J . Chem. SOC., Faraday Trans. 1 , 1982, 78, 3595-3603 Electron Transfer in Solids Temperature Dependence of Dielectric Relaxation and Conductivity in Mixed-valence Potassium Manganate-Permanganate BY DAVID R. ROSSEINSKY* AND JAMES S. TONGE Department of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD Received 18th March, 1982 The site-transfer conductivity expression o = ne2a2 v/6 kT has been further tested for K,(MnO,), by observations of the d.c. conductivity, o, and of the electron-transfer frequency, v, from dielectric relaxometry, over a temperature range, providing good agreement both individually and in activation energies E: v / v o = exp [-(6080 K)/T] and o/oo = exp [-(5908 K)/T] with v, = 4.8 x 10l2 Hz and oo = 84.7 l2-l cm-l. The Marcus-Hush semi-classical formulation for E as a sum of Ein = 3 k, k,Ar2/(kl + k,) and Eout = fe2(+r;’ +&l- r;;) (ny2 -&;I) together with summary adoption of v, = kT/h in the site-transfer expression, gwes a complete theoretical formulation for the conductivity as o = (ne2a2/6h) exp [ - (Ein + Eout)/ka or numerically (204 l2-l cm-l) x exp [ -(5300 K)/T], acceptably according with experiment. KMnO, and K,MnO, have lower conductivities; dielectric relaxometry on KMnO, does not show the Cole-Cole behaviour which we associate with simple electron transfer in such materials, and K,MnO, is not amenable to dielectric relaxometry. The combination of two approaches has recently thrown further light on the site-transfer (hopping) mechanism of d.c.conductivity in mixed-valence solids and in donor-acceptor adducts.The first involves the application of a phenomenological relation’ between the natural frequency v of electron transfer between the transfer sites and the value of the d.c. conductivity = ne2a2 v/6 kT where n is the number density of charge carriers, equated to the donor concentration, and a is the distance between the transfer ~ i t e s . l - ~ The second is the recognition that in such systems electron transfer represents a polarisation mechanism, the dielectric relaxation time of which will be the inverse of the electron-transfer frequency v.2-4 An array of relaxation frequencies about the most probable value gives Cole-Cole beha~iour,~ and in either case the frequency v is inferred from the value of the applied frequency at the maximum of a semicircle or circular-arc plot of the imaginary permittivity E” against the real, E’, where E” is calculated from the frequency-dependent conductivity less the d.c.value, 0, when appreciable. An alternative is a linear plot (arising from straightforward modification5 of the functions generating Cole-Cole arcs) as used in our original study2 of K,(MnO,),. K,(MnO,), is an alternating MnO, - MnOi- mixed-valence solid stoichiometrically comprising equal parts of KMnO, and K2Mn04. We report here extensions of our measurements, with minor modifications, to temperatures other than ambient and to 35953596 ELECTRON TRANSFER IN SOLIDS the parent compounds. In more detail we compare the K,(MnO,), frequency values with the electron-transfer rates for aqueous reaction, MnO; + MnOi-, prima facie a closely similar process to site transfer in the solid.EXPERIMENTAL The reagents and methods were much as in ref. (2), but in addition to bridge measurements (10 Hz-100 kHz) a phase-sensitive detector (Brookdeal 9505) and, once, a Solartron 1172 were used for confirmation. An Oxford CF4 cryostat was used down to -70 OC for cooling, and a helium-filled thermally insulated vessel with heater and thermocouple, for heating of compacted discs, in o measurements. The dielectric measurements were performed at or above ambient temperature because the circuitry associated with the cryostat introduced irresolvable interference. A variety of contacts for conductimetry were tried for the three solids, including a four-probemercury-contact cell for K3(Mn04),.6 Platinum or gold platings vacuum-deposited on the compacted discs served best, but even these decayed because of oxidation by all three highly reactive solids.Pressed-on platinum discs were the most constant, but not very reproducible. Special treatment of the data was thus devised (as below) to avoid errors arising from these problems. Values now reported replace ref. (2). A one-off relaxation measurement on K3(Mn04), was made on a Solartron 1172 frequency- response analyser, to which we had momentary access. Controlled by an Apple computer, the frequency range lou4 Hz-1 MHz is scanned automatically, and real and imaginary impedances outputted ; for a simple resistive-capacitive circuit this gives Cole-Cole-like response and an identical means of estimating v (see fig.3 later). RESULTS AND DISCUSSION Gold-plated electrodes and guard rings were used in measurements treated by the linear method in ref. (2). However, as o in the present measurements was measured after dielectric studies, to avoid polarisation, the values obtained were a lower limit because of electrode deterioration. This uncertainty made the low-frequency part of the Cole-Cole plot uncertain (hence the use2 of the linear plot which emphasizes high frequencies). Heavy Pt electrodes gave good Cole-Cole plots but erroneous E, and E, values (as shown by the non-retrieval of vacuum values with a void replacing the sample.) In order to make use of low-frequency data, it was decided to treat o in the relaxation analysis as an adjustable constant giving the best circularity of (minor) arc.This resulted in increases in o of up to 30% from the finally observed value, but these were more usually ca. lo%, which led to 10% to three-fold increases in the derived values of v. (With other materials and conditions, no such dependence on fitted o was found.) The assumption of arc circularity in fitting o proved a convenient procedure, and OFIT was always shown to be within a small or moderate (30%) interval of oOBS. Further confirmation of the benefit of this procedure is the generally improved fit of v derived via eqn ( 1 ) to oOBS or gFIT (table 1). The Cole-Cole plots are shown in fig. 1. $2-l cm-l. The four-probe mercury contact gave a value 1.4 x R-l cm-l, which was not sustained because of rapid Hg oxidation, as shown by a grey-white colouration.While possibly an upper limit for o, it could comprise the result of numerous surface effects, and we have preferred to take the through-disc values as being representative of the bulk conductivity . The direct comparison of oDIEL calculated from vet and eqn (1) with oOBS or oFIT, when d.c. and dielectric-relaxation data were measured within short intervals of time (< 24 h) on the same sample, gave entirely satisfactory agreement (table 1). For 296-298 K, deviations from average in one (oOBS) closely follow deviations in the other The final 298 K value of oOBS found over 7 samples was (2.1 0.5) xD. R. ROSSEINSKY AND J. S. TONGE 3597 75 r E 0 50 100 150 200 E l FIG. 1.-Typical Cole-Cole plots from K,(MnO,), at various temperatures: (a) 296, (b) 319, (c) 325, ( d ) 337 K.The numbers on the curves are the applied frequencies in kHz. (oDI EL), each being an intrinsic property of the sample [according to eqn (l), the same property]. The agreement now found between oOBS and (TDIEL provides strong support for eqn (1). This agreement greatly exceeds that originally found2 for a variety of materials, for which sample-by-sample comparisons were mostly not available ; thus a single perylene+hloranil sample3 yields oOBS = 3.2 x and oDIEL = 2.7 x Temperature dependences of both vet and oOBS in fig. 2 give activation energies presented in table 2. To these results are added the relevant data from electron-transfer studies of the reaction MnO, + MnOi- -+ MnOi-+ MnO, R-l cm-l, showing better agreement than do the averages.23598 ELECTRON TRANSFER I N SOLIDS TABLE 1 .-D.C.CONDUCTIVITY, DIRECTLY OBSERVED, FITTED TO COLE-COLE ARCS AND CALCULATED FROM DIELECTRIC RELAXATION FREQUENCY lo7 oll2-l cm-l ~ ~~~ sample TIK OBS FIT DIEL 1 2 3 4 5 6 337 325 318 296 282 330 292 298 298 307 298 296 27.5 10 5.3 1.75 0.78 1.75 1.60 2.15 5.4 2.3 3.5 19.0 29 13.2 6.1 2.1 0.83 22.4 2.0 1.8 2.45 6.1 2.5 3.85 21.4 12.9 6.5 2.07 (1.8 * 1)" 20.5 1.77 1.62 2.26 5.77 2.33 3.81 a Poorly defined arc. 2.5 3.0 3.5 4.0 4.5 5.0 1 0 3 IT FIG. 2.-Log ooBS and log v plotted against 1/T for K,(MnO,),. Sample A: a, T decreasing; 4, from start; 0, T increasing; giving o/SJ-l cm-I = 546 exp [ -(5592 K)/A. For sample B (+ and 0, respect- ively); o/SJ-' cm-' = 84.7 exp [-(5908 K ) / q and v/Hz = 4.8 x lo*, exp [-(6080 K)/q.D. R.ROSSEINSKY AND J . S. TONGE 3 599 TABLE 2.-vALUES OF 298 K ELECTRON-TRANSFER FREQUENCY (V/HZ) FROM AQUEOUS-SOLUTION RATE CONSTANT, FROM d.c. CONDUCTIVITY AND FROM DIELECTRIC RELAXOMETRY, AND THE CORRESPONDING ACTIVATION ENERGIES (E/kJ mol-I) FOR KMn0,-K,MnO, a Same sample. in aqueous s~lution.~ We quote here the intramolecular transfer frequency for 298 K in water (based on a juxtaposition constant2 of 0.1 dm3 mol-l), which is within a factor of five of those observed for d.c. conductivity [from eqn (l)] and for dielectric relaxometry, so confirming the original comparison.2 Furthermore, the activation energies are almost identical within experimental error ; that for the solution reaction has had an estimated value of -2.2 kJ mo1-1 for the simple cou.lomb interaction subtracted from it.This appears to be the first establishment of an identity of electron-transfer rate in both solid and liquid phases. The observation furthermore emphasizes the discrete nature of the transfer acts, which can be viewed in the solid as occurring within a narrowly defined, just half-filled density of states in the band gap, which overlap so weakly as to be virtually localised. If a common mechanism, discrete site-to-site transfer, underlies all three phenomena, then a similar analysis of the activation energy applies. This is quite well understood for the aqueous p r o ~ e s s , ~ - ~ ~ the activation energy containing a coulombic approach term not applicable in the solid and two terms involving 'reorganisation'.The inner reorganisation energy Ein is due to prior bond-length adjustments in the reactant coordination sphere, here in Mn-0 bonds, to values intermediate between the equilibrium values (a consequence of the Franck-Condon principle). The activation energy Ein attributable to a bond-length adjustment, Ar, is1, where k , and k , are the breathing-mode force constants. For the present system, E., has been estimatedlO as 12 kJ mol-l for Ar = 0.05 A and 6.8 kJ mo1-1 for Ar = 0.02 A, which an EXAFS study shows to be accessible at room temperature.12 The higher value is adopted here, partly on the grounds that in the solid (rather than in solution, for which the estimates were made) greater adjustment will be called for in achieving the electronic overlap required, in view of the immobility of the MnO, ions on lattice sites relative to the solution state.The 'outer' activation energy term Eout arises in essence from the polaron trap, the Franck-Condon principle disallowing electrostatic equilibration of ambient dielectric with the transferring charge, apart from the electronic polarisability part represented by nf, the square of the refractive index. The semiclassical theory of Marcus8* 11* l3 and Hushs gives for Eout14 (3) where rl and r2 are the radii of MnOy and MnOi-, r12 is the distance between the anion centres, and E , is the static permittivity ( E , and e are in e.s.u.). The use of the observed E , as the bulk permittivity about spherical anions in a lattice has proved quite realistic for alkali halides,15 such a model yielding calculated ion sizes in good accord3600 ELECTRON TRANSFER IN SOLIDS with experiment.With Y, x r2 x 2.8 A, r12 x 8.4 A and E, = 175 from our Cole-Cole plots, only n: is required. For KMnO, n, = 1.59, and from comparisons of n, for MHSO, and M2S0, (M = Na, K, Rb) n, for K2Mn0, should be ca. 1.62 and for K,(MnO,), 1.6. These values give Eout = 31.7 kJ mol-l and with Ein = 12 kJ mol-1 a total of 44 kJ mol-l, which is in accord with observed values (table 2). The pre-exponential of the transfer frequency (fig. 2) is found to be v, = 5 x 10l2 Hz, a magnitude often deemed typical of phonon frequencies16117 or an electron delo- calisation frequency in the pretransfer stage. A naive classical interpretation would predicate the slower of these alternatives, but v, clearly has a more complex origin involving processes akin to spin-lattice n.m.r.relaxation,ls requiring a more detailed mode119 of electron-phonon interaction than we have available. For the present pis a h we summarily insert the value kT/h = 6 x 10l2 Hz, taken from transition-state statistical mechanics. The conductivity can thus be expressed as kT h 0 = (ne2a2/6 kT) - exp [ - (Ein + Eou,)/RT] (4) where Ei, and Eout are given in eqn (2) and (3). A firm theoretical basis for v, is thus all that is required for a comprehensive understanding of the site-transfer d.c. conductivity. It is worthwhile to note that complex impedance measurement, fig. 3, on the independent Solartron frequency-response analyser, and involving a different but consistent numerical analysis, yielded v = 6.3 kHz and oDIEL = 2.2 x lo-' R-l cm-l, in total agreement with the point-by-point bridge and phase-sensitive detector permittivity measurements.It was of further interest that a second relaxation with a characteristic frequency of 6 Hz was also observed, conceivably due to ionic motion (K+). Clearly the corresponding conductivity contribution would be very small, but FIG. 3.-Complex impedance plot for K,(MnO,), at 296 K, from Solartron frequency-response analyser, indicating two relaxation processes. Applied (oscillation) frequencies shown. with an activationenergy sufficiently different from theelectronic, the ionic conductivity could predominate in favourable temperature ranges, verifiable by temperature- dependence studies on the low-frequency relaxation.K2Mn0, AND KMnO, Attempts at studying K,MnO, were particularly bedevilled by oxidation of contacts. Many were tried, the better o results being summarised in table 3 for a variety of contacts, most being vacuum-deposited. Gold contacts usually failed more rapidly than Pt, although that quoted survived untarnished for several hours. The higher values never persisted, but ca. 2 x L2-l cm-l may be accepted as the value for the material, probably free of electrode-deterioration effects. However, since grinding ofD . R. ROSSEINSKY A N D J. S. TONGE 360 1 -7.5 -- - 8 . 0 - s d I C -? - 8 . 5 - ---. v 0 -9.0 TABLE 3.-cONDUCTIVITY COBS OF K,MnO,: EFFECT OF CONTACTS AND OF TIME - - vac. dep." Pt +pressured leads < 1 2.0x 10-8 vac.dep. Pt + Ag-paint leads vac. dep. Au+vac. dep. Cu < 1 5 x 10-8 vac. dep. Pt + Ag-paint leads < 1 3.0 x lo-* 15 (evacuated) 1.2 x 10-10 Pt disc, pressure < I 4.8 x all the above > 12 8 x a Vacuum-deposited. s 2.9 3.1 3.3 3.5 lo3 KIT FIG. 4.-Temperature dependence of c7 for KMnO,. K,MnO, results in some disproportionation, traces of consequent K,(MnO,), in the compaction could be responsible for some enhancement of 0. It was fruitless to attempt dielectric relaxometry, because of the pronounced instability (although rapid scanning, as with the Solartron apparatus, within minutes of electrode application might work). KMnO, was much more amenable to study, oOBS values being steady for several weeks with Pt or Au/Cu contacts for any one sample at between 7 x and 0-1 cm-l, depending on the sample.Regardless of whether typical or highest values are considered, o varies as K,(MnO,), > KMnO, > K,MnO, (the latter inequality requiring revision of an earlier footnoted conclusion based on limited dataso). The activation energy for a KMnO, sample having oOBS (298 K) = 1.2 x Dielectric-relaxation studies on KMnO, are obscured by a critical dependence on the value of o employed, and it is not unequivocal that Cole-Cole arcs apply; at 298 K 0-l cm-l was found to be 56 f: 3 kJ mol-1 (fig. 4).3602 ELECTRON TRANSFER I N SOLIDS 0 50 100 150 200 250 f ’ FIG. 5.-Cole-Cole plots for KMnO, at 337 K, for CT = (3.4k0.4) x lop8 Q-l cm-l . (%BS = 3.4 x lop8 l2-l cm-l gives the middle curve; the other two curves for f 10% values emphasize the doubt in inferring conformity with an arc.) only continued increase of E” with E’ is observed, although arc dependence can be induced at higher temperatures by slight (ca.10%) adjustments of a; only at 337 K is a definite arc found (fig. 5). There is no simple evidence of a site-transfer mechanism, although it is not precluded. CONCLUSIONS The agreement between d.c. conductivity of K,(MnO,), and the calculated value from dielectric relaxometry has now been greatly enhanced. These both agree with the MnOL-MnOi- electron-transfer rate in solution, and all three activation energies (d.c. conductivity, relaxometry and aqueous reaction) also agree closely. The details of the aqueous reaction being already largely understood, a quite complete account of the conductivity process is now available.As expected, the conductivities of K,MnO, and KMnO, are lower, and of unresolved mechanism. 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