3598
J . Phys. Chem. 1988, 92, 3598-3604
Thermodynamics of Mixed-Valence Intercalation Reactions: The Electrochemical Reduction of Prussian Blue James W. McCargar and Vernon D. Neff* Department of Chemistry and The Liquid Crystal Institute, Kent State University, Kent, Ohio 44242 (Received: April 20, 1987; In Final Form: January 2, 1988)
The reduction of Prussian blue on gold and indium-tin oxide electrodes is considered in detail. It is established that films well-aged in KC1 solution accept potassium ions on reduction to Everitt's salt and eject them on oxidation to Berlin green. Our previous proposal that partially oxidized, or reduced, PB can be treated as a solid solution is reexamined, and the electrochemicalenthalpy of mixing is defined. It is found that the half-widths of the reversible voltammogramsvary dramatically with the nature of the injected cations. This behavior is explained in terms of the enthalpy of mixing parameter. The expression for the cell potential in terms of the composition variables is carefully defined and compared with other formulations which employ statistical models.
Introduction In the past few years interest has developed in the electrochemistry of Prussian blue (PB) and several of its transition-metal analogues. The subject has been reviewed recently by Itaya, Uchida, and Neff.] Although a number of studies have been devoted to the fundamental reactions, it cannot be claimed that they are yet fully understood.2-6 Since these materials show considerable promise for practical applications, there is ample justification for a thorough understanding of the prototype compound. For example, different reaction schemes have been proposed for both the oxidation and the reduction, and other possibilities may e x i ~ t . A ~ .variety ~ of factors which affect the shape, and peak separation, of the voltammograms have not yet been fully explained. Also, kinetic factors such as ion and electron diffusion and migration need further consideration. Recent kinetic studies have focused on electron conduction in dry films and in Little is yet known about the ion transport mechanisms. A major point of confusion regarding the whole class of insoluble, mixed-valence, transition-metal hexacyanometalates is the complicated and variable stoichiometry which often depends on the conditions of their synthesis. PB is well-known to exist in at least two forms known as soluble KFeFe(CN)6 and insoluble Fe,(Fe(CN),),. Soluble PB is prepared from a solution containing excess potassium ion." The details of the stoichiometric composition become particularly important in considering the electrochemistry because reduction, or oxidation, requires transport of both electrons and ions. In this sense, PB, and its analogues, should be classified as intercalation compounds. The presumed structure of soluble PB is based solely on powder patterns for which the space group has been determined to be Fm3m.12313Ludi et al. have obtained a complete structure of insoluble PB single crystals grown by slow precipitation from 12 M HCI ~ o l u t i o n . ' The ~ ~ ~structure ~ is disordered, and ca. onefourth of the ferrocyanide sites are vacant and there are no po(1) Itaya, K.; Uchida, I.; Neff, V. D. Acc. Chem. Res. 1986, 19, 162. (2) Ellis, D.; Eckhoff, M.; Neff, V. D. J . Phys. Chem. 1981, 85. 1225. ( 3 ) Rajan, K. P.; Neff, V. D. J . Phys. Chem. 1982, 86, 4361. (4) Itaya, K.: Uchida, 1.: Toshima, S. J . Electrochem. Sor. 1982, 129, 1498. ( 5 ) Itaya, K.; Ataka, T.; Toshima, S.J . A m . Chem. SOC.1982, 104,4767. (6) Ohzuka, T.; Sawai. K.; Harai, T.J. Electrochem. SOC.1985,132, 1369. (7) Tennakone, K.; Dharmaratne, W. B. D. J . Phys. C 1983, 16, 5633. (8) Viehbeck, A.; DeBerry, D.W. J . Electrochem. Sor. 1985, 132, 1369. (9) Chidsey, C. E.; Feldman, B. J.; Lundgren, C.; Murray. R. W. Anal. Chem. 1986, 58, 601. (10) Feldman, B. J.; Murray, R. W. Inorg. Chem. 1987, 26, 1702. (11) Wilde, R . E.; Ghosh, S. N.; Marshal, B. J. Inorg. Chem. 1970, 9. 2512. (12) Keggin. J. F.; Miles, F. D. Nature (London) 1936, 137, 5 7 7 . ( 1 3 ) Weiser. H. B.: Milligan, W. 0.:Bates, J. R. J . Phys. Chem. 1942, 46, 99. (14) Ludi, A.; Gudel, H. C . Struct. Bonding (Berlin) 1973, 14, 1. (15) Herrin. F.; Fisher. P.: Ludi, A.; Halg, W. Inorg. Chem. 1980. 19. 956.
0022-3654/88/2092-3598$01.50/0
tassium ions on the interstitial sites. On the other hand, the normal conditions for the synthesis of PB are quite different from those employed by Ludi and there is no requirement that the structures be the same. Indeed, it was observed long ago that powder patterns obtained from microcrystals grown from 12 M HCI contain additional lines that cannot be indexed on the basis of a face-centered unit celI.l3 These lines do not appear in the powder patterns of soluble PB. In this paper we are primarily concerned with the thermodynamic aspects of the reduction of PB on gold and indium-tin oxide (ITO) substrates under reversible conditions. We shall be particularly concerned with the proper formulation of the expression for the cell potential in terms of the composition variables (Le., the Nernst equation). The fully reduced form of PB, known as Everitt's salt (ES), can be chemically synthesized as a stable compound (in the absence of oxygen).I6 We shall emphasize that partially reduced PB may be regarded as a mixture or solid solution of PB and ES. Although we have previously presented this view, we shall here attempt to elaborate on the concept of electrochemical, as opposed to mechanical, mixing.2 We will also present evidence that the enthalpy of mixing is the fundamental thermodynamic parameter which determines the shape of the voltammograms obtained in the presence of different metal cations. Finally, we shall compare the views presented here with other recent attempts to explain the thermodynamic behavior of PB, and other intercalation compounds, based on lattice models invoking electrochemical potentials for individual species on the lattice.
Experimental Section Two types of electrode were employed in these experiments: gold and indium-tin oxide (ITO) on glass. The 1.O-mm-diameter Au wire electrodes were polished with p-alumina and cleaned in an equal volume of mixture of concentrated sulfuric and nitric acid. They were reduced for 10 min in 0.1 N HCI at 0.1 mA/cm2. The I T 0 electrodes were obtained from LXD Inc., Bedford, OH. The deposition of PB on various substrates has been described p r e v i ~ u s l y . ~We , ~ used the galvanostatic procedure of Itaya et aL3 We wish to emphasize, however, that all electrodes used in this study were prepared with extreme care with regard to the purity of materials and the deposition conditions. All of the group I metal salts used as electrolytes were Alpha Ultrapure grade. Potassium ferricyanide was obtained by triple recrystallization from TDW. Ferric chloride solutions were obtained by dissolution of pure iron wire in HCI. The films were deposited at constant current of 10 A/cm2 from solutions which were 0.01 M in FeCI,, 0.01 M in K3Fe(cN),, 0.1 M in KCI, and 0.01 M in HCI. The average film thickness was estimated by coulometry and ranged between 300 and 1000 A." (16) Duncan, J . F.: Wigley, P. W. R . J . Chem. Soc. 1963, 1120
0 1988 American Chemical Society
Electrochemical Reduction of Prussian Blue
The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3599
-3OOb
I 0.60
I
'I
1
040
020
00
I -020
E (Volts vs SCE)
E (Volts vs SCE)
Figure 1. Voltammogram of a well-cycled PB film on 10 ohm/(" of ITO) glass in 1.0 N KC1. The scan rate was 1.0 mV/s and the p H was 3.0. T h e total charge for the reduction reaction corresponds to 5.13 X C / c m 2 with an average film thickness of 1040 A.
All electrodes were thoroughly rinsed in TDW and allowed to stand for 10 min in 0.1 N F&04 in order to remove coprecipitated ferricyanide.2 They were again rinsed and dried in air for 24 h. The electrochemical behavior is affected by the adherence of the film to the substrate. A good film should not be removed from the substrate by gentle rubbing. Even under identical deposition conditions some films did not adhere well and were rejected. The voltammetric measurements were camed out with the usual three-electrode configuration in an H cell. The working cell compartment, which contained the calomel reference electrode in close proximity, was flushed with purified nitrogen. All electrochemical measurements were made with a PAR Model 273 potentiostat. Optical spectra were obtained with I T 0 electrodes in a cell designed for a Cary Model 118 spectrophotometer.2 The potassium ion concentrations in HCl solution were measured with an IL Model 157 AA spectrometer.
Results and Discussion I . Basic Reactions. The voltammogram of a carefully prepared PB film on 10 ohm/(cm2 of ITO) is shown in Figure 1. This film was cycled 50 times between -0.2 and 1.2 V in 1.O N KCl solution. We note the sharpness of the curves, particularly for the reduction reaction, at a scan rate of 1 mV/s. Very low scan rates are necessary in order to obtain reversible conditions for PB on ITO. We have found that the half-width of the voltammograms begins to increase at scan rates greater than 5 mV/s. For the reduction reaction in Figure 1 the half-width (full width at half-height) is 25 mV for the reverse (cathodic) scan and 3 1 mV for the forward (anodic) scan. This slight difference in half-width is characteristic and always observed. Aside from this, the main voltammetric peaks are remarkably symmetrical. Murray et al. have suggested that the voltammogram for the reduction contains three features at 0.33, 0.20, (the main peak), and 0.1 V.I8 We find that the hump at 0.33 V does not appear in carefully prepared films and believe that it is associated with unreacted ferricyanide ions coprecipitated during the deposition reaction.2 There is always, however, a minor feature on the cathodic side of the main peak. This accounts for ca. 10% of the charge consumed in the reduction reaction. Unlike the main peak, it is somewhat pH sensitive becoming sharper, and moving toward more negative potentials, in strong acid solutions (>0.1 M). If there are some ferrocyanide vacancies in this compound (as has been determined by Ludi for insoluble PB), we might expect that the ferric ions surrounding the vacancies would be reduced at a different po-
+
(1 7) Average film thickness was determined from the total charge for
complete reduction of a well-cycled film and from the unit cell dimension (10.2 A) for soluble Prussian blue. We assumed four ferric ions per unit cell. (18) Chidsey, C. E.: Feldman, B. J.; Lundgren, C. L.; Murray, R. W. Anal. Chew. 1986, 58, 601.
Figure 2. Voltammogram for the reduction of PB on a gold wire electrode in 1.0 N KCI, in the presence (dotted curve) and absence of dissolved molecular oxygen. T h e scan rate was 5.0 mV/s.
tential. This point requires further study, but our concern here will be with the main reduction peak. Many factors can influence the shape of the voltammogram. Among these are scan rate, cycling history, the presence of dissolved 02,the resistance of the substrate, and particularly the nature of the cation in the electrolyte solution. The effect of substrate resistance is easily demonstrated with PB on I T 0 electrodes. The peak separation for the reaction on 10 ohm/(cm2 of ITO) (Figure 1) is only 5 mV whereas the peak separation for a PB film on 100 ohm/(cm2 of ITO) was found to be over 200 mV. High substrate resistance also led to considerable broadening of the voltammogram. Dissolved oxygen will react slowly with Everitt's salt in aqueous solution to give PB. This reaction has little effect on the voltammograms at high scan rates but becomes significant at low scan rates, as shown in Figure 2. The rising tail for the voltammogram obtained in the presence of oxygen is a direct indication of the irreversible loss of ES in the cathodic region of the scan. This result emphasizes the need for the exclusion of oxygen in voltammetric studies. There has been considerable discussion comparing the amount of charge consumed in the oxidation and reduction reactions.] Our original observations were confined to Au and Pt electrodes where, for the oxidation reaction, the coulometry is complicated by large residual currents and a gradual deterioration of the film.2 Itaya et al. have shown that the oxidation is well-defined on the I T 0 substrate, leading to the formation of a stable yellow film.5 Our original reaction scheme suggested only two-thirds of the ferrocyanide ions were oxidized, but on the I T 0 substrate we find that the conversion is higher, although still not 100%. For example, the total charge for the oxidation of PB in Figure 1 is 87% of that for the reduction. The additional charge for the reduction arises because we integrate the current out to -0.2 V, including the second cathodic peak mentioned previosly. If this were excluded, there would be good agreement on the amount of charge consumed in both reactions. Thus, there appears to be a one-to-one correspondence between ferric ion, which is reduced at 0.2 V, and oxidizable ferrocyanide ion. This also suggests that there may be reducible ferric ion in the lattice which is not coordinated to ferrocyanide ion. Allowing for such structural complexities, we still find that the main features of the reactions, for well-cycled films, are described by the following equations: KFeFe(CN), = K+ + e-
+ FeFe(CN),
(2)
The reaction scheme proposed by Itaya involves the oxidation and reduction of insoluble PB Fe,(Fe(CN)6),.5 This was based on the observation that the initially deposited PB film did not contain potassium ion. The oxidation step requires that anions are injected into the film. We have previously rejected this
3600 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988
t
9
" t
/
0
mechanism on the basis of the dependence of the thermodynamic potential on the concentration of the KC1 electrolyte.2 Rosseinsky et al. have suggested that potassium ion accumulates in the film after cycling to the reduced state in the presence of the electrolyte.Ig We have obtained further evidence that well-cycled PB films contain potassium ion. This is based on a nondestructive in situ analysis for potassium ion released in reaction 2. A 25-cm2 film of PB was deposited on I T 0 and cycled repeatedly in 1.O N KCI. After thorough rinsing the film was oxidized to 1.0 V in 0.1 N HC1 solution. In this single-step oxidation the reference electrode consisted of a PB film on Pt. After oxidation the cell solution was analyzed for potassium ion by atomic absorption spectrometry. An average of five such measurements gave a potassium ion concentration that was 94% of the expected amount based on eq 2. This direct evidence that potassium ion is released on oxidation strongly supports the presumption that electroactive PB is in the soluble form. The structure of the film, and the disposition of potassium ions in the lattice, are still not well-understood. II. Electrochemical Mixing. The electrochemistry of the mixed-valence intercalation compounds is unique in several respects. First of all, the electrodes do not fall nicely into any of the established categories. They resemble redox electrodes in that they involve two or more oxidation states in a single homogeneous phase. They also resemble electrodes of the second kind in that they involve ions from a separate solution phase. Unlike the latter, however, the solid phase undergoes a continuous change in composition with change in applied potential. This uniqueness has been recognized by a number of investigators who have formulated expressions for the cell potential as a function of the composition variables.2~6~20~21 None of these formulations are complete, and there are misconceptions, and incorrect equations, which will shortly be discussed in detail. Because of the increasing interest in these reactions, we believe that careful consideration of the thermodynamic relations is justified and that misconceptions should be corrected. The discussion will apply explicitly to PB, but the main features apply equally to any of the mixed-valence intercalation reactions. We begin by discussing an experiment which can be clearly, and unambiguously, defined in thermodynamic terms. We prepared several PB films containing n moles of PB on 3-cm X 7-cm I T 0 plates. They were rinsed and vacuum-dried, and their absorbance was measured at 710 nm. They were then partially immersed into a cell containing KCI electrolyte, and the immersed portion was reduced completely to ES. The molar amounts n l of PB and n2 of ES on each plate were determined by the distance of immersion into the electrolyte. On open circuit the electrodes were then completely immersed in the KCl solution whereupon (19) Mortimer, R. J.; Rosseinsky, D. R. J . Chem. SOC.,Dalton Trans. 1984, 2059. (20) Atlung, S.; Jacobsen, T. Electrochim. Acla 1981, 26, 1447. (21) Chidsey, C. E. D.; Murray, R. W. J . Phys. Chem. 1986, 90, 1479.
McCargar and Neff they quickly (less than 1 s) became entirely uniform in color. They were again removed, rinsed, and vacuum-dried, and their absorbance was measured at 710 nm. The results are shown in Figure 3 where we have compared the ratio of absorbances A ( x l ) / A owith the mole fraction x , of PB. Here A. represents the absorbance of the PB film before reduction (no base line correction). Assuming the validity of Beer's law and no submicroscopic phase segregation, it is apparent that the resulting compounds can be considered to be a homogeneous mixture of pure PB and The experiment demonstrates what we shall call electrochemical (as opposed to mechanical) mixing. During this process electrons are transported through the I T 0 substrate and the KCI solution acts as a source, or sink, for the transport of potassium ions. By this procedure one could, in prinicple, prepare an infinite number of nonstoichiometric compounds varying over a continuous range of the composition variables. It is important to emphasize that these compounds represent a two-component mixture. Although one can identify a number 'K etc., the mixing of ionic species such as Fe3+,Fe2+,Fe(CN),", , process is completely described by only two independently variable quantities, namely, n and n l (or n2). We have carefully avoided identifying the components at the molecular or ionic level which, of course, is not required by thermodynamics. We shall refer to the "components" of the mixture as PB and ES with the mole fractions x1 and x2, respectively. The free energy of mixing is the driving force for this process, and in order to consider it further we define the chemical potentials in terms of the components as follows: p(PB) = po(PB) R T In a(PB) (3)
+ p(ES) = po(ES) + R T In a(ES)
(4) The superscripts refer to the molar free energies of the pure components for which the activities are unity by definition. The molar free energy of mixing is AG, = AH, - T U , = x , R T In a(PB) + x,RT In a(ES) ( 5 )
In order to proceed further, this exact equation must be simplified by suitable assumptions allowing us to express the activities in terms of the mole fractions. A simple phenomenological model which has proved useful for many types of solutions, including solid solutions, was pioneered by the work of Hildebrand.23 This theory of regular solutions is applicable when the volume of mixing is negligible and the enthalpy of mixing is symmetrical with a maximum at x = 0.5. The molar free energy of mixing is given by AG, = w x l x 2 + R T ( x l In x , + x 2 In x 2 ) (6) where the enthalpy of mixing LW= wxlx2could, in principle, be determined from the heat absorbed or evolved in the experiments described above. As we shall show, AG, can also be obtained from the coulometric curve for the reduction of PB. In fact, the reason we have discussed the mixing process is that it allows us to relate the electrochemical data to a measurable thermodynamic parameter. The main disadvantage with regular solutions is that the entropy change is for random mixing, thus excluding the possibility of ordered arrangements in the mixed phase. Such possibilities can be included, however, and will be briefly discussed in a later section. I l l . The Nernst Equation. In formulating the most general expression for cell voltage in terms of chemical potentials, Guggenheim abandons the method of equating electrochemical potentials across interfaces and considers instead the reversible work of a well-defined cell reaction.24 We shall adopt this method in this section. We consider a cell in which one electrode is the mixed-valent film containing n , moles of Pb and n2 moles of ES. The other electrode is a couple such as Ag/AgCI capable of
-
(22) By this we mean that the oscillator strength of the 2T,, *TZg ligand-to-metal transition depends only on the number of ion pairs and not on their disposition in the lattice. (23) Hildebrand, A. J . Am. Chem. SOC.1929, 51, 66. (24) Guggenheim, E. A. Thermodynamics; North-Holland: Amsterdam, 1967; p 311.
The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3601
Electrochemical Reduction of Prussian Blue reducing the film. The chemical reaction is written symbolically as Ag
+ PB(m) + K+(aq) + Cl-(aq) = ES(m) + AgCl
(7)
where PB(m) and ES(m) represent the components in the mixture of fixed composition. It is necessary to imagine that both electrodes, and the electrolytic solution phase, are present in very large (approaching infinite) amounts. The alternative to this assumption would be to introduce the chemical affinity. By opposing the cell voltage with a suitable external EMF, it is possible to allow 1 mol of charge to flow reversibly, at constant T and P,under conditions where there is essentially no change in composition in any phase. The negative of the free energy change for the process represents the reversible work, and we can write -AG = FE where E is the cell voltage. If we express the free energy change in terms of the chemical potentials defined by eq 3 and 4, we obtain the following equation for the half cell potential for the reduction reaction: E = Eoo
R T In a(PB)a(K+) +F a(ES)
where EOo =
+ POPB) + P"(K+(aq))
-p0(ES)
F
Equation 8 is exact and could be used to measure the ratio of activities from measured cell potentials. It also emphasizes the important role of the activity of the potassium ion in the aqueous phase. It is convenient to rearrange this equation as follows: E = Eo
T a(PB) + RIn F a(ES)
(9)
where Eo = Eoo
RT +In a(K+(aq)) F
The cell voltage is directly related to the free energy of mixing introduced in section 11. For convenience, we define the mole fraction of PB as x, = x and that of ES as x2 = (1 - x). If we now make use of the Gibbs-Duhem equation xRT d In a(PB) + (1 - x) d In a(=) = 0 and take the derivative of AG,,, with respect to x at constant n (note that this derivative is not a chemical potential), we obtain
where AE = E - EO. Thus, we see that AE is directly proportional to the derivative of the free energy of mixing with respect to mole fraction for PB and any other mixed-valence intercalation compound of two components. The inverse relation
Figure 4. Determination of the free energy of mixing (left scale) by integration of the coulometric curve (right scale). Total charge was assumed to be zero for E = -200 mV and maximum for E = +200 mV. The coulometric curve was obtained from a 1100-A PB film on Au in 1.0 N KCI solution. The scan rate was 1.0 mV/s. Curve a represents the coulometric data obtained experimentally. Curve b was obtained by numerical integration of curve a. Curve c represents the free energy of
mixing for an ideal solution. the greatest contributions to the integral in eq 11 come from the extremes of the coulometric curve. In these regions small residual currents, or possible additional electroactivity such as discussed in section I, make a large contribution to the integral. It also follows directly from eq 10 that the molar current for a reversible voltammogram is given by
i = F2S(a2G,,,/a~2)-1 where S is the scan rate. That is, the voltammetric current is directly proportional to the inverse curvature of the free energy of mixing curve. These general relations are useful in helping to visualize the effect of the enthalpy of mixing on the shape of the coulometric or voltammetric curves. Again we point out that they are exact and that no special assumptions have been made relating the activities to the composition variables. In order to proceed further, we introduce the activities and chemical potentials for strictly regular solutions: a(PB) = x1 e x p ( w ~ ~ ~ / R T ) w(PB) = po(PB)
+ xz2w+ R T In xI
a(ES) = x2 exp(wxI2/RT) AG,(x) = F S '0A E ( x ) dx
(11)
may be used to determine the free energy of mixing from coulometric data, as shown in Figure 4. The coulometric curve for the reduction of PB on a gold wire electrode in 1.0 N KC1 was integrated numerically. The voltage limits for determination of the total charge were set at +0.4 and 0.0 V, respectively. The mole fraction x at a given value of AE(x) was determined from q ( A E ) / q owhere qo was the total charge at 0.0 V. For PB films on Au and Pt electrodes the additional current on the cathodic side of the main reduction peak is generally not resolved as a distinct shoulder. Hence, this does not appear as a distinct feature in the free energy of mixing curve. The experimental curve b may be compared with the value of AGm for an ideal solution c (Figure 4). In this case the experimental free energy is less negative than the ideal value because the enthalpy of mixing is positive for the potassium salt of PB. The flat portion of curve b gives rise to the sharp feature of the voltammogram. It should be pointed out that the absolute value of AG,,, cannot be considered to be very accurate. This is because
k(ES) = ko(ES)
+ xI2w + R T In x2
(12) (13) (14) (15)
where w is the enthalpy of mixing parameter. Substituting eq 12 and 14 into eq 9, we obtain E = Eo
T RT + Rz(x2 - xl) + - In (x1/x2) F F
(16)
where we have introduced z = w/RT. For future reference we also point out that the enthalpy terms in the chemical potentials are quadratic in the composition variables and that eq 14 can also be obtained directly from the negative difference of the chemical potentials, -(p(ES) - k(PB))/F. From eq 16 we have previously shown that the reversible voltammetric current for the reduction of PB is given by
where n is the total number of moles and S is the scan rate.2 The shape of the current-voltage curves depends only on z. When z
3602 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988
McCargar and Neff
TABLE I"
1.0 N KC1 1.0 N NH4CI 1.0 N
RbCl
1.0 N CSCl
PB, IO-* mol/cm2
mV
Z
4.13 5.12 4.98 5.04
198 236 3 00 425
1.19 0.10 -1.90 -3.92
EO,
E', mV
i,(calcd), pA/cm2
i,(obsd), pA/cm2
1, lo-* cm
27
478 253 120 80
450 240 135
654 81 1 788 798
85 195
300
90
"The value of z was chosen to give the best fit of the half-width AE' to the experimental voltammograms. The calculated peak current i, was determined from AE'and the amount (mol/cm2) of PB on the electrodes. The amount of deposit, and the average film thickness, were determined coulometrically after the films were cycled many times in the presence of the appropriate cation. In all cases the pH was adjusted to 3 with HCI.
t',,t
'Oo0
E
4001
Iil
k
300
A E ("4)
Figure 5. Calculated current-voltage curves for different values of z = w / R T . The values, in order of increasing half-width, are z = 1.0, 0.5, 0.0, -0.5, and -1.0. The peak current was normalized to 1 mA for z = 1.o.
--300[ 400 ~
07
= 0 we obtain the voltammogram for an ideal thin film with a half-width (full width at half-height) of 90.6 mV. In this case the free energy of mixing is driven by the entropy term alone. For positive enthalpy of mixing, the free energy is less than the ideal value and the voltammogram is narrowed. When z approaches 2, the peak current tends to infinity and the half-width to zero, suggesting phase separation. That is, for z > 2 there would be no spontaneous mixing. For negative enthalpy of mixing the voltammogram becomes broader with no lower limit. For any value of z, the half-width AE'can be obtained from the composition xl' determined from the condition i / i , = 1 / 2 where i , is the peak current. This gives XI'
= (1
+ (2 - z)/(4
- z)l/2)/2
(18)
The shapes of the current-voltage curves for several values of the enthalpy of mixing are shown in Figure 5 . IV. Experimental Justification of Regular Solution Behavior. In studying the voltammograms for the reduction of PB in solutions containing different cations, we have found that the nature of the cation has a dramatic effect on their shape. This can be clearly seen in Figure 6 where we have shown the current-voltage curves for the reduction of PB in the presence of K', NH4', Rb', and Cs'. The curves shown are for the chlorides, but different anions have little effect on this behavior. Similar results have been obtained for solutions of the sulfates and the nitrates. The PB films were deposited on gold wire electrodes, and they were cycled many times in KCI solution in order to ensure the formation of the potassium salt. Different films, of approximately equal thickness, were prepared for each cation. After standing for 24 h in the appropriate cation they were again cycled many times in order to ensure the injection of a given cation. In Figure 6 we first note the increase in the Eo values, toward more positive potentials, starting with K+ and ending with Cs'. These values are listed in Table I. This type of behavior has been previously noted by ourselves and It has been considered
06
05
04
03
02
01
00
-01
-02
E (Volts vs SCE)
Figure 6. Voltammograms for the reduction of PB on gold wire electrodes in different cation solutions. The scan rate was 5 mV/s. (a) 1.0 N KCI, (b) 1.0 N NH4C1, (c) 1.0 N RbCI, (d) 1.0 N CsCI. Pertinent experimental parameters are shown in Table I.
in greatest detail by Bocarsly et al. in a series of investigations of nickel ferricyanide films obtained by oxidizing nickel electrodes in the presence of ferricyanide ion^.^^**^ Basically, it suggests that Everitt's salt is more stable, toward oxidation, when it contains the heavier group I cations. In fact, the reduction potential for the ferric ion in CsFeFe(CN), is approaching the value for ferric ion in 1.0 N aqueous KC1 solution. The second major feature in Figure 6 is the large increase in half-width in the sequence from K' to Cs'. Experimental values of the half-width are shown in Table I. The enthalpy of mixing parameter z was adjusted to give the experimental value of the half-width. The theoretical peak current was calculated from z and the amount (mol/cm2) of PB on the electrodes. As is shown in Table I, the calculated values of the peak current are in quite good agreement with the experimental values. We regard this as good evidence that simple regular solution theory can be used to understand the main features of the thermodynamic behavior. On this basis we see that the sharp voltammogram for potassium ion is related to the positive enthalpy of mixing (w = 1.19R7'). On the other hand, the z value for the ammonium salt is small and the half-width of 85 mV is close to ideal behavior. For Rb+ and Cs' the enthalpy of mixing becomes increasingly negative as might be expected from the fact that these ions are known to displace potassium ions from PB.27 The very broad curve for the cesium salt still varies smoothly over the entire range of potentials. There is no indication of an abrupt change which might suggest the onset of long-range order in the mixed phase. A model involving random mixing appears to apply well to the PB system although this may not be true for other intercalation compounds. (25) Bocarsly, A. B.; Sinha, J. J . Electroanal. Chem. 1982, 140, 167. (26) Sinha, S.; Humphrey, B. D.; Bocarsly, A. B. Inorg. Chem. 1984, 23, 203. ( 2 7 ) Seifer, G. B. Russ. J . Inorg. Chem. (Engl. Transl.) 1962, 7 , 621.
The Journal of Physical Chemistry, Vol. 92, No. 12, 1988
Electrochemical Reduction of Prussian Blue
50
3603
charge (Le., a single electron transfer at the metal-lattice interface). For the reduction of PB these species are FeL3+and FeL2+ where L indicates that they are located on a lattice. These species are to be regarded as acceptor and donor in the same sense as that defined by Gerisher in his treatment of electron-transfer kinetics.30 In the conducting substrate (designated by M) the donor species may be regarded as the electrons, and the acceptors as the vacant energy levels, in the conduction band. These are designated as eM-and OM,respectively. We also need to include the transfer of potassium (or other) ions between the lattice and the aqueous solution. These are designated as KLf and Kaq+. We can, in principle, define the electrochemical potentials for all of these species. One might object to the concept of the chemical potential of vacant energy levels (or holes), but it is necessary for the consistency of the formulation. It is analogous to the chemical potential of the vacant sites (or holes) in a lattice gas. The electrochemical potentials are defined as
t
I-
2 W
IT
a
3
0
-50 I
I
03
L3
, fi2
01
POTENTIAL ( V vs SCE)
Figure 7. Voltammograms of NH4FeRu(CN), in 1.0 N (NH&SO4 (solid line) and KFeRu(CN)6 in 1 .O N K2S04(dashed line) on a gold wire electrode. T h e scan rate was 1 mV/s.
p(eM-) = pa(eM-) + R T In a(eM-) - FdM; ~ ( 0 , )= OM)
+ RT
In 4 0 ~ )
p(FeL3+)= pa(FeL3+)+ R T In a(FeL3’) + 3F4,; p(FeL2’) = wo(FeL2’) R T In a(FeL2+) 2F4L
+
+
It has been suggested that the ability of the PB film to accept cations is directly related to the hydrated ionic radius.’ One might and the conditions of equilibrium are also infer that the increase id half-width (negative enthalpy of p(eM-) = p(FeM2’) mixing) would also be so related, and this appears to be true for the sequence shown in Figure 6. That this is not generally true is shown in Figure 7 for the reduction of the ruthenium analogue of PB. Here the half-width of the ammonium salt of NH4FeR u ( C N ) ~is only 18 mV whereas that for the potassium salt is Subtracting eq 20 from 21, and substituting for c $from ~ eq 22, 83 mV. This corresponds to z values of +1.68 and +0.10, rewe finally obtain spectively. We see that the half-widths are reversed relative to the PB electrode even though the hydrated ionic radii of N H 4 and K+ are ca. the same (1.25 A).34 A similar situation occurs with Bocarsly’s data on nickel ferricyanide films.25 These films admit Li+ and Na+ ions as well as the other group I cations. The absolute where E o = - A p a / F . The activities of the donor and acceptor values of the half-widths of the current-voltage curves are probably species in the metal substrate have been retained in the Aha term not significant because the voltammograms were obtained at since the metal is a pure one-component phase. Equation 23 is relatively high scan rates (100 mV/s). Comparatively, however, correct assuming we have properly identified the individual species it is found that the half-width for Li+ (160 mV) is larger than on the lattice. The previous papers on this subject all neglect the for Na+ (1 10 mV) and K+ (120 mV) which are ca. the same. equilibrium condition (eq 21) between the electron acceptors in Thus, there is no direct correlation between the half-width and the lattice and the metal. Without this condition it is impossible the hydrated ionic radius. to obtain the cell potential in terms of a ratio of the activities of V. Lattice Models. As previously mentioned, there have been the oxidized and reduced states in the lattice. This argument is a number of papers concerned with the thermodynamic formualso true for the simpler example of a redox couple in aqueous lation of the cell potential for PB and related mixed-valence solution. The reason that the authors cited above obtain, in most compounds. Aside from the strictly phenomenological approach cases, a plausible final result is that they all use an incorrect discussed above, all other treatments have employed models based equation for the chemical potential, in the mean field approxion lattice statistics in the mean field approximati~n.~,~~,~’,~~,~~ The mation, which leads to a kind of compensation of errors. This theory is based on two principal procedures. First, one introduces will be shortly discussed in detail. the chemical potential for individual species on the lattice and Equation 23 may be compared with eq 9, which does not obtains the cell voltage by equating electrochemical potentials identify individual ionic species. It should not be presumed that across the phase boundaries. Then the chemical potentials are the product of activities in the denominator of (23) can be reduced calculated from a simplified (mean field) partition function. to a product of concentrations as is done, for example, in ref 6 Neither of these procedures is carried out correctly in the papers and 21. The concentration of ferrous ions in the lattice is not listed above. For this reason we shall formulate the problem in independent of that of the potassium ions if electroneutrality is the most general way and point out misconceptions along the way. to be maintained at equilibrium. That is, as we have previously We would like to emphasize that we have postponed this approach stated, for the reduction of PB the mixed phase should be regarded to the problem, not because we disdain the use of statistical as a two-component system. Equation 23 becomes identical with thermodynamics, but because of our poor current understanding eq 9 if we define a(PB) = a(FeL3+)and a(ES) = U ( F ~ ~ ~ ~ ) U ( K ~ + ) . ~ * of the nature of the independent “species” and the energetics of The final justification for this can be determined by experiment their “interactions”. since a product of concentrations, even if they are equal, would The mixed-valence phase represented by a redox thin film lead to distinctly different coulometric and voltammetric curves contains ions in two, or more, oxidation states. We shall specify a compound in which there are only two states differing by unit ~
(28) Berlinsky, A. J.; Unruh, W. G.; Mckinkon, W. R.; Hearing, R. R. Solid State Commun. 1979, 31, 135. ( 2 9 ) Coleman, S . T.; McKinnon, W. R.; Dahn, J. R. Phys. Rev. E Condens. Matter 1984, 29, 4147.
(30) Gerisher, H. Physical Chemistry, An Advanced Treatise: Eyring, H., Denerson, D., Jost, W., Eds.; Academic: New York, 1970. (31) By analogy we could, in principle, define the activity of a collection of hydrogen molecules in terms of the product of the activities of the hydrogen atoms in the molecules. It by no means follows that the hydrogen molecule activity is equal to the product of the hydrogen atom concentrations.
3604
J. Phys. Chem. 1988, 92, 3604-3612
which are not observed in the case of the PB electrode. For the statistical lattice model of a two-component system we consider the Bragg-Williams theory which, in its most general form, includes an order parameter s.32 For our purposes we shall assume that s = 0 since this has been done by the authors cited above and since, for the reduction of PB, there is no experimental indication of an abrupt transition in the mixed phase. More general lattice models for intercalation compounds, going beyond the mean field theory, have been discussed by McKinnon and H a e r i ~ ~ gIn. ~the ~ Bragg-Williams model the molar configurational energy (enthalpy) is given by
u = X 1 2 W l l + 2XlX2Wl2+ X22W2,
(24) where wI1, and w i 2 represent the interaction energies of nearest-neighbor 11 atoms, 22 atoms, and 12 atoms, respectively. The sign of the energy parameters is opposite to that used conventionally. The reasons for this is so that the energy term W,, can be directly compared with the enthalpy of mixing parameter introduced in section 11. It also explains why the sign of the energy parameter in section I1 differs from that used by others in the conventional lattice model. The chemical potentials of the two types of atoms (or ions) are equal to pl = pI0
+ x1(2 - x I ) w I I+ 2x,*w1, - x22w22+ R T In x1 (25)
p2
+
+
= pLZo x2(2 - x2)wZ2 2x12wl,- x I 2 w l l+ RT In x2
(26) (32) Fowler, R.; Guggenheim, E. A. Statistical Thermodynamics; Cambridge University Press: New York, 1952; p 570. (33) McKinnon, W. R.; Haering, R. R. Mod. Aspects Electrochem. 1983, IS, 235. (34) As suggested by one of the reviewers, this comparison may be oversimplified in that NH4+ is capable of hydrogen-bonding interactions with Ru-CN sites whereas K+ is not and the Ru-CN sites, as bases, are not the same as Fe-CN sites.
where bl0 and p20 are the molar free energies of the pure compounds excluding the configurational energy terms. If w1 = w2* = 0, the equations above reduce to those given previously for strictly regular solutions with 2w,, = w. If w I 2 = w2, (or w l l ) = 0, we have the lattice gas which has been used in all of the other mean field treatments of this problem. In the latter case the equations given for the chemical potential (eq 16 in ref 21, eq 27 and 28 in ref 6, eq 16 in ref 20, eq l a in ref 28, eq 1 in ref 29) are incorrect. In the mean field approximation the energy terms in the potentials are quadratic in the composition variables and the logarithmic entropic terms contain only x1 or x,, not the ratio xI/xz. Equation 25, for example, is obtained from -(a In Z/dnl),,, where Z is the configurational partition function. The equations given in the references cited above can be obtained from the difference of the chemical potentials for the two components. If we subtract eq 25 from eq 26, we obtain Ab = Abo
+ 2[x2wzZ- x l w l l + ( X I - x2)w121 + R T
In (x2/x1) (27)
If we further assume that Ab = -FE and that w l i = w22= 0, we obtain eq 16 in section 111. Thus, we see again that the Nernst equation is obtained from the difference of the chemical potentials for the two components in the mixed phase. This is also true for the lattice gas for which we obtain the Nernst equation as given in ref 20, 28, and 29. Even in the lattice gas we have to consider the difference of the chemical potentials because, although the vacant sites have no energy, they do contribute entropy. Acknowledgment. This research was supported, in part, by a T.A. Edison Grant from the state of Ohio. Registry No. PB, 12240-15-2; ITO, 50926-1 1-9; ES, 15362-86-4; NH4FeRu(CN),, 83017-33-8; KFeRu(CN),, 83017-32-7; Au, 7440-575 ; KCI, 7447-40-7; 01, 7782-44-7; NH4+, 14798-03-9; K, 7440-09-7: Rb. 7440-17-7; Cs, 7440-46-2; Berlin green, 14433-93-3.
Ionic Distributions and Competitive Association in DNA/Mlxed Salt Solutions Russell J. Bacquett and Peter J. Rossky* Department of Chemistry, The University of Texas at Austin, Austin, Texas 78712 (Received: July 22, 1987; In Final Form: December 22, 19871
The hypernetted chain integral equation is used to study ionic distributions near a simple model of DNA in aqueous mixtures of NaCl and MgCI2. A wide range of both relative and absolute concentrations of the salts is considered, including cases where only one salt is present. In dilute systems (low ionic strength), the solution composition near the polyion is more responsive to changes in the bulk solution composition, and competition by the divalent counterions is more effective. It is found that for reasonable definitions of the polyion vicinity the displacement of Na' by Mg2+in this region is approximately one for one, with a corresponding increase in the degree of polyion neutralization at short range, with increasing Mg2+concentration in the bulk. Calculated estimates of mean-square electric field gradients experienced by sodium nuclei in these solutions manifest a very good correlation with measured NMR line widths, indicating that the observed structural phenomena are realistically described.
I. Introduction The interaction of rodlike polyions with small ions in mixtures of charged species is of great interest. For example, synthetic polyions can act to concentrate like-charged reactive species with a consequent enhancement of reaction rate.[ Nucleic acid polymers function naturally in a mixture of salts, and changes in solution composition are one means by which their biochemical 'Present address: Department of Chemistry, University of Houston, Houston, TX 77004.
0022-3654/88/2092-3604$0 1SO10
behavior can be regulated. The conformational transitions2 and binding interactions3 of both natural and synthetic polyions have a well-known sensitivity to solution composition. An understanding of these phenomena requires a detailed knowledge of how charged species are distributed near the polyion and how those distributions (1) Morawetz, H.; Vogel, B. J . A m . Chem. Soc. 1969, 91, 563. (2) Saenger, W. Principles of Nucleic Acid Structure; Springer-Verlag: New York, 1984. (3) Rich, A,; Seeman, N. C.; Rosenberg, J. M. In Nucleic Acid-Protein Recognition; Vogel, H. J., Ed.; Academic: New York, 1977.
0 1988 American Chemical Society
