~ . ~ -In ~ such cases the formulas for the time lag given in Barre? (calculated on the assumption of a constant diffusion coefficient) no longer apply. The difficulty in estending the method used in that reference to obtaining the time lag for a concentration dependent diffusion coefficient is’ that it requires explicit knowledge of the concentration of the diffusing species throughout the membrane for all times which can be obtained only by solving the appropriate Fickian diffusion equation. In most instances of physical interest such solutions are unknown and because of the non-linearity of the equation no general techniques for finding such solutions are known. Indeed attempts have been * This research was supported by the United States Air Force Office of Scientific Research of the Air Research and Development Command under contract no. A F 18 (603) 122. (1) Bell Telephone Laboratories, Murray Hill, N. J. (2) For details of such a calculation refer to R. M. Barrer, “Diffusion In and Through Solids.” Cambridge a t the University Press, 1951,p. 18
ff. (3) R. J. Kokes and F. A. Long, J . Am. Chem. Soc., T 5 , 6142 (1953). (4) A. Aitken and R. M. Barrer, Trans. F a r a d a ~SOC., 51, 116 (1955). ( 5 ) R. Waack, N. H. Alex, H. L. Frisch, V. Stannett and M. Szwarc, I n d . Eng. Chem., 41, 2524 (1955).
made to find L roughly from approximate solutions of the diffusion equation obtained by perturbation method^.^ The Time Lag.-In this paper we will derive our explicit expressions for the time lag which apply also to systems with a concentration dependent diffusion coefficients without explicitly solving the diffusion equation. I n view of our own interest in the permeation of gases and vapors through high polymer membranes we will choose a simple experimental arrangement which is often used to study these systems.4-6 In doing this we lose little of generality since other arrangements can be handled just as easily by the method developed. We imagine the left membrane surface in contact with a reservoir of the gas or vapor a t some fixed pressure, while the right membrane surface is in contact with essentially a vacuum. The gas or vapor dissolves at the left membrane surface a t x = 0 to give a concentration co then undergoes activated diffusion to the right membrane surface a t x = 1 where it evaporates and is immediately removed from contact with the membrane surface so that the concentration a t x = 1 is zero. The membrane is assumed to be sufficiently thin so that diffusion occurs only along the x-axis. If c(x; t ) denotes the concentration of the gas a t a distance x from the left membrane surface a t time t then the appropriate boundary value problem is completely specified by
ac -
;(a(c)
=
o in o < 2 < 1, t > o
(1)
with c(r;O) = 0 for z > 0, (with c(0;O) bounded) c(0;h) = co for t > 0 c(1;t) = 0 for t > 0
The differential diffusion coefficient, 9 ( c ) is assumed to be a given, single-valued, a t least once continously differentiable function of c and 5. For example for the diffusion of paraffins in rubber4 D(c) = Do(1 bc) while for the diffusion of paraffins i n polyisobutylenee B(c) = Do exp (bc,/2) (I bc/2) where b is a constant depending only on the temperature and the chemical make-up of the diffusion system. In all cases investigated a steady state in the flow of the gas or vapor is finally attained, Le.
+
+
lim c(z;t)
t+
=:
cJz)
(2)
m
where c.(x) is the solution of eq. 1 with bc/bt = 0, viz. (3) (6) 6. Prager and F.
A. Long, J . Am. Chem. SOC.,73, 4072
(1051).
H. L. FRISCH
94 with CdO) =
c.(l) = 0
Co
In what follows we shall assume that D(c) is such that eq. 2 holds. In Appendix I we show that the time lag L is given by ca(z)dzdx
Jzl
L=-
c0
sg”” D(u) du
(4)
where the steady-state concent,ration c,(x) can in principle always be explicitly found as a solution of t.he quadrature
In view of eq. 5 one may rewrite eq. 4 in terms of quadratures of D(c) only as L=
;(so”
w D(w)
[ I :
%(u) d u ] d w j
[loco (4 du]
3
(6)
a
showing thak L is a function only of 1, co and the constant parameters of D, L e . , Bo and b in the foregoing examples. Since the functional form of D(c) is known the measurement L for various values of co determines the constant paramet,ers of and hence 9 itself. Applications and Discussion.-To illustrate the use of eqs. 4-6 we will apply them to two cases which have been discussed in the literature. Case 1: D(c) = DOa constant We find from eq. 5 that
cdx) = co (1
I n order to compare this result with Aitken and Barrer’s4 approximate value of L obtained from an approximate solution of the diffusion equation we expand eq. 8 in powers of b or preferably substitute the series expansion for ca(x) in eq. 4 to obtain
JO1 x C J X ) dx
D(u)du
12
Vol. 61
This agrees well with Aitken and Barrer’s result for small b
The method developed in the appendix to this paper applies to other diffusion problems with different symmetries and boundary conditions (see e.g., Appendix 11). A variant can even be applied to the study of time lags of the more general Smoluchowski equation. The author is indebted to his wife for the series expansion, eq. 9, and to Dr. W. No11 of the Department of Mathematics, Un.iversity of Southern California, for his interest in this problem. Appendix I We shall deriv,e eqs. 4 and 5. From eq. 3 we find (on integration) that there exists a constant qa such that Integrating eq. 11 over z from 1 to 0 we find
- ;)
and substituting this result into eq. 4,noting that
Hence
- ps = l me obtain for L the known result2 L = 12/6Do Case 2: B(c) = Do(1
(7)
+ bc)
c8(z) =
b
{ - 1 + [ I + (2bcO + b2c02)( 1 - !)]”’/
+ g1 b4c04(Gua -
u2
- 5 u 4 ) + O(b6c06)
;u = 1
-1
Substituting this c8(x) into eq. 4 we find the exact L in this case =
{ 4A6/2 - (4A +15B26B1)( A - Bl)’12
Do(A
with
- 1)/2
co
D(u) du
(12)
The total flow of gas (per unit volume through the right membrane surface) up to time t a t steady state is given by &.(t)
Again from eq. 5 we find that
Jo
=
-
E
p,dt =
soco
B ( u ) du
Similarly for the non-steady state flow there exists a flux function q = q(t) given by such that the total flow time t is given by the ana1% of eq. 13 &(O = - Jot dt (14) To find &(t) integrate both sides of eq. 1 over 2 from 1 to 5
(8)
Jz2
dz - q(t)
C + D(c) bb= 0 X
Integrating this equation again over x from we find on rearranging B=%(c
o
+ 2 bco2)
(13)
- p(t)
=
{Joco P ( u ) du
- JzJzb*at
I to 0
dz d z t
(15)
.
FLUORINE MAGNETIC RESONANCE STUDIES OF SOLIDFLUOROETHANES 95
Jan., 1957
Finally integrating over t from 0 to t we find uniformly. For then by virtue of eq. 1 bc(x;t) lim ___ at +h ( z )
