10164 J. Phys. Chem. 1991, 95, 10164-10168
a[(2n
+
I)ao
+
nul+
( n-
I)a2+
(n-
2)a3+ ... +
a , , ] / a ~ = a2[(2n+
I)ao+
nul+
( n-
l ) a z+
( n-
2)a3+
...
+
a , , ] / 8 y 2Therefore, at any time
Thus, during the anodic potential step, the current varies between two limits corresponding to complete reversibility ( [ u ~ ] , , = ~ = 0,
[all,-o = 1, ajs2...,, = 0):
i ( t )
-
1 FSPD’12 ( * t ) ’ / 2--
and complete irreversibility ([ao]y=o = 0, [ a j = l . . . n l y = ~ = 0):
2 n + 1 1
=-- i(t)
FSPDII~ n
+
1 ( T t ) l I 2In other words, the number of electrons per monomer passes from
1 to (2n
+
I)/(n
+
I ) , i.e., to 2 when n is large (not taking accountof the additional charging of the polymerI2). The same is true in cyclic voltammetry.
Two types of calculations were performed for each mechanism. We first set all A’s equal to zero except XI, then all A’s were made equal to zero except XI and A,, X2 was made equal to XI, finally all A’s were made equal to zero except XI, X2, and X3, and X2 and
X, were made equal to XI. The resulting R-k, C“0 curves are
represented in parts Aa and Ba of Figure 4 for t i e CR-CR and CR-S-irr mechanisms, respectively, while the resulting R-k,,@02 curves are represented in Figure 5a for the CR-S-rev mechanism. The calculations could have been pursued for XI = X2 = A, = X4 and (Xj),,4 = 0 and so forth, but we noticed that the R curves
cease to vary appreciably in between the last two calculations (Figures 4Aa,Ba and 5a).
In the second series of calculations we apply the condition XI
<<
(Xj)j=2,0n and considered the case where n-
m . That amounts to replacing the preceding sets of partial derivative equations, initial and boundary conditions, by the following ones.C R - C R duo a2ao a7 ay2
-
=-
+ nXla12
CR-S-irr a2ao ay29
=-
+
nX,aoal aa, a2a,- -
2nXlaoal ay2 CR-S-rev O I r I O : a o = O 7 2 0 : a l = OCalculation
of
Lag Time
for
Convective-Reactive Diffusion
Jenn-Shing Chen*
Department of Applied Chemistry, National Chiao Tung University, Hsin Chu, Taiwan, ROC
and Franz Rosenberger
Center for Microgravity and Materials Research, University of Alabama in Huntsville, Huntsville, Alabama 35899 (Received: July 15, 1991)
Closed form solutions for the steady-state permeability P and the lag time L of a linear diffusion system with concurrent convection and reaction were obtained by two methods. In the first method, we identify the singularity at s = 0 of the Laplace transform of the total amount of diffusant, Q(s), released into the receiver as representaiion of the asymptotic diffusion behavior. P and L are then obtained from the time-independent coefficients of an expansion of Q(s) about s = 0. In the second method, we transform the convective-reactive diffusion equation into a form that contains only first and second derivatives of the concentration distribution function. By comparison of the resulting equation with that for a heterogeneous diffusion system, relationships of the convection velocity and rate constant with the position-dependent partition coefficient in heterogeneous diffusion is found. Taking advantage of the known solutions for permeability and lag time in heterogeneous diffusion, the corresponding expressions for P and L in convective-reactive diffusion are then obtained by transcription. These methods have the advantage over earlier approaches in that solutions in an infinite form are avoided.
Introduction impulse transmission: and colloid flocculation.s In recent years Diffusion plays an important role in numerous processes, such some attention has been paid to the mathematical description of as chemical reactions in condensed phases,’ nucleation,2J nerve transient diffusion.- This requires complete solutions to diffusion
( I ) For a recent review, see: Hanggi, P.; Talkner, P.; Borkovec, M. Reo. (4) Dudel, J. Exciration, irs Conducrion and Synapric Transmission, in Biophysics; Hoppe, W . , et al. Eds.; Springer-Verlag: Berlin, 1983; Chapter 15-1.
(3) Kelton, K. F.; Greerand, A. L.; Thompson, C. V. J . Chem. Phys. 1983, ( 5 ) Vold, R. D.; Vold, M. J. Colloid and Inrerface Chemistry; Addision- Mod. Phys. 1990.62, 251.
79, 626 I . Wesley: London, 1983.
(2) Frisch, H. L.; Carter, C. C. J. Chem. Phys. 1971, 54, 4326.
Lag
Time
for Convective-Reactive Diffusion The Journal of Physical Chemistry, Vol. 95, No. 24, 1991 10165 equations, which, however, apart from some simple cases, cannotbe obtained in closed form. Therefore, one usually contents oneself with asymptotic solutions which provide information on the permeability and lag time, Le., the steady-state rate of mass transport and time required for the attainment of steady state. With these two parameters, one is able to estimate the total amount of diffusant released into the receiver over long times. Such an estimate is adequate for most practical applications.
The lag time in simple homogeneous diffusion assumes a value of h 2 / 6 D , 9 where D is the diffusivity, and h the length of the diffusive path (thickness of membrane, etc.). Expressions for lag times have also been derived for more complex systems. For instance, Jaeger,Io by employing a Laplace transform, gave a routine (instead of an explicit expression) for the calculation of the lag time for thermal conduction across a multilayer slab. His procedure was followed by Barrie et aLI1 to calculate the lag time of mass diffusion in a three-layer slab. The pioneering work of Frisch12 yielded an elegant method to obtain the lag time for a system with concentration-dependent diffusivity without solving diffusion equations. This strategy was also used by PaulI3 to obtain the lag time for the dual absorption diffusion in polymer films. Later Frisch and
collaborator^"^^
and Chen and Fox18 formulated the lag time for a heterogeneous diffusion system characterized by a position-dependent diffusivity and partition coefficient. Chen and Rosenberger have recently extended the treatment to diffusion systems with serial and/or parallel paths19 and multiple and pe- riodic laminates.20 Earlier, employing Frisch's concept, Petro- poulos and Roussis studied the lag time in diffusion systems with time- and position-dependent anomalies2'.22 and the "directional diffusion lag time a ~ y m m e t r y " . ~ ~The lag time in a diffusion system with a linear reaction and sorption process has been studied by Ludolph et Their result is now superseded by that of Leypoldt and G o ~ g h , ~ ~ obtained by employing a finite Fourier transform. Keister and Kasting26 treated convective diffusion in a constant electrical field (elec- trodiffusion) by a separation of variables method. They obtained solutions for the lag time in the form of infinite series, which require simplification based on the residue theorem in complex variable theory if compact forms for P and L are desired.
In this article we present two other schemes to obtain closed form solutions for
P
and L.In
the first method we utilize Laplace transforms as originally suggested by Carslaw and Jaeger27 and solve the diffusion equation in the Laplace domain to obtain the total diffusant amount Q(s) released into the receiver. The as- ymptotic part of Q(s) is identified to correspond to the singularity at s = 0. The permeability and lag time arc then obtained in closed forms from the expansion coefficients of Q(s) around s = 0. The second scheme makes use of the method of change of variables(6) Agmon, N.; Hopfield, J . J. J . Chem. Phys. 1983, 78, 6947.
(7) Grzywna, Z. J.; Petropoulos, J . H. J . Chem. Soc., Faraday Trans. 2 1983. 79. 571. 585.
( 8 ) Amarantos, S. G.; Tsimillis, K.; Savvakis, C.; Petropoulos, J. H. J .
( 9 ) Crank, J . The Mathematics of Diffusion, 2nd ed.; Oxford University
Membr. Sci. 1983, 13, 259.
. .. Press: Oxford, 1975.
Soc. 1963, 59, 869.
(IO) Jaeger, J . C. Q. Appl. Math. 1950, 8, 187.
( I I ) Barrie, J. A.; Levine, J. D.; Michaels, A. S.; Wong, P. Trans. Faraday (12) Frisch, H . L. J . f h y s . Chem. 1957, 62, 93.
(13) Paul, D. R. J . f o l y m . Sci. 1969, A-2, 7, 181 1 .
(14) Frisch, H . L.; Prager, S. J . Chem. Phys. 1971, 54, 1451.
(15) Frisch, H. L.; Bdzil, J . B. J . Chem. Phys. 1975, 62, 4804.
(16) Frisch, H. L.; Fargacs, G.; Chui, S. T. J . Phys. Chem. 1979,83,2787.
(17) Frisch, H . L. J . Membr. Sci. 1978, 3, 149.
(18) Chen, J . S.; Fox, J . L. J . Chem. Phys. 1978, 89, 2278.
(19) Chen, J . S.; Rosenberger, F. Chem. Eng. Commun. 1991, 99, 77.
(20) Chen, J . S.; Rosenberger, F. Chem. Eng. Commun., in press.
(21) Petropoulos, J . H.; Roussis, P. P. J . Chem. Phys. 1967, 47, 1496.
(22) Petropoulos, J. H.; Roussis, P. P. J . Chem. Phys. 1969, 51, 1332. (23) Petropoulos, J. H.; Roussis, P. P.; Petrous, J. J. J . Colloid Interface (24) Ludolph, R. A.; Vieth, W. R.; Frisch, H. L. J . Phys. Chem. 1979, 83, (25) Leypoldt, J . K.; Gough, D. A. J . Phys. Chem. 1980,84, 1058. (26) Keister, J. C.; Kasting, G . B. J . Membr. Sci. 1986, 29, 155.
(27) Carslaw, H . S.; Jaeger, J. C. Conduction of Heat in Solid, 2nd ed.;
Sci. 1977, 62, 144.
2793.
Oxford University Press: Oxford, 1959; p 402.
to transform the original convective-reactive diffusion equation into the form which contains only terms of first and second de- rivatives of the concentration distribution function. By comparison
of the resulting diffusion equation with that for a heterogeneous diffusion system,lel* relationships of the convection velocity and rate constant with the position-dependent partition coefficient in heterogeneous diffusion is found. Taking advantage of the known solutions for permeability and lag time in heterogeneous diffusion, the corresponding expressions for P and L in convective-reactive diffusion are then obtained by transcription.
Laplace Transform Approach
Consider a convective-reactive diffusion process taking place in a membrane extending from x = 0 to x = h. The region x
<
0 is designated as donor compartment, and the region x>
h as receiver compartment. During an experiment, the concentration of diffusant in the donor is kept at a constant level, po, while that in the receiver is kept a t zero. Thus, an "infinite constant supply and sink condition" is imposed. The membrane is void of diffusant a t the onset of the experiment.The mathematical description of the problem involves a dif- fusion equation of the form
supplemented with the initial and boundary conditions
P ( X , O ) = 0 (2)
P ( O 4 = KPO (3)
p(h,t) = 0 (4)
Here, p ( x , t ) is the diffusant concentration a t location x and time
t ; D, u, and K are the diffusivity, convective velocity, and rate
constant of the first-order reaction, respectively. We assume that
D,
u, and K are positive and constant. K ( 0 ) in eq 3 represents thepartition coefficient characterizing the distribution of diffusant
on both sides of the interface a t x = 0 and accounts for the continuity of the diffusant's chemical potential a t this interface. The foregoing linear differential equation with constant coefficients can effectively be solved by a Laplace transform technique. In
the Laplace domain, eqs 1-4 take on the forms
a
a
ax2 ax
Sfi(X,S) = D-fi(X,S)
-
U-fi(X,S)-
K ~ ~ ( X , S ) (5)P(O,s) = p o K / s ( 6 )
fi(h,s) = 0 (7)
Here, a circumflex over a function designates its Laplace transform
on t , e.g.
p ( x , s ) L ( p ( x , t ) ) S-exp(-st)p(x,t) 0 dt (8)
where s is a Laplace variable. Note that the initial condition (2)
has been incorporated into eq 5, and the partial differential equation has been transformed into an ordinary differential equation. The solution to eqs 5-7 is found to be
with
q = 4 ( ~ / 2 D ) ~
+
(s+
K ) / D (10)An experimentally measurable quantity, Q(t), the total amount of diffusant released into the receiver within time t , is defined as
l'[
D F-
u p ( x , r )Q ( t ) = f J ( h , r ) 0 d r =
-
d r =1.x-h
-If[
D?] d r (11)10166 The Journal of Physical Chemistry, Vol. 95, No. 24, 1991 Chen and Rosenberger where J(h,t) is the diffusion
flux
into the receiver, and theboundary condition (4) has been used. The Laplace transform of Q ( t ) in eq 11 acquires the form
Q ( s ) =
--
x-h
On substituting eq 9 into eq 12, one obtains
exp(
$)
csch (qh) (13) PoDK9Q ( s ) =
-
S2
In general, the total release Q(r) in a linear diffusion system
subject to an infinite constant supply and sink condition consists of two parts: ( 1 ) The transient part, which decays with time eventually to zero, is a linear combination of terms in the form of exp(-Xit), where the As’s are the eigenvalues of the Sturm- Liouville differential equation associated with the method of separation of variables9 (2) The long-time asymptotic limit, which depends linearly on time, defines the permeability P and lag time
L, respectively, as the slope of the asymptote and its intercept with the time axis. A combination of these two parts together gives Q ( t ) in terms
P
and L as(14)
To obtain explicit expressions for P and L, we take the Laplace
DD
Q ( t ) = POP(?
-
L )+
E c i exp(-Xit)i= I
transformation of Q ( t ) in eq 14:
Equations 13 and 15 are different expressions for the same entity. Equation I3 contains the parameters D, u, and K to characterize
the convective-reactive diffusion, while eq 15 contains P and L to characterize its asymptotic behavior. Thus one can deduce relations of
P
and L-with D, u, and K from the following consid-eration. Note that Q(s) possesses a pole at s = 0, and an infinite number of poles a t s =
-Xi.
The pole a t s = 0 is of second order and accounts for the asymptotic part. The rest of the poles, onthe other hand, account for the transient part. To obtain ex- pressions for P and L, the singularity a t s = 0 is removed by multiplying eq 18 by s2 to obtain
-
s2ci+IS
+
Xis2Q(s) = pop
-
p0PLs +c-
(16)In the infinitesimal neighbor_hood around s = 0, where the as- ymptotic part dominates, s*Q(s) acquires the form of a Taylor series (instead of a Laurent expansion as in eq 15). Accordingly
pop = Iim s2&) (17)
S-0
On substituting eq 13 into eqs 17 and 18, one obtains P =
(20) Thus, we have derived expressions for P and L in convective- reactive diffusion without resorting to any solutions in forms of infinite series. All manipulations were performed in closed form. The advantage of using a Laplace transform in the formulation of
P
and L results from the fact that the asymptotic part corre-sponding to the singularity of order 2 a t s = 0 can be singled out from the transient part.
Change-of-Variables Approach
In this section we introduce a change of variables to transform eq 1 into a form containing only terms of first and second de- rivatives of the distribution function. The resulting equation is then compared with the governing equation for a heterogeneous diffusion system to obtain an expression for position-dependent partition coefficient in terms of
D,
u, k, andK
in eqs 1-4. Such a relation will enable us to formulate permeability and lag time for convective-reactive diffusion system simply by transcription from their previously derived counterparts in heterogeneous diffusion. To this end, p(x,t) is factored into a purely position- dependent part and position- and time-dependent part, namelyp ( x 4 = A x ) flx,t) (21)
Substitution of eq 21 into eq 1 results in
Up to this point, the functional forms of f(x,t) and g(x) are unspecified. If g(x) is set to satisfy the differential equation (23)
a 2
a
D&x)
-
u,,g(x)-
Kg(X) = 0then eq 22 reduces to
a x
For conciseness, we refer henceforth to a diffusant whose distributionflx,?) obeys eq 24 ‘asfdiffusant, and, correspondingly, to one that follows eq 1 and eq 23, respectively, as p and g dif- fusants. Note that in view of eq 30 the diffusivity remains un- changed on transformation from p diffusant to f diffusant.
Now, imagine an f diffusant undergoing heterogeneous diffusion characterized by a constant diffusivity D and a position-dependent partition coefficient K(x). The diffusion equation for such a heterogeneous diffusion system is14*1s*18
T - ( x , t )
a
= D&( K(x)& (25)or upon expansion
a
$x,t) =
On comparing eq 24 with eq 26 term by term, one obtains
a
a
u-
a x In K(x) = -2-ax
In g(x)+
-
Da 2
-
ax2 In K(x) = 0 The solution for K(x) in eqs 27 and 28 is
(29) exp(ux/ D)
[ d X ) l 2 K(x) = B
where B is a constant to be determined. A combination of eqs 27 and 28 gives
Lag Time for Convective-Reactive Diffusion The Journal of Physical Chemistry, Vol. 95, No. 24, 1991 10167
V - d G
d x ) = A ~ X P ( 20 x ) (31)
where A is a constant to be determined. Without loss of generality, we set A = 1, which, in turn, implies that the boundary conditions for g(x) in eq 23 are
g(0) = 1 (32)
On
substituting the boundary conditions (32) and (33) into eqs 2-4, one sees that f(x,r) in eq 24 possesses the same initial and boundary conditions as p(x,r). Specificallyf(X,O) = 0 (34)
f ( 0 J ) = KPO (35)
f ( h , t ) = 0 (36)
Substitution of g ( x ) in eq 31 with A = 1 into eq 29 yields
K ( x ) = B e x p
ITXI
(37)At the interface x = 0, the partition coefficient K is given by eq
35 and, thus, from eq 37, B = K. As a result, the partition coefficient for the f diffusant in heterogeneous diffusion takes on
the position-dependent form
K(x) = Kexp( d C D
x)
(38)To obtain relations for the permeability and lag time in con- vective-reactive diffusion, we compare the time development of the total diffusant released into the receiver in the long-time limit for a p diffusant, Q ( P ) ( t ) , with that of an f diffusant, @ ( t ) . This yields
where the boundary condition (4) is used and J’P)(x,t) denotes the diffusion flux (at location x and time t). Substitution of eq
21 into eq 39, with the help of eq 36, yields then
This simple proportionality relationship between Q(P)(t) and Q @ ( t ) holds for any t
>
0. Specifically, in the long-time limit, we haveIim Q ( p ) ( t ) = poP(P)(t - L ( p ) ) =
I--
U - d G
~ X P ( 20 h )
!iy
Q@(4
=which indicates that
L(P) =
La
(43)Expressions for the permeability and lag time in heterogeneous diffusion, in which both the diffusivity and partition coefficient are position dependent, have been derived previo~sly.’~*’~*l* For constant diffusivity the expressions are
(44)
On substitution of eq 38 for K(x) into eqs 44 and 45 one obtains for the permeability for the f diffusant and lag time for thefand
p diffusants
= 1 -exp(-
d
z
h ) ) -I (46)pf’
= L(P) =(47)
Substitution of eq 46 into eq 42 results then in
p(P) = - ~ K D X
2 0
exP
E[exp( h
&h)(
d m h )csch ( d m h ) /As expected, P and L obtained through the change of variables are identical to those resulting from Laplace transform, eq 20.
Discussion
The system dealt with above represents a prototype interplay between convection, reaction, and diffusion processes, of which convective diffusion, reactive diffusion, and simple diffusion systems are special cases. Hence, P and L for each subsystem should be derivable from our general expressions. For reactive diffusion, with u = 0 eqs 48 and 47 respectively give
Equation 50 was derived earlier by Leypoldt and GoughZs using a finite Fourier transform.
For convective diffusion, we chose, for comparison, an ionto- phoresis (electrodiffusion) system, in which an constant external
10168 The Journal of Physical Chemistry, Vol. 95, No. 24, 1991 Chen and Rosenberger electrical field E is applied to enhance the permeability for ions.
The convection velocity in this case is
v = D Z F E / R T (51)
where Z is the valence of the ions, F the Faraday constant, R the universal gas constant, and T the absolute temperature. Sub- stitution of eq 51 into eqs 48 and 47 with K = 0 results in the
solution obtained by Keister and Kasting:26
P =
"I(
h @h) csch ( z h ) l (52)ZFE
With respect to simple diffusion, which corresponds to v = 0 and K = 0, both eqs 47 and 48 yield indeterminate forms of
O/O.
This difficulty, however, can be removed by applying the following series expansion:28 x 7x3 6 360 "' x c s c h X = 1
-
-
+
-
-
(54) (55) x2 x4 3 45 x coth x = 1+
-
- -
+
...
withSubstitution of eq 54 into eq 48 and eq 55 into eq 47, respectively, and taking the limit x
-
0, the well-known expressions P = K D / h and L = h 2 / 6 D are recovered.As revealed by eq 47, L is symmetric with respect to the di- rection (or the sign) of the convection velocity u and is independent of the partition coefficient K. The reciprocity of the lag time in a general convective-reactive diffusion has been proved mathe- matically in terms of the symmetry of the related Green's function with respect to the exchange of coordinate variables.29 On the other hand, according to eq 48,
P
is proportional to K and dis- symmetric with respect to the direction of u. Thus the permea- bilities for forward and reverse direction, pC+) andPC-)
can be related as(28) Spiegel, M. R. Mathematical Handbook, Schaum's Outline Series;
(29) Chen, J . S.; Rosenberger, F. Chem. Eng. Commun., in press. McGraw-Hill: New York, 1978.
In
(fi+)/p-))
= v h / D (57)which results from eq 48 on altering the sign of u.
In
the ion- tophoresis experiment, eq 57 can be transformed with eq5
1 toThis relation can be used to determine the thickness of the membrane from the measurement of pC+) and pC-).
Siege130 also used a Laplace transform, though in a different approach, to derive a lag time expression for a diffusion system governed by a linear equation with source terms. H e obtained, written in our notation
(59)
where Jsjs the steady-state diffusion flux. With J, = pOp and
J ( s ) = sQ(s), eq 59 is essentially identical to our eq 18. The expression for lag time in eq 47 also reveals an interesting formal similarity between diffusion lag time and electric polar- ization. With eq 56, eq 47 can be rewritten to
where L b ) is the Langevin function encountered in the theory of induced electric polarization. Though the connection between these two physical entities may be remote, eq 60 is still useful for mnemonic purposes.
In
conclusion, we have obtained closed form solutions for the permeability and lag time of a diffusion system with convection and reaction. The Laplace transform and change-of-variables techniques have the advantage over earlier approaches that involve explicitly solving the relevant diffusion equation and manipulating the solution (usually in the form of infinite series) into a closed form. However, there is no guarantee that closed forms for lag time can be extracted from such infinite series solutions.29Acknowledgment. This research was supported by the State of Alabama through the Center for Microgravity and Materials Research a t the University of Alabama in Huntsville, and the starting fund provided for faculties in the Department of Applied Chemistry, National Chiao Tung University, Hsin Chu, Taiwan.
J.-S.C. also expresses his gratitude to Professor Jeffrey L. Fox, Department of Pharmaceutics, University of Utah, for helpful discussions.
(30) Siegel, R. A. J . Membr. Sci. 1986, 26, 251. Note that q 9 of this reference is dimensionally incorrect and should be read as q 59 in this work.
