THERMOELECTRIC POWER OF IONIC CRYSTALS BY R. E. HOWARD AND A. B. LIDIARD Clarendon Laboratory, Oxford ; A.E.R.E., Harwell, Berkshire Received 14th January, 1957 The thermoelectric power of an ionic conductor MX fitted with electrodes of metal M is discussed from the point of view of the lattice defects existing in the salt. This power is composed of two parts : (1) a “ homogeneous ” part (coming from the thermal diffusion potential caused by the temperature gradient in the salt) which is related to the “heats of transport ” of the defects, and (2) an “ inhomogeneous ” part (coming from the dependence of M/MX contact potential on temperature) which is related to the en- tropies of formation of the defects. Detailed results are given for the case where MX is a cationic conductor showing Frenkel disorder, and the influence of impurities of type NX;! in MX is discussed.When the lattice disorder is dominated by the presence of impurities the total power contains an important concentration dependent term -(k/e) In [c(l - p ) ] , where c is the molar fraction of impurity and p is the degree of association between impurity N2+ ions and M+ vacancies (k, e are respectively, Boltzmann’s constant and the electronic charge). The power of an impure crystal is found to be un- altered by the establishment of a Soret concentration gradient. Agreement between present theory and the existing, rather scant, data on AgBr is adequate but further experi- ments on AgBr and AgCl are desirable. 1. INTRODUCTION The properties of lattice defects in solid ionic conductors have been extensively studied through electrical conductivity and diffusion measurements.In the present paper it is shown how the thermoelectric power of a crystal MX fitted with metallic electrodes M is also related to the properties of these defects. The experimental arrangement is imagined to be as in fig. 1. We define the thermoelectric potential as the difference in potential V B - VA ; this is opposite to the convention generally employed in connection with metallic thermocouples 1 but of the same sign as that used by de Groot 2 , 3 and Holtan 4 in their thermodynamic treatments of salt and electrolyte thermocells. The potential difference VB - VA is made up of (i) the “ homogeneous ” potential difference Vp - VA due to the temperature difference between P and A; this term is small by comparison with the other terms and will be dropped ; (ii) the contact potential difference VQ - V p z -@(T) ; (iii) the homogeneous potential difference VR - VQ due to the temperature gradient in the salt ; (iv) the contact potential difference Vs - VR = @(T+AT) ; (v) the homogeneous term in the metal electrode and wire, which we shall neglect.The total is VB - VA = (VR - VQ) f @ ( T f AT) - @(T). The quantity ( V B - VA)/AT is the thermoelectric power 8 and is composed of a “ homogeneous ” part Oh, = ( VR - VQ)/AT and a “ heterogeneous ” part Ohet = A@/AT. These sign conventions will be observed throughout.* Owing to the different types of lattice disorder that may exist a general treatment is not possible. We shall therefore deal only with the case where the salt displays predominantly cationic Frenkel disorder, but it will be assumed that this may be modified by the presence of impurity cations of valency differing from M * A recent paper by Patrick and Lawson,s to which we shall refer below, takes - ( VB - V&/AT as the thermoelectric power.113 Our 0 is thus minus their c114 THERMOELECTRIC POWER (aliovalent). This is the type of disorder existing in AgBr 6 and we shall thus be able to apply our equations to the existing data for this compound.5 The prin- cipal results of this theoretical discussion are (i) eqn. (2.6) for Ohorn for a pure crystal (previously given by Patrick and Lawson) in terms of the “ heats of transport ” (qi*, qu*) of the defects, (ii) eqn.(3.10) for the inhomogeneous power of a pure crystal showing the correction to the Wagner7 estimate (square bracket term), (iii) eqn. (3.12) for 8het for an impure crystal, of which the Wagner term may be only a small part, lastly, (iv) eqn. (4.1) for the total power of an impure crystal, applicable whether a Soret con- T T+AT centration gradient is established or not. As would be expected our equations are compatible with the thermodynamic equations developed by HoltanP The agreement with the existing experimental data on the thermoelectric power of AgBr 5 is satisfactory but these data are not very extensive and further experiments, particularly on AgBr with varying CdBr2 contents, would be valuable. Another system for which there already exists a sufficient knowledge of the lattice defects to make thermoelectric studies profitable is AgCl + CdC12.8 1 1 M A 6 FIG.l.-Schematic diagram of the ex- 2. HOMOGENEOUS THERMOELECTRIC POWER perimental arrangement for the measure- ment of the thermoelectric power of an OF AN IONIC CONDUCTOR is necessary to include some results which are not new and this fact will be in- dicated by giving the appropriate reference. We have to deal with a crystal containing two types of intrinsic lattice defects, namely, interstitial ions and lattice vacancies; in addition aliovalent foreign ions may also be present. First, how- ever, we consider the range of intrinsic behaviour in which foreign ions may be neglected. (i) INTRINSIC RANGE In keeping with the approach which treats the system of lattice defects as an “ ideal solution ” we write phenomenological equations for the current densities of interstitials and vacancies as follows : 2 ~ 3 and in which nj and n, are the numbers of interstitials and vacancies per unit volume and qi* and qv* are their “ heats of transport ”.* Di and Du are the diffusion coefficients for interstitials and vacancies which, it is assumed, carry effective charges + e and - e respectively. E is the electric field strength.*They are heats of transport appropriate to the thermodynamic fluxes and forces employed by Prigogine.9 In 9 52 of ref. (3) they are denoted by a double asterisk.l i . E . HOWARD A N D A . B . LlDIAKU 115 Now in the steady state in the absence of an external field the system will be everywhere electrically neutral and there will be no electric current flowing ni == n,, (2.3) j i = j w .(2.4) In addition, since the departures from " thermostatic " equilibrium are always small we can employ the relation (2.5) where N and N' are respectively the numbers of normal and interstitial (cation) sites per unit volume and gF is the Gibbs free energy of formation of a Frenkel defect pair. By use of (2.1)-(2.5) we get an equation for the thermal diffusion field E,5 ninw = NN' exp (- gF/kT), where 4 = Di/Dv is the ratio of mobilities, and is the enthalpy of formation of a Frenkel defect pair. For small AT (fig. 1) the eXPreSSiOn (2.6) gives the homogeneous thermoelectric power, ohom. The quantity 7 (2.7) 4(4i + 3hF) - ( 9 w + 4hF) + + I QM* = is the overall heat of transport for the cations appearing in the thermodynamic treatments of de Groot 2 and Holtan.4 (ii) INTRINSIC RANGE, EXPERIMENTAL Patrick and Lawson5 measured the thermoelectric power of pure AgBr and of AgBr + 0.25 mole % CdBr2 fitted with Ag electrodes.The results are shown in fig. 2. The impure system will be discussed later. In order to analyse these data on the basis of eqn. (2.6) it is necessary to subtract the inhomogeneous power resulting from the dependence of Ag/AgBr contact potential on temperature. Following an earlier estimate by Reinhold and Blachny,7, 10 Patrick and Lawson took 6het =- 140pVldeg. They then found that the remaining power could be satisfactorily analysed on the basis of (smoothed) values for as found pre- viously by Teltow 6 provided that they took qw* =- 0-385 eV, qi* = 0.017 eV, the heat of formation hF for Frenkel defects in AgBr having been taken as 1.27 eV -also from Teltow's work.The negative value for the vacancy is understand- able since the flow of ions (which carry the heat) is opposite to the flow of vacancies. (iii) KINETIC INTERPRETATION OF HEATS OF TRANSPORT The magnitudes and signs of qw* and qi* are qualitatively understandable on the basis of an intuitive generalization by Wirtz 11 of the usual rate process formula116 THERMOELECTRIC POWER for atomic jump frequency. The isothermal expression for the jump frequency of, say, an interstitial ion is 12,139 14 (2.9) where v is the frequency of vibration of the ion in the (thermal) average potential energy field of all the other ions of the lattice and Ag is the Gibbs free energ of activation.Wirtz supposed that Ag may be decomposed into three parts, Agl, Ag2 and Ag3, being energies of activation respectively supplied (i) at the w = Y exp (- Agjkr), Temperature O C FIG. 2.-The thermoelectric power oh,, + ohet as a function of temperature for AgBr fitted with Ag electrodes. The upper curve is for pure AgBr, the lower curve for AgBr + 0.25 mole % CdBrz. Note that the power is negative with the convention of 5 1 (after Patrick and Lawson). initial position of the interstitial-to initiate the jump ; (ii) at the plane of normal atoms between the initial and final positions of the interstitial ion-to move them apart and facilitate the passage of the interstitial through them; (iii) at the final position of the interstitial-to push apart the surrounding ions so that they can accommodate the interstitial.The point of this decomposition is that the com- ponent energies are supplied at slightly different temperatures in an anisothermal system. In place of (2.9) we would thus have for a jump in the same direction as the temperature gradient. equation for the net flow of interstitials, ji, can be derived kinetically. found to have the same form as (2.1) with On this basis an This is qi" = Ahil - Ahi3, (2.11) where Ah is the enthalpy part of the corresponding free energy As. The expression for vacancies is (2.12) qv* =- (Ah,, - &2>. The results (2.11) and (2.12) require the heats of transport to be numerically less than the corresponding heats of activation. The heat of activation for Ag+ interstitials in AgBr from Teltow's results 6 is 0.18 eV ; the value of qi* (2.8) isR .E . HOWARD A N D A . B . LIDIARD 117 only about a tenth of this. Such a result might be expected for an interstitial since a large part of Ahi must be supplied at the midway position whether the ion moves by the direct mechanism or the indirect (" interstitialcy ") mechanism.15 For a vacancy jump Teltow's data give the heat of activation as 0-37 eV which is numerically nearly equal to q,* (2.8) so that here most of the heat of activation appears to be supplied to the ion which jumps. This again is reasonable. Further use has been made of the kinetic approach in connection with the Soret effect in impure systems.16 (iv) RANGE OF IMPURITY CONTROLLED CONDUCTION We imagine now that the ionic conductor is not in its intrinsic temperature range but contains lattice defects other than those introduced thermally.We idealize this situation by supposing that all the defects are introduced by the presence of aliovalent impurities. To be specific we may take AgBr containing a sufficient concentration of CdBr2 that the Ag+ interstitials may be neglected: the two principal defects are now the Ag+ vacancies (effective charge -e) and the substitutionally incorporated Cd2+ ions (effective charge e). However, since the Cd2 ions are on normal lattice sites, they can move only when they are ad- jacent to a vacancy, i.e. when they are " associated." 6 Hence when we consider the flow of impurity ions, attention must be focused on the movement of the (electrically neutral) impurity-vacancy pairs.17 Let nk be the number of impurity- vacancy pairs per unit volume then the current density of pairs (equal to the current density of impurity ions) is (2.13) in which D k is the diffusion coefficient of the pairs (assumed tightly bound) and qk* is their heat of transport.The relation between nw and nk is given by the mass action equation for the association reaction, namely, Nnk -- - 12 exp ((/kT), nW2 (2.14) where 5 is the Gibbs free energy of association. In the true steady state both j, and j k are zero : eqn. (2.2), (2.13) and (2.14) then yield where x = 5 - T(35/3T), is the enthalpy of association. We have written this power as @hom(a) to emphasize that it is established only after a long time since the condition j k = 0 requires the establishment of an impurity concentration gradient (Soret effect).In practice, measurements of the thermoelectric power will be made in a comparatively short time, and there will be no impurity con- centration gradient although there will be a gradient of complexes due to the increased dissociation at the hot end. To find ohom we therefore set j ,= 0 and relate n, and nk through (2.14) ; the result is (2.16) where p 1. nk/(nk -I- n,) is the degree of association. The expression (2.16) may be evaluated for Cd2+ ions in AgBr by using Teltow's results for the association rcaction (x = 0.16 eV and association constants as given in table 2 of ref. (6)) and assuming q,* = - 0-385 eV as found by Patrick and Lawson from their analysis of pure AgBr.At 200" C a Cd2+ concentration of 0.25 mole % gives p = 0.36 and ohom = - 720 pV/deg. The observed total power is - 290 pV/deg : hence118 THERMOELECTRIC POWER the inhomogeneous power in this case must be 430 pV/deg. This is very different from the Wagner estimate - 140 pV/deg. found to be consistent for the pure AgBr, but will be shown to be understandable when we analyse oh& in 0 3 : the inhomogeneous power is strongly dependent on the purity of the crystal. Another important feature of our results for ohet, which we may anticipate here, is the occurrence of a term which just cancels with Xp/eT(l + p ) in (2.16) when no Soret effect is allowed for and with (qk* - x)/2eT in (2.15) when time is allowed for the Soret effect to develop.The total thermoelectric power is the same whether there is sufficient time for a true steady state to be reached or not ; consequently the Soret effect in these systems cannot be studied via the thermoelectric power. The magnitude of the concentration gradient established may be studied directly and is related to qk*. We have accordingly analysed qk* on the basis of the Wirtz kinetic approach by an extension of the method used for studying Dk.17 Since these calculations are not directly relevant to the thermoelectric power we shall not discuss them further in this paper. 3. INHOMOGENEOUS THERMOELECTRIC POWER OF AN IONIC CONDUCTOR We have seen in 8 2 that a consistent analysis of the existing data on pure AgBr can be obtained using the Wagner estimate of the inhomogeneous power: we also saw that the theory of the power of an impure crystal demanded a different value for the inhomogeneous power.In this section we shall examine the in- homogeneous power from the point of view of the theory of lattice defects and show how the Wagner estimate must be corrected. The contact potential between the ionic conductor MX and the metal M may be evaluated formally by setting the electrochemical potential ji of ion M+ in MX equal to the electrochemical potential of ion M+ in M : or, the electrical potential in M relative to that in MX, @(T), is e-1 times p(M+ in MX) - p(M+ in M), where the ps are ordinary chemical potentials. It follows that ehet is 1 c ) e 3T = - -{p(M+ in MX) - p(M+ in M)} = - (s(M+ in MX) - s(M+ in M)}/e, (3.1) where the ss are the corresponding partial entropies.Wagner’s 7 analysis assumes that s(M+ in M) is:almost identical with the total entropy of the metal-the electrons making only a very small contribution on account of their Fermi-Dirac degeneracy. That is s(M+ in M) = C,(M) dT/T. (3.2) F s: Wagner also suggests that s(M+ in MX) is one-half the total entropy of MX, thus s(M+ in MX) = 3 C,(MX) dT/T. (3.3) On the basis of (3.2) and (3.3) one obtains the estimate - 140pV/deg. for AgBrJAg. 10 In order to look critically at the assumption (3.3) we return to the definition of p(M+ in MX) and enquire what is the free energy change on adding one M+ ion to a MX crystal. Since the salt is crystalline the added ion can only be accommodated on surface sites (or dislocation jog sites) or by going into an inter- stitial position or by filling an already existing vacancy.Since the interactions in an ionic solid are primarily electrostatic and nearest neighbour repulsion forces one would expect the addition of one M+ ion (but no X- ion) to the surface toR . E . HOWARD AND A . B . LIDIARD 119 lower the energy of the crystal by about one-half the lattice energy per ion pair. Hence p(M+ in MX)=+g(MX), where g is the free energy per ion pair of the salt. If differentiation of this approximate relation is allowed then (3.3) follows. Thus, if all the ions transferred to or from the metal were removed or accom- modated at surface sites of the salt then one might expect (3.3) to be a good ap- proximation. Under such conditions the surface electrical double layer would have two components, one formed by an excess (or deficiency) of electrons in the metal, the other formed by an excess (or deficiency) of M+ ions on the salt surface. The double [ayer will not, however, be of this type but will be diffuse in a way already described by several authors in other connections.19~ 20921.The reason is that the salt can lower its free energy if the added ions are distributed more randomly in the crystal, since in this way the (configurational) entropy is increased. The contact potential (from which we derive OheJ is then the potential difference between the metal and the interior of the salt beyond the space charge region near the boundary. We thus require the chemical potential of an interstitial ion in the interior of the salt.Let the work required to bring a cation from a state of rest at infinity into a particular (but arbitrary) interstitial position in the crystal (at constant T, P) be gi. Similarly, let g, be the work required to remove a normal cation from the crystal to a state of rest at infinity. The chemical potential of an interstitial ion in a region of the crystal containing ni interstitials per unit volume is then (3.4) This expression is easily obtained as the change of Gibbs free energy on addition of one interstitial. Likewise the change of free energy on creation of one addi- tional-vacancy is (3 * 5 ) In passing, we note that eqn. (2.5) with gF = gi + gv, is obtained from the equation pi = gi + kT In (nJiV’). pv = gv + kT In (n,iN).pi + /-b = 0, interstitial + vacancy -+ 0. corresponding to the quasi-chemical reaction : We may now substitute (3.4) into (3.1) to obtain &t, thus (3.6) 1 3Pi 1 ohet = - - + - s(M+ in M). e 3 T e The explicit form of pi depends upon the impurity content of the crystal. We deal separately with the region of intrinsic conduction where only thermally produced defects need be considered and with the region of impurity controlled conduction. (i) INTRINSIC RANGE Here the numbers of interstitials and vacancies are equal and given by H i = nv = (NN’)$ exp [- (gi + gW)/2kT]. pi = Hgi - gv) + (kT/2) In (NIN’), Hence where Si =- 3gi/3T, s, =- 3gv/3T. In order to bring out the relation of (3.9) to Wagner’s expression we write si = si’ + s(M+/MX), SV = sV’ - s(M+/MX),120 THERMOELECTRIC POWER where s(M+/MX) is the entropy of a M+ ion added to the surface of MX from the standard state at infinity and which may be assumed to be given by (3.3).The quantity si’ is the increase of entropy on taking a surface M+ ion and putting it in an interstitial position; a similar definition is made for s,’. Both sj’ and s,’ originate from the influence of the defects on the vibrations of the crystal. We have therefore 1 k 2e = (- si’ + s ~ ’ ) + - In (NIN’) (3.10) + [i s(M+ in M) - - 1 s(M+/MX)] . e The first two terms constitute the correction to the Wagner estimate. However, for pure AgBr the correction is probably small, because of the consistency of the analysis based on the Wagner estimate. The term (k/2e) In (NIN’) is only - 30 pV/deg. so that si’ and s,’ are thus nearly equal.(ii) IMPURITY CONTROLLED RANGE Here we suppose that the number of thermally produced defects is small and that we have a large number of vacancies introduced by the presence of divalent cations. If the number of substitutionally incorporated divalent cations is Ni per unit volume then n, = Ni(1 - p ) and by (2.5) and (3.4) (3.1 1) Therefore 1 + - (s(M+ in M) - s(M+/MX)). (3.12) In arriving at (3.12) we have assumed that no impurity concentration gradient has been established. For AgBr containing 0.25 mole % CdBr2 we have (cf. $ 2 (iv)) p = 0.36 at 200” C, whence is found to be 465 pV/deg., whereas the Wagner term is - 140 pV/deg. This is consistent with the value 6het = 430 pV/deg. inferred in 2 (iv) if s,’ is about lO-4eV/deg. or about k : such a value is to be expected from the influence of a vacancy on the lattice vibrations.At the present time it is difficult to be more conclusive in our comparison with experiment since the available data are so scanty. We believe that further study of these systems would be valuable, especially studies of thermoelectric power as a function of impurity content (eqn. (2.16) and (3.12)). 4. SUMMARY AND CONCLUSION The explicit expressions which we have obtained for Ohhom and Ohet may be added together to give the total thermoelectric power 8. For an impurity con- trolled system there is some cancellation of terms and we have 1 e + - (s(M+ in M) - s(M+/MX}. (4.1)R. E. HOWARD AND A . B. LIDIARD 121 The concentration dependent term is large and should be detectable experimentally.Although we have not calculated e h e t ( 0 ) it is not difficult to do so and hence to show that ohom( co) + &,et( co) is also given by (4. l), i.e. the Soret effect does not lead to a change in thermoelectric power. It is possible to rewrite our expressions for 0 in such a way as to make contact with the thermodynamic analysis of Holtan.4 We write 0 as the sum of two parts, one of which (Holtan’s “ thermostatic ” part) is (l/e)(s(M-+ in M) - s(MX)) or (l/e)(s(M) - s(MX)) to the approximation that the homogeneous terms in the electrodes can be neglected; s(M) and s(MX) are the entropies (per molecule) of the metal and the salt respectively. The remaining parts are - Q M*/eT (eqn. (2.6) and (2.16)) with (i) in the intrinsic range, (- si’ + sV’)/2e +(k/2e) In (N,”’)+ s(X-/MX)/e, where s(X-/MX) is the entropy of an X- ion added to the surface (ii) in the impurity range (s,‘/e) f- s(X-/MX) + (k/e> In [N/Ni(1 - p ) ] -, Xp/eT(l + p ) .By studying the small amount of disorder which exists in the anion sub-lattice it is easy to show that each of these expressions is just e-1 times the partial entropy of an anion sx. The thermodynamic treatment of this system shows that QM* =- ex* (Holtan 4 3 9) ; hence the non-thermostatic part of our result is (Qx* + Tsx)/eT rsx*/e, which is Holtan’s thermodynamic result for this case (his eqn. (4.22)) ; sx* is called the “ entropy of transfer ”. From a com- parison of his thermodynamic equations with experimental data on a number o salts Holtan concluded that the thermostatic part of the power was often nearly the whole power.In conclusion we suggest the desirability of further measurements of the thermo- electric power of ionic conductors particularly AgBr and AgCl containing Cd2+ ions, as the lattice disorder in these salts is already well studied. The examination of thermoelectric power as a function of impurity concentration should disclose a concentration dependence brought about by the In [c(l - p ) ] term. Our equations can offer no interpretation of this result. 1 Domenicali, Rev. Mod. Physics, 1954, 26, 237. 2 de Groot, L’Efet Soret (North Holland Publishing Co., Amsterdam, 1945). 3 de Groot, Thermodynamics of Irreversible Processes (North Holland Publishing CO., Amsterdam, 1951). 4 Holtan, Electric Potentials in Thermocouples and Thermocells. Thesis (University of Utrecht, 1953) ; alternatively see Proc. K. Akad. Wetensch, B, 1953, 56, 498 and 510: also 1954, 57, 138. 5 Patrick and Lawson, J. Chem. Physics, 1954, 22, 1492. 6 Teltow, Ann. Physik, 1949, 5, 63 and 71. 7 Wagner, Ann. Physik, 1929, 3, 629. * Ebert and Teltow, Ann. Physik., 1955, 15, 268. 9 Prigogine, Etude Thermodynamique des PhCnomtnes Zrrkversibles. Thesis (University of Brussels, 1947). 10 Reinhold and Blachny, 2. Elektrochem., 1933, 39, 290. 11 Wirtz, Physik. Z., 1943, 44, 221. 12 Wert, Physic. Rev., 1950, 79, 601. 13 Seeger, Handbuch der Physik (Springer-Verlag, Berlin, 1955), vol. 7, part 1, p. 383. 14 Lidiard, Handbuch der Physik (Springer-Verlag, Berlin, 1957), vol. 20, 5 15 et seq. 15 Friauf, Physic. Rev., 1957, 105, 843. 16 Howard, to be published. 17 Lidiard, Phil. Mag., 1955, 46, 815 and 1218. 18 Schone, Stasiw and Teltow, Z. physik. Chem., 1951, 197, 145. 19 Verwey and Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, 20 Grimley and Mott, Faraday SOC. Discussions, 1947, 1, 3. 21 Lehovec, J. Chem. Physics, 1953, 21, 1123. Amsterdam, 1948).