J. Phys. Chem. 1983, 87, 1984-1990
1984
between the platinum and the TiOz.
all cases the behavior becomes fairly ohmic with appreciable reverse currents flowing. While the nature of the TiOz material here under investigation is very different from the rutile and anatase particles used in photoelectrochemical experiments, the measurements should have some relevance to the platinized powders and colloids. The Ti02 there is usually formed under conditions where a high density of interface states would be produced. Thus, the Pt/Ti02junctions would be more ohmic and the role of the Pt would be that of a catalyst for the reduction reaction rather than contributing to formation of a Schottky diode. This view is supported by recent finds by Heller and co-workers18that hydrogen absorption on Pt decreases its work function and tends to make its junctions with semiconductors ohmic.
Conclusion Surface preparation of TiO, and treatment of the sample following Pt contact formation has been shown to be highly important in the fabrication and properties of Pt/Ti02 junctions. I-V curves demonstrate significant variation with surface preparation; the best Schottky barriers were obtained with Pt deposited onto etched Ti02. Abraded and polished samples exhibited substantially greater reverse currents. The curve shape was also dependent upon the degree of reduction of the Ti02 and the direction of the potential scan and sample history. These results, in conjunction with evidence from capacitance, ac resistance, and the variation in reverse resistance with sweep speed, indicate the presence of very slow surface states at the Pt/TiOz interface which can be filled when the junction is forward biased and emptied when the junction is reverse biased. Thermal treatment of the Pt/Ti02 junctions can lead to interdiffusion of the materials which produce ohmic, low-resistivity contacta. However, even with etched samples the behavior only approaches that of an ideal Schottky barrier (e.g., linear C2vs. V plots) on the first scan. Changes in the barrier occur with time and in almost
Acknowledgment. Dr. Gregory A. Hope is on leave from the School of Science, Griffith University, Nathan, Queensland, Australia. The support of this research by the National Science Foundation (CHE8000682) is gratefully acknowledged. Registry No. Rutile, 1317-80-2; R,7440-06-4 'EOz, 13463-67-7. (18)Heller, A.,private communication.
Nonideal Multlcomponent Mixed Mlcelle Model P. M. Holland' and D. N. Rublngh The Procter and Oembk Company. Mlaml Valley Laboratorks, Cincinnati, Ohio 45247 (Received: SeptemLmr 17, 1982; I n Final Form: December 27, 7982)
A generalized multicomponent nonideal mixed micelle model based on the pseudo-phase separation approach is presented. Nonideality due to interactions between different surfactant components in the mixed micelle is treated via a regular solution approximation. The problem is addressed in a way designed to avoid the algebraic complexity that would arise from a direct extension of the binary surfactant model to multicomponent mixtures, while allowing for arbitrarily accurate numerical solution of the appropriate relationships. For example, in the (limiting) ideal case numerical solution for a single unknown is sufficient to give both the micellar composition and individual monomer concentrations in a surfactant system consisting of an arbitrary number of components. For nonideal systems the approach is designed to allow for a straightforward extension to systems of increasing complexity by taking advantage of the Nelder-Mead simplex technique for function minimization. Comparison of the multicomponent model with experimental critical micelle concentration (cmc) results on ternary nonideal mixed surfactant systems is reported, using net interaction parameters determined independently from cmc measurements on binary systems. The results show good agreement with the predictions of the multicomponent model.
Introduction Micelle formation in aqueous solutions of surfactants leads to striking changes in the behavior of their physical properties such as surface tension and light scattering. A critical micelle concentration (cmc) where such changes occur can also be measured in mixed surfactant systems. The resulting mixed micelles contain two or more surfactant components in equilibrium with surfactant monomers. Pseudo-phase separation models have been developed to treat mixed micellization in binary surfactant mixtures. Generally these assume ideal mixing of the (1)H. Lange, Kolloid 2.131, 96 (1953). (2)K.Shinoda, J . Phys. Chem., 58, 541 (1954). (3)H.Lange and K. H. Beck, Kolloid 2.2.Polym., 251, 424 (1973). 0022-3654/83/2087-1984$01.50/0
surfactants in the micelle and allow calculation of the mixed cmc as a function of the overall composition of the mixture and the cmc's of the individual components. Explicit treatment of the monomer concentrations and micelle composition can also be i n ~ l u d e d .The ~ pseudophase separation approach has been quite successful in predicting the behavior of both binary nonionic and binary ionic mixtures of surfactants, particularly when the surfactants have the same hydrophilic group. More recently ideal mixed micelle models have been extended to twophase systems.6 In the case of binary mixtures of nonionic and ionic surfactants or of surfactants with different hydrophilic (4)J. Clint, J . Chem. Soc., 71, 1327 (1975). ( 5 ) F.Harusawa and M. Tanaka, J. Phys. Chem. 85, 882 (1981).
0 1983 American Chemical Society
The Journal of Physical Chemlstry, Vol. 87,No. 11, 1983 1985
Nonideal Multicomponent Mixed Micelle Model
groups, significant deviations from the ideal model are often observed. This has led to the development of a pseudo-phase separation model where nonideal mixing in the micelle is treated via a regular solution approximation,3*6p7 thus allowing interactions between different surfactant components in the micelle to be included. This approach has been successfully applieds in predicting the nonideal behavior of a wide variety of binary surfactant mixtures including mixtures of nonionic and ionic surfactants (even though the effects of counterions are not explicitly taken into account). The purpose of this paper is to report work generalizing the approach of the pseudo-phase separation model to treat multicomponent nonideal mixed micellar systems. In this work, we develop a thermodynamic model for nonionic surfactant mixtures which can also be applied empirically to mixtures containing ionic surfactants. The problem is addressed in a way designed to avoid the algebraic complexity that would arise from a direct extension of the binary models to multicomponent mixtures, while allowing for (arbitrarily) accurate numerical solution of the appropriate relationships. For example, in the (limiting) ideal case, iterative solution for a single unknown in a generalized relationship is sufficient to give both the micellar composition and individual monomer concentrations in a surfactant system consisting of an arbitrary number of components. For nonideal systems the approach is designed to allow for a straightforward extension to systems of increasing complexity by taking advantage of the Nelder-Mead simplex technique for function minimization. 73 The ability to treat multicomponent surfactant systems, particularly when they deviate from ideal behavior, is of both practical and theoretical interest. In such systems, micellization is a controlling factor in establishing the monomer concentrations of the individual surfactant components. Knowledge of the monomer concentrations is important since they specify the chemical potential of individual species in solution. The chemical potential in solution provides the driving force for processes of practical importance such as interfacial tension lowering and contact angle changes besides establishing a thermodynamic relationship between species distributed among different phases once equilibrium is reached.
Theory The chemical potential of the ith free monomeric component of the mixed micellar solution of n total components can be written as p i = pio
+ RT In Ci"
(1)
where the activity coefficient of the free monomer is assumed to be unity (see list of symbols). Within the mixed micelle itself the chemical potential of the ith component can be expressed as
+ RT In fixi
piM= piMo
(2)
The use of a pseudo-phase separation model for micellization gives piMo= p i o
+ RT In Ci
(3)
for pure ith-component micelles. ~~~
(6) D.N. Rubingh in "Solution Chemistry of Surfactants", Vol. 1, K. L. Mittal, Ed., Plenum Press, New York, 1979, p 337. (7) R. F. Kamrath and E. I. Franses, Ind. Eng. Chem. Fundam., in press. (8) J. A. Nelder and R. Mead, Comput. J., 7, 308 (1965).
Since at equilibrium piM= p i , eq 1-3 can be combined to yield cim
= xifici
(4)
as a generalized result. A t the mixed cmc (C*) the monomer concentration of each component is given simply by
Ci" = QiC*
(5)
which together with eq 4 and the constraint n
cxi=1 is1
can be combined to give a generalized relationship for the mixed cmc of a system with an arbitrary number of surfactant components.
(7)
It should be noted that for activity coefficients f i of unity the expression reduces in form to that previously derived by Lange and Beck3 and Clint4 for ideal mixed micelles. Mass balance relationships allow the mole fraction of the ith component in the mixed micelles to be directly expressed as aic - Ci" xi
=
(8)
n
c - ic= l
Cj"
Substituting eq 4 for Ci" in the numerator, rearranging, and then applying the constraint 6 allows one to obtain the generalized expression
where
M =
cn Ci"
i=l
for multicomponent micelles of n total components. Solving for M allows the mole fractions in the micelle and the individual monomer concentrations to be determined for each component via the generalized relationships xi = Q i C / ( C fit; - M) (10)
+
.
Ci" = QifiC,C/(C + fjCi - M) (11) obtained from eq 8 and 4. In the ideal mixed micelle case, where the fi's are equal to unity, this generalized expression can be solved directly (to arbitrary accuracy) for the total monomer concentration M. This approach inherently avoids the significant increase in algebraic complexity that would result from a direct extension of the binary model to multicomponent mixtures. In particular, application to mixtures consisting of large numbers of different components is not significantly more difficult than for those with only a few components. In the case of multicomponent mixtures which contain surfactants nonideal with respect to each other, it is first necessary to determine the appropriate activity coefficients for use in the generalized equations 9-11. Several useful relationships can be derived which related the mole fraction in the micelle for any one component, x i , to that of any other, x j . At the cmc, eq 4 and 5 can be combined to give fj
= QjC*/(XjCj)
(12)
198% The Journal of Physical Chemistry, Vol. 87, No. 11, 1983
Holland and Rubingh
which in turn yields the relation xi =
criCjfjXj/(CiCujfi)
(13)
valid a t the cmc. Above the cmc eq 8 can be applied separately to two different components and by eliminating the sum of the monomer concentrations when one combines them the general relation (14) xi = CuiC/(C,fi - C j f j + crjC/Xj) results. Together with the constraint given by eq 6 these epxressions provide basic relationships useful in solving for the activity coefficients. Using a simple regular solution approximation, one can express the activity coefficients as functions of the mole fractions of each of the components in the mixed micelle x i and appropriate interaction parameters &j. The parameters Pij are constants related to net (pairwise) interactions in the mixed micelle of the form (15) P i j = N ( Wii + Wj; - 2 Wij)/RT In the case of binary nonideal mixtures of components a is required and the regular single interaction parameter approximation for the activity coefficients in the mixed micelle gives fl
= exp Pl&1 - x1I2 f2
= exp
P12X12
(16)
(17)
The net interaction parameter P12 can be readily determined when the mixed cmc for the binary system, C12*, is known. This requires solving iteratively for x1 at the cmc using a relationship such as
obtained from eq 12, 16, and 17. P12 can then be directly obtained by combining eq 1 2 with eq 16 to give
In the generalized (multicomponent) case, the activity coefficients can be expressed in the form n
n
j-1
( i # J # k)
where Pij represents the net (pairwise) interaction between components i and j , and x j the mole fraction in the micelles of the j t h component. In general, for n different nonideal components (or groups) there will be a total of
Czn = n!/l(n- 2)!2!]
(21)
different (pairwise) parameters (e.g., 3 for n = 3, 6 for n = 4, 10 for n = 5 , etc.). There will also be an equivalent number of terms in the general expression of the activity coefficiences f i , with n - 1 direct terms containing x j 2 (i # j ) and c~~ - n + 1 cross terms containing x j k (i # j # k ) . Given the appropriate P parameters (obtained independently from binary mixtures), the nonideal multicomponent problem requires solving for the xi's a t the total composition and concentration of interest in order to obtain the activity coefficients. This requires solution of the appropriate simultaneous equations obtained by various substitutions of eq 20 into eq 14 generally, or into eq 13 a t the mixed cmc, and by employing the constraint 6 that the sum of the xi equals unity. Efficient numerical solution of such multiple equations for multiple unknowns can be
m I
0 E u
t 0
.2
.4
.6
.8
1.0
Mole Fraction C120SO: Na+
Flgure 1. Cmc's of mixtures of Clo(CH3),P0 and C,20S03-.Na+ in 1 mM Na,CO, (at 24 "C). The plotted points are experimental data, the solid line is the prediction of the nonideal mixed micelle model with /3 = -3.7, and the dashed line is the prediction for ideal mixing.
difficult. We have chosen to employ the Nelder-Mead simplex technique for function minimization8 for this aspect of the problem due to its considerable generality and excellent efficiency. In particular, the Nelder-Mead simplex procedure allows for a rather straightforward extension to cases containing many independent variables. Examples of the Nelder-Mead procedure are given in detail in l i t e r a t ~ r e . ~ Once the activity coefficients f i have been determined, the generalized nonideal expressions (eq 7 and 9-11) can be readily solved for the multicomponent case. It should be pointed out that, since the P parameters used in determining the activity coefficients are independently determined from binary mixtures, the multicomponent result is obtained without adjustable parameters.
Experimental Section To provide a test of the nonideal multicomponent mixed micelle model, experimental cmc measurements were carried out on three different mixed ternary surfactant systems. These measurements included each (binary) pair of the surfactants involved (in order to independently determine the net interaction parameters, P) as well as the cmc's of the individual surfactants. The cmc's were determined from surface tension vs. concentration data measured by using a tensiometer with du Nouy ring. All compounds used in this study were pure single specie surfactant determined to be of >98% purity by thin-layer or gas chromatography. Surface tensions vs. concentration data for the single components did not show the presence of surface-activeimpurities with the exception of tetraoxyethylene glycol monooctyl ether (CBE,), CB(OCH2CH2)40H, which exhibited a slight dip at the cmc. In the case of the CBE4,however, gas chromatographicanalysis showed the sample to be >99% pure. Measurements on the decylmethyl sulfoxide (C,,SO), Clo(CH,)SO/decyldimethylphosphine oxide (C,,PO), (9) D. M. Olsson, J. Qual. Tech., 6,53 (1974).
Nonideal Multicomponent Mixed Micelle Model
The Journal of Physical Chemistry, Vol. 87,
5.0
7.0
r
No. 11, 1983 1987
-
----
4.0
--3.0
-e
2.0
"O
t 0
.2
.4
.6
.8
1.0
Mole Fraction C12OSO; Na+ Flgure 2. Cmc's of mixtures of C,,(CH,)SO and CI,OSO3--Na+ in 1 mM Na2C0, (at 24 "C). The plotted points are experimental data, the solid llne Is the prediction of the nonideal mlxed micelle model with p = -2.4,and the dashed line is the prediction for ideal mixing.
1
7.0
-e-
t
L
I
I
1
I
I
A
B
C
D
E
Flgure 4. Cmc's of various ternary mixtures of C,,(CH3),PO/Clo(CH3)SO/C1,OSO3-.Na+ in 1 mM Na,C03 (at 24 "C). The molar ratios of the mixtures were respectively for A-E 0.201/0.646/0.153, 0.354/0.378/0.260,0.622/0.221/0.157,0.231/0.246/0.523,and 0.136/0.145/0.719.The plotted points are experimental data, the solid lines the prediction of the multicomponent nonideal mixed micelle model using p's independently determined from the binary systems (Figures 1-3), and the dashed lines the prediction for ideal mixing.
8.0
7.0
6.0
I
6.0
I I
I
f I 1
/
l'O
t I
0
I
0
I
I
I
1
.2
.4
.6
.8
I
Mole Fraction C120S03 Na+ I
.2
.4
I
.6
.8
Mole Fraction C10(CH3)2P0 Flgure 3. Cmc's of mixtures of Clo(CH3)S0 and C,,(CH&PO in 1 mM Na2C03(at 24 "C). The plotted points are experimental data and the solid line is the prediction of the nonldeal mixed micelle model with (? = 0 (equivalent to ideal mixing).
Clo(CH3)2PO)/sodium dodecyl sulfate (SDS, Cl2OSO3-. Na+) system were carried out a t 24 "C in 1 mM sodium carbonate (pH 10). Measurements were also carried out on the dodecyldimethylamine oxide ( ClzAO), C12(CH3)2NO)/tetraoxyethyleneglycol monodecyl ether (C1OE4, Clo(OCHzCH2),0H)/SDS system at 23 "C in 0.5 mM sodium carbonate (pH 9.8). The final ternary system was sodium decyl sulfate (SDeS), CloOS03-.Na+/decyl-
Flgure 5. Cmc's of mixtures of C,,(CH&NO and C,,OSO,-.Naf in 0.5 mM Na,CO, (at 23 "C). The plotted points are experimental data, the solid line is the prediction of the nonideal mixed micelle model with p = -4.4,and the dashed line is the prediction for ideal mixing.
trimethylammonium bromide (CloTAB), CloN+(CH3)2Br-/C8E4 in 0.05 M NaBr (at 23 "C). Results and Discussion Results for the CloPO/CloSO/SDS system are shown in Figures 1-4, where the experimental points are compared with calculated results from the nonideal mixed micelle model (solid line) and the ideal mixed micelle model (dashed line). It can be readily seen in Figures 1 and 2 that the binary C120/SDS and CloSO/SDS systems deviate significantly from ideality with (? values of -3.7 and -2.4, respectively. The CloPO/CloSO system (Figure 3)
I988
Holland and Rubingh
The Journal of Physical Chemistry, Voi. 87, No. 11, 1983 8.0
7 0
i 1
I I I
I I
I
/ /
-
a
- - e +
/
1
I
I
0
.2
.4
I
.8I
.6
Mole Fraction Cl20SO;
Na+
Figure 6. Cmc's of mixtures of Clo(OCH2CH2),0H and Cl20SO3-.Na+ in 0.5 mM Na2C0, (at 23 "C). The plotted points are experimental data, the soHd line is the predictkn of the nonldeal mixed mlcek model with B = -3.6, and the dashed line is the prediction for ideal mixing.
I
I
I
1
A
B
C
D
I E
Figure 8. Cmc's of various ternary mixtures of C,2(CH3),NO/Clo(OCH2CH2),0H/C120S03--Na+in 0.5 mM Na2CO3 (at 23 OC). The molar ratios of the mixtures were respectively for A-E 0.459/0.342/0.199, 0.201/0.451/0.348, 0.383/0.286/0.331, 0.548/0.136/0.316, and 0.165/0.123/0.712. The plotted points are experimental data, the s o l i lines the prediction of the multicomponent nonideal mixed micelle model using p's Independently determined from the binary systems (Figure 5-7), and the dashed llnes the prediction for ideal mixing.
-
7.0
6.0
5.0
4.0
,
I
0
.2
.4
.6
.8
Mole Fraction C12(CH3)2N0 Flgure 7. Cmc's of mixtures of Clo(OCH2CH2),0H and C12(CH3)2N0 in 0.5 mM Na2C0, (at 23 OC). The plotted points are experimental data, the soli line is the predicUon of the nonldeal mixed micelle model with @ = -0.8, and the dashed llne Is the predlction for ideal mixing.
was found to behave ideally ( p = 0). Using the P parameters determined independently from the binary systems, we then applied the nonideal model to the ternary system giving the results for various mixture ratios shown in Figure 4. It can be seen that the ternary system deviates significantly from ideality and that the nonideal model gives good agreement with the experimentally measured cmc's.
.2
..4
.6
.8
Mole Fraction C,,N+(CH,),.BrFigure 8. Cmc's of mixtures of CloN+(CH3),.Br- and C,oOS03-.NaC in 0.05 M NaBr (at 23 OC). The plotted points are experimental data, the solid line is the predlction of the nonideal mixed micelle model with p = -13.2, and the dashed line is the prediction for ideal mixing.
Similar results for the system C12AO/CloE4/SDSare shown in Figures 5-8. Figures 5 and 6 show the binary C,,AO/SDS and CloE4/SDSsystems which also deviate significantly from ideality with (3 values of -4.4 and -3.6,
The Journal of Physical Chemistry, Vol. 87,
Nonideal Multicomponent Mixed Micelle Model
No. 11, 1983 1989
0
I
I
I
1
.2
.4
.6
.8
1.0
.2 .4 .6 .e Mole Fraction CIoN+(CH&.Br-
Mole Fraction C,oOS03-.Na+ Figure 10. Cmc's of mixtures of C8(OCH2CH2),0H and CloOS03--Na+ in 0.05 M NaBr (at 23 OC). The plotted points are experimental data, the sdii line is the prediction of the nonideal mixed micelle model with 6 = -4.1, and the dashed line is the prediction for ideal mixing.
respectively. In Figure 7, the behavior of C12AO/C10E14 is seen to be more nearly ideal with p = -0.8. Using the p parameters determined independently from these binary systems, we then applied the model to the ternary system. It can be seen in Figure 8 that the nonideal model again gives generally good agreement with the experimental cmc's for various ratios in the ternary system. A more extreme example of a nonideal mixed ternary system is that provided by the anionic/cationic/nonionic system SDeS/CloTAI3/C&, shown in Figures 9-12. Here the binary system SDeS/ClqTAB (see Figure 9) shows striking deviation from ideality resulting in a value of -13.2. It should be noted that, in spite of the extreme deviation from ideality observed in this system, the regular solution approximation provides a good description of its behavior over a wide range of composition. The binary systems C8E4/SDSand C8E4/CloTAB,shown in Figures 10 and 11, give less extreme deviations from ideality with p values of -4.1 and -1.8, respectively. Results calculated for the ternary system (Figure 12) are again based on the p parameters determined from the binary systems and again good agreement is seen with the experimentally measured mixed cmc's. Together, these results on ternary systems demonstrate that a nonideal multicomponent mixed micelle model based on the pseudo-phase separation approach and a relatively simple regular solution approximation can be successfuly applied to a variety of nonideal mixed surfactant systems. Since these results are obtained by using net interaction parameters independently determined from binary systems, the multicomponent model does not depend on adjustable parameters. This increases the usefulness of the model in predicting the properties of mul-
1.0
Flgure 11. Cmc's of mixtures of C8(OCH2CH2),0H and C,oNf(CH3)3.Brin 0.05 M NaBr (at 23 OC). The plotted points are experimental data, the solid line is the prediction of the nonideal mixed micelle model with (3 = -1.8, and the dashed line is the prediction for ideal mixing.
i
- 20 7
-I
E
m I
0
T1
Y
0
- - - - a A
I
I
B
C
D
Flgure 12. Cmc's of various ternary mixtures of CloOS03-.Br+/ C,oN+(CH3)3~Br-lC8(OCH2CH,),0H in 0.05 M NaBr (at 23 OC). The molar ratios of the mixtures were respectively for A-D 0.224/ 0.20710.569, 0.36010.334/0.306, 0.62810.19410.178, and 0.216/ 0.601/0.183. The plotted points are experimental data, the solid lines the prediction of the multicomponent nonideal mixed micelle model using 6's independently determined from the binary systems (Figures 9- 1l), and the dashed lines the prediction for ideal mixing.
ticomponent mixed surfactant systems. It is interesting that the regular solution approximation, originally developed for liquid mixtures, is so successful in treating nonideality in mixed micelles. Here, the micellar structure is ignored and the interactions between different surfactant molecules are accounted for by a single generalized parameter which represents an excess heat of mixing. This suggests that the utility of the regular so-
J. Phys. Chem. 1983, 87, 1990-1997
1990
lution approximation (with its single parameter) is not very dependent on the geometry of the interacting system and the consequent “coordination number” of nearest neighbors. For example, the same approach has been applied with some success to nonideal interactions in mixed adsorbed monolayers of surfactants.1° It is also interesting to note that ionic surfactants, and particularly mixtures of ionic and nonionic surfactants, seem to be adequately treated even though the model neglects effects due to the binding of counterions. This suggests that deviations due to each effects either are small or can be empirically (and successfully) accounted for in the net interaction parameter of the regular solution approximation.
List of Symbols chemical potential of monomeric surfactant i standard chemical potential of monomeric surfactant i M chemical potential of i in mixed micelles chemical potential of i in pure micelles Ci” monomer concentration of surfactant i fi activity coefficient of surfactant i in mixed micelles Xi mole fraction of surfactant i in mixed micelles Ci cmc of pure surfactant i mole fraction of surfactant i in total mixed solute cmc of mixed system C total concentration of mixed surfactants M total surfactant monomer concentration Pij net interaction parameter between surfactants i and
Acknowledgment. We acknowledge D. F. Etson for experimental cmc work and other valuable contributions in the laboratory of P.M.H.
Wi., Wii, pairwise interaction energies between surfactant b i j molecules in micelle N Avogadro’s number Registry No. SDS, 151-21-3;Clo(CH3)S0,3079-28-5; Clo(CH3)P0,2190-95-6; C12(CH3),N0,1643-20-5; Clo(OCH2CH2)40H, 5703-94-6; CloN+(CH3)2-Br-,2082-84-0.
(10) B. T. Ingram, Colloid Polym. Sci., 25, 191 (1980).
Pi Pio
cMo ?*
j
Application of Moments to the General Linear Multicomponent Reaction-Diffusion Equation John Savchlk, Brltton Chang, and Herschel Rabltz” Department of Chemistty, Princeton University, Princeton, New Jersey 08544 (Received: JuV 13, 1982; In Final Form: November 17, 1982)
A sequence of spatial moments to the solution of the linear multicomponent reaction-diffusion equation in the infinite domain with general nondiagonal diffusion and rate matrices can be established without prior determination of the solution. This is significant for several reasons. First, while the solutions to full reaction-diffusion equations cannot generally be found in closed analytical form, a simple recursion relation can be established for the moments. Secondly the lower moments can be given a clear physical meaning and they can be explicitly represented in terms of their contribution from the kinetic and diffusion portions of the problem. These results suggest that conversion of raw experimental concentration profiles into moments would be useful for physical interpretation as well as analysis in terms of the underlying diffusion and rate matrices. As a particular point of illustration,it is shown that the lower moments can be combined to yield expressions for the mean-squared diffusion length for each component of the mixture. It is shown that the time behavior of the mean-squared diffusion length in a reaction-diffusion system deviates from that in simple Fick’s law diffusion only if there is both chemical reaction and a nondiagonal diffusion matrix. As a specific illustration the mean-squared diffusion lengths are calculated for a two-solute system undergoing general diffusion and a reversible isomerziation reaction.
I. Introduction Multicomponent diffusion with chemical reaction is an important aspect of many physical and biological proce~ses.l-~However, in even the simplest linear multicomponent reaction-diffusion equations with nondiagonal diffusion and rate matrices, explicit solutions can be ex~ this pressed only in terms of the transformed s ~ l u t i o n .In paper, we will not attempt to solve the linear multicomponent reaction-diffusion equation but instead determine (1) J. Hearon, Bull. Math. Biophys., 12, 135 (1950). (2) R. Aria, ‘The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts”. Vol. I and 11. Clarendon Press, Oxford, 1975. (3) G. Nicolis and I. Prigogine, ‘Self-Organization in Nonequilibrium Systems”, Wiley, New York, 1977. (4) J. M. Hill, I. M. A . J. Appl. Math., 27, 177 (1981).
certain spatial averages of the solution in the infinite domain. These averages, in particular, are a sequence of spatial moments which can be determined without the explicit solution in time and space. This approach has the virtue of providing rather simple relationships between the moments and the diffusion and rate constant matrices. This observation may be of ultimate use for reducing experimental reaction-diffusion data which can easily be converted into spatial moments. Before we proceed, it is important to review certain aspects of diffusion theory, in-particular general multicomponent diffusion with nondiagonal diffusion matrices. Classical diffusion theory is based on Fick’s first law5 (5) W. Jost, “Diffusion in Solids, Liquids, and Gases”, Academic Press, New York, 1960.
0022-3654/83/2087-1990$01.50/00 1983 American Chemical Society
