Anal. Chem. 1999, 71, 183-195
Concentrations of Electroactive Solutes, during Cyclic and Other Voltammetries, at Points Away from the Electrode Surface. 1. Fundamental Relationships and Their Validation Jan C. Myland and Keith B. Oldham*
Department of Chemistry, Trent University, Peterborough, Ontario K9J 7B8, Canada
There are circumstances in which it is useful to know the concentration, or its gradient, of an electroactive product or reactant at some distance from an electrode. Here it is demonstrated that this information is available, in principle, under conditions of semiinfinite planar diffusion, provided that the surface concentration and/or the current are known as functions of time. Several methods of calculation are explored, using the example of linear sweep voltammetry. Some of these methods rely on analytical formulations; others are purely numerical. Concordance of the results of the various methods provides validation of all. A method based on the convolution of the current is especially versatile. We shall be concerned with the voltammetry of the electrode reaction
R(soln) - ne- f P(soln)
(1)
occurring at a planar electrode of a large enough area A that edge effects may be ignored. Notice that a positive n corresponds to an oxidation, a negative n to a reduction. The solution is quiescent and excess supporting electrolyte is present, so that planar semiinfinite diffusion is the sole transport mechanism. At time t ) 0, some electrical event initiates reaction 1. Historically, theoretical studies of such transient voltammetry1 have sought a relationship of the form
f{E, I, t} ) 0
(2)
interrelating the three prime voltammetric variables: the electrode potential, the faradaic current, and time. On the way to this relationship, the concentration(s) and/or concentration gradient(s) of one or more electroactive species are often invoked as secondary variables but, because they are rarely accessible to experimental measurement, these variables are generally eliminated from the final result. Concentrations at the electrode surface (1) For many examples, see: Macdonald, D. D. Transient techniques in electrochemistry; Plenum Press: New York, 1977. 10.1021/ac980769i CCC: $18.00 Published on Web 12/03/1998
© 1998 American Chemical Society
c(0,t) are more relevant than concentrations elsewhere because the former can be directly related to potential via the Nernst, Butler-Volmer, or some similar equation. Likewise, it is the concentration gradients (∂/∂x)c(0,t) at the electrode surface that are most pertinent because of their direct relevance to the faradaic current. Nevertheless, there are instances in which concentrations or concentration gradients at a specific finite distance X from the electrode surface are important variables. When absorption spectrophotometry2,3 is used to supplement voltammetric measurements, for example, it is values of c(X,t) over a range of X values that are relevant. On the other hand, the promising new technique of probe beam deflection,4-8 involving the bending of a narrow light beam by a refractive index gradient, responds to the variable (∂/∂x)c(X,t) at some essentially constant value of X. Nowadays, a further important use to which c(X,t) data can be put is in debugging or validating digital simulation.9 Having an exact solution for the concentration(s) of electroactive species at points within the diffusion field can provide a valuable check on digital data in the heart of a computed concentration vector. Classical studies have provided some pertinent c(X,t) and (∂/∂x)c(X,t) results, especially for cases in which transport is solely by planar semiinfinite diffusion. In the case of an n-electron reversible electrode reaction, for example, the concentration and concentration gradient of an initially absent product obey the equations10
xDR/DP cR
b
cP(X,t) )
erfc{X/2xDRt}
1 + exp{nF(E1/2 - E)/RT}
(3)
and (2) DeAngelis, T. P.; Heinemann, W. R. J. Chem. Educ. 1976, 53, 594. (3) Bard, A. J.; Faulkner, L. R. Electrochemical methods: fundamentals and applications; Wiley: New York, 1980; pp 577-583. (4) Barbero, C.; Miras, M. C.; Kotz, R. Electrochim. Acta 1992, 37, 429. (5) Vieil, E.; Meerholz, K.; Matencio, T.; Heinze, J. J. Electroanal. Chem. 1994, 368, 183. (6) Weaver, J. K.; McLarnon, F. R.; Cairns, E. J. J. Electrochem. Soc. 1991, 138, 2579. (7) Brisard, G. M.; Rudnicki, J. D.; McLarnon, F.; Cairns, E. J. Electrochim. Acta 1995, 40, 859. (8) Ericksson, R. Electrochim. Acta 1995, 40, 725. (9) Brett, C. M. A.; Oliveira Brett, A. M. Electrochemistry: principles, methods, and applications; University Press: Oxford, 1993; pp 412-415. (10) Delahay, P. New instrumental methods in electrochemistry; Interscience: New York, 1954; p 53.
Analytical Chemistry, Vol. 71, No. 1, January 1, 1999 183
∂ c (X,t) ) ∂x P
-cRb exp{-X2/4DPt}
xπDPt[1 + exp{nF(E1/2 - E)/RT}]
2I nAF
x
(11)
or sometimes, as when the electron transfer is quasireversible, a single equation that is a hybrid of eqs 10 and 11. Linkages between the concentrations of the two electroactive species, and their gradients are provided by the relationships
in response to a potential step, or11
cP(X,t) )
∂ c (x)0,t>0) ) a specified f(t) ∂x i
(4)
[ { }
t 1 -X2 exp DP xπ 4DPt X 2xDPt
erfc
{ x }] X
2 DPt
R, P
R, P
∑xD c (X ,t) ) ∑ xD c
(5)
i i
i
b
i i
i)
(12)
i)
and
and
{ }
∂ X -I c (X,t) ) erfc ∂x P nAFDP 2xDPt
R, P
BOUNDARY CONDITIONS Because it involves three levels of differentiation, the solution of Fick’s second law for planar geometry
∂2 ∂ ci(x,t) ) ci(x,t) ∂t ∂x2
i ) R or P
i
i
i
(7)
which are easily proved. The two X values are interrelated by XP/DP1/2 ) XR/DR1/2. In the present research, we sometimes make use of eqs 10 and 11 as the spatial boundary conditions, eq 9 being ignored. This is not to say that condition 9 is invalid; rather the case is that it has been incorporated into eqs 10 and 11. Of course, when both (10) and (11) are employed as boundary conditions, these equations are mutually interdependent; they cannot both be chosen arbitrarily. GENERAL SOLUTION A general solution of Fick’s second law, eq 7, will be sought in this section, giving explicit expressions for ci(X,t) and (∂/∂x) ci(X,t) when conditions 8, 10, and 11 apply. Specifically, we shall be concerned with the product P of an electrode reaction, P being absent initially. Accordingly, condition 8 will take the form
cP(x,0) ) 0 requires that three boundary conditions, two spatial and one temporal, be specified. In most electrochemical studies, and in all of those to be discussed here, these conditions are given by eqs 8-11 below. The temporal boundary condition specifies the uniformity of concentration at the instant, t ) 0, at which the experiment commences.
ci(xg0,t)0) ) cib, often 0 for i ) P
(8)
jP(0,t) I(t) ∂ ) c (0,t) ) ∂x P DP nFADP
(15)
(9)
and either
ci(x)0,t>0) ) a specified f(t)
(14)
though, in fact, this condition is not invoked explicitly, being implicit in the functional form of boundary condition 10. In accordance with condition 10, the surface concentration of P is regarded as a known function cP(0,t) of time. However, instead of specifying the product’s concentration gradient at the electrode surface, as required by condition 11, it is more convenient to specify P’s surface flux density jP(0,t), which Fick’s first law
-
The spatial boundary conditions are
ci(xf∞,tg0) f cib
(13)
i)
in response to a current step. See Appendix A for a comprehensive list of the definitions of all symbols. One purpose of the present study is to provide general formulas capable of yielding such specific results as eqs 3-6. A second purpose is to employ these general expressions to derive formulas or algorithms for concentrations and concentration profiles, at any value of X, during an electrochemical experiment such as linear-potential-sweep or cyclic voltammetry. These two objectives are addressed, respectively, in parts 1 and 2 of this pair of articles.
Di
∂
∑D ∂xc (X , t) ) 0
(6)
(10)
links directly to its surface concentration gradient. Expression 15 also reports the link, via Faraday’s law, to the current I(t). Note that substitution of subscripts P by R requires a sign change in the first and second, but not the third, component of relations 15. Maclaurin’s theorem12 permits the writing of an expression for the concentration of the electroproduct at a finite distance X from
or (11) Karaoglanoff, Z. Z. Elektrochem. 1906, 12, 5.
184
Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
(12) Spanier, J.; Oldham, K. B. An atlas of functions; Hemisphere: Washington; Springer-Verlag: Berlin, 1987; p 90.
the electrode surface as the convergent infinite series
cP(X,t) )
B0 B1X B2X2 B3X3 + + + + ... 0! 1! 2! 3!
may be written as the sum ∞
(16)
cP(X,t) )
∑
k)0
X2k
∂k
(2k)!DPk ∂tk
cP(0,t) X2k+1
where the B coefficients are successive derivatives of cP(x,t), with respect to x, evaluated at x ) 0.
Bk )
∂k cP(0,t) ∂xk
k ) 0, 1, 2, 3, ...
(17)
The zeroth coefficient B0 is simply the temporal function, namely cP(0,t) serving as boundary condition (10) while the first coefficient B1 arises directly from eq 15 as -jP(0,t)/DP. Specialization of Fick’s second law to x ) 0 leads immediately to the second coefficient
B2 )
∂2 1 ∂ cP(0,t) ) c (0,t) DP ∂t P ∂x2
(18)
∂2 ∂k ∂k ∂2 1 ∂k ∂ c (x,t) ) c (x,t) ) c (x,t) ) P P DP ∂xk ∂t P ∂x2 ∂xk ∂xk ∂x2 1 ∂ ∂k c (x,t) (19) DP ∂t ∂xk P after specialization to x ) 0. Thus, invoking eq 15 once more, the third coefficient is found to be
B3 )
∂3 1 ∂ ∂ -1 ∂ c (0,t) ) cP(0,t) ) 2 jP(0,t) (20) 3 P D ∂t ∂x ∂x DP ∂t P
while the fourth is
B4 )
∂4 ∂2 ∂2 1 ∂2 c (0,t) ) c (0,t) ) jP(0,t) (21) P P ∂x4 ∂x2 ∂x2 DP2 ∂t2
Repeating this procedure indefinitely, one sees that for even k
Bk )
k/2
1 ∂ cP(0,t) DPk/2 ∂tk/2
k ) 0, 2, 4, ...
(22)
whereas when k is odd
Bk )
-1 DP(k-1)/2
∂(k-1)/2 ∂t
j (0,t) (k-1)/2 P
(2k + 1)!DPk+1 ∂tk
jP(0,t) (24)
involving two series. Note that this expression contains all the nonnegative integer orders of temporal derivatives of both the concentration and the flux density of P at the electrode surface. The presence of the denominatorial (2k)! and (2k + 1)! factors will generally ensure rapid convergence. Corresponding to (24) is the following expression ∞
∂
cP(X,t) )
∂x
∑
k)0
X2k+1
∂k+1
cP(0,t) (2k + 1)!DPk+1 ∂tk+1 X2k
All other B coefficients can be evaluated by recognizing that Fick’s second law asserts that two differentiations with respect to x, when applied to (∂k/∂xk)cP(0,t), are equivalent to one differentiation with respect to t accompanied by division by DP. That this is so follows from the equivalences
∂k
∂k
(2k)!DPk+1 ∂tk
jP(0,t) (25)
for the concentration gradient at points where x ) X. It can be derived in a similar manner and it, too, involves two series. These derivations are perfectly general. Their implementation usually requires that both the surface concentration and the surface concentration gradient (or the surface flux or the current) be known functions of time. The derivations have been for the initially absent product of the electrode reaction. Equations similar to (24) and (25) apply to the reactant, without sign changes. APPLICATION TO STEPS The purpose of this section is to validate eqs 24 and 25 by showing that they give correct results when applied to well-studied voltammetric experiments. As an example of the application of eq 24, consider the case of a current suddenly imposed on the electrode at the instant t ) 0 and maintained constant at a value I thereafter. The flux density of the product P at the electrode surface is, by Faraday’s law, proportional to the current, as in eq 15 and, since the latter is constant in the chronopotentiometric example, so is the flux density at the electrode surface.
jP(0,t) ) I/nAF ) constant
(26)
Accordingly
∂k jP(0,t) ) 0 ∂tk
k ) 1, 2, 3, ...
(27)
x
(28)
The well-known result13
k ) 1, 3, 5, ...
(23)
cP(0,t) )
2I nAF
t πDP
describes the surface concentration of a product during currentHence, with redefinition of k, the Maclaurin expansion 16 for the concentration of the product at a distance X from the electrode
(13) Oldham, K. B.; Myland, J. C. Fundamentals of electrochemical science; Academic Press: San Diego, 1994; p 405.
Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
185
step chronopotentiometry. Multiple differentiations are found to give
(2k - 2)!I ∂k c (0,t) ) for k P ∂t (k - 1)! xπDPt nFA(-4t)k-1
concentration gradient at a distance X from the electrode, rather than the concentration itself. Though in fact (35) only is critical, substitution of these four equations into eq 25 yields
∂
k ) 1, 2, 3, ... (29)
cP(X,t) )
∂x
-cRb When these last four results are substituted into eq 24, there emerges
cP(X,t) ) -2I nFA
x
t
[
DP
X
-
2xDPt
1
1
+
xπ
∞
∑
(-X/2xDPt)k
xπ k)1
k!(2k - 1)
]
(30)
This summation can be expressed as an error function complement integral14
cP(X,t) )
2I nAF
{ }
x
t X ierfc DP 2xDPt
(31)
after some algebra. The identity of this expression to formula 5 can be readily demonstrated. In this example, it was unnecessary to specify the degree of reversibility of the electrode reaction. As a second example, we treat a potential step from a preexisting ineffective value to a more extreme (more positive for an oxidation) potential E. This causes a step in the concentration of the product P from zero to the constant value
cP(0,t) )
cRbxDR/DP
1 + exp{nF(E1/2 - E)/RT}
xDR/πt
1 + exp{nF(E1/2 - E)/RT}
∂k cP(0,t) ) 0 ∂tk
cP(0,t) - cPb )
k ) 1, 2, 3, ...
∂ jP(0,t) ) ∂tk k!(-4t)k[1 + exp{nF(E1/2 - E)/RT}]
(37)
(38)
∂1/2 cP(0,t) ∂t1/2
(39)
showing the surface flux density to be proportional to the temporal semiderivative of the surface concentration. Substitution of eq 39 into eq 24 gives ∞
cP(X,t) )
(2k)!cRbxDR/πt
(36)
Application of the semidifferentiation operator to both sides of this equation leads to
∑
k)0
k
d-1/2 I(t) -1/2 nFAxDP dt 1
∂-1/2 jP(0,t) ) xDPcP(0,t) ∂t-1/2
(34)
The corresponding derivatives of jP(0,t) are easily found to be
k!
∑
Our prime interest is in voltammetry from which the product is initially absent. On setting cPb to zero in (37) and combining this result with Faraday’s law, embedded in eq 15, one finds
(33)
The constancy of the c(0,t) means that all its derivatives are zero
k)0
INCORPORATING THE FRACTIONAL CALCULUS Under conditions of planar semiinfinite diffusion, the excursion in the surface concentration of a reactant or product is proportional to the semiintegral of the faradaic current. Applied to a product species P, the relationship is16,17
jP(0,t) ) xDP
cRb
(-X2/4DPt)k
which is evidently identical to eq 4, as it should be. These two examples validate eqs 24 and 25. It appears that when a step in surface flux density, or a step in surface concentration, is applied to an electrode, one of the dual series in (24) and (25) contracts to a single term. In general, however, both series are infinite.
) constant (32)
if the electrode reaction is reversible, which is assumed in this example. The ensuing current obeys the Cottrell equation15 and therefore
jP(0,t) )
xπDPt[1 + exp{nF(E1/2 - E)/RT}]
∞
X2k
∂k
cP(0,t) (2k)!DPk ∂tk X2k+1
∂k+1/2
cP(0,t) (40) (2k + 1)!DPk+1/2 ∂tk+1/2
k ) 1, 2, 3, ... (35)
when the “composition rule” of the fractional calculus18 is invoked. Redefinition of the summation index k now permits the two series
For variety’s sake, eqs 32-35 will be used to determine the
(16) Oldham, K. B. Anal. Chem. 1969, 41, 1904. (17) Reference 13, p 254. (18) Oldham, K. B.; Spanier, J. The fractional calculus; Academic Press: New York, 1974; p 82.
(14) Reference 12, Chapter 40. (15) Reference 13, p 410.
186
Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
to be conjoined as
this equation becomes
(-X)k ∂k/2
∞
cP(X,t) )
∑
cP(0,t)
(41)
k! DPk/2 ∂tk/2
k)0
Equation 25 may be converted to
∂ ∂x
(-X)k-1
∞
cP(X,t) ) -
∑
k)1
∂k/2
(k - 1)! DPk/2 ∂tk/2
cP(0,t)
(42)
∂
∑
nFADP k)-1 (k +1)! D
cP(X,t) )
∂x
(-X)k+1
∞
1
-1
∞
∑
k/2 P
dtk/2
k
dk/2
k/2 P
dtk/2
(-X)
nFADP k)0 k! D
dk/2
I(t)
I(t)
{
}
(47)
where / denotes the operation of convolution, defined below in formula 53. The analogous equation for the reactant R is given in Appendix C as eq C1. Differentiate eq 45 with respect to x, and specialize to x ) 0. Then combine this simple result with the original eq 45, to eliminate jci(0,s). This gives
∂ jc (0,s) ) ∂x i
x[
] {x }
b s ci - jci(x,s) exp x Di s
s Di
(48)
(44) Next, after specialization of (48) to x ) X and to a product P that is absent from the bulk solution, replace (∂/∂x) jcP(0, s) by -Ih(s)/nFADP, which the transform of eq 15 allows. One finds
CONVOLUTIVE APPROACH In Laplace space, all solutions to Fick’s second law (7), subject to initial condition 8 and boundary condition 9 are embodied in the well-known formula1
[
(46)
in which the right-hand side is the product of two functions of s. Such products can be inverted via the convolution property of transforms,1 as the convolute of the two functions resulting from inverting each member of the product individually. In the present case, we get
(43)
Tables19 exist to aid in the successive semidifferentiations called for in formulas 41-44. Notice that, on adopting these semicalculus formulations, we have now returned full circle to the original set of boundary conditions in which either (10) or (11) is required, but not both. The preceding four formulas permit the concentration of P, or its gradient, to be found as a function of time, at any distance X from the electrode, provided that an analytic expression exists for either the current or the product’s surface concentration. In principle, these formulas could also employ experimental data, but the process of finding successive (semi)derivatives numerically is so error-enhancing that such a route is inferior to the approach discussed in the next section.
] { x}
cib cib jci(x,s) ) exp -x + jci(0,s) s s
s DP
X X2 cP(X,t) ) cP(0,t)/ exp 4DPt 2xπDPt3
by similar procedures. Via eq 38, the cP(0,t) variable in eqs 41 and 42 may be replaced by jP(0,t). Rather than reporting those versions, the similar but more experimentally relevant equations involving the current are reported below
cP(X,t) )
{ x}
jcP(X,s) ) jcP(0,s) exp -X
s DP
nFA xDP jcP(X,s) ) hI (s)
{ x}
1 exp -X xs
s DP
(49)
after rearrangement. Again, the right-hand side of this equation is the product of two functions, each of which may be inverted separately. Consequently the convolution
nFA xDPcP(X,t) ) I(t)/
{ }
1 -X2 exp 4DPt xπt
(50)
inverts the product. Equation C2, in an appendix, addresses the case of the reactant. By use of convolution 47 or 50, respectively, the product’s concentration at x ) X can be determined from either P’s surface concentration or the current. The corresponding convolutions for the concentration gradient at x ) X are 2
{ }
(51)
{ }
(52)
X - 2DPt ∂ -X2 -xDP cP(X,t) ) cP(0,t)/ exp ∂x 4DPt 4DPxπt5
(45) or
where an overbar denotes the transform of the unbarred quantity and s is the “dummy” variable of Laplace transformation. After specialization to x ) X and to an initially absent product i ) P, (19) Oldham, K. B. J. Electroanal. Chem. 1997, 430, 1.
-nFADP
∂ c (X,t) ) I(t)/ ∂x P
X 2xπDPt3
exp
-X2 4DPt
Notice that, in all four of these convolutions, the temporal function Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
187
for calculating f(t)/g(t) at the instant t ) K∆ is
Table 1a
f(t)/g(t) )
fKi2g1 ∆
[
+ f0 igK -
∆
+
i2gk+1 - 2i2gk + i2gk-1
K-1
∑f
]
i2gK - i2gK-1
K-k
∆
k)1
(54)
where K ) 1, 2, 3, .... Here igk and i2gk denote the first and second integrals of the g(t) function evaluated at t ) k∆:
igk )
∫
i2gk )
∫
k∆
0
g(τ) dτ
(55)
ig(τ) dτ
(56)
and
a Some of the functions listed are usefully convolved with other functions of time to determine cP(X,t). Other listed functions are employed in the convolution algorithms presented here. Each tabled function is the semiintegral of the function immediately above it.
with which c(0,t) or I(t) is convolved is one of those listed in Table 1. The list is a hierarchy in which each entry is the semiintegral of the one immediately above.20 Note the pattern established toward the end of the listing, which enables extension downward indefinitely. Upward extension requires easy differentiation, with respect to time and, repeatedly, of the second listed function. The generic symbol g(t) can be associated with any one of these function, all of which depend also on the characteristic time X2/ DP. The analogues of eqs 51 and 52 will be found in Appendix C. The temporal convolution of any two functions, f(t) and g(t), is defined by
f(t)/g(t) )
∫ f(t - τ) g(τ) dτ t
0
(20) Roberts, G. E.; Kaufman, H. Table of Laplace transforms; W. B. Saunders Co: Philadelphia, PA 1966.
Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
Linear interpolation between consecutive f data is the basis for this algorithm. Its derivation, in a slightly less general version that requires f0 to be zero, is described in the literature.21 Those g(t) functions that are most likely to be needed in the present context were assembled in Table 1. This is a list of functions arranged such that each entry is the semiintegral of the one above it. Hence the i2g(t) function will be found four levels below g(t), i3/2g(t) three levels below, ig(t) two levels below, and i1/2g(t) one level below. Algorithm 54 is perfectly satisfactory for many electrochemical purposes, and it has found diverse applications in convolutive modeling.22 In the present studies, it has been observed to behave superbly when executing convolution 47. However, because it requires a value of f0, it is unsuitable for convolving a current when, as in a Cottrellian experiment, I(t) exhibits a spike at t ) 0, with a subsequent t-1/2 falloff. For then, f0 is theoretically infinite and experimentally immeasurable. Nevertheless, we have found that algorithm 54 can be modified appropriately, and as will be demonstrated later in this article, it then gives surprisingly good values. The modification, which requires a single value of the semiintegral of the g function, is
f(t)/g(t) ) xπ∆f1i K-2
∑
k)1
[
1/2
[ ] ]
gK + fK -
fK - k -
f1
i2g1
xK
∆
+
f1
i2gk+1 - 2i2gk + i2gk-1
xK - k
∆
(57)
(53)
If f(t) has known values f0, f1, f2, ..., fk, ... at the equally spaced time instants t ) 0, ∆, 2∆, ..., k∆, ..., and integrals of g(t) can be accurately computed at those instants, then an efficient algorithm
188
k∆
0
The basis of this modification is explained in Appendix C. With use of the “i” and “i-1” prefixes to denote integration and differentiation, it is evident from the Laplace transformed equivalence of a convolution, that f(t)/g(t) may be replaced by i1/2f(t)/i-1/2g(t) or if(t)/i-1g(t) or indeed by many other similar combinations. Because if0 is zero and may therefore be excluded, (21) Oldham, K. B. Anal. Chem. 1986, 58, 2296. (22) Mahon, P. J.; Oldham, K. B. J. Electroanal. Chem. 1998, 445, 179.
The pioneering work of Reinmuth25 showed that the voltammetric current in a reversible linear-sweep experiment obeys the equation
the algorithm
1 f(t)/g(t) ) if(t)/i-1g(t) ) [ifK ig1 + ∆ K-1
∑ if
K-k
(igk -1 - 2igk + igk + 1)] (58)
I(t) ) -nFAcRb xbDR
∞
∑
(-)m xm exp{mb(t - t1/2)}
m)1
(62)
k)1
appears to provide an attractive solution to the problem of convolving Cottrellian-like currents. It is, indeed, applicable when the form of those currents is known as an integrable formula. However, for purely numerical, e.g., experimental, current data it, too, is unsatisfactory because of the need to integrate f(t) without knowledge of f(0). Perhaps the best resolution of this problem is to use the “G1 semiintegration algorithm”,23,24 in the convenient nested form
i 1/2fk ) x∆
[[[ ... [[f kk--3/21 + f ] kk--5/22 + 3/2 1/2 f ] ... ] +f ] + f ] (59) 2 1 1
2
3
k-1
k
to operate on numerical values of f1, f2, f3, ..., fk to produce a set of i1/2f1, i1/2f2, i1/2f3, ..., i1/2fk data that can then be smoothly extrapolated versus k to a horizontal i1/2f0 intercept at k ) 0. Then these data can be used in the
f(t)/g(t) ) i1/2f(t)/i-1/2g(t) ) i
1/2
[
f0 i 1
1/2
gK -
]
i3/2gK - i3/2gK-1 ∆
+
i1/2fK i3/2g1 ∆
+
K-1
∑ f (i ∆
3/2
k
gK-k+1 - 2i3/2gK-k + i3/2gK-k-1) (60)
k)1
algorithm. In effect, this procedure follows the route I(t) f cP(0,t) f cP(X,t). The original algorithm 54, and its modified form (57), will be used in the following section to carry out the convolutions given in eqs 47 and 50, respectively. The concordance with other methods of calculating the concentration cP(X,t) at distance X from the electrode will provide validation of the algorithms, too. The early stages of reversible linear sweep voltammetry will provide the test bed for these algorithms and for other ways of predicting cP(X,t) and its gradient. LINEAR SWEEP VOLTAMMETRY In linear-potential-sweep voltammetry, the electrode potential changes linearly with time at a rate v, which is positive for an oxidation, negative for a reduction. If we use t1/2 to denote the time at which the swept potential reaches the half-wave point E1/2, then
E(t) ) E1/2 + v(t - t1/2) (23) Oldham, K. B. J. Electroanal. Chem. 1981, 121, 341. (24) Reference 18, Section 8:2.
(61)
when the product is absent initially. In this, and many subsequent, equations, we use the abbreviation
b ) nFv/RT
(63)
Note that, because n and v share the same sign in linear-scan voltammetry, b is always positive. Strictly, Reinmuth’s equation is valid only in the t1/2 f ∞ limit, i.e., when the potential sweep starts from a potential that is indefinitely ineffectual (indefinitely negative for an oxidation). However, as we shall find, provided that t1/2 is large enough compared with RT/nFv, no significant error is incurred, especially in the later stages of the voltammogram. Unfortunately, the series in Reinmuth’s equation diverges if t g t1/2, so (62) can predict the voltammetric current only up to the half-wave potential; however, an analytic continuation exists26,27 that is valid for all positive t and that will be employed in part 2 of this bipartite series of articles. The form of the reversible current versus time relationship, when the sweep starts at a potential that is not sufficient ineffectual to validate Reinmuth’s equation, depends on the length of time that the electrode lingers at the starting potential, E1/2 - vt1/2. We shall treat the case in which this time period is infinitesimal. This would be the case for an experiment in which the potential ramp commences prior to t ) 0 but is electrically connected to the electrode only at that instant. In this scenario, the surface concentration experiences a sudden step at t ) 0 from zero to the value
cP(0,0) )
cRbxDR/DP
1 + exp{bt1/2}
(64)
before adhering to the Nernstian condition
cP(0,t) )
cRbxDR/DP
1 + exp{b(t1/2 - t)}
(65)
This elicits a spike of current, theoretically infinite at t ) 0, followed in turn by a t-1/2 falloff, passage through a minimum, and a slow climb. In light of eqs 15 and 39, the current versus time relationship can be obtained by semidifferentiating eq 65 with respect to time. Formulas for this purpose are available in the literature28 and lead to (25) Reinmuth, W. H. Anal. Chem. 1962, 34, 1446. (26) Oldham, K. B. J. Electroanal. Chem. 1979, 105, 373. (27) Oldham, K. B. SIAM J. Math. Anal. 1983, 14, 974.
Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
189
I(t) )
nFAcRb 1 + exp{bt1/2} ∞
∑
nFAcRbxbDR
x
DR πt
-
(-)m xm exp{mb(t1/2 - t)} erf xmbt
m)1
(66)
a complicated expression, but one that is easily evaluated numerically. For a starting potential of E1/2 - 7RT/nF, Figure 1 shows the discrepancies between the initial voltammetric currents predicted by eqs 62, for an indefinitely remote start, and (66), for a “jump start”. Of course, the latter current exceeds the former, but the difference steadily decreases and, beyond the current minimum, never exceeds 0.1% of the eventual peak current for the sweep rate in question. We now turn to the question of how cP(X,t), the concentration of the product at a distance X from the electrode, evolves in time early during reversible linear sweep voltammetry. Three approximate methods of predicting this concentration, and its gradient, will be compared with two methods that we believe to be exact. The comparison is made numerically in Table 2, the following constants being used throughout: |n| ) 1, cRb ) 1.000 mol m-3, DR ) DP ) 1.000 × 10-9 m2 s-1, X ) 1.000 × 10-4 m, A ) 1.000 × 10-5 m2, T ) 298.4 K, v ) 0.02000 V s-1, t1/2 ) 9.000 s, ∆ ) 0.01000 s, and b ) 0.7778 s-1. Exact Method for Short Times. A binomial expansion of [1 + exp{b(t - t1/2)}]-1 converts eq 65 into
x
DR exp{b(t - t1/2)} DP 1 + exp{b(t - t1/2)}
cP(0,t) ) cRb
x
DR
) -cRb
(67)
∞
∑ (-)
DP m)1
m
exp{mb(t - t1/2)}
an expression that converges for t < t1/2. Another binomial expansion leads to
x
cP(0,t) ) -cRb
DR
∞
∑
DP m)1
(-)m exp{-mbt1/2}
∞
(mbt)k
k)0
k!
∑
(68)
Combination of the Laplace transform of this last equation with eq 46 produces
jcP(X,s) )
x
-cRb
DR
∞
∑ [-exp{-bt
DP m)1
∞
1/2}]
m
∑
(mb)k
exp -X
s
DP
(28) Oldham, K. B. J. Electroanal. Chem. 1997, 430, 1.
190
and inversion gives
x
cP(X,t) ) -cRb
DR
∞
∑i
2k
DP k)0
erfc
{x } X
2 DPt
∞
∑ [-exp{-bt
m 1/2}]
(4mbt)k (70)
m)1
after interchanging the order of summation. This is the formula used to calculate the data in the third column of Table 2. To calculate numerical values of the i2k erfc{X/2(DPt)1/2} function, we used a standard algorithm29 for the error function together with sufficient applications of the recursion
in erfc{z} )
-z n-1 1 erfc{z} + in-2 erfc{z} i n 2n
(71)
where i0erfc{z} ) erfc{z} ) 1 - erf{z} and i-1 erfc{z} ) (2/π1/2) exp{-z2}. Exact Method for Long Times. We again start with eq 67, which combines with eq 41 to give the double summation
x
cP(X,t) ) -cRb
DR
∞
∑
(-X)k
DP k)0 k! D
∞
∑
k/2 m)1 P
(-)m
dk/2 dtk/2
exp{mb(t - t1/2)} (72)
sk+1
{ x} k)0
Figure 1. Linear-potential-sweep voltammetric currents for (upper curve) a starting potential 180|n| mV less extreme than the half-wave potential and (lower curve) an infinitely early start. See text for other conditions.
Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
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Appendix D is devoted to a discussion of repeated semidifferentiation of functions akin to exp{mb(t - t1/2)}. From that discussion, (29) For example: Oldham, K. B. J. Electroanal. Chem. 1982, 136, 175.
Table 2. Results of Five Methods of Calculating cP(X)100µm,t)/mol m-3 during Reversible Linear Sweep Voltammetry time t/s
(E - E1/2)/mV
eq 70
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.6 0.7 0.8 0.9 1 1.25 1.5 1.75 2 2.5 3 3.5 4 5 6 7 8 9
-180.00 -179.00 -178.00 -177.00 -176.00 -175.00 -174.00 -173.00 -172.00 -171.00 -170.00 -168.00 -166.00 -164.00 -162.00 -160.00 -155.00 -150.00 -145.00 -140.00 -130.00 -120.00 -110.00 -100.00 -80.00 -60.00 -40.00 -20.00 0
0 1.3893 × 10-26 1.4046 × 10-15 7.1165 × 10-12 5.2780 × 10-10 7.1680 × 10-9 4.1502 × 10-8 1.4733 × 10-7 3.8479 × 10-7 8.1843 × 10-7 1.5070 × 10-6 3.8254 × 10-6 7.5688 × 10-6 1.2808 × 10-5 1.9523 × 10-5 2.7656 × 10-5 5.3794 × 10-5 8.7813 × 10-5 1.2983 × 10-4 1.8066 × 10-4 3.1482 × 10-4 5.0797 × 10-4 7.8770 × 10-3 1.1948 × 10-3 2.6556 × 10-3 5.7487 × 10-3 1.2151 × 10-2 2.4771 × 10-2 4.7567 × 10-2
eq 76
eq 77
eq 78
eq 79
3.7255 × 10-10 7.1853 × 10-12 5.2480 × 10-10 7.1673 × 10-9 4.1502 × 10-8 1.4733 × 10-7 3.8479 × 10-7 8.1843 × 10-7 1.5070 × 10-6 3.8254 × 10-6 7.5688 × 10-6 1.2808 × 10-5 1.9523 × 10-5 2.7656 × 10-5 5.3794 × 10-5 8.7813 × 10-5 1.2983 × 10-4 1.8066 × 10-4 3.1482 × 10-4 5.0797 × 10-4 7.8770 × 10-3 1.1948 × 10-3 2.6556 × 10-3 5.7487 × 10-3 1.2151 × 10-2 2.4771 × 10-2 4.7567 × 10-2
5.6051 × 10-5 5.8273 × 10-5 6.0584 × 10-5 6.2985 × 10-5 6.5482 × 10-5 6.8078 × 10-5 7.0777 × 10-5 7.3582 × 10-5 7.6499 × 10-5 7.9531 × 10-5 8.2684 × 10-5 8.9369 × 10-5 9.6593 × 10-5 1.0440 × 10-4 1.1284 × 10-4 1.2196 × 10-4 1.4812 × 10-4 1.7988 × 10-4 2.1845 × 10-4 2.6528 × 10-4 3.9112 × 10-4 5.7649 × 10-4 8.4933 × 10-4 1.2505 × 10-3 2.7016 × 10-3 5.7875 × 10-3 1.2184 × 10-2 2.4799 × 10-2 4.7592 × 10-2
1.3893 × 10-26 1.4046 × 10-15 7.1165 × 10-12 5.2780 × 10-10 7.1680 × 10-9 4.1502 × 10-8 1.4733 × 10-7 3.8479 × 10-7 8.1844 × 10-7 1.5070 × 10-6 3.8254 × 10-6 7.5689 × 10-6 1.2808 × 10-5 1.9523 × 10-5 2.7656 × 10-5 5.3794 × 10-5 8.7814 × 10-5 1.2984 × 10-4 1.8066 × 10-4 3.1482 × 10-4 5.0798 × 10-4 7.8771 × 10-3 1.1948 × 10-3 2.6556 × 10-3 5.7488 × 10-3 1.2151 × 10-2 2.4771 × 10-2 4.7567 × 10-2
1.4100 × 10-26 1.4226 × 10-15 7.1902 × 10-12 5.3223 × 10-10 7.2179 × 10-9 4.1747 × 10-8 1.4808 × 10-7 3.8652 × 10-7 8.2172 × 10-7 1.5125 × 10-6 3.8368 × 10-6 7.5881 × 10-6 1.2836 × 10-5 1.9561 × 10-5 2.7703 × 10-5 5.3868 × 10-5 8.7902 × 10-5 1.2994 × 10-4 1.8078 × 10-4 3.1496 × 10-4 5.0812 × 10-4 7.8786 × 10-3 1.1950 × 10-3 2.6558 × 10-3 5.7489 × 10-3 1.2151 × 10-2 2.4771 × 10-2 4.7567 × 10-2
and after reversing the order of summations, it follows that
cP(X,t) )
x
-cRb
DR
∞
∑ (-)
m
DP m)1
∑
cP(X,t) ) -cRb
exp{mb(t - t1/2)}
∞
[1 - Φk{mbt}]
k)0
( ) mb
k/2
(-X)k
DP
the second part loses half its terms:
k!
(73)
x
DR
∑ (-)
{
exp mb(t - t1/2) - X
x
DR
exp{-mbt} (mbt)k/2Γ{1 - 1/2k}
(74)
when k is an odd integer, with
Φ-1{mbt} ) erfc{xmbt}
(75)
Further discussion of the Φk{ } and Γ{ } functions will be found in Appendix D. It is valuable to split expression 73 into two parts, because one of those parts can then be reduced to a single summation, while
∞
x} mb
∑ (-)
m
[ x]
∞
Φk{mbt}
k)1,3
k!
∑
+
DP
DP m)1
exp{mb(t - t1/2)} Φk{mbt} ) Φk-2{mbt} -
m
DP m)1
cRb where Φk{mbt} is zero when k is even but is given by the recursion formula
∞
-X
mb
k
DP
(76)
Though it is complicated, this expression is exact. However, it experiences numerical difficulty when t is small, as evident in the italicized tabular entries. It was used to calculate the fourth column of data in Table 2. Because the original binomial expansion yields a convergent series only when t < t1/2, eq 76 applies only in the potential range up to the half-wave potential. Early-Start Approximation. If a large value is given to bt1/2, corresponding to a linear-sweep experiment that starts at a very ineffectual potential, then interest is confined to rather large values of bt, also, because negligible electrolysis occurs until b(t - t1/2) reaches about -10. As demonstrated in Appendix D, Φk{mbt} becomes progressively smaller as bt increases. If bt is small enough that all Φk{mbt} functions can be ignored, eq 76 reduces Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
191
Convolving the Current. The current and the corresponding g(t) function are convolved in eq 50. The current appropriate to reversible linear sweep voltammetry is given in eq 66. Putting these togther gives
x
DP cP(X,t)
bDR
)
cRb
[
1
∞
∑ (-)
m
xm exp{mb(t1/2 - t)}
m)1
Figure 2. Concentrations at a distance of 100.0 µm from the electrode during a linear potential sweep experiment that started at (a) an infinitely negative potential and (b) -210, (c) -180, and (d) -150 mV. See text for other conditions.
to its first part:
cP(X,t) ) -cRb
x
DR
∞
∑ (-)
m
DP m)1
{
exp mb(t - t1/2) - X
x} mb DP
(77)
There are evidently close similarities between this equation and the Reinmuth eq 62, from which it may alternatively be derived. Note also its reduction to the exact eq 67 as X f 0. The fifth column in Table 2 shows, on comparison with the third and fourth columns, the error introduced by using this approximation. The error in making this approximation is quite substantial. In other words, the starting potential has a marked effect on the growth of concentration at a point away from the electrode. This is brought out in Figure 2, which shows the concentration growth for linear sweep voltammograms that start at -∞, -210 mV, -180 mV, and -150 mV with respect to E1/2. Curves a and c use data from the fifth and third columns of Table 2. Convolving the Surface Concentration. Equation 47 shows how to convolve the concentration of product P, at the electrode surface, to obtain cP(X,t). We used numerical values of cP(0,t), calculated via eq 65, in convolution 54:
x
{ }
DP cP(X,t) -X2 1 X exp ) / b DR c 4DPt 1 + exp{b(t1/2 - t)} 2xπDPt R (78)
As specified in Table 1, a g(t) function of [X/2(πDPt)1/2] exp{-X2/4DPt}, as here, corresponds to an i2gk of 4k∆i2 erfc{X/ 2(kDP∆)1/2}, while igK is erfc{X/2xDPt}. Therefore, these were the functions used in evaluating the convolute. Using the values of various constants as specified earlier, the required numbers were calculated and the convolution was executed. The sixth column in Table 2 contains data calculated by eq 78. The agreement with exact values in the third column is seen to be superb. 192
Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
-
xπbt[1 + exp{bt1/2}]
erf{xmbt}
] [ { }] 1
*x
πt
-X2
exp
4DPt
(79)
as the equation showing how to evaluate the concentration of P at a distance X from the electrode. Because the initial behavior of the current is “Cottrellian-like”, we chose the modified algorithm, eq 57, as the tool by which to carry out this convolution numerically. The functions two and four levels below {1/(πt)1/2} exp{-X2/4DPt} in Table 1 were used, respectively, to calculate ig and i2g. It is the seventh column in Table 1 that lists the results of this convolution. In regions where both perform efficiently, eqs 70 and 76 give essentially identical results, giving confidence that the unitalicized data in the third and fourth columns are correct. Notice how well the values in the sixth column, those from convolving surface concentration data via eq 78, agree with the exact values. The seventh column, based on convolution 79, using the modified algorithm, does not fare quite as well in a comparison, but never shows a discrepancy greater than 0.26 nM. Of course, the departure of the fifth column of data, based on the simple eq 77, has well understood physical causes and does not represent miscalculation. A Sixth Method. When, as in reversible linear sweep voltammetry, both cP(0,t) and jP(0,t) are functions whose numerical values are accurately calculable, use of eq 24 is perhaps the most direct method for finding cP(X,t). We attempted to use this approach. From numerical values of cP(0,t) and jP(0,t) at equally spaced time instants, the first 11 derivatives were calculated via custom-created formulas akin to published analogues,30 and used to determine cP(X,t). In most cases, the computed values agreed with those found by other methods. However, at some points, discordant results were found, even differing in sign. We attribute the unreliability of this method to loss of arithmetic precision, even though our calculations were carried to 15 decimal digits. Accordingly, we cannot recommend this method and present no further details. Concentration Gradients. Values of (∂/∂x)cP(X,t) can be found by methods that are strictly analogous to those used for cP(X,t) itself. Because no new principles are involved, the equations required to carry out the five principal methods will be presented
(30) Abramowitz, M.; Stegun, I. A. Handbook of mathematical functions; National Bureau of Standards: Washington, DC, 1964; p 914.
without commentary. The following five equations
∂
cRb
cP(X,t) )
∂x
x
DR
2DP
t
∞
∑i
2k-1
erfc
k)0
{x } X
2 DPt
∞
∑ [-exp{-bt
m 1/2}]
(4mbt)k (80)
m)1
∞ cRb cP(X,t) ) - xbDR (-)mxm ∂x DP m)1
∂
∑
{
exp mb(t - t1/2) - X
x} mb
+
DP
DP
∑
k!
k)0,2
∂
cRb
cP(X,t) )
∂x
DP
xbDR
∞
xbDR
∑ (-)
cRb
∑ (-)
m
m)1
mb
-X
k
DP
(81)
xm
m
m)1
{
exp mb(t - t1/2) - X -DP
∞
[ x]
Φk+1{mbt}
∞
exp{mb(t - t1/2)}
cRb
x
x} mb DP
{ }
X - 2DPt 1 -X2 exp / 4DPt 1 + exp{b(t1/2 - t)} 4D xπt5 P
(82)
(83)
and
-DP cRb
x
∂
1
bDR ∂x
[
cP(X,t) ) ∞
1
-
xπbt[1 + exp{bt1/2}]
][x
exp{mb(t1/2 - t)} erf{xmbt} /
∑ (-)
xm
m
m)1
X
2 πDPt3
{ }] -X2
exp
4DPt
ACKNOWLEDGMENT This work was inspired by discussions with Professor Rob Hillman (University of Leicester, U.K.) and assisted by Dr. Peter Mahon (C.S.I.R.O., Australia). The financial assistance of the Natural Sciences and Engineering Research Council of Canada is acknowledged with gratitude. APPENDIX A
1 ∂ c (X,t) ) DR ∂x P 2
The most versatile of the quintet of methods is the fifth: convolving numerical current data with an easily calculated function of time. The advantages of this method are as follows: It may be applied equally to controlled-current and controlledpotential voltammetries. It may be applied to current data from an analytical expression, as in our example, or equally well to experimental current data. It is not restricted to reversible processes. Though the standard convolution algorithm encounters difficulty when the current displays an initial spike, the simple modification advanced here overcomes it satisfactorily. Consequently, convolving the current will be our chief tool in part 2 of this pair of articles, in which we decipher the concentration profiles of product and reactant during cyclic voltammetry.
(84)
are the analogues of eqs 70 and 76-79. SUMMARY Four methods of predicting the concentration and concentration gradient of an electroproduct at a finite distance from an electrode have been presented. A fifth method is restricted to an indefinitely early start. All four methods give similar results when applied to the early stages of linear sweep voltammetry, which we take to confirm them all. Our purpose in undertaking part 1 of this research was not to study linear sweep voltammetry per se, but to derive and validate general methods for modeling concentrations remote from the electrode.
R,β
arbitrary constants
Γ{ }
gamma function
γ*{ ; }
entire incomplete gamma function
∆
a small time interval (s)
φk
a difference function defined in Appendix C
Φk{ }
a function defined in Appendix D
τ
subsidiary time variable (s)
A
electrode area (m2)
b
abbreviation for (nFv/RT) (s-1)
Bk
k-th coefficient in Maclaurin expansion
ci(x,t)
concentration of species i (mol m-3) as a function of x and t
cib
bulk concentration of species i (mol m-3)
Di
diffusivity (diffusion coefficient) of species i (m2 s-1)
E
constant value of electrode potential (V)
Eo
conditional or standard potential (V)
E1/2
half-wave potential (V) ) Eo - (RT/2nF) ln{DP /DR}
E(t)
time-dependent electrode potential (V)
e-
an electron
erfc{ }
error function complement
exp{ }
exponential function, exp{z} ) (2.71828...)z
F
Faraday’s constant (96 485 C mol-1)
f{ }
an unspecified function
hf(s)
Laplace transform of f(t)
fk, gk, ilgk values of f(t), g(t), ilg(t) at t ) k∆ g(t)
arbitrary function of time, specifically one in Table 1
i
either of the electroactive species, R or P.
I
constant value of faradaic current (A)
I(t)
time-dependent faradaic current (A)
in erfc{ } n-th integral of the error function complement ilg(t)
l-th integral of g(t), l ) 0, (1/2, (1, (3/2, (2, ... Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
193
ji(x,t)
flux density (mol m-2 s-1) of species i at distance x and time t
Hence
K
maximum current value of k, often equal to t/∆
f(t)/g(t) )
k, l, m, n summation indexes ln{ }
natural logarithm
n
signed number of electrons, positive for oxidation
N
optimal value of final summation index in an asymptotic series
p
an arbitrary power
P
solution-soluble product of electrode reaction
R
electroreactant
R
gas constant (8.3145 J K-1 mol-1)
s
“dummy” variable of Laplace transformation (Hz)
t
time from commencement of experiment (s)
T
thermodynamic temperature (K)
t1/2
elapsed time (s) before potential reaches E1/2
v
signed sweep rate (V s-1)
x
distance measured normal to electrode (m)
∫ f(t - τ) g(τ) dτ ≈ ∫ t
0
x∆f(∆)
∫
x∆f(∆)
∫
g(τ) dτ
t
)
xt - τ t-∆ g(τ) dτ
t-∆
∫
f(t)/g(t) ) xπ∆f(∆) i1/2g(t) +
∫
X
a specific constant value of x (m) distances (m) from the electrode such that cP(XP, t) - cPb ) cRb - cR(XR, t)
z
arbitrary variable
/
convolution operator (s)
[
t-∆
f(t - τ) -
APPENDIX B
K-2
The following equations for the reactant species are analogous to (47) and (50)-(52) of the main text.
}
X X2 cRb - cR(X,t) ) [cRb - cR(0,t)]/ exp 4DRt 2xπDRt3 (B1)
∂
xDR ∂x cR(X,t) ) [cR
nFADR
b
X2 - 2DRt
- cR(0,t)]/
4DRxπt5
exp
(B2)
{ }
-X2 4DRt (B3)
{ }
∂ -X2 X cR(X,t) ) I(t)/ exp ∂x 4DRt 2xπDRt3
(B4)
APPENDIX C If it is assumed that, during at least the 0 e τ e ∆ initial period of the voltammetry, the current is dominated by the Cottrellian dependence on time, so that I(t) ∝ 1/t1/2, then during this interval f(τ) is proportional to 1/t1/2 and therefore equal to (∆/τ)1/2 f(∆). 194
Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
∫
t
0
g(τ) dτ
xπ(t - τ)
(C1)
xt -∆ τf(∆)]g(τ) dτ (C2)
If we define φ(t) ) f(t) - (∆/t)1/2f(∆), then the integral remaining in eq C2 resembles φ(t)/g(t) except that the upper limit in the convolution integral is t - ∆ instead of t. It may be evaluated through exactly the same procedure by which the right-hand side of eq 54 was used to evaluate the right-hand side of eq 53. In the same subscript notation used previously, the result is
f(t)/g(t) ) xπ∆ f1 i1/2gK + φK
{ }
f(t - τ) g(τ) dτ -
The final integral will be recognized19 as the semiintegral of g(t), which we denote by i1/2g(t). The first two integrals may be combined, so that
XR, XP
-X2 1 nFAxDR[cRb - cR(X,t)] ) I(t)/ exp 4DRt xπt
t-∆
0
f(t - τ) g(τ) dτ +
+ xπ∆f(∆)
xt - τ
0
0
{
t-∆
0
∑φ
i2g1 ∆
+
i2gk+1 - 2i2gk + i2gk-1
K-k
k)1
∆
(C3)
at t ) K∆. This formula is equivalent to eq 57 of the main text. APPENDIX D A generalized differintegration formula,31 valid for any order p, positive or negative, integer or fractional, is
dp exp{R + βz} ) z-p exp{R + βz} γ*{-p;βz} p dz
(D1)
where γ*{ ; } denotes the entire incomplete gamma function.32 When p, is a nonnegative integer, γ*{-p;βz} reduces to the simple power (βz)p and therefore
dk/2 exp{R + βz} ) βk/2 exp{R + βz} dzk/2
k ) 0, 2, 4, ... (D2)
However, when β is positive and p equals -1/2 or is a moiety of an odd positive integer, γ*{-p;βz} adopts a more complicated formulation, namely, (βz)p erf(βz)1/2 + exp{-βz}Σ(βz)n/Γ{n - p + 1}, where the summation runs from n ) 0 to p - 1/2. After (31) Reference 18, p 94. (32) Reference 12, Chapter 45.
some algebra, this leads to
dk/2 dzk/2 β
k/2
Φk{βz} ) erfc{xβz} - exp{-βz}
(k-1)/2
∑
n)0
exp{R + βz} )
[
exp{R + βz} erf{xβz} + exp{-βz}
(k-1)/2
∑
n)0
(βz)-n-1/2 1
Γ{ /2 - n}
]
k ) -1, 1, 3, 5, ... (D3)
The (complete) gamma function33 Γ{ } simplifies, in this case, to any of the following formulations
Γ{1/2 - n} )
(-2)nxπ (-2)nxπ ) ) (1)(3)(5)...(2n - 3)(2n - 1) (2n - 1)!!
xπn!(-4)n (D4) (2n)!
It is convenient to merge eqs D2 and D3 into the single equation
dk/2 exp{R + βz} ) βk/2 exp{R + βz}[1 - Φk{βz}] dzk/2 k ) -1, 0, 1, 2, 3, 4, ... (D5)
irrespective of the parity of k. For odd k, (33) Reference 12, Chapter 43. (34) Reference 12, eq 40:6:4.
(βz)-n-1/2
Γ{1/2 - n}
k ) -1, 1, 3, 5, ... (D6) When k is even, Φk{βz} is zero. When βz is large, the standard asymptotic expansion34 of the error function complement, namely, erfc(βz)1/2 ∼ exp{βz}Σ(βz)-n-1/2/Γ{1/2 - n}, may be used, which transforms eq D6 into
Φk{βz} ∼ exp{-βz}
∞
(βz)-n-1/2
n)(k+1)/2
Γ{1/2 - n}
∑
N
(βz)-n-1/2
n)(k+1)/2
Γ{1/2 - n}
∑
≈
k ) -1, 1, 3, 5, ... (D7)
It is evident from this result that Φk{βz} tends rapidly toward zero with increasing βz, especially for large odd k. Numerical values of Φk{βz} may be calculated from either (D6) or (D7). However, because of the series’ asymptoticity, care must be exercised in using eq D7 by making a judicious choice of the upper summation limit N. Summing should not be continued beyond the point at which the (N + 1)th summand exceeds the N-th in magnitude; instead, add half the (N + 1)th term and quit. Received for review July 14, 1998. Accepted October 13, 1998. AC980769I
Analytical Chemistry, Vol. 71, No. 1, January 1, 1999
195
