5306
J. Phys. Chem. 1988, 92, 5306-5313
Electrochemistry of Organic Redox Liquids. Reduction of 4-Cyanopyridine Rae1 B. Morris, Kurt F. Fischer, and Henry S. White* Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455 (Received: January 22, 1988)
Electrochemical studies of liquid 4-cyanopyridine (4-CNPy; melting point = 78 "C) are reported. In solutions containing only tetra-n-butylammonium perchlorate (TBAP) as supporting electrolyte, 9.6 M 4-CNPy can be reduced at a steady-state mass-transport rate (10 A/cm2) yielding a highly concentrated solution layer of the product, TBA+.4-CNPy0-, adjacent to the electrode surface. The transient voltammetric response is distorted by the slow coupled diffusion/migration of the supporting electrolyte cation (TBA') necessary to maintain electroneutrality within this layer. Transport-related phenomena unique to concentrated solutions are presented and discussed with regard to the observed electrochemical behavior of 4-CNPy. These include the effects of (1) counterion dilution and migration, (2) fluid convection induced by interdiffusionof reactant (4-CNPy) and product (4-CNPy'-), and (3) concentration-dependent diffusivities. General flux equations (diffusion, migration, and convection) are presented in terms of the partial molar volumes of each solution component for a constant density fluid. A theoretical analysis of the current-dependent ohmic solution resistance, Rs(i),at a spherical electrode is also presented for the special case where the concentration of supporting electrolyte is significantly less than the redox active species. This current-dependent resistance generates a 200 V/cm electrical potential gradient, d$(r)/dr,associated with the diffusion-limited reduction of undiluted 4-CNPy.
Introduction Recently we described the electrochemical behavior of several organic redox liquids in the absence of an inert solvent.'s2 For example, the electroreduction of undiluted nitrobenzene (NB) to the corresponding radical anion (NB'-) has been quantitatively examined at small Pt disks. Mass-transfer-limiting voltammetric currents and apparent diffusion coefficients of nitrobenzene as an undiluted liquid and in concentrated mixed solutions have been reportedS2 These preliminary investigations have prompted an examination in our laboratory of several fundamental issues regarding mass-transport effects on heterogeneous electron-transfer reactions in highly concentrated solutions. In particular, limiting current plateaus observed in the voltammetric response of undiluted redox liquids suggest depletion of the parent liquid to near zero levels at the surface and electrogeneration of a thin solution layer comprised of molar quantities of product anion and supporting electrolyte cation. The structure and properties of this ionic transport layer may be markedly different from the neutral liquid, imparting electrochemical behavior not anticipated based solely on the parent compound properties. Although several literature reports describe electrolysis of neat ~ o l v e n t s (e.g., ~.~ methanol), this paper, to our knowledge, represents the first attempt to quantitatively resolve key issues regarding transportlimited reactions of electroactive liquids. In contrast to diffusion of a dilute redox species, transportlimited currents observed in undiluted redox liquids can be visualized as the intertransport of two very concentrated liquids, with the electrode surface acting as a source and sink of the product and reactant, respectively. Accumulation of molar quantities of the supporting electrolyte cation within the transport layer is required to maintain electroneutrality, resulting in a significant decrease in the concentrationof electroactive species (Scheme IA). This dilution by the charge-balancing ion, hereafter referred to as counterion dilution, is always negligible in reactions involving redox-active species at dilute concentrations but is generally significant in highly concentrated solutions and can effect the chemical reactivity. Furthermore, if the partial molar volume of the product (e.g., NB-) is different from that of the reactant (e.g., NB), then migration and diffusion necessarily create a net convective fluid flow resulting in either a decrease or increase in voltammetric currents. The magnitude of this effect is proportional to the concentration of electroactive species and is expected to be significant in undiluted redox liquids. In addition to this
diffusion-engendered fluid convection, the buildup of a concentrated product layer may alter fluid properties (e.g., viscosity, ionic conductivity, etc.) within the transport layer that determine the electrochemical behavior. For example, due to ion-ion interactions, the diffusivity of the parent compound within the transport layer may be considerably smaller than in the bulk liquid, thereby decreasing the observed transport-limited rate. Because the chemical composition changes across the transport layer, key parameters, e.g., diffusivity, cannot be described by bulk fluid constants (Scheme IB). In this report, we described the electrochemical behavior of 4-cyanopyridine (4-CNPy) at temperatures slightly above its melting point (78 "C). The viscosity of this liquid is relatively low, resulting in current densities as large as 10 A/cm2 at 12.5pm-radius Pt disk electrodes. In discussing these results, we focus initially on the roles of migration and diffusion in concentrated redox solutions under several different experimental conditions. In latter sections, analytical models are developed to explore effects of migration, counterion dilution, diffusion-engendered fluid convection, and concentration-dependent diffusivities, on the observed steady-state transport-limited reduction of 4-CNPy.
(1) Malmsten, R. A.; White, H. S. J . Electrochem. SOC.1986, 133, 1067. (2) Malmsten, R. A.; Smith, C. P.; White, H. S . J . Elecfroanal. Chem. 1986, 215, 223. (3) Sundholm, G.J . Elecfroanal. Chem. 1971, 31, 265. (4) Scholl, P. C.; Lentsch, S. E.; Van De Mark, M. R. Tetrahedron 1975, 32, 303.
Results General Voltammetric Behavior in Liquid 4-CNPy. All experimental studies reported here were made at temperatures between 80 and 100 "C, slightly above the melting point of 4CNPy (78 "C). Figure l a shows the voltammetric response at
0022-3654/88/2092-5306$01.50/0
Experimental Section Chemicals. 4-Cyanopyridine (Aldrich) and bis(pentamethylcyclopentadieny1)iron (Strem Chemicals, Inc.) were sublimed under reduced pressure. Tetra-n-butylammonium perchlorate (TBAP) was recrystallized from acetone-ether and stored under vacuum at 100 "C. HPLC grade toluene (Aldrich) was used as received. Electrochemical Measurements. Pt microdisk electrodes (1 2.5-km radius) were prepared by the previously described procedure.2 Electrochemical measurements were made by using a Princeton Applied Research Corp. (PAR) Model 173 potentiostat and PAR Model 175 programmer. Positive feedback circuitry was not used to compensate for ohmic polarization losses. A 2-mL single-compartment cell equipped with a Ag quasireference electrode (0.2 f 0.1 V versus saturated calomel electrode) and Pt wire counterelectrode were used throughout these investigations. A description of the Ag electrode has been given elsewhere.* The electrodes were sealed in the cell by using ground-glass joints. Weighed amounts of solid 4-CNPy (typically 1 g) and TBAP were placed in the cell at room temperature and heated to melting. Temperatures are reported to within il OC.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92. No. 18, 1988 5307
Reduction of 4-Cyanopyridine
SCHEME I: (A) Sketch Showing the Variation in the Depletion Layer Chemical Composition during the Reduction of a Neutnl Redox Lquid and (B)the Resulting Variation in Molecular Diffusivity (B). Concentration Dependent Diffusivity (A). Counterion Dilution
0
0
Product anion
Electrolyte cation
4-CNPy/0.15M (n-butyl),
0
Neutral Reactant
4-CNPy
NCIO,
93"c
no supporting electmlyte
I I IC
S=5x
I -3.0
-2.0
-1.0
1.0
0.0
2.0
3.0
-4.0
V vs Ag. Figure 1. Voltammetric response of a 12.5-pm-radiusFt' disk in 4-CNPy (96 "C) containing 0.15 M tetra-n-butylammonium perchlorate; scan rate = 20 mV/s. Note the change in current scale between positive (B) and negative (A) potentials. (C) Voltammagram of 10 mM DMFc under same solution conditions.
a 12.5-@m-radiusPt disk in an unstirred 4-CNPy solution containing 0.15 M TBAP as supporting electrolyte. The large reduction wave beginning at -1.50 V versus Ag corresponds to the steady-state one-electron reduction of the redox liquid to the radical anion:
4-CNPy + e-
* 4-CNPy'-
A
(1)
Although this wave is somewhat drawn out due to uncompensated ohmic potential drop, the onset of the cathodic current occurs at a potential ca. 0.1 V positive of the reported half-wave potential of the 4-CNPyf4CNPy- couple in dilute acetonitrile solutions? The chemical stability of the product radical anion in the parent liquid was not explored in these studies, other than to note that no fouling of the electrode surface occurred over prolonged experimentation and that the bright red electrogenerated radical anion could be observed streaming away from the electmde surface in the otherwise colorless solution. The anodic process beginning at +2.5 V was also not investigated, but we have noted the formation of bubbles on the electrode surface at potentials positive of 3.5 V. No limiting current for the anodic reaction was obtained over a 5-V-wide scan to positive potentials. (Note the 1/100 times difference in the current scale between anodic and cathodic scans.) (5) Kowert. B. A,; MBICOUX, L.;Bard, A. J. J. Am. Chem. Sm. 1972.94. 5538.
-3.0
-2.0
-1.0
0.0
V vs Ag Figure 2. Voltammograms of 4-CNPy at a 12.5-pm pt disk as a function of electrolyte concentration: (A) no supporting electrolyte (20 mV/s); (B) 50 mM TBAP (50 mM V/s); (C) same as (B) except potential stopped and held at -2.45 V.
Voltammetry of electroactive species present at low concentrations can be performed in the 4-V-wide range set by the oxidation and reduction of 4-CNPy. For instance, the dashed curve in Figure 1 shows the voltammetric wave corresponding to the oxidation of IO mM bis(pentamethyIcycl~ntadieny1)iron (DMFc; E,,2 = 0.25 versus Ag). The voltammetric wave for this reaction is considerably less drawn out than the larger solvent reduction wave due to flow of smaller currents resulting in a smaller ohmic potential drop.6 At high potential sweep rates or in solutions of low TBAP concentration, the 4-CNPy reduction wave is strongly affected by the slow coupled diffusionfmigration of the supporting electrolyte cation TBA+ into the depletion layer to balance the excess negative charge associated with the product anion. For instance, in the absence of an intentionally added electrolyte, the voltammetric current is suppressed by more than 3 orders of magnitude (Figure 2A). Addition of TBAP to the solution at millimolar concentrations restores the normal flow of Faradaic currents, but considerable hysteresis is observed in the voltammetric response even at moderate sweep rates. Figure 2B, for instance, shows the response obtained at a sweep rate of 50 mV/s in a solution containing 50 mM TBAP. The current increases very slowly with (6) Wightman,
R. M. A w l . Chcm. 1981,53, 1125A.
5308
The Journal of Physical Chemistry, Vol. 92, No. 18, 1988
decreasing potential, reaching the limiting current value near the end of the negative scan. This limiting current, however, persists on the return scan, resulting in the normal sigmoidal-shaped voltammogram expected for diffusional flux to a microdisk electrode. The hysteresis is reduced at slower scan rates and at higher electrolyte concentrations (e.g., Figure 1A). Additionally, employing positive feedback circuitry in the experiment shown in Figure 2B has an insignificant effect on the i-Y hysteresis and creates large instabilities in the current response. In contrast to the true steady-state curves of Figure 1 , the i-V behavior shown in Figure 2B is dominated by a time-dependent process. For example, if the potential is stopped on the forward-going scan (curve C, Figure 2), the current immediately but slowly increases until the limiting current is obtained. We attribute the general time-dependent behavior shown in Figure 2 to the slow transport of the supporting electrolyte cation, TBA', from the bulk solution, where it is present at relatively low concentrations ( 1/200 times less than the redox-active liquid), to the surface to balance the negative charge of the electrogenerated product anion. The minimum theoretical quantity of electrolyte cation, N(6) (mole), required to balance this Faradaic charge can be estimated by integrating the steady-state product concentration profile over the depletion layer thickness.' Assuming radial diffusion of 4-CNPy'- away from the electrode surface N
N(6) = rC*r:(s2 - 1)
(2)
where ro is the microdisk radius and 6 is the normalized depletion layer thickness, equal to r / r o . With 6 = 10, approximately 3 X mol of TBA' is required to migrate from the bulk solution inward to the surface. Alternatively, partial charge balance may be obtained by migration of the electrolyte anion, C l o t , out of the depletion layer. This process, however, does not contribute significantly to maintaining a charge balance since the total amount of C10,- within this volume element mol) is 1/30th of the required value. An approximate experimental value of N(6) can be obtained by integrating the current under the rising part of the 4-CNPy wave prior to reaching the limiting current response. In Figure IA, this area (indicated by gray shading) corresponds to 7 X mol of electrons and is twice the approximate theoretical value (eq 2) found necessary to achieve a steady-state product distribution profile associated with the limiting current response. Interestingly, although the voltammetric wave is distorted greatly at low electrolyte concentrations (Figure 2B), the quantity of mol of electrons) before charge passed (approximately 5 X the limiting current is obtained is nearly equal to that obtained at higher electrolyte concentration (Figure 1A). The difference between these curves, therefore, is the rate at which TBA' is transported inward to the surface. The charge obtained by integration of the rising current overestimates the actual value of N(6) since a fraction of the product ion diffuses out of the depletion layer before steady-state conditions are reached. However, rearrangement of eq 2 suggests that the integrated charge per bulk concentration of reactant, N(6)/C*, should be independent of the electrode reaction for voltammograms obtained at the same potential scan rate. Experimentally, we find N(6)/C* to be approximately 4-5 times larger for the oxidation of 10 mM DMFc relative to the reduction of 4-CNPy. This result indicates that less charge is required to establish the steady-state concentration profile in the redox liquid, a consequence of either (1) a decrease of the depletion layer thickness or (2) a uniform decrease in product concentration throughout the depletion layer, a point discussed in more detail in later sections. The limiting current density resulting from 4-CNPy reduction at the Pt disk is ca. 10 A/cm2, resulting in substantial ohmic potential loss, A 4 = iR,. The resistance between a disk microelectrode and a point at infinite distance is given by8
-
R, = ( 4 ~ r ~ ) - '
(3)
(7) Delahay, P. New Instrumental Methods in Electrochemisfry;Interscience: New York, 1980; p 59.
Morris et al.
;1 5.2
5.3
2.75
2.80
2.85
T - ~ X I O -( O~K - ' ) Figure 3. Diffusion coefficient of 4-CNPy and DMFc (10 mM) as a function of temperature. Values of D obtained from i,,, values in solutions containing 0.1 5 M TBAP.
where K is the solution conductivity (0 cm)-', assumed to be constant throughout the solution. However, under experimental conditions where the redox active species is present in solution at concentrations comparable to or exceeding the supporting electrolyte concentration, the solution conductivity near the electrode surface will vary in relation to the concentration profile of ionic species produced by Faradaic chemistry. For instance, the discharge of 4-CNPy'- (and corresponding inward migration of TBA') must increase the solution conductivity nonuniformly within the depletion layer relative to the zero-current condition. It is interesting to note that the solution resistance should in fact decrease with increasing current, an unusual effect due to the expected dependency of K on the product (ion) concentration. Thus, this current-dependent resistance, R,(i),produces an ohmic potential drop, iR,(i),that is less than values based on the bulk K value (eq 3) and that cannot be compensated for by use of current feedback through an external and constant-valued resistor, a conclusion noted in the experiments described above. A theoretical expression for R,(i) is derived below. Analysis of Limiting Currents. Limiting currents obtained for the reduction of 4-CNPy are briefly analyzed in this section by assuming a purely diffusion-controlled response. In later sections, we present flux equations that are based on more realistic descriptions of transport in concentrated solutions. The limiting diffusional current to a microscopic disk is given by9 ili, = 4nFDC*ro (4) where C is the bulk concentration of the redox active species (c' = 9.6 M at 90 "C), D is the apparent diffusion coefficient of 4-CNPy, and F is the Faraday constant. Equation 4 is derived by assuming that D is constant and independent of the local concentration of 4-CNPy or 4-CNPy'- (Le., dilute solution approximation) and that convection is negligible. Values of D for 4-CNPy and DMFc are shown in Figure 3 over the temperature range 74-90 "C. Qualitatively, the shape of the voltammetric wave for 4-CNPy reduction over this range is similar to that shown in Figure 1A. Two points are worth noting in Figure 3. First, diffusion coefficients obtained from ih values for 4-CNPy reduction (e.g., 6 X cm*/s at 80 "C) are approximately one-half as large as the corresponding values for DMFc. This ratio is the same as the ratio of diffusion coefficients previously measured for nitrobenzene reduction and DMFc oxidation in undiluted nitrobenzene solutions,2 DDMFc/DNB = 2. Second, it is worth noting that the diffusional activation energies for DMFc and 4-CNPy (2.2 f 0.5 kcal/mol) are equal within error. A general difficulty of transport measurements in concentrated solutions is the inability to vary the electroactive species concentration without affecting the solution viscosity. In the present studies, we have found that values of DDMFc are nearly independent of solution composition in 4-CNPy/toluene mixtures (Figure 4), indicating that these two liquids and mixtures prepared from them (8) Newman, J. J . Electrochem. SOC.1966, 113, 501 (9) Saito, Y . ReG. Polarogr. 1968, 15, 177.
The Journal of Physical Chemistry, Vol. 92, No. 18, 1988 5309
Reduction of 4-Cyanopyridine r
w
SCHEME II: Inward Diffusion of Parent Redox Liquid and Outward MigratiodDiffusion of Electrogenerated Product Assisted by an Electrical Field, d 9 (r)/dr
6 c
4t l
I
0
,
r
.
2
[4-CNPY
r
4
'
l
,
6
I, F;?
Figure 4. Diffusion coefficient of 4-CNPy and DMFc (10 mM) as a function of solution composition in mixed toluene/4-CNPy solutions containing 0.15 M TBAP.
have similar viscosities ( Q = 0.35 ~ CP~at 70 ~ OClO).~ This~ finding ~ has been used to explore the dependence of DCCNPy on 4-CNPy concentration. Values of DCCNPY were obtained from ilimmeasurements (eq 4) for mixed solutions containing between and 4.7 M 4-CNPy and are plotted in Figure 4. The higher resistivity of toluene prevented analysis of i1,, data in mixed solutions of higher toluene content. However, within error, D4.cNpy is independent of 4-CNPy concentration over a surprisingly large range (Figure 4).
Discussion Voltammetric Analysis in Concentrated Solutions. The steady-state behavior of the 4-CNPy system and that of the nitrobenzene system previously reported is entirely consistent with diffusional transport of the parent neutral to the electrode surface. It is worth noting again that the magnitudes of limiting currents are also consistent with near quantitative reduction of 4-CNPy. If in fact this is the case, the depletion layer volume would consist of a highly concentrated salt solution, with the product 4-CNPy'and counterion TBA' approaching molten salt levels at the electrode/solution interface. Two effects of this extreme concentration profile are expected. First, because electroneutrality within the depletion layer requires the presence of one supporting electrolyte cation for every electrogenerated anion, the volume occupied by supporting electrolyte represents a significant fraction of the depletion layer volume (Scheme I). As shown below, this effect significantly decreases the depletion layer concentration of the product 4-CNPy'-. This effect is referred to as counterion dilution. Second, the fluid viscosity within the depletion layer may be significantly different from the bulk value. With the exception of simple alkali halides, molten salts tend to be relatively viscous fluids.]' For instance, the viscosity of tetra-n-hexylammonium benzoate, which is structurally similar to TBA+.4CNPy'- (no hydrogen bonding), is reported to be similar to that of glycerol.12 An anticipated effect of this concentration-dependent viscosity is that the diffusivity should vary across the depletion layer, decreasing with increasing product ion concenro). In addition, because the local solution tration (Le., r chemistry is a function of the instantaneous current, the local diffusivity should also vary with electrode potential over regions where the current is changing. We refer to this as concentration-dependent diffusivity. An implicit assumption of the analyses presented below is that the product salt is completely soluble in the parent liquid. Evidence for the miscibility of TBA'q4-CNP'- in 4-CNPy is directly provided by the sigmoidal voltammetric response, which is characteristic of a soluble redox couple. Phase separation or precipitation
-
(10) Handbook of Chemistry and Physics: Weast, R. C . , Ed.; The Chemical Rubber Co.: Cleveland, 1970-1971; Vol. 51. (11) Blander, M.; Newman, D. S.; Mamantov, G.; Saboungi, M.-L.; Johnson, K. Proc.-Electrochem. SOC.1984, 84-2, and references therein. (12) Swain, C. G.; Ohno, A,; Roe, D. K.; Brown, R.; Maugh, T. J . Am. Chem. SOC.1967, 89, 2648.
~
of a salt layer on the electrode surface would likely manifest itself in the form of a transient reoxidation wave upon reversal of the potential scan. This is not experimentally observed. Counterion Dilution and Migration. In this section, we consider the effects of migration and convection on the mass-transferlimited reduction rate. The model employed here is sketched is Scheme 11. We consider the one-electron reduction of the undiluted liquid (species 1 ) to the corresponding anion (species 2), requiring migration of the supporting electrolyte cation (species 3) to the surface to maintain electroneutrality. At steady-state, the charge of electrogenerated anion (z2 = -1) within the depletion layer is balanced by an equal number of supporting electrolyte cations ( z 3 = + l ) , Le., C2 = C3 and dC2/dr = dC3/dr. This approximation breaks down at the fringe of the depletion layer where the product anion concentration decreases to the same order of magnitude as the background supporting electrolyte concentration. The concentration profiles of 4-CNPy and CNPy'- are required to obtain the steady-state flux equations. This is accomplished by considering the total molar solution volume, V (cm3/mol), expressed as the sum of the product of the partial molar volume (vi) and mole fraction (xi) of each solution component:
v = U l X l + 02x2 + 03x3
(5)
In writing eq 5 , we ignore the contribution of the supporting electrolyte anion, Clod-, which everywhere is negligible in our system. (In dilute redox systems the mole concentrations of electrolyte and redox components are generally small compared to the background solvent. Under this condition, eq 5 is replaced by V i= Usolvcnt.) In the bulk solution, V = vl. Equating this to the left-hand side of eq 5 and noting that x2 = x3 by electroneutrality yields 1=
x1
+ x2(u2 + u 3 ) / u I
(6)
for constant solution density. Division of eq 6 by u, yields the relationship between C1and C2 within the depletion layer:
CI* = CI
+ C2(~2+ ~ 3 ) / ~ 1
(7)
where the concentration of the ith species is defined as Ci = xi(ul. Equation 7 describes the essence of counterion dilution. Assuming that the partial molar volumes of 4-CNPy and 4-CNPy'- are approximately equal (Le., u1 = u2), then for any finite value of u3, the concentration of product, 4-CNPy'-, will be reduced throughout the depletion layer by a factor of (1 u3/uJ. In the limiting case of v3 0 ( i s . , counterion does not occupy any volume), and with u2 = vl, eq 7 yields the result expected for dilute solutions: C1* = C1 + C2. The boundary conditions necessary to evaluate the flux equations for 4-CNPy and 4-CNPy'- are obtained from eq 7:
-
C1 = C1*
+
r
C,S = C1*ul/(u2 + vj)
-
r
-
(8) ro
where the superscript s denotes surface concentration of the
5310 The Journal of Physical Chemistry, Vol. 92, No. 18, 198r8
Morris et al. n1
= D,C,*(ro/r?
(17)
+ u3)1Ci*(ro/rZ)
-n2 = (02 + Dd[ui/(Uz
(18)
From eq 17 and 18, the steady-state solution n, = -n2 yields
c/c;
- 0.0 --1.0 -- 5.0
Figure 5. Steady-state concentration profiles of 4-CNPy and 4-CNPy'showing the effects of counterion dilution and diffusion-engendered convection. CI*is the bulk redox liquid concentration and (u2 - ul) is the difference in partial molar volumes of 4-CNPy'- and 4-CNPy, re-
spectively. product, C,. The concentration profiles of 4-CNPy, 4-CNPy'-, and TBA+ are plotted in Figure 5 for the special case u, = u2 i= 03.
The net fluxes of 4-CNPy (n,), 4-CNPy- (nJ, and TBA+ (n3) are obtained by using the Nernst-Planck relationship: d C1 Cluo (9) dr dC2 z ~ F d 4 nz = -D2-C D - C2u0 (10) dr RT 'dr dC3 z ~ F d 4 n3 = -D3-C D - C3u0 (11) dr RT 3dr where d@/dr is the local electric field (V/cm). The volume-average velocity, uo, is defined as
+
+
+
+
+
uo = nlul
+ n2u2+ n3u3
(12) At steady state, the flux of TBA' is zero (n3 = 0) and the flux of 4-CNPy'- is equal to and opposite from 4-CNPy (n2 = -nl), allowing eq 12 to be written as uo = nl(ul - u2)
(13)
Combining eq 9 and 13 yields the reactant flux, n,: dC1 Clnl(ul - u,) dr and combining eq 10, 11, and 13 yields the product flux, n,: nl = -D,-
+
dC2 + 2C2n2(u2- u , ) n2 = -(Dz + D3)(15) dr For small differences in the product and reactant partial molar volumes, i.e., ( u , - u l ) / u l < 0.1, eq 14 and 15 can be combined to yield -Dl dCl/dr = (D2 + D,) dC,/dr
UI/(U,
+
(19)
Although there is no a priori physical basis for this unanticipated constraint, eq 19 will be approximately valid for many electrochemical systems. For example, assuming approximately equal diffusivities and partial molar volumes of each solution species, both sides of eq 19 are equal to -0.5. In terms of the limiting current at a hemispherical electrode, eq 17 and 18 can be written as
r/ro
nl = -DI-
DI/(D2 + 03) =
-
(16)
+
The factors 2 and (D, D,) in eq 15 and 16 are a result of electroneutrality being maintained by the two migrating species, 4-CNPy'- and TBA', a consequence of the electrogenerated product concentration in the depletion layer being much larger than the background supporting electrolyte concentration. Equation 16 is appropriate regardless of the absolute redox concentration and has been applied by several researcher^'^.'^ in the analysis of dilute solution electrochemistry in the absence of a supporting electrolyte. The steady-state flux is obtained from the continuity equation by using the boundary conditions containing counterion dilution effects, eq 8, to yield (13) Amatore, C.; Deakin, M. R.; Wightman, R. M. J. Elecfroanul.Chem. 1987, 225, 49. (14) Bond, A. M.; Fleischmann, M.; Robinson, J. J. Elecfroanal. Chem. 1984, 172, 11.
ilim= 27rnFDlCl*ro
(20)
Equation 20 is identical with the expected steady-state diffusional limited-current response in dilute solution, indicating that if D, remains constant and if the partial molar volumes of reactant and product are approximately equal, the steady-state limiting current associated with reduction of the redox liquid is purely diffusion controlled and the current can be predicted from the bulk diffusivity. This fortuitous result is a consequence of the effects of counterion dilution (decreased product concentration) being accommodated by the effect of migration (increased product flux). Application of eq 20 is limited to the one-electron reduction (or oxidation) of undiluted liquids only where the solution chemistry meets the criterion stated in eq 19. The concentration profiles shown in Figure 5 (solid lines) indicate that the quantity of electrolyte cations per unit concentration of redox-active species, N ( 6 ) / C ,for liquid 4-CNPy is anticipated to be lower by '/2 than for the dilute species, DMF, in agreement with experiment. The factor of 2 difference in the flux per unit concentration for the redox liquid and dilute species however cannot be accounted for by migration and counterion dilution since eq 20 should hold for any concentration. We show below that this discrepency can be accounted for by allowing D to vary as a function of ion concentration across the transport layer. Electrical Field and Current-Dependent Solution Resistance. Diffusion and migration contribute roughly equally to the net flux of 4-CNPy'- from the electrode surface. The electrical field at the surface, (d4/dr)r=ro,associated with the migrational flux can be derived by subtracting eq 11 from eq 10 and using the approximations D2 = D3 and u1 = u2 is given by
-
n2 = 2(z,F/RT)C2D2 d4/dr or in terms of the limiting current ili,/nF2rro2 = ~ ( Z ~ F / R T ) C ~ ~ D ~ ( ~ ~(21) /~~),=,,
+
Using C,S = [uI/(u2 u3)]Cl*= '/,C,* = 4.8 M, z2F/RT = -10.5 V-' at 90 OC, D, = 5 X 10" cm2/s, and the experimental value of 10 A/cm2 for ilirn/27rr,2yields (d4/dr),=, = 200 V/cm. This field decreases as r-l through the depletion layer (e.g., d+/dr = 100 V/cm at r = 2r0, etc.; Figure 6a). The total potential drop across the depletion layer, A4, can be obtained by rearranging eq 11: d 4 = (RT/Z,mdC,/C,) and integrating from the surface to the bulk to obtain
A 4 = (RT/z3fl In (C3/C3s)
(22)
C3sis the surface concentration of electrolyte cation, which by electroneutrality is equal to '/,C1*. C3* is the bulk electrolyte concentration, typically 0.1 M. With these values, A 4 = -0.39 V, slightly smaller but in reasonably good agreement with the magnitude of the ohmic potential loss observed in the voltammetric reduction of liquid 4-CNPy (Figure 1). The difference in these two values, -0.2 V, may represent the effects of slow heterogeneous electron transfer associated with eq 1. However, we believe this small difference is more likely due to errors resulting from approximations used in the analysis concerning equal partial molar volumes of reactant, product, and counterion.
The Journal of Physical Chemistry, Vol. 92, No. 18, 1988 5311
Reduction of 4-Cyanopyridine At potentials corresponding to the rising part of the reduction wave, the steady-state voltammetric current is given by i = 2mFDlro(Cl* - CIS)
(23)
which can be combined with eq 7, 20, and 22 and the approximation u1 ir. u2 ir. uj to yield the ohmic resistance as a function of the reactant surface concentration and bulk electrolyte concentration: R,(i) = ( ( R T / z 3 F )In (2C3*/(C,* - ClS)))/2?mFDlrO(C1*- Cls) (24) Equation 24 predicts that the solution resistance should decrease as the limiting current is approached, a result of the depletion layer conductivity increasing at higher currents. This dependence is plotted in Figure 6B for a 12.5-rm hemispherical electrode for 0.1 < i/ili,,, < 1.0. Over this range, Rs(i)decreases by a factor of 4, theoretically eliminating the use of iR, compensation through a constant valued resistor. Diffusion-Engendered Conuection. In the previous sections, we assumed that convection of the redox-active species was negligible. If the difference between the partial molar volumes of the reactant and product is not small, however, then the flux relations, eq 14 and 15, must be expressed as n, =
n2 =
1
-D1
dC1
-202
dC2 -
1
2
4
3
5
6
7
8
9
10
rho
P
0
4
2
2
0 '1
02
03
0'4
05
06
07
08
09
10
"ll,m
Figure 6. (A) Electrical field as a function of distance at an applied potential corresponding to the limiting reduction current plateau of 4CNPy. (B) Ohmic resistance plotted as a function of voltammetric current (eq 24).
+ C,(U,- U ] ) dr
1 - C2(u2- u l ) d r
The equation of continuity, for a spherical coordinate system, applied to species 1 yields (27) which can be integrated by using the boundary conditions CI= C1* as r -
C, = 0
"
0.01
at r = ro
010
100
c :(vy
10.0
"1 )
Figure 7. Normalized flux, nl/nld", of 4-CNPy as a function of the product of the difference in partial molar volumes of 4-CNPy'- and 4-CNPy, (u2 - u l ) , and the bulk concentration of redox liquid CI*.
to yield
with a corresponding flux -Dlro In [ l nl =
--
(02
+ CI*(u2- u , ) ] - u1)r2
(29)
-
In the limit Cl*(u2 - ul) 0, which corresponds to either the dilute solution limit (C1* 0) or equimolar counterdiffusion (ul u2), eq 29 reduces to 'eq 17, the diffusion-migration equation. The concentration profile for species 2 can be found by a similar procedure, by using the boundary conditions
C2-0
asr-m
n2 = -nl This yields the result
The deviation of the steady-state flux from the dilute solution limit is given by _n, -- In [ l + CI*(uz - u l ) ] (31) ,,]dil
CI*(U2- 01)
The effect of the diffusion engendered flow on the flux is shown in Figure 7, where nl/nl"' is plotted against C1*(u2- uI) (eq 31). The species concentration profiles have also been plotted (eq 28
and 30) in Figure 5 , for different values of Cl*(u2 - u , ) . The deviation from eq 17 is generally expected to be small, unless exceptionally large differences between reactant and product partial molar volumes occur. For instance, if the partial molar volume of 4-CNPy'- were twice that of 4-CNPy, the product C1*(u2- u,) is approximately equal to 1 (note that Cl* ir. u,-l). Under these conditions, the steady-state flux is expected to decrease by only 20% (Figure 7). Concentration-Dependent Diffusiuity . Intuitively, one expects the mass-transport resistance within the depletion layer to vary as a function of distance if the fluid properties of the pure product and reactant phases are significantly different. In considering the voltammetric behavior of such a system, we employ two general types of empirical mixing rules commonly used to estimate the diffusivity of binary systems. We first consider the solution diffusivity, Dmix,to be given by the geometric average of diffusivities of the pure reactant and product;I5 Dmix= DI'ID2'2
(32)
where xI and x2 represent mole fractions of species 1 and 2. The second mixing rule we have considered is the arithmetic average of the diffusivities:I6 Dmix = XIDI+ ~ 2 D 2
(33)
(15) Hartley, G. S.; Crank, J. Trans. Faraday SOC.1949, 45, 801. (16) Vignes, A. Indust. Eng. Chem. Fundam. 1966, 5, 189.
5312
Morris et al.
The Journal of Physical Chemistry, Vol. 92, No. 18, 1988 -0.2 1.0
1
'
o
-0.1
0
1
0.2
0.1
I
'
t
/
E-E", I
v
I
I
D D,
0.5
arithmetic I
0
'
,
0
0.5
1 .o
c,/ c ; Figure 8. Arithmetic and geometric averages of diffusion coefficients for mixtures of liquids 1 and 2. Values plotted for Dl/Dz = 5 .
The subscripts 1 and 2 refer again to the parent liquid (4-CNPy) and the electrogenerated product (4-CNPy'-), respectively. Ignoring molar volume differences, we can replace x1and x2 by the concentrations of 1 and 2 normalized to the bulk 4-CNPy concentration (Le., x1 = C1/CI*and x2 = C2/C1*). The variation of Dmixacross the depletion layer is shown graphically in Figure 8 for both the geometric and arithmetic mixing rules for D , = 5D2. The diffusion-limited current and reactant concentration profile, C l ( r ) ,are readily obtained by solving the continuity equation
by using either mixing rule for the local diffusivity, D~ Expressed in terms of the surface concentration of parent reactant, CISand the ratio of diffusion constants in pure solutions of 1 and 2, 6 = D,/D2, the steady-state current and reactant concentration profile for the geometric average are given by
and for the arithmetic average
Equations 34-37 are exact for a hemispherical electrode geometry and are used below to approximate the qualtitative behavior of small planar disks. Normalized currents for both geometric and arithmetic averages of the diffusion coefficients are plotted in Figure 9 as a function of potential by assuming that the surface concentrations are governed by the Nernst equation. In both cases, results are plotted for D I / D 2 = 5 . The voltammogram obtained by assuming a constant diffusivity equal to the bulk solution value, D1,is included for comparison. The graded variation in diffusivity across the depletion layer has two effects on the expected voltammetric response. First, for D 1 / D 2> 1, the limiting current is always less than that predicted for a constant diffusivity. However, the inset of Figure 9 shows that for D 1 / D 2> 3, the limiting current is rather insensitive to the depletion layer diffusivity and approximately equal to the limiting current that would be obtained for a constant parent diffusivity. In the case of the arithmetic average, ih asymptotically
/,/
constant D
D./D,
i,,;
,2a n FD,C; I,
Figure 9. Normalized voltammetric current (i/ZmFD,C*) obtained by assuming either a (a) constant, (b) arithmetic, or (c) geometric average diffusivity. Values in (b) and (c) were obtained for D1/D2= 5 . The inset shows normalized limiting currents corresponding to the arithmetic and geometric average of the diffusion coefficient as a function of D1/Dz.
1
2
3
4
5
6
r/r, Figure 10. Concentration profiles of 4-CNPy obtained by assuming either a constant, arithmetic, or geometric average diffusivity. The diffusion layer thicknesses, 6, for each case are noted on the figure.
approaches as D 1 / D 2 goes to infinity. For the geometric average, ih decreases slowly as In (D1/D2).Thus, limiting currents observed for reduction (or oxidation) of undiluted solvents are expected to be nearly, independent of the variations of fluid properties within the defiletion layer for D1/D22 3 and l/zas large, per unit concentration, as observed for dilute redox species. This result is in agreement for the measured diffusivity of undiluted 4-CNPy and the previously measured value for nitrobenzene. Electroreduction of both redox liquids yields apparent diffusivities that are ca. as large as diffusivities of DMFc in the same solutions at relatively dilute concentrations. Variation in the diffusivity across the depletion layer also affects the shape of the voltammetric curve, shifting Ellz to slightly more positive values for reductions. In Figure 9, for instance, the curves for both the geometric and arithmetic averages are about 20 mV positive of for the constant D case. This shift increases very slowly for larger values of D I / D z and, in view of the residual iR, losses, is probably not a measurable quantity. The near independence of diffusion currents on the local fluid properties for D 1 / D 2> 3 is a result of the decreased diffusivity being offset by larger concentration gradients. Figure 10 shows concentration profiles of the parent liquid for geometric average, arithmetic average, and constant D case. The former two are plotted for D 1 / D 2 = 5 . The depletion layer thickness for the constant D case is approximately 200%and 170% larger than for the geometric and arithmetic averages, respectively. These values become correspondingly larger for larger values of D1/D2,yielding a flux that is weakly dependent on the fluid properties within the depletion layer. Similar to the effects of counterion dilution, one consequence of the shortened depletion layer is a reduction in the charge required to establish a steady-state concentration profile. In the Results section, we noted that the amount of charge (normalized
The Journal of Physical Chemistry, Vol. 92, No. 18, 1988 5313
Reduction of 4-Cyanopyridine to the bulk reactant concentration) required to obtain the steady-state diffusion current for 4-CNPy reduction was ca. 25% of that required for DMFc. This value can now be used to evaluate the ratio of bulk to surface diffusivities. Integration of the concentration profile (e.g., Figure 10) yields the required product charge "(6)
= 2rJd(CI* - C,(r))? dr 0
for either the geometric and arithmetic average, where C ( r ) is given by eq 35 or 37. Combining these integrals with eq 2 allows D,/Dz to be determined for an experimentally measured Nt(6)/N(6). Using N'(6)/N(6) z 0.25, we calculate that Dl/Dz = 4 for the geometric average and D,/Dz = 5 for the arithmetic average. From the inset of Figure 9, these relative values correspond to an il,, ca. 50%lower than the value based on the bulk diffusivity (Dl). Thus, experimental values of DeCNPy shown in Figures 3 and 4, obtained by assuming a constant D1, eq 4, may be as much as 50% lower than the actual bulk values.
Conclusion In this and previous reports, we have demonstrated that quantitative electrochemical studies are possible in concentrated and undiluted redox systems. However, the 4-CNPy/4-CNPy0redox couple is representative of a number of organic electrochemical systems that may be studied by heating an ambient solid above its melting point. This strategy should allow investigations of compounds that are relatively insoluble or unstable in other solvents. In addition to fundamental electrochemical studies of the redox liquid itself, these investigations may prove fruitful in identifying solvents for electrochemical studies. For instance, liquid 4-CNPy has a 4-V-wide potential range and a relatively low viscosity and readily dissolves at least one common organic electrolyte, TBAP. The electroreduction of a redox active liquid results in transport layerfluid properties that differ from that of the parent liquid. In this paper we have identified and discussed three effects of nonbulk fluid properties. First, migration of the supporting electrolyte cation inward to the surface to maintain charge balance decreases the local concentration of product anion within the depletion layer. Migration of the cation also results in a more ionically conductive solution surrounding the electrode, reducing the net ohmic potential losses but requiring the solution resistance to be described as a function of the steady-state current magnitude. This latter result is a consequence of the redox-active species being present at larger concentration than the supporting electrolyte. An apparent reduction of the ohmic drop with increasing current should also be observable in the voltammetric response of dilute redox-active species in low ionic solution^.^^-^^ I ~~
~
~
~
(17) Bond, A. M.; Fleischmann, M.; Robinson, J. J . Electroanal. Chem 1984, 168, 299.
Second, the interdiffusion of a concentrated product and reactant creates a fluid convection that decreases the voltammetric currents whenever the molar volume of the product is larger than that of the reactant. This situation is generally anticipated whever an ionic species is obtained from reduction or oxidation of a neutral redox species. Similar to the current-dependent iR,(i) drop, the relative magnitude of fluid convection depends on the net current and is more pronounced in less viscous solutions and at higher redox concentrations. Third, inclusion of a concentration-dependent parent diffusivity into the transport theory yields predictions in good agreement with the overall current-voltage behavior of 4-CNPy. The model presented is based on a monotonic increase in fluid viscosity with increasing product ion concentration. Although it is unlikely that either the arithmetic or geometric average is a particularly accurate description of the local diffusivity, the resulting equations yield useful predictions of the current magnitude that do not depend strongly on the choice of model. The two key results predicted and experimentally observed are that (1) the limiting current per unit bulk concentration, irm/C*,in redox liquids will be reduced by ca. 50%relative to limiting currents of dilute redox species and (2) the depletion layer thickness monotonically decreases with an increasing ratio of the parent molecule diffusivity in the bulk to that at the surface, D1/D2. These results are generally applicable to other electrochemical systems where the diffusivity of a reacting species may be a function of concentration (e.g., in redox polymers2IJ2),temperature(e.g., in frozen s o l ~ t i o n s ~or~ )distance , away from the electrode surface (e.g., electrodes of molecular dimens i o n ~ ~albeit ~ ) , for different chemical and physical reasons than discussed here. For instance, many of the underlying concepts employed here are suggested from transport studies in supercritical fluids,z5 an area of current interest in electrochemistry.26~27 Acknowledgment. Financial support was provided by the Shell Companies Foundation. Registry No. 4-CNPy, 100-48-1; 4-CNPy'-, 34536-53-3; TBA'a4CNPy'-, 115652-50-1; TBAP, 1923-70-2; DMFc, 12126-50-0; Pt, 7440-06-4; toluene, 108-88-3. (18) Bond, A. M.; Fleischmann, M.; Robinson, J. J . Electroanal. Chem. 1984, 172, 11.
(19) Howell, J. 0.; Wightman, R. M. Anal. Chem. 1984,56, 524. (20) Chen, J.-W.;Georges, J. J . Electroanal. Chern. 1986, 210, 205. (21) Hunter, T. B.; Tyler, P. S.; Smyrl, W. H.; White, H. S. J . Electrochem. SOC.1987, 134, 2198. (22) Chidsey, C. E. D.; Murray, R. W. J. Phys. Chem. 1986, 90, 1479. (23) Bond, A. M.; Fleischmann, M.; Robinson, J. J . Electroanal. Chern. 1984, 180, 257. (24) Morris, R. B.; Franta, D. J.; White, H. S.J . Phys. Chem. 1987, 91, 3559. (25) Cussler, E. Diflusion: Mass Transfer in Fluid Solution; Cambridge University Press: New York, 1984. (26) (a) Flarsheim, W.; Tsou, Y.; Trachtenberg, I.; Johnston, K.; Bard, A. J. J . Phys. Chem. 1986, 90, 3857. (b) McDonald, A.; Fan, F.; Bard, A. J. Ibid. 1986, 90, 196. (27) Phillips, M. E.; Deakin, M. R.; Novotny, M. V.; Wightman, R. M. J . Phys. Chem. 1987, 91, 3943.
