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(C-
(10) B. T. Ingram, Colloid Polym. Sci., 25, 191 (1980).
Pi Pio
cMo ?*
j
H3)P0,2190-95-6; C12(CH3),N0,1643-20-5; Clo(OCH2CH2)40H, 5703-94-6; CloN+(CH3)2-Br-,2082-84-0.
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; I n 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
General Linear Multicomponent Reaction-Diffusion Equation
Si
= -DVci
si
The Journal of Physical Chemistry, Vol. 87, No. 11. 1983
(1)
In eq 1 the flux of the ith species is (mol/(cm2s)), ci is the concentration of the ith soluie (mol/cm3), D is the diffusion coefficient (cm2/s),and V is the spatial gradient operator. Previous research has shown that eq 1 does not adequately describe the diffusion of multicomponent mixtures.6 In particular, eq 1 does not allow for the diffusive interaction between the various components of the mixture. A widely used generalization of Fick‘s law assumes the flux for a given species i is dependent on the concentration gradient of species i along with the concentration gradients of the other solutes in the m i ~ t u r e . ~ J For a mixture composed of N solutes it follows that N
si = -CDij?cj
i = 1, N
j=l
In this paper, the fluxes are measured with respect to the volume-fixed frame of reference defined as8 (3)
i=O
where Oi is the partial molar volume of species i and the subscript i = 0 corresponds to the solvent. It is important to note that, in the diffusion representation_ of eq 2, the solvent flux, Jo, does not directly appear. Jo can be obtained, however, from eq 3. An alternative representation of diffusion can be defined where the solvent flux is explicitly coupled to the fluxes of the remaining components in the ~ y s t e m . ~ N
si = -CDij*?cj
i = 0, N
j=O
(4)
The diffusion matrix elements Dij of eq 2 can be related to the elements Dij* of eq 4.1° This paper will adopt the diffusion formulation defined by eq 2. In general, the diffusion coefficients of eq 2 are concentration-dependent quantities. In particular, the offdiagonal elements of the diffusion matrix D must satisfy the following limits to ensure species conservation:”J2
si
-
lim Dij = 0
(5)
Ct-O
since 0 as ci 0. A continuity equation can be obtained for each component of the mixture by considering the flux of particles entering and leaving an elemental control volume (Fick’s second law). -+
a
--
-ci = -V.Ji at
If first-order chemical reactions occur between the solute components, a reaction-diffusion equation results of the form8 (7)
The quantities Kij are elements of the reaction matrix K for the set of coupled first-order chemical reactions. As pointed out by Gosting: the presence of chemical reactions (6) E. Cwler, “MulticomponentDiffusion”,Elsevier, New York, 1976. (7) R.Baldwin, P. Dunlop, and L. Gosting, J.Am. Chem. SOC., 77,5235 (1955). (8) L. Gosting in “Advancesin Protein Chemistry”,Vol. XI, M. Anson, K.Bailey, and J. Edsall, Ed., Academic Press, New York, 1956. (9) L. Onsager, Ann. N.Y. Acad. Sci., 46, 241 (1945). (IO) M. Dole, J. Phys. Chem., 26, 1082 (1956). (11) I. O’Donnel and L. Gosting, “The Structure of Electrolyte Solutions”, W. Hamer, Ed., Wiley, New York, 1957. (12) F. Shuck and H. Toor, J.Phys. Chem., 67, 540 (1963).
1991
only affects the continuity equation, eq 6, and not the equation for the fluxes, eq 3, since the effects of chemical reactions create and remove species from a given point but do not transport material. Equation 7 with the definition for the fluxes in eq 2 constitutes the general reactiondiffusion equation with effects of solute-solute diffusive interactions and first-order chemical reactions. Since the diffusion coefficients are concentration dependent, in general eq 7 represents a coupled set of nonlinear partial differential equations. If one assumes the matrix elements of D are constants, eq 7 reduces to
a
= C[DijV2cj+ Kijcj] atci The assumption of constant D reduces the reaction-diffusion equation to a coupled linear partial differential equation. In view of the concentration limits on the offdiagonal matrix elements of D in eq 5, care must be taken in the interpretation of the solutions to eq 8a. In general, the constant D assumption limits eq 8a to the study of reaction-diffusion systems that involve only small changes in concentration. In this paper, the effects of solute-solute diffusive interactions on reaction-diffusion systems will be explored within the limitations of eq 8a. This equation has been the source of considerable interest as a mathematical model of systems of biological chemical, and geophysical interest. The stability of the nonequilibrium stationary states for nonlinear reaction-diffusion equations also leads to eq 8a.3J3 Not surprisingly, for general reaction-diffusion systems, stability is dependent on both the reaction and diffusion matrices. If the diffusion and reaction matrices, D and K , do not commute, explicit solutions for a general N solute component system are obtainable only in a formal sense. (The case where D and K commutes is a particularly restrictive condition which does not appear to have physical significance.14) Formal solutions to eq 8a with appropriate initial and boundary conditions can be obtained through standard transform techniques.15J6 In the next section, it is shown that these solutions are in terms of an exponential matrix with an argument of the form Mt = -(w2D - K ) t . A first step in obtaining the explicit solution involves the determination of a formula for the eigenvalues of M as a function of the parameter w . In general, this becomes very tedious for systems consisting of more than two or three s o l ~ t e s . ~ J ~ - ~ ~ In this paper, we examine an alternative approach to the study of multicomponent linear reaction-diffusion equations by solving for various spatial averages over the species concentrations. Attention is focused on the sequence of moments of ci(?,t) defined as
Al(t) =
IJ?? ...?In ci(?,t) d3r
where Al(t)is the nth tensor moment of species i over a spatial volume V. The symbol {??...?In denotes n powers of the vector 7. Although the remainder of this paper could be simply generalized to this three-dimensional case (cf. the central formula in eq 11would then contain a three(13) J. Gmitro and L. Scriven in “Intracellular Transport”, K. B. Warren, Ed., Academic Press, New York, 1965. (14) H. Toor, Chem. Eng. Sci., 20, 941 (1965). (15) H. Toor in “Intracellular Transport”, K. B. Warren, Ed., Academic Press, New York, 1965. (16) G. DeLancey and S. G. Chiang, Ind. Eng. Chem. Fundam., 9,334 (1970). (17) J. Hill and E. Aifantis, Q.J. Appl. Math., 33, 23 (1980). (18) J. Albright, J. Phys. Chem., 67, 2628 (1963). (19) H. Othmer and L. Scriven, 2nd. Eng. Chem. Fundam., 8, 302 (1969).
1992
Savchik et ai.
The Journal of Physical Chemistry, Vol. 87, No. 11, 1983
dimensional Fourier transform), there is no loss in content by restricting the remainder of the paper to one dimension. For the special case of diffusion in an infinite medium, the moments can be explicitly determined without the prior knowledge of ci(x,t)for complex linear reaction-diffusion systems. The practical experimental realization of an effective infinite medium is easily achieved by considering measurements over a time interval such that the initial spatial disturbance does not appreciably diffuse to the vessel walls. Several of the lower order moments have a simple physical interpretation and analytical expressions for these moments are useful in examining the coupling of reaction and diffusion in a system. The focus of the remainder of this paper is on the moment analysis of eq 8a in the one-dimensional infinite domain. 11. Theory
In this section, we consider some aspects of the solution to the general multicomponent one-dimensional linear reaction-diffusion equation in an infinite medium and show how moments of the solution are obtained without prior determination of the explicit solution. We use eq 8 as our basic reaction-diffusion equation which in matrix notation can be written as
a
a2
D-c
-C
at
ax2
x-fm
The integration over w in eq 11 requires the diagonalization of the matrix M ( w ) = w2D- K for all w. The evaluation of eq 9 for multicomponent systems quickly becomes a complex procedure where there is little chance for a closed form analytical solution. Complicated expressions for c(x,t) have been obtained for the 2 X 2 case for isomerization kinetics by Albright18 and Hill and Arfanti~.~," Moments of the solution to the linear reaction-diffusion equation in the one-dimensional infinite domain can be determined without the explicit solution c(x,t). The moments are a sequence of spatial averages of c(x,t). We define the nth moment of c(x,t) as A"(t) = ~ -m + D r ' c ( x , tdx )
(12)
A general moment equation can be formulated by multiplication of eq 8b by n" and integration over the spatial domain.
+ KC
a
-A"(t) = n(n - l)DAn-' +KA" at
The boundary conditions are20 lim JxmJc(x,t) =0
The inverse Fourier transform of eq 10 results in a formal expression for the solution to the linear reaction-diffusion equation. 1 c(x,t) = -jmexp{-(w2D - K)t) exp(-iwx) f(w) dw 2 a -m (11)
for all m = 0, 1, ...
and a general initial condition is specified by c(x,O) = f(x) For a mixture composed of N solutes, c is a concentration vector of length N a n d the matrices D and K are of order N . The formal solution to eq 8b can be obtained with the use of standard integral transform methods.21 After Laplace transforming eq 8b in the time domain and then Fourier transforming the resulting equation in the spatial domain, one obtains a set of coupled algebraic equations for the transformed concentration, c(w,s):
+
C(W,S)= (SI w 2 D - K)-' f ( w )
(9)
where L(c(x,t))= j m e x p ( - s t )c(x,t) dt = c(x,s)
(13)
Equation 13 assumes that, at the boundaries, c and its spatial gradient approach zero faster than any integral power of x . For finite and semiinfinite boundaries, such conditions are not generally valid and a more complicated moment equation results which requires the a priori determination of c(x,t) for the determination of its moments. Equation 13 can be integrated over time to yield an expression for the nth moment in terms of the ( n - 2)th moment. AYt) = n(n - 1 ) l t e x p ( - K ( t ' - t))DA"-2(t? dt'+ 0
exp(Kt)A"(O)
n = 0, 1, ... (14)
Equation 14 is an integral recursion relation for the determination of the moments. One observes that when f(x) is an even function of x , the quantities A"(0) vanish for all odd n and it follows from eq 14 that all odd moments of c(x,t) vanish. The first three moments of c(x,t) can be written as
0
S(c(x,t)}= l+mexp{iwx) c(x,t) dx = c(w,t) L(S{c(x,t))1=
C(W,S)
and I is the unit matrix. The inverse Laplace transform of the general matrix (SI+ A)-l can be written as22
L 1 { ( s I+ A)-l) = exp{-At) where exp(-At) is the exponential matrix. It follows that the inverse Laplace transform of eq 9 can be written as c(w,t) = exp(-(w2D - K ) t ) f(w) (10) (20) M. J. Lighthill, "Introduction to Generalized Functions, Fourier Analysis and Generalized Functions",Cambridge University Press, Cambridge, 1978. (21) J. Crank, "The Mathematics of Diffusion", Clarendon Press, Oxford, 1975. (22) Y. Takahashi, M. Rabins, and D. Auslander, "Control and Dynamic Systems", Addison-Wesley, Menlo Park, 1972.
Ao(t) = exp(Kt)AO(0)
(154
A1(t) = exp(Kt)A'(O)
(15b)
A2(t) = 2Jtexp{K(t'-
t ) ) Dexp(Kt1 dt'AO(0)
+ exp(Kt)A2(0) (15~)
Explicit evaluation of the moments for each component of the concentration vector c(x,t) requires the similarity matrix S and ita inverse which will diagonalize the constant K matrix. As a consequence of the moment recursion relation, eq 14, c(x,t) is not required to determine the sequence of moments An(t). In the next section, interest will be focused on the evaluation of the second moment. It is convenient to decompose the second moment into a sum of terms contributed by each individual matrix element of D . This is accomplished by decomposing the matrix D into a sum of elementary matrices. Any arbitrary square matrix P can be decomposed into a sum of ele-
General Linear Multicomponent Reaction-Diffusion Equation
The Journal of Physical Chemlstry, Vol. 87, No. 11, 1983
mentary square matrices that contain only one nonzero element.
P=
c c P,@Eu@
a=l@=l
The order of the matrix P is 1 and the matrices E"@are matrices of order 1 with elements defined as
E$ = Gai6Bj It follows that for an N solute system, the diffusion matrix can be decomposed as
1993
retical and experimental consideration. The moments of c(x,t) can be related to the formal solution, eq 11, through a moment generating function +(t,a). Multiplication of eq 11by the factor exp(iax) and integration over the spatial domain results in an expression for the moment generating function. 1 +(t,a)= -x.fmx:mexp(iax) 2* exp(-(w2DK ) t Jexp{iwx)f(w) dw dx (17a) With the definition of the delta function
N N
D=
CD,,Eu@ a=l@=l
In Appendix A it is shown that the second moment, A2(t), can be written as A2(t) = 2CCD,,R"SA0(0) . B
+ exp(Kt)A2(0)
+(t,a)= xImexp(-(w2D- K ) t J f(w) 6(w - a) dw
(16)
= exp{-(a2D - K)t) f(a)
where the kth element of Ra@is
r
Equation 17a can be rearranged to yield
(17b)
Repeated differentiation of the moment generating function +(t,a)with respect to a and evaluation at a = 0 results in the sequence of moments for c(x,t). Thus
r
J
and the matrix that diagonalizes K is S defined by the similarity transformation
I t is assumed that the matrix K contains no degenerate eigenvalues. At this point it is useful to compare the formal solution for the concentrations in eq 11 and the explicit moment expressions in eq 15. The ultimate aim of any theoretical or experimental reaction-diffusion analysis is invariably directed toward understanding or determining the roles of the elements in the D and K matrices. Although these matrices are clearly evident in eq 11, they only appear in the integrand of an integral which must be evaluated numerically. Such a numerical quadrature will obscure the roles of K and D. In contrast the easily calculated moments in eq 15 are seen to have a much simpler dependence on K and D . In particular Ao and Al depend exclusively on K , and the similar forms in eq 15a and 15b suggest a convenient consistency check on experimentally derived moments Ao and A'. This could also prove useful when using an experimental moment Ao(t) to extract K (or elements of it) and then employing these rates to predict the moment A '(t) for comparison with its measured value. The second moment A2(t) contains D in a linear fashion and eq 16 could be readily manipulated to solve for D in terms of sufficiently available experimental moment data. Such an "inversion" could also be achieved through eq 13, but now dA 2/at would have to be evaluated from the experimental data. Numerical differentiation requires considerable care when the data are appreciably contaminated with noise, but this problem may not be serious since the process of evaluating the integral in eq 12 acta as a filter for noise. The optimal mode of applying eq 15 remains to be established, but the physical meaning of the moments clearly makes them attractive for theo-
This differentiation involves the use of the well-known formula for the differentiation of an exponential operator with respect to a parameter as shown by W i l ~ o x . Suc~~ cessive differentiation of eq 18 results in the recursion relation in eq 14. Expressions for the moments can be obtained for any experimentally realizable initial spatial distribution of chemical species c(x,O) = f(x), since only A"(0) = S-,+"f(x) xn dx is affected. It is especially instructuve to consider
f(x) = Q6(x) (19) since the moments can be readily interpreted in terms of mean diffusion lengths when the initial spatial distribution is well localized. The vector Q has elements corresponding to the total initial amount of each species. Mean diffusion lengths for the system can be defined in terms of the moments, An@). ( P ) i= A l ( t ) / A ? ( t )
(20)
It follows that ( x ) and ~ ( x ~ are ) ~ the mean and meansquared diffusion lengths for species i. Since f(x) is symmetric about the origin, it is necessary that ( x ) = ~ 0. For an initial condition of the form of eq 19 the zeroth and second moments are Ao(t) = exp(KtJQ A2(t) = 21texp[-K(t'- t)]D exp(Kt? d t ' Q 0
(21a) (21b)
Inspection of eq 20 and 21 allows the deduction of certain limiting behavior for the mean-squared diffusion length of a given species in a reaction-diffusion system. (1) In classical Fick's law diffusion, Dij = Daij and it ~ regardless of any chemical reacfollows that ( x ~ =) 2Dt tions occurring between the solutes. This is a well-known result for no reactions in Brownian motion diffusion. It is convenient to define a nondimensional mean-squared diffusion length as
( P ) i = (x2)/(2Dt) (23) R. Wilcox, J.Math. Phys., 8 , 962 (1967).
(22)
I994
Savchik et al.
The Journal of Physical Chemistry, Vol. 87, No. 11, 1983
where D is a characteristic diffusion constant for the system. Thus in a multicomponent reaction-diffusion system obeying classical Fick's law diffusion
( P ) i= 1
(23)
for D chosen as the Fick's law diffusion constant. In this particular case, the nondimensional mean-squared diffusion length is a constant independent of time, the initial condition Q, and any effects from chemical reactions. (2) For the general diffusion of an N solute mixture without chemical reaction (z2)i
l N = -ED..Q. DQij=1 v
I
(24)
This is a time-independent quantity which depends on the initial condition Q and the ith row of the diffusion matrix
D. (3) For general diffusion with first-order chemical reaction, ( x ~will ) ~ be a complicated timedependent function involving all elements of the diffusion matrix and effects from the chemical reaction. This is evident from the general expressions for the second moment in eq 16. A conclusion from this analysis is that the time dependence ( x ~ provides ) ~ a useful indicator of the degree to which complex kinetics and diffusion are simultaneously operative. For example, if ( x ~ is)found ~ to grow linearly with time, then either no significant reaction occurred over that time or the diffusion is dominated by simple Fick's law behavior, D = D I . In the next section, the time behavior of mean-squared diffusion lengths is determined for a simple example of the diffusion of a mixture containing two solutes which undergo a reversible isomerization reaction.
111. Diffusion in an Infinite Medium with Two-Solute-Component Isomerization In this section expressions are formulated for the mean-squared diffusion lengths for a two-component system with diffusion and simultaneous reversible chemical isomerziation. The two solutes in the system undergo the reversible reaction K,
and the reaction-diffusion equation takes the form of eq 8b
a
-C
at
+ KC
a2
= D-c
ax2
where c is a two-component vector of the concentrations for species 1 and 2 c= L
J
The initial condition is specified by eq 19 where the twocomponent vector Q specifies the initial amount of each species in the system. The boundary conditions correspond to those of section I1 for an infinite medium. Specific moments of c(x,t) can be generated from eq 14 and the initial condition on the solution c(x,t) provides the initial condition on the sequence of moments.
n=O n#O Expressions for the zeroth and second moment vector are given by eq 21. The mean-squared diffusion lengths are expressed in terms of the individual components of the zeroth and second moment. One can determine these components with use of the similarity matrix S that diagonalizes K. When standard matrix techniques are used,24S and its inverse are =Q =O
where t = K2/K1. It follows that
The expressions for the zeroth and second moments, eq 21, can be written in terms of the diagonal matrix A. Ao(t) = S exp(At)S-'Q (264 A2(t) = 2 1 ' s exp{-A(t'- t ) ) S I D Sexp(At3S-' d t ' Q 0
(26b) Simple matrix multiplication results in the two components of the zeroth moment vector 1 AP(t) = -[(e + exp{-.))Q1 + t ( l - exp{-7))Q2] (27a) l + t
1 A'$t) = -[(l
l + t
- expl-.r1)Ql
+ (1 + E exp (-71)Q21
(27b) where 7 = ( K , + Kz)t. The sum of the two components of the zeroth moment is time independent since the total amount of species in the system is a conserved quantity. AP(t) + A$'(t) = Qi + Qz The ith component of the zeroth moment represents the total amount of species i in the system. To obtain the components of the second moment, it is convenient to express A2(t) as a sum of contributions from the various elements of the diffusion matrix (see Appendix A). The diffusion matrix D is decomposed into D = D1lZ1 + D,,Z2 + D21iP' + D22P2 where
the matrix D is a general nondiagonal diffusion matrix with elements
and the matrix K is the chemical reaction matrix describing the isomerization reaction. The elements of K are given by
The second moment vector A2((t)can now be expressed as a linear combination of fundamental matrices, R'@, weighted by the diffusion matrix coefficients. A2(t) = 2 x E D a 8E@& a 0
(24).R. Frazer, W. Duncan, and A. Collar, "Elementary Matrices", Cambridge University Press, 1965.
General Linear Multicomponent Reaction-Diffusion Equation
The Journal of Physical Chemistry, Vol. 87,No. 11, 1983 E ' T - E T ~ ~ ~ { - Tt } ~
+ ET exp{-T} ET + ET exp{-T} ET
- ET
T
) )
E'T ET
(1-~)(1--exp{-.r}j - 241 -exp{-T})
+
t ( 1 €')(I - e x p { - T } ) - (1 - € ) ( I - e X p { - T } )
eXp{-T}
1905
]
+ E'T e x p { - 7 ) - 2 e 2 ( 1 - e x p { - T } ) - E'T e x p { - T } - E ( I - € ) ( I - e x p { - T } )
E T - E ' T ~ ~ ~ { - T } - E ( ~ - E ) ( ~ T + E'T e x p { - T } + 2.41 - e x p { - T } )
For the isomerization reaction the elements of S and S-' are given by eq 25a and the eigenvalues of K by eq 25b. The evaluation of the R"b matrices (eq 28) is straightforward and follows from eq A2 in the Appendix. With the expressions for the zeroth and second moments, the nondimensional mean-squared diffusion lengths can be determined from eq 22. Here we analyze the special case where Q1 = Q2 = Q (i.e., the initial state contains equal amounts of both species). Thus it follows from eq 26 that
(9), =
exp(-7)
E)
+ (2 - + t2)( 1 - e:(+)} t
interactions, the system must reduce to simple Fick's law behavior. In addition, for the two-solute system, under the transformation t l / t it follows that
-
i zj and the ailk are only functions of the dimensionless variables t and 7. It is important to examine certain limiting cases of the interaction coefficients. (1)No Chemical Reaction. In this case, K 1 = K 2 = 0 and thus the interaction coefficients can be evaluated in the limit as 7 0 for finite t since 7 = ( K , + K 2 ) t . One finds that the interaction coefficients for 7 0 become independent of t and have the form aik
ai,
-
-
Thus the nondimensional mean-squared diffusion length for species i becomes
+
2
(P)i=
EDij j=l
which is the nonreactive general diffusion result for Q1 = Q2 = Q in eq 24. The short time behavior is coincident with the limit 7 0, and all the interaction coefficients have the following form
-
a7
Uik
(9), =
exp(-.r) + (1 -
t
+ 2t2)(1 - e
0)
where Dij = D i j / D (with D being a characteristic diffusion constant) and 7 = (Kl K2)t. As a notational convenience, one may rewrite eq 29 as
+
ailsll+ ui2Dlz+ a h l ~ z+l a6,D2,
=
1
= ai2 = -
=
l + t
The leading time behavior in the long time limit approaches
y 9 }+
(29b)
( a 2 ) i=
-
(7
where a is a constant dependent on t. (2) Long Time Behavior. As 7 m, the interaction coefficients asymptotically approach the constant values given below.
4 2
E)
+ ail
(29c)
where aik is the interaction coefficient corresponding to the lkth element of the diffusion matrix for species i. In general, the interaction coefficients are functions of time, chemical rates, and initial conditions. The a& terms for each species are related by the equation xahk = 1 k
This relationship arises since, in the limit of no diffusive
where
p=-
€
k = l l+€ --1 k = 2 1+ t and a' is a constant dependent on t. With the values from eq 30, the nondimensional mean-squared diffusion lengths become
(9) =, -[t(Dll
1 l + t
+ B2J + B12+ Bz2]
1 -[t(Dll l + t
+ B2J + B12+ B22]
(X2)Z
=
(31)
In comparing eq 31 with the general nonreactive diffusion
1996
Savchik et al.
The Journal of Physical Chemistry, Vol. 87,No. 7 1, 7983
termine a sequence of moments for the linear reactiondiffusion equation with general nondiagonal D and K matrices without having to solve explicitly for the time and spatial dependence of the solution vector c(r,t). Since I I a2, .............. aZ2 ______ explicit expressions for the moments require only the K2/Ki = 2 diagonalization of the reaction matrix K ,the determination of moments for large multicomponent linear reac-I ......................................................................... tion-diffusion systems can be accomplished with little ... ........ difficulty. In addition, it was argued that the moments -0- i, \ have a clear physical meaning and they contain an exs,. 0.4-; plicitly identifiable dependence on the rate matrix K and * _ _ - ._- . --. __. __. __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _the - - -diffusion - - - ~ - ~matrix - ~ D . These points should ultimately i ,._---:, be of benefit in analyzing experimental data, since the y concentration profiles can be easily converted into moments. Another possible alternative not elaborated on here would involve Fourier transforming the spatial dependence I l l I I , I , of the laboratory concentration; although this would lead 0.0 1.0 2.0 3.0 4.0 5.0 to a measured degree of simplification, the roles of K and TxlO D would still be intertwined. Figure 1. Interaction coefficients of species 1 vs. T for two-compoThe moments derived from delta function initial connent isomerization reactions; T = ( K ,+ K,)t and t = 2; ( n 2 ) ,= ditions can be used to formulate expressions for the a l p , , + al2bI2+ a$,, + a:2822. mean-squared diffusion lengths for a given species in the system. The analysis of these diffusion lengths offers Interaction Coefficients useful insight into the reaction-diffusion coupling. In 2 particular, any time dependency of the nondimensional a,, - af2 - - mean-squared diffusion length is a measure of the devia..... ... .............. a&, ------K 2 / K i = 2 tion from Fick's law diffusion behavior for a given species. .......... .......................................................................... Sections I1 and I11 showed that such a deviation is the result of two combined effects which must be present in the system: (1)solute-dependent diffusion interactions; and (2) chemical reaction. Without both processes existing in the system, the time behavior of the mean-squared diffusion length for each species follows that for Fick's law diffusion. When both effects exist, the nondimensional mean-squared diffusion length exhibits a complicated time behavior. Depending on the magnitude of the various matrix elements of the diffusion matrix, one can infer that the general diffusion time scale for a system may be appreciably altered by the presence of the solute-solute 0.0 1.0 2.0 3.0 4.0 5.0 diffusive interactions and chemical reaction. T x IO Moments can be generated in the infinite domain for Figure 2. Interaction coefficients of specles 2 vs. T for two-compoany well-behaved initial condition. As well as providing nentiqnsri"srizareaction;~=(K1+K,)tandt=2; (~2)2=a:1D1, an attractive framework for analyzing the roles of K and + a:,D12 + a&D21 + @,2. D , the sequence of moments for a given distribution could also be used to bound other spatial averages. This apresult of eq 24, one observes that the final asymptotic proach has been thoroughly discussed by Wheeler and behavior of (x^2)i, after all chemical reaction has ceased, Gordonz5in the context of bounds for various averages of is dependent on all the diffusive interactions in the system. a general distribution function but it is also applicable to It is not a function of only the ith row of the diffusion diffusion phenomenon. matrix. Figures 1 and 2 are plots for species 1and 2, respectively, Acknowledgment. The authors acknolwedge the Deof the interaction coefficients as a function of T for t = 2. partment of Energy for support of this research. H.R. The general linear short time behavior and the inverse T thanks Drs. M. Demiralp and R. Larter for several interlong time behavior is clearly evident in the plots. Of esting discussions on material related to this research. particular interest is the behavior of the interaction coefficient ai1. For Q1= Q2 = Q and t = 2, there is a net Appendix A positive conversion of species 2 to species 1. Although species 2 is being depleted, the interaction coefficient a:,, The second moment of the solution to the linear reacwhich is a measure the effects of t$e coupling of diffusive tion-diffusion equation can be expressed as a sum of terms and chemical interactions between the two species, ineach weighted by one of the matrix elements of the difcreases in magnitude due to the production of species 1. fusion matrix D . In general, any square matrix A can be The general time behavior of the two plots indicates that decomposed into a sum of elementary matrices each of mean-squared diffusion lengths, which are linear combiwhich contain only one nonzero element. nations of the interaction coefficients weighted by the 1 1 matrix elements of the diffusion matrix, can deviate subA = CAijE'I ;=1j=1 stantially from simple Fick's law behavior. Interaction Coefficients all - ai2
~
.a.
_,_,
-,-,-
I
I
g;
a i,
I
17'1
IV. Discussion It was shown that, in an infinite medium, one can de-
(25) J. C. Wheeler and R. G. Gordon in "The Pad6 Approximant in Theoretical Physics", G. Baker, Jr., and J. Gammel, Ed., Academic Press, New York, 1970.
J. Phys. Chem.
where 1 is the order of the matrix. BJis a matrix with elements defined as
Egm =
aikajm
Defining S as the matrix which diagonalizes K by the transformation
s-'m = A =
["
1
1997
1983, 87, 1997-2003
with the ijth element written as exp(-Xit jSi;1S8j exp(Xjt 1 Integration of the expression above over time results in the quantity
*.%A,]
The second moment vector can now be written as the second moment in eq 15c can be written in terms of A.
A%) = 2CCD,,S exp(AtJMa@S'Ao(0) a @
We define the fundamental matrices R as
A2(t) = 2 J t S exp(-A(t'-
= S exp(At)Ma@S1
t ) ) S ' D Sexp(At)S' dt'AO(0) + exp(KtJA2(0) (We will leave the last term in the above expression in terms of K since it does not involve the diffusion matrix.) The general diffusion matrix can be decomposed into a sum of elementary matrices, E"@
D = CCDa,E"@
so that the second moment is expressed as A2(t) = 2CCD,$@A2(O)
+ exp(Kt)A2(0)
(Al)
a P
By performing the various matrix multiplications we obtain for the klth element of Rap
a P
and the second moment vector can be written as A2(t) = 2CCD,,S exp(htJS'exp(-At)S'E"eS 0
X
. P
exp(At) dt' SIAo(0)+ exp(Kt)A2(0) The integrand in the expression for A2(t) is the matrix exp(-At )SIE"PSexp(At1
Electron Spin Resonance Studies of Electron Localization In Molten Alkali Metal-Alkali Halide Solutions N. Nlcoloso and W. Freyland' Fachberelch Physlkallsche Chemle, Phlllpps-Unlversitlit,03550 Marburg, West Germany (Recelved: September 27, 1982; In Flnal Form: December 21, 1982)
Electron spin resonance (ESR) has been investigated in different concentrated liquid alkali metal-mixed alkali halide solutions at temperatures up to 700 O C . The ESR data strongly indicate that alkali metals are not dissolved in atomic-likestates in the molten salts. From the g-factor shift of the resonances observed with varying salt matrix, we deduce that localized electron states very similar to that of the F center in the corresponding crystals exist even in nondilute solutions. In agreement with the concentration behavior of the static magnetic susceptibility, the magnitude of the observed spin densities shows, however, that spin-paired states in equilibrium with the paramagnetic F centers are important in concentrated metal-molten salt solutions. At high metal concentrations approaching the metal-nonmetal (M-NM) transition region, a first indication for spin delocalization-possibly via cluster formation-is given by the occurrence of narrow ESR signals with g values close to those of bulk alkali metals.
Introduction Liquid alkali metals form true solutions with their molten salts with a complete miscibility of the two liquids a t elevated temperatures.' Thus, with increasing ratio of metal to salt (M-MX) a transition from nonmetallic to (1) M. A, Bredig in -Molten Salt Chemistry", M. Blander, Ed., wiley-Interscience, New York, 1964.
0022-3654/83/2087-1997$01.50/0
metallic states may be continuously studied in a permanent thermal equilibrium state. Concerning the concentration range where this metal-nonmetal (M-NM) transition occurs, the measured concentration dependence of the electrical conductivity1 may be taken as a first clue. The conductivity, for example, of some sodium and potassium solutions is less than IO0 E'cm-' if the metal mole fraction zM lies below about 0.1-0.15. On the basis of the concepts 0 1983 American Chemical Society
