C. E. Vallet and J. Braunstein
2438
Solution of Electrochemical Flux Equations with Variable Diffusion Coefficient and Transference Numbert C. E. Vallet" and J. Braunsteln Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 (Received May 27, 1977) Publication costs assisted by the Department of Energy
Finite difference solution of the nonlinear partial differential equation of diffusion and migration for constant current electrochemical processes in binary mixtures of molten salts has been carried out for several important cases. Concentrations at the electrode surface as a function of time, and concentration-distance profiles between the electrode surface and the bulk electrolyte, have been computed for the following model cases, the first two of which have analytical solutions that could be used to test the computation (D is the interdiffusion coefficient, and M is a migrational term whose form depends on the mobilities of the ions and on the electrode reaction): (I) constant D ,M = 1; (11) constant D, constant M # 1; (111)D varying exponentially with composition, M = 1; (IV) D constant, varying M, (V) D and M varying. The results demonstrate that chronopotentiometric measurements can be applied to the evaluation of diffusion coefficients of concentrated electrolytes, in cases where the Sand equation does not apply, by analysis of the early portion of the chronopotentiogram.
Introduction Nonlinear partial differential equations with derivative boundary conditions arise in a variety of electrochemical problems such as electroanalytical chemistry, the study of diffusion and migration in molten salts and solid electrolytes, and the modeling of batteries and fuel cells. In most electrochemical studies of diffusion and migration, only the linear cases with analytical solutions have been treated, corresponding to constant diffusion coefficients and the absence of migrational terms, severely limiting the applicability of these electrochemical methods. In this paper we consider the more general case of the diffusion-migration equation with composition dependent diffusion coefficient and transference number in binary molten salt mixtures. Electrochemical methods such as polarography, chronopotentiometry, and voltammetry have been widely applied to the determination of self-diffusion coefficients of very dilute solute species in a supporting electrolyte of high concentration which suppresses composition dependence of the transport properties of the dilute species.l We have s h o ~ that n ~ chronopotentiometry ~ ~ can also be applied to the estimation of interdiffusion coefficients of the major constituents in binary molten salt mixtures even in cases of strong composition dependence of the interdiffusion coefficient. The r e ~ u l t showever, ,~ have shown the need for a more direct analysis based on numerical solution of the nonlinear diffusion-migration equation. Here we present the results of an explicit scheme of numerical solution; we consider most of the cases of composition dependence of diffusion coefficient and transference number observed in binary ionic mixtures, including polymeric and network melts, as well as the effect of differing electrode reactions. The cases considered include: test of the algorithm with (I) constant interdiffusion coefficient and unity migrational term and (11) constant interdiffusion coefficient and constant migrational term; (111) strongly varying interdiffusion coefficient and unity migrational term; (IV) constant interdiffusion coefficient and varying migrational terms appropriate to the molten salt systems in the Li/S battery with LiCl-KC1 'Research sponsored by the Materials Science Branch of the Division of Physical Research of the U S . Energy Research and Development Administration under contract with Union Carbide Corporation. The Journal of Physlcai Chemistry, Vol. 81, No. 25, 1977
electrolyte, to the carbonate fuel cell with Li2CO3-K2CO3 electrolyte, to metal dissolution in a polymeric melt (A1 dissolution in A1C13-NaC1; (V) both interdiffusion coefficient and migrational term varying as in metal dissolution in binary melts with a network forming component (Be dissolution in LiF-BeF2). These computations can be applied to the estimation of composition profiles in the modeling5 of molten salt batteries and fuel cells as well as to the simulation of chronopotentiograms.
The Diffusion-Migration Equation We have shown6 that, during the polarization under constant current of a metal anode M' in a molten salt binary mixture MA-M'A the flow of cations, M, relative to the common anions A may be written
J&
=
where is the mutual diffusion coefficient of the cations M and M', 4 is a distance variable from the anode such that increments d t contain equal numbers of anions A, CM the MA concentration in equivalents/cm3, I the constant current density, and M = tb the transference number of cations M relative to common anions A.16 The differentiation of eq 1 with respect to 4 gives the nonlinear diffusion-migration equation
The boundary and initial conditions depend upon the type of experiment to be simulated. In chronopotentiometric measurements, the polarized indicator electrode of small area is relatively far from the counterelectrode, so that a good approximation consists of treating the problem in terms of semiinfinite diffusion from a planar electrode. The boundary condition a t the indicator electrode is obtained by writing that the flux of cations M in eq 1 is zero since a t the boundary the nonelectroactive cation M does not cross the electrolyte-electrode surface
M I
t > 0,t = 0
2439
Solution of Electrochemical Flux Equations
0 co
The initial condition is uniformity of composition as the experiment is initiated t = 0, t; > 0 C M = C(bu1k) = C o (2c)
A
CA ( b o u n d a r y )
c t + co I
When the calculation is applied to a battery or fuel cell, the initial condition applies over the space 0 5 4 IZ between the two electrodes
C M = C o = C(bu1k) t = 0, 0 G < P Pd) Since the two electrodes are of similar size and close to one another, the boundary condition (2b) must be replaced by a condition analogous to eq 2a given by the flux condition a t the second electrode
I
I
I
'
U
i
--+--'-i'
II -
At
I
M I
I
I
I
t>O,[=Z
When both interdiffusion coefficient and transference number are constant, eq 2 has an analytical solution. In actual cases, where both are composition dependent, numerical solution must be employed to solve eq 2 which may be written
a2 c M yFiM't$)2 + DM-M, at
-- --at
;Et:)
-
(3)
Numerical Solution In the finite difference scheme the continuous functions ( a C ~ / a t )(,a C ~ / a t ) ,and a2CM/at2are finitized, with centered finite differences for the space derivatives
0
At
2AE
DISTANCE FROM ELECTRODE
Figure 1. Sequence of computations with forward explicit scheme (semiinfinite problem): (0) Co; (A) Cot(electrode boundary); (0) d
#
c?
satisfy eq 2b; they are used to calculate the last composition differing from the bulk composition in the next time step. In the case of two electrode boundaries, a t time zero, uniform composition (equal to the bulk composition) is placed on the entire space mesh of width 2 = N A t . The composition at the second electrode is calculated from an equation analogous to eq 6
(7) This scheme is not unconditionally stable; for stability the fixed time step and distance step must fulfill8
The calculation follows a forward explicit scheme in which the composition profile at time (t At) is computed from the profile calculated a t time t with
+
The sequence of calculations is illustrated in Figure 1 for the case of semiinfinite boundary conditions. Three points with composition of the bulk melt are placed on the time/distance mesh, in accord with eq 2c, for starting the calculations. Composition 1 (equal to the bulk composition in the first cycle) is calculated with eq 5. Then composition 2, which is different from the bulk composition, is calculated by using the finite difference form of eq 2a
Before the next time step, two points with compositions equal to the bulk composition are added on the mesh to
For our calculations, we took At = A(2/3D(c). Combining eq 5 and 8, it can be shown that (Cb - Ci:At) is always negative. Consequently, since CaAtis smaller than Cg , Cb is closer to the value of C a A tto be calculated than Ctp+FAt and is used in eq 6 to evaluate M and D at the boundary. The numerical calculation converges toward the true solution as the distance step approaches zero. Figure 2 shows the relative compositions at the polarized electrode vs. the square root of time calculated with different disto 5 X cm and composition tance steps from 1 X dependent diffusion coefficient. The ordinate, [ 1 CM*/C0] / I is a relative concentration, normalized to the current density, which has been used3s4 to interpret chronopotentiometric data. Its value is zero at time zero and 111as the concentration of MA becomes zero at the electrode surface. The slope of the line is proportional to the reciprocal of the square root of interdiffusion coefficient and independent of current density. The solid lines correspond to Cf, , while the dashed lines correspond to C&. As expected, differences between Cb and C& decrease as A[ decreases, but the relative compositions C& seem to converge more rapidly toward the true solution. Therefore, we chose to place the electrode surface on the mesh a t At, using the image point ( E = 0) as discussed by Newmang when applying the boundary condition with eq 6.
Application of the Numerical Computation Figure 3 shows curves computed for a number of composition dependences of interdiffusion coefficient and The Journal of Physical Chemistty, Vol. 81, No. 25, 1977
2440
C. E. Vallet and J. Braunstein
I -
9
'
I
'
I
'
I
'
I
/
-c;
N -
5 0.3
c
"0 \
0.2
I
1
\
-t/
I 0
--
0.1
z 2 + a 0.005 cc
L1
5
I
-1
/
l
z
I 0.001
0 0.001
0.005
0.01
DISTANCE FROM ELECTRODE ( c m )
Figure 4. Concentration profiles at 3.3 and 20 s with constant diffusion Coefficient D = 1 X lo-' cm2/s and unity migrational term. Current density = 0.18 A (-) numerical solution; (0)with eq 9.
E
\
X
0"
2.5
\
2.0
0.85
0.5
0.80
0
J ' i.5 1 .o
the transition calculated from the analytical solution by the Sand equation.1° Concentration profiles are shown in Figure 4 at short time and at times near the transition time calculated from both the numerical solution and the analytical solutionlo
L
Y
(9) 1.0
2.0
3.0
4.0
m (S''2)
5.0
6.0
Figure 3. Numerically calculated time dependence of the composition at the electrode surface with varying diffusion coefficient or migrational term: (0) with constant diffusion coefficient (1 X lo-' cm2/s) and unity transference number; (0)represents the transition (C, = 0 at electrode surface) from the analytical solution; (1) exponentially varying diffusion coefficient D(cm2/s) = 1.5 X 10-6exp[-0.094/(1.02 - X,J]; (2 and 2') migrational term equal to transference number: M = 1 for C > CT= C0/3; M = 1 - [2.52 X lO'(CT- ch,1)3for ] CM < C T = C 13;
v
(3) constant and nonunity migrational term M = 1 - X o ; (3a and b) migrational term equal to transference number, T, M = T = uM/uW/([(l uM/uMt\(a) uM/uM'= 2; (b) UM/UMl= 1; (4) migrational term equal to T - X M , U M / u M f = 2.
xM)/xM]+
migrational terms which will be discussed below, including a test of the algorithm for constant interdiffusion coefficient and migrational terms. I. Test of t h e Algorithm with Constant D and Unity M. The algorithm has been tested in the case of a constant interdiffusion coefficient, DM-M,, and unity migrational term M , for which an analytical solution is known10 (the concentration a t the electrode is [l - c M / c O ] / I= 2t1I2/ FCO(TDM-M,)~/~). With a distance step A t = 1 X cm, curve 0 in Figure 3 shows the relative concentration at the electrode surface'^^ proportional to the square root of time up to the transition time (at which time the concentration falls to zero) and with slope equal to that calculated from the analytical solution. Curve 0 has been calculated with DM-Mt = 1 X lo4 cm2/s and current density 0.18 A/cm2 in a mixture MA-M'A2 of composition 0.778 mol fraction MIA2. (The numerical values correspond to those for LiF-BeF2 mixtures, but without considering the composition dependence of D.) The filled circle corresponds to The Journal of Physical Chemistry, Vo/.8 1, No. 25, 1977
where CM is the concentration of MA in equivalents/cm3. 11. Constant D, Constant M # 1. The migrational term occurs directly in the boundary condition (eq 6 ) , but only through its derivative in the computation of the composition profile at time ( t A t ) from the profile a t time t (eq 5). In the case of a constant, but nonunity migrational term M in a mixture with constant interdiffusion coefficient, we have shown4 that with the transformations of D M - Mand ~ 5:
+
D'M-Mv = DMVI-M'/M2 $' = t1M (10) the analytical solution for the concentration at the electrode surface applies
The numerical solution obtained with D M - M , = 1 X lo4 cm2/s and constant migrational term, M = (1- XO),gives curve 3 in Figure 3. The slope is in the ratio (1- X o ) to the slope of curve 0, as expected from eq 10. 111. Interdiffusion Coefficient Exponentially Dependent on composition, M = l . Composition Profiles. Strong composition dependence of the interdiffusion coefficient is observed in binary melts with network forming components such as silicates and fluoroberyllates. In LiF-BeF2 mixtures diffusivity, like fluidityll or electrical conductivity,12 is found to decrease strongly as the BeFz concentration increases. In a first approximation we interpreted our chronopotentiometric data assuming
2441
Solution of Electrochemical Flux Equations
o~o'l-"---I .-.. >
DECREASING 0,
0.3
0
'-2
v
0.00f
0 002
0
IR is the ohmic loss, n the number of electrons taking part in the faradaic process, X the mole fraction of M'A, at the anode, Xothe mole fraction of M'A, in the bulk melt, t& the transference number of M+ relative to anion A, FM'A, the chemical potential of M'A,. Curves 0 and 1, re-
2
4
6
8
10
12 14 TIME ( s )
16
18
20
22
24
Figure 6, Chronopotentiograms calculated for hypothetical melt 0.778 mole fraction BeF,-0.222 LiF with current density 0.18 A/cm* assuming: (0)constant diffusion coefficient D = 1 X lo-' cm2/s; (1) varying D , B = 1.5 X 10-sexp[-0.094/(1.02 - XMF)];(2) M = 1 for CM> Co/3, M = 1 - [2.52 X lO'(CT- CM)3] for < CT = Co/3.
6,
spectively, in Figure 6 show simulated chronopotentiograms (corrected for the IR drop) calculated with constant D (1X lo4 cm2/s) and with the varying D given by eq 11. In both calculations, X o = 0.978, the current density is 0.18 A/cm2, and the transference number t& = 1. For times up to 3 s the simulated chronopotentiograms coincide, but at longer times, as the varying D becomes smaller, they diverge. For the case considered here, a transition time of 13.1 s would be obtained from the chronopotentiogram corresponding to varying D, while the chronopotentiogram with constant D has a transition time of 23.3 s. Ignoring the composition dependence of the interdiffusion coefficient and basing the calculation on the apparently sharp transition would have resulted in a lower interdiffusion coefficient, 5.6 X lo-' cm2/s, a 44% error. In Figure 3, however, it can be seen from curves 1 and 2 that the limiting slope at zero time of the relative concentration function (1- CM/Co)/Ivs. the square root of time leads to a diffusion coefficient in the bulk melt 1 X 10" cm2/s. These calculations confirm our earlier conclusion4that in chronopotentiometric measurements at high concentration, the transition time is difficult to interpret and that the analysis of the early parts of chronopotentiograms is preferable. Furthermore, reliance on the sharpness of a chronopotentiometric transition as a diagnostic procedure is treacherous, since the effect of a strongly decreasing composition dependent D is, in fact, to sharpen the transition. IV. Constant D , Varying M . Little variation with composition of the interdiffusion coefficient is found in simple ionic mixtures (e.g., nitrates, chlorides, carbonates) and in some mixtures with a polymeric component (e.g., aluminum chloride-alkali chloride mixtures). Migrational terms take forms depending on the electrode reaction and on the physical properties of the melt. When one of the cations (M') is produced or consumed by the electrode reaction, the migrational term is equal to the transference number of the other cation relative to the common anion ( M = tb).2i6When the common anion reacts at electrode, it has been shown6from the appropriate flux equation that the migrational term has the form ( t b - XM). The transference number, in terms of the ionic mobility ratio r = UM/UM/, is given by tb = r / { [ ( I- xM)/xM]+ r];the mole fraction is related to the equivalent concentration as XM = C M ~ M (-t / [c~~ ( v-~v M 1) ] where the are the partial equivalent volumes of the components. In mixtures with a polymeric component, the transference number of the mobile cation is unity over a wide range of composition, decreasing as its composition approaches zero. Curves 2, 3a, 3b, and 4 in Figure 3 are calculated with the assumption of constant interdiffusion coefficient and different comThe Journal of Physical Chemistry, Vol. 8 1, No. 25, 1977
C. E. Vallet and J. Braunstein
2442
position dependent migrational terms. A situation not far removed from Li/S battery systems with molten lithium chloride-potassium chloride as the electrolyte5 may be simulated with the transference number of potassium relative to chloride written in terms of the mobility ratio r and the migrational term equal to t h . In Figure 3 curve 3a corresponds to r = 2, and curve 3b to r = 1 ( t h = XM), cm2/s. both with interdiffusion coefficient of 1 X In carbonate fuel cells where the common ion, A (CO,2-), crosses the electrode surface, the migrational term5 in eq 1 is (t& - XM). Curve 4 in Figure 3 is calculated with a mobility ratio of 2 in the transference number. (With r = 1,M = 0, there is no migrational contribution and the composition would remain unchanged at X = 0.778.) In each of these cases, the initial slope is proportional to M/(DM-Ml)1/2.Thus when the migrational term is known for the melt of initial composition, interdiffusion coefficients may be calculated from the initial portion of a chronopotentiogram. In network and polymeric systems, the transference number of the nonelectroactive species is unity over a wide composition range, but may then decrease rapidly when M is nearly depleted. The effect of such composition dependence of the transference number is isolated from the effect of varying D in the calculations shown in curve 2 in Fi ure 3 and curve 2 in Figure 6 for a system in which M = tM. 9 These computations simulate the case of the dissolution of an aluminum anode in mixtures with the polymeric component AlC13 where interdiffusion coefficients may have little dependence upon composition. The equation used to represent the transference number in calculating these curves gives t& = 1 for M’A concentrations, up to 92 mol % M’A, decreasing to tb = 0.01 near pure M’A. As the transference number decreases, downward curvature appears in the relative concentration plot (Figure 3, curve 2) which leads to a longer transition time in the chronopotentiogram (Figure 6, curve 2). In the analysis of our experimental data in LiF-BeF2 mixtures, since the variation of transference number with composition was not known near pure BeF2,we calculated the composition at the electrode surface by solving eq 12 with the assumption of a transference number equal to unity over the entire range of composition. This assumption, applied to the potential/time curve generated by the numerical solution with decreasing transference number, led to the dotted curve 2’ in Figure 3; curve 2’ is lower than curve 2 and would lead to longer transition time, but the effect is not large for a transference number decreasing only near pure BeF2. V. Diffusive and Migrational Terms Composition Dependent; Comparison to Chronopotentiometric Data. We have reported chronopotentiometric experiments in an LiF-BeF2 melt of initial composition 53 mol % BeF,. The experimental curve of the relative LiF concentration a t the anode surface vs. the square root of time has a sigmoid shape which has been qualitatively related to composition dependence of both the interdiffusion coefficient and the transference number, as the concentration at the electrode surface varies from 53 mol % BeF2 to near pure BeF2. Independent mea~urementsl~ of the transference number of lithium relative to fluoride give the value unity in the composition range 30-85 mol % BeF2. This suggested the calculation of the interdiffusion coefficient in melt of initial composition from the short time linear portion of the relative concentration-(time)1’2 plot, a procedure justified by the computations shown in Figure 3, curve 1. The upward curvature at intermediate times was attributed to a decreasing interdiffusion coefThe Journal of Physical Chemistry, Vol. 8 1, No. 25, 1977
“ !.
0.50 0.45
::$/, 0.0
0.2
I
/
0.15
, 0.4
0.6
0.8
,
,
, 1.0
,o;;~o~020
1.2
1.4
0.024
(sed”)
Flgure 7. Effect of decreasing D and decreasing Ton the time dependence of composition at electrode surface, comparison with experimental results (ref 4): (0)calculated trom experimental data4 D = 7.4 X exp(-16Xw,) 4- 5 X (-) calculated by numerical solution with D , T = 1 for cL> c r = c0/4; I - [2.66 x i o 6 ( c r - CL)3] for CL < CT= C0/4; (---) calculated assuming T E 1 from the chronopotentiogram calculated with the numerical solution.
r=
ficient, and the later downward curvature was attributed to a decrease of the transference number as the composition approached pure BeF,. Values of the interdiffusion coefficient obtained in melts of varying initial c o m p o ~ i t i o n ~show , ~ J ~the strong decrease in diffusivity in 53 mol % BeF2 to 1 X loT7near pure from 1.6 X BeF2 which is expected as the melt gets richer in the network forming component. These results suggested an exponential composition dependence of the interdiffusion coefficient with a decrease of about two orders of magnitude between 0.53 BeF2 and nearly pure BeF2. In order to test whether sigmoid plots similar to those found experimentally could be generated with simple functions representing plausible decreases of D and tr, the following functions were employed:
lo-’ e X p ( - 1 6 X ~ ~+~ ,5) x 0.53 < XBeF, < 0.86
D
=
7.43 x
(13)
t:
=
1
(14)
t:
= 1- [2.66 X 106(C0/4- C L ) 3 ]
0.86 < X B e ~ < , 1 Figure 7 shows a simulated relative concentration plot generated with the above D and t f , together with the experimental points. The general features of the experimental curve are well represented, but the downward curvature and flattening take place earlier in the simulation than in the experimental plot. This suggests that the transference number of lithium relative to fluoride may remain unity up to compositions closer to pure BeF2 than the composition X B e F z = 0.86 (CT = C&/4) at which the decrease was assumed to occur in the function given by eq 14. This example shows that in melts with network structure (BeF2,silicates), analysis of the entire chronopotentiogram can provide insights into the composition dependence not only of the interdiffusion coefficient but of the transference number as well. Conclusion Numerical solution of the diffusion migration equation at a polarized electrode extends the use of electrochemical scanning methods such as chronopotentiometry, in studies of transport properties and possibly for analytical applications, over a wide composition range in binary molten
2443
Computational Analysis of a Chemical Switch Mechanism
salt mixtures. Our calculations demonstrate that chronopotentiometric data obtained from solutions concentrated in the electroactive ion cannot be reliably interpreted in terms of the transition time alone. On the other hand, the very early part of chronopotentiograms can be used in a straightforward manner for the determination of transport properties in melt of initial composition. Furthermore, the entire chronopotentiogram may be analyzed in terms of the composition dependence of interdiffusion coefficients and transference numbers over a wide composition range, and the numerical simulation can provide the basis for fitting equations to chronopotentiometric measurements. With modification of the boundary conditions to represent two electrodes of comparable area a t finite separation, the computation is applicable to the simulation of mass transport in molten salt batteries and fuel cellsa5 In some of the cases to which the calculations have been applied, the interdiffusion coefficient changes by two cm. orders of magnitude over a distance of about 1 X The strong variation of transport properties over a small distance introduces complexities into the computation analogous to the complexities introduced by a large number of reaction steps or rapid reactions. This variation dictates the choice of small distance steps and concomitant small time steps for stability. Furthermore, since the purpose of these calculations is to provide an accurate estimate of the time dependence of concentration a t the boundary, and since the boundary condition is given by the distance derivative of composition, the distance step near the boundary must be as small as possible. Distance cm were found necessary in most of the steps of 1 X cases treated here to obtain convergence and computational stability. For diffusion coefficients near cm2/s the corresponding time step calculated from the stability condition was 2 X s. In the present form, the computation is rather time consuming for use in a fitting
process. As an example, the computation of the curve in Figure 7 using about 8500 time steps required 9.5 min with the IBM 360/91 computer and used a region of 332K. A computation with constant D and varying M (curves 2, 3, and 4 in Figure 3), by contrast, required about 1 min. Varying the time and distance steps on the mesh, together with alternative schemes, to decrease the computational time while increasing stability are being investigated. References and Notes (1) A. Weissberger and B. W. Rossiter, Ed., "Physical Methods in Electrochemistry", Wiley-Interscience, New York, N.Y., 1971. (2) J. Braunstein, H. R. Bronstein, and J. Truitt, J. Hectroanal. Chem., 44, 463 (1973). (3) C. E. Vallet and J. Braunstein, J. Am. Ceram. Soc., 58, 5-6, 209-14 (1975). (4) C. E. Vallet and J. Braunstein, J. flectrochem. Soc., 124, 1, 78-83 (1977). (5) J. Braunstein and C. E. Vallet, "Proceedings of Symposium on Electrode Materiils and Processes in Energy Conversion and Storage", The Electrochemical Society, Princeton, N.J., in press. (6) C. E. Vallet, H. R. Bronstein, and J. Braunstein, J. f/ectrochem. Soc., 121. 11. 1430-39 (19741. (7) J. Crank, "The Mathematbs of Diffusion", Clarendon Press, Oxford, 1975, pp 203-209. (8) K.Rektotys, Ed., "Survey of Applicable Mathematics", The MIT Press, Cambridge, Mass., 1969. (9) J. S.Newman, "Electrochemical Systems", Prentice-Hall, Englewood Cliffs, N.J., 1973, pp 414-425. (10) P. Delahay, "New Instrumental Methods in Electrochemistry", Interscience, New York, N.Y., 1954, pp 179-181. (11) S. Cantor, W. T. Ward, and C. T. Moynihan, J . Chem. Phys., 5 0 , 7, 2874 (1969). (12) G. D. Robbins and J. Braunstein in "Molten Salts Characterization and Analysis", G. Mamantov, Ed., Marcel Dekker, New York, N.Y., 1969, p 443. (13) C. E. Vallet and J. Braunstein, Silic. Ind., 41, 3, 161 (1976). (14) K. A. Romberger and J. Braunstein, Inorg. Chem., 9, 1273 (1970). (15) As pointed out by Crank' a single mutual diffusion coefficient can be defined in binary systems, even if the volume is not constant, by choice of a suitable frame of reference and consistent distance and concentration units. Here, the reference volume must be defined as containing a constant number of common anions. I n the case of LiF-BeF2 mixtures, this effect is very small since the partial equivalent volumes V, and V, are very close.
A Computational Analysis of a Chemical Switch Mechanism. Catalysis-Inhibition Effects in a Copper Surface-Catalyzed Oxidation D. L. Allara" and D. Edelson" Bell Laboratories, Murray Hill, New Jersey 07974 (Received April 26, 1977) Publication costs assisted by Bell Laboratories
A previously reported, catalysis-inhibition switching phenomenon in the copper-catalyzed oxidation of hexadecane has been modeled. The sharp switching between inhibition and autocatalysis as a function of the copper surface site to hydroperoxide ratio has been reproduced. The model consists of coupled homogeneous and surface mechanisms with a total of 87 reactions. The homogeneous set is based on the well-known reactions of free-radical oxidation and is consistent with the extensive theoretical and experimental information on these processes. The surface mechanism is based on a proposed competition for trapping of hydroperoxide at inhibition and/or catalysis surface sites. The computational modeling results provide a clue to specific experiments which would allow efficient proof (or disproofs) of the validity of the proposed reaction steps.
1. Introduction 1.1. T h e Computational Modeling Approach t o Reaction Mechanism. The technique of computational modeling of chemical reaction systems can be of great value in determining quantitative mechanisms, particularly in
cases where reaction complexity nearly or completely precludes traditional analytical approaches. The validity of the models and mechanistic conclusions derived therefrom should be based on the completeness of the reaction sets, the confidence in the associated rate paThe Journal of Physical Chemistry, Vol. 81, No. 25, 1977
