Faraday Discuss. Chem. SOC., 1989, 88, 43-54 Effect of Ion Association on Transport in Polymer Electrolytes Peter G . Bruce” Department of Chemistry, Heriot- Watt University, Riccarton, Edinburgh EH14 4AS Colin A. Vincent Department of Chemistry, University of St Andrews, St Andrews, Fife KY16 9ST Numerous techniques have been employed in an attempt to elucidate details of the conductivity mechanism in polymer electrolytes, and in particular to determine whether cations, anions or both are mobile. Until recently, most of these studies have assumed a ‘strong electrolyte’ model. Here we examine the effect of ion association on the interpretation of different techniques, including Hittorf, concentration cell, centrifugal cell, ratio-tracer, pulsed field gradient n.m.r. and d.c.polarisation methods. The circumstances under which certain of these experiments yield individual ionic transport numbers are described. it is also shown that, in general, measured diffusion coefficients may not be sensibly compared with mobilities from conductivity measurements using the Nernst-Einstein relationship. The d.c. polarisation technique is examined in some detail and the value of the steady-state conductivity in assessing polymer electrolytes for battery applications is described. Finally, some experimental evidence is presented which appears to support the existence of ion association in polymer electrolytes. High-molecular-weight polar polymers such as poly( ethylene oxide), ( -CH2CH20-)), , can offer an attractive alternative to conventional low-molecular-weight organic solvents as media for non-aqueous electrochemistry.Polymer electrolytes generally have the form of flexible solids, but their behaviour is more closely related to liquid systems than to classical solid electrolytes such as sodium p-alumina. It is instructive to compare solid polymer electrolytes and non-aqueous liquid electrolytes. Similarities: (i) Both have conductances that are in general lower than those of aqueous solutions of comparable salt concentration; (ii) owing to the low relative permittivity of the host (solvent), ion association leading to the formation of long-lived ion pairs, triples efc. is considered to occur; (iii) both exhibit a wide electrochemical stability window, which in practice is further enhanced in the case of polymer electrolytes because of the slow kinetics of decomposition in the solid state. Differences: (i) Entanglement of long polymer chains, or chemical crosslinking in the case of networks, precludes long-range motion of the ion-coordinating atoms, in contrast to the situation in low-molecular-weight liquids where a solvation sheath can move with the ions.One consequence of this is that ions which can undergo long-range motion in low-molecular-weight systems may be immobile in polymer electrolytes. A second and more subtle consequence concerns the significance of the transport numbers as measured by, say, the Hittorf method. In liquid solutions in order to define a transport number it is necessary to establish an imaginary reference plane fixed with respect to the solvent, since transport of molecules of the latter in the ionic solvation sheaths makes it impossible to determine ‘absolute’ or ‘true’ transport numbers.(Attempts to evaluate the latter by relating ionic migration to a non-conducting reference substance, or by using membranes have not been successful’). In polymer electrolytes since there is no net transport of solvent by the ions, absolute transport numbers may be determined. 4344 Transport in Polymer Electrolytes (ii) In polymer electrolytes there is no significant convective flow, so that stable concentra- tion gradients can be established. This is of considerable importance since in practical devices (batteries, displays, etc.), where polymer electrolytes are generally used in the form of thin films, a concentration gradient may extend throughout the thickness of the film.While considerable progress has been made in the past decade both in the synthesis of new polymer electrolytes and in the understanding of their properties, many detailed questions concerning mass and charge transport remain unanswered. In particular, there is still little direct evidence for ion association in polymer electrolytes, nor has there been any rigorous treatment of the effect of such a phenomenon on the interpretation of the various transport experiments. The low relative permittivity of ether-based polymers suggests the likelihood of strong ion-ion interactions, and there is indeed some support for the idea of intimate cation-anion contact from Raman spectroscopy.”’ Conductivity measurements on short-chain liquid polyethers with low concentrations of salt have been interpreted on the basis of ion a~sociation.~ It is very likely that similar simple ion pairs, ion triples or other small ion clusters exist in solid polymer electrolytes, but it is not clear whether they play a significant role in determining transport properties over the range of salt concentrations which may be encountered in these systems.A variety of techniques has been employed to probe transport processes in polymer electrolytes, ranging from experimentally simple measurements such as d.c. conductivity to more sophisticated methods such as radio-tracer diffusion studies, Hittorf transference number determinations, e t ~ . ~ Results derived from these techniques often appear to be in conflict; this we consider may be attributed, in part, to the simple model of a fully dissociated polymer electrolyte which is often used to interpret the measurements. The presence of ion association considerably complicates the interpretation of transport processes.For simplicity we here restrict our attention to a solid polymer electrolyte based on a monovalent salt MX, which contains only the species M+, X-, MX, M2X+ and MX2-. In some cases we will further restrict our attention to electrolytes containing only M+, X- and MX. Extension of the treatment to more complex salts such as MX2 or to higher associated species makes the analysis more tedious, but does not introduce significant fundamental differences. We examine the effects of ion association on the methods used to study ion transport, showing that each technique measures the motion of a combination of species and that the various techniques are influenced in different ways by the motion of the species involved.The different methods may be divided into four classes. Class I: Small Signal A.C. and D.C. Conductance Measurements For low fields, transport of each charged species is directly proportional to the field strength, the mobility and the concentration of the species. The latter may be written in terms of the total salt concentration, c, as follows: [M+] = a , c ; [X-] = a,c; [M2X+] = a4c and [MX2-] = a5c. (The concentration of the neutral species MX, which does not contribute to the conduc- tivity is given by [MX] = a3c.) Thus the total conductivity (T is given by l O O O a = ( a l u l + a2u,+ a4u4+ a#g}c where ui is the mobility of species i in unit field strength.Class I1 As in the case of conductance measurements (class I ) , methods in this class probe only the transport of charged species.P. G. Bruce and C. A. Vincent 45 HittorfITubandt Experiments This method involves the passage of a measured quantity of charge through a cell and subsequent determination of changes in the composition of the electrolyte in the vicinity of the anode and cathode. For a cell of the type M( s) Ipolymer-MX( s)/polymer- MX( s) Ipolymer-MX( s)lM( s) in which all three electrolyte compartments are identical, on passage of 1 F of charge, 1 mol of M is stripped from the anode and is deposited on the cathode.Current is carried by the motion of M' and M2X+ towards the cathode and of X- and MXT towards the anode. Provided that the central compartment remains invariant throughout the experiment, neutral ion pairs are not involved in the flux between the electrode compartments and the central compartment. Measurement of the change in salt con- centration in the cathode Compartment gives directly the change in the concentration of X. In the case of a completely dissociated electrolyte, this would lead to a direct determination of the anion transport number. For an associated electrolyte, analysis of the cathode compartment leads only to the determination of the net transfer of the X-constituent of the salt due to the transport of MIX' into the compartment and the transport of X- and MX, out.For the system considered, the transference number for the X-constituent, Tx (given by the change in the number of moles of X in the cathode compartment) may be related to the individual transport number for X-containing species by Similarly Also, Quite generally the transference number of a constituent cation or anion is defined' as the net number of Faradays carried by that constituent in the direction of the cathode or anode, respectively, on the passage of 1 F of charge through the cell. To date there have been few reliable measurements of Hittorf transference numbers in solid polymer electrolytes because of experimental difficulties in applying the tech- nique. Cheradame and co-workers6 have, however, successfully applied it to highly cross-linked networks where cells could be formed using a series of non-adherent thin sections.When the anion was immobilised by attachment to the polymer host, the transport number of the cation was shown to be equal to unity. E.M.F. Measurements on Electrolyte Concentration Cells A cell of the form M(s)lpolymer-MX(s)l Ipolymer-MX(s)lM(s) for c1 > c2 has an e.m.f. due to the tendency of the cell to equalise the concentration of the salt on either side of the electrolyte junction. Passage of 1 F of charge through such a cell has the effect of forming a mole of M' at the anode, depositing it at the cathode and transferring TM moles of M-constituent across the junction towards the cathode, and Tx moles of X-constituent in the opposite direction. The net result is the transfer of Tx moles of salt across the junction.In the situation where c1 and c2 differ by an infinitesimal amount, the e.m.f. of the cell is given by c1 c2 dE,,,, = -( RT/ F ) Tx d In a46 Transport in Polymer Electrolytes where a is the combined activity of the ionic species. It is thus possible to determine Tx by measurement of the e.m.f. of suitable cells ( i e . without involving the actual passage of current) using the relationship Tx = -( F / R T ) dECel,/d In a It is, of course, necessary to know how the activity of the ions varies with concentration. In their application of this method to PEO/LiI systems, Armand and CO-workers' determined this by making e.m.f. measurements of the cell Li(s)lPEO- LiI(s)(PbI,(s)lPb(s). In a later paper,' the same authors circumvented the problem by making e.m.f.measurements on equivalent cells with and without transference, namely Li(s)lPEO-LiC104(8 : 1)1 IPEO-LiC1O4( M : 1 for M > 8)ILi(s) (1) and Li(s)IPEO-LiC104(8 : 1)1 poly ( DV2+, 2C104-), poly (DV+, C1O4-)/PE0-LiC1O4( M : l)ILi(s) ~c,04- = dE,/dE2 for which (DV = decaviologen). E.M.F. Measurements on Cells in Force Fields If a symmetrical cell of the form M(s)lMX-polymer host (s)lM( s) is placed radially on a spinning rotor, an e.m.f. is developed due to the fact that there is a difference in the kinetic energy of the reagents at the two electrodes. The kinetic energy of a mass W at a distance r from the centre of rotation is given by E = 22"' Wr2 where f is the frequency of rotation. Hence the energy difference involved in moving the mass from rl to r2 is A& = 27r2f2 W ( r i - r ; ) .Thus, by considering the transfer of mass occurring in the cell during the passage of 1 F of charge in a reversible manner, an expression for the cell e.m.f. may be obtained: W = ( M , - V , p ) - T ~ ( M M . - V , X p ) . Here MM and MMx are the relative molar masses of M and MX, respectively, vM and vMx their partial molar volumes, and p is the density of the polymer electrolyte. Note that this equation assumes a relatively fluid polymer electrolyte. The cell potential is given by E~~~~ = 2.rr2f2 W<rf - r : ) / F and Tx may be obtained by measuring Ecel, as a function of rotation speed. This technique was developed by MacInnes for aqueous solution electrolytes but has not been widely applied. It does, however, seem to be particularly suited to solid polymer electrolytes and is currently being studied by the present authors.Unlike the Hittorf method it does not depend on the analysis of relatively small concentration changes; nor is it necessary to establish reproducible electrolyte junctions as in the concentration cell method.P. G. Bruce and C. A. Vincent 47 Class 111 The methods in this group are concerned with the measurement of diffusion coefficients of species in the electrolyte. Unlike the situation for class I and class 11, these methods are affected by the flux of both charged and electrically neutral species containing the particular constituent under investigation. Radio-tracer Methods Experiments using serial-sectioning techniques have been applied by Chadwick et aZ?' I to study, for example, PPO-NaSCN polymer electrolytes by measuring the distribution of "Na and I4C following diffusion into a solid sample from a thin layer deposited on its surface.In an associated electrolyte, the total flux of a radioactive species *M is given by d[*M'] d[*MX] dx dx D*MX ~ J*, 1 -D*M+ -- d[ * M (M )X+] dx The concentration of each species may be expressed as a function of the total (radioactive) salt concentration, *c: [*M+] = a,*c; ["MX] = a,*c; [*M(M)X+] = a4*c and [*MX,-] = as*c d[*MXJ dx - D * M X , - D*M(M)Xi so that Similarly, where DLM and DLx are the diffusion coefficients obtained experimentally for con- stituents *M and *X, respectively, and D*MX = DM*x, D*M(M)X+ = DM2*X+, D*MX2- = DM*x(x)-.It is evident that if the sum of second and subsequent terms in the expressions for D L M and DLx are large in comparison with the initial terms, then the cation component flux will tend to equal the anion component flux. It is possible to consider that the drift velocity of ions under the influence of unit force is the same, no matter what the origin of the force. This enables the mobility of a species such as M', under unit electric field, u+ (as noted in class I above) to be related to its diffusion coefficient D, by the Nernst-Einstein relationship: D+ = RTu+/F. For an ideal electrolyte containing a fully dissociated salt, application of this relationship enables the individual transport numbers to be defined in terms of their diffusion coefficients: and If the electrolyte is associated, such that DA= DL then t+ and t - as calculated on this basis will both approach 0.5.Clearly it is not generally valid to apply the Nernst-Einstein equation to such composite diffusion coefficients. A number of workers have attempted to make a rough estimate of the degree of ion pairing by comparing the experimental conductivity with that calculated from the Nernst- Einstein relationship, on the assumptions that the correlation coefficient is unity t+ = D+/( D, + D-) t- = D-/( D, + D-).48 Transport in Polymer Electrolytes and that there are no associated species higher than ion pairs. Lack of information on the ion pair diffusion coefficient makes this very difficult. Note, too, that if relatively immobile ion pairs are present, the flux in both the diffusion and the conductivity measurements would be lowered, and thus no information is obtained on the concentra- tion of the free ions.We have so far ignored the fact that the radio-tracer technique leads to the determina- tion of the tracer diffusion coefficient for any given constituent. In the absence of ion association, this is equal to the self-diffusion coefficient (as used in the Nernst-Einstein relationship) only in the absence of correlation. At high salt concentrations where polymer electrolytes may resemble molten salts more closely than non-aqueous solutions, mechanisms of ion motion may be envisaged in which the correlation coefficient f in the equation is less than unity. For example, if a vacancy mechanism for ionic conduction is dominant or if a cooperative rearrangement of charges occurs in each step of the conduction process, values off which depart significantly from unity may be expected.Furthermore, in molten salts it is well known'* that deviations from Nernst-Einstein behaviour occur since favoured motion in the same direction by a neighbouring pair of oppositely charged ions contributes to diffusional flux, but not to conductivity. In such systems pair correlation need only last for periods of ca. 2 x s, i.e. there is no need for the formation of long-lived ion pairs, Pulsed Field Gradient N.M.R. Experiments This is similar to the radio-tracer diffusion method in that a small proportion of a salt constituent is labelled, in this case magnetically.Generally ca. 10'' atoms are labelled so that the experimental conditions are to all intents and purposes identical to those of the radio-tracer method. Measurements of component diffusion coefficients have been made in polymer electrolytes by a number of workers, notably Berthier and co-worker~'~ and Whitmore and c o - w ~ r k e r s ' ~ , ' ~ by following the motion of 7Li and 19F in polyether- LiCF3S03 systems. As in the case of radio-tracer derived measurements, interpretation of the results of diffusion coefficient measurements is difficult if the latter are indeed composite quantities because of ion association. A comment is appropriate here on the distinction between the self-diffusion coefficient of ions D+ and D- and the salt diffusion coefficient which in the case of a fully dissociated salt may be written as Dsa,, = 2D+D-/( D, + 0-).In macroscopic measurements of salt diffusion, maintenance of electrical neutrality requires that cations and anions move at the same speed with a flux determined by Dsalt. In contrast, in radio-tracer and n.m.r. pulsed field gradient methods, the diffusion potential is negligibly small, i.e. the motion of an ion is not tied to that of other ions. We therefore find it difficult to understand the inference of Bhattacharja et al.14 that the conductivity should be related to D,,,, rather than to ( D , + D-) in a fully dissociated system. Cottrell Equation-based Methods Recently an electrochemical method based on the Cottrell equation has been described.I6 Here polymer- NaC F3S03 systems were doped with millimolar levels of electroactive cations such as Ag+ or anions such as I - and the diffusion behaviour of these was studied by chronoamperometry. Again, the conditions of these experiments are, in principle, similar to those of the radio-tracer method described above, a compositeP.G. Bruce and C. A. Vincent 49 diffusion coefficient for the electroactive constituent being measured. Note that in this experiment there is likely to be association between the electroactive ion and the ions of the supporting electrolyte (e.g. AgNaCF,SO,+ triples). Thus the measured diffusion coefficient could depend markedly on the nature of the supporting electrolyte. Class IV Methods in this class involve the transport of charged and electrically neutral species, but the presence of both electrical and chemical potential gradients distinguishes them from class I11 methods in which an electric field is absent.This distinction has an important influence on the way in which the charged species contribute to the net flux. D.C. and A.C. Polarisation Methods Let us consider first a fully dissociated electrolyte. If a cell of the form M(s)lMX-polymer host(s)(M(s) in which the electrode reactions are reversible and the electrolyte is ideal, is polarised by the application of a constant potential difference between the electrodes, the current, whose initial value, I o , is determined by the conductivity of the electrolyte, is found to fall with time until a steady-state value, I+, is eventually observed. The fall in current is due to the establishment of a linear salt concentration gradient across the cell which reduces and finally stops the net motion of anions.A linear relationship is predicted between the steady-state current and the applied voltage only for small concentration gradients and hence only for small values of the latter. Under these circumstances, and assuming the Nernst-Einstein relationship to hold, the transport number of the cation t+ is given simply by I + / I o . The theory of this method for completely dissociated electrolytes was developed by the present authors.” The technique was first applied to polymer electrolytes by Shriver and co-workers.” The basis of the method is exactly the same as an earlier technique suggested by Sgrensen and Jacobsen” who applied a variable frequency a.c.signal to the same type of symmetrical cell. At very low frequen- cies the (alternating) current is affected by concentration gradients which give rise to a characteristic feature in the complex plane called a diffusional impedance. Again, for fully dissociated electrolytes, it is straightforward, in principle, to evaluate the cation transport number from such measurements. This a.c. method is by far the most widely used technique for the practical estimation of transport numbers in polymer electrolytes, on the implicit assumption that the electrolyte is completely dissociated. (Note that both this method, and the d.c. polarisation method require correction for the effects of finite electrode kinetics, electrode passivating layers, e t ~ .’ ” * ~ In many cases allowance for such electrode phenomena has not been properly made.) However, as in the situation considered in I11 above for direct diffusion coefficient measurements, it is not possible to derive transport numbers directly from d.c. or a.c. polarisation experiments when mobile associated species are present. On the other hand, the understanding of cation transport across a thin film of polymer electrolyte under the combined influence of an electric field and a concentration gradient is of great practical importance, since such circumstances are likely to be encountered regularly in practical devices such as polymer electrolyte batteries under load. We have therefore examined in some detail the theory of the steady-state current response of the cell M(s)lMX-polymer host(sj)M(s) where the polymer electrolyte contains mobile ion-associated species, such as ion pairs and triples, and have undertaken a full analysis for model systems containing simple mobile ions and ion pairs” A qualitative description of the establishment of a steady state following the polarisa- tion of the cell M(s)lMX-polymer host)M(s) in which the ideal electrolyte contains50 Transport in Polymer Electrolytes mobile M', X- and MX species and the electrodes are reversible to M+ may be given as follows.Following the application of a constant voltage, the current is initially carried by migration of cations and anions in the electric field. While cations are continuously supplied by oxidation of M(s) at the anode, and consumed at the cathode by reduction, this is not the case for anions which do not react at either electrode.As a consequence of electroneutrality, a concentration gradient of M+ and X- ions develops across the electrolyte. Since the ion pairs are assumed to be in fast dynamic equilibrium with the free ions, a concentration gradient of MX is also established. Eventually a steady state is achieved where the net flux of X due to field-driven migration from cathode to anode is balanced by the sum of diffusional fluxes of free X- ions and MX ions pairs in the opposite direction. No further changes in concentrations now occur. The steady-state current I+ is due to the migration of M' cations in the field, together with the diffusion of M+ cations and MX ion pairs down their respective concentration gradients.Expressions may be derived for the steady-state current and voltage as follows. In the steady state, we have for the anions K l ' dx dx anion anion ion-pair migration diffusion diffusion d 4 dx where D- and Do are the anion and ion-pair diffusion coefficients, respectively, - is the electric field gradient and we assume that u- is related to D- by the Nernst-Einstein relationship, u = D- F/ RT. Hence d + RT d[X-] Do d[MX] - dx-F[X-] [ dx + ( K ) T ] . Considering now the cations, we can write for the steady-state current: d[MX] F2D+[Mf] d+ dx dx RT dx I+ = - FD+ -- d[M+I FDO-- - cation ion-pair cation diffusion diffusion migration where D, is the diffusion coefficient of the cation and u+ = D+F/ RT. By introducing the equilibrium constant for ion association, K = [ MX]/ [ M'][X-], and integrating, where [M'Ia and [M'Ic represent the steady-state concentration of M+ adjacent to the anode and cathode, respectively.Similarly, by noting that [M+] = [X-] and [MX] = KIM']' an expression may be obtained for the potential drop across the electrolyte: 2 KDo A + = RT [ In (s) + - ([ M'Ia - [ M+]J} . F D- Now for reversible electrodes, the electrode potential difference is so that the potential difference across the cell is given by A V = A+ -+ AE.P. G. Bruce and C. A. Vincent 51 It is now possible to combine the equations for I+ and AV to obtain an expression for I+/AV. Under certain conditions this ratio is a constant and may therefore be called the effective conductivity, C T , ~ , of the cell at steady state.An explicit expression for ueff may be derived as follows.21 When the difference between the cation concentration at the cathode and the anode is small, and then, where [M'l0 is the initial concentration of free ions in the electrolyte and is equal to ([M+Ia+[M+lc)/2. [M'l0 may also be expressed as ac where CY is the degree of dissociation of the salt, and c is the total salt concentration. Two special cases may be considered. First, when the concentration and/or the diffusion coefficient of the ion pairs is small compared with those of the anions, KDo[ M+],/ D- << 1, the equation for the effective conductivity simplifies to aeff = F2D+[ Mi],/ RT and the steady-state current is seen to be principally due to the flux of M+.Secondly, and in contrast, when the ion pairs dominate, KDo[M+],/D- >> 1, and ueff = F'[ M+],( D+ + D-)/ RT which is identical to the (initial) electrolyte conductivity, u. Here the anion flux has been balanced almost exclusively by the flux of ion pairs. This latter situation has been described by Ingram et al.22,23 A number of conclusions may be drawn from the above analysis. (i) The presence of ion pairing in polymer electrolytes has important implications for the operation of practical devices such as solid-state batteries of the type M( s) Ipol ymer- MX( s) lintercalation host. On discharge, the electroactive constituent, M, is carried by both M+ and MX. Provided that the total flux of M from anode to cathode is high, it is not important which species carries the current, so that apparently, significant ion pairing is of no disadvantage in a battery electrolyte provided that Do is high.This, however, is not necessarily the case since a high concentration of ion pairs implies a low concentration of ions, and hence a low ( T , ~ (since ( T , ~ = (T in this situation as noted above). In the exceptional circum- stances where cations are completely immobile, then ion pairs would, nevertheless, provide the only means of transporting M. (ii) In contrast to the case of the completely dissociated electrolyte, the ratio (~,*/u does not yield the transport or transference number of M+, since by definition, a transference number measures the contribution of charged species to transport under the influence of an electric field alone.Furthermore, this ratio does not even furnish information on the transport parameters of all the species contributing to the steady-state flux; as noted above, the equation for (T,~ for an electrolyte dominated by mobile ion pairs contains neither the concentration nor the diffusion coefficient of this species. Despite this, trends in the ratio of aeff/u with variations in temperature, salt, concentra- tion, etc., provide a useful insight into the nature of the transport process, as is demonstrated by fig. 1 which presents data for a PEO-LiC104 electrolyte at a number52 I I I I I I Transport in Polymer Electrolytes 2.0 - - 1.5 - - 1 - 0.0 1 .o 2.0 3 .O concentration/mol dm-3 Fig. 2. Total and 'effective' conductivities as a function of concentration for the cell Li(s)lPEO- LiClO,(Li(s); (- - -) 100 "C, (-) 140 "C.of temperature^.^^'^^ In fig. 2 the total conductivity of the electrolyte is compared with the effective conductivity of the cell Li(s)lPEO-LiClO,(s)lLi(s) (corrected for electrode effects) at two temperatures. At 140°C despite the significant rise in total conductivity with concentration, it is seen that nee rises only marginally. This is probably indicative of changes in the association equilibria as the salt concentra- tion is increased.P. G. Bruce and C. A. Vincent 53 Table 1". total final d.c. corrected applied steady-state electrode electrode applied voltage current ( I , ) resistance iR drop voltage (AV) A V / I+ ueff / m v 1 PA /a / m v /mv la /S cm-' 20.0 136.5 24 3.3 16.7 122.3 2.21 X 40.0 226.7 54 12.2 27.8 122.6 2.20 x 60.0 354.0 46 16.3 43.7 123.4 2.19 x 80.0 498.0 37 18.4 61.6 123.6 2.19 x lop4 100.0 684.0 27 18.5 81.4 119.0 2.26 X lop4 a Cell : LilPEO-LiC104(8: 1)ILi at 120 "C.Cell constant = 0.0270 cm-'. (iii) Extending the equations to include triple ions introduces further variables, namely an association constant and a diffusion coefficient for each triple ion. Since X-containing species are transported in both directions by an electric field, the concentra- tion gradient which develops due to the non-discharge of X may run from anode to cathode, or vice versa, depending on the direction of the net flux of X following initial application of the field. In either case, the net flux of X is zero and ueff remains a measure of the net flux of the M constituent in the steady state.Where the concentration and mobility of ion pairs are high, the comments made in (ii) above are valid, mutatis mutandis. The model may be further simplified in practice since as Cheradame has suggested,26 the mobility of MX2- is likely to be much higher than that of M2X+ in practical polymer electrolytes. The reason for this is that the latter species are likely to be solvated by two polymer chains and thus will be 'anchored' to the matrix. Note that for MX2-, diffusion down the concentration gradient will be opposed by migration in the opposite direction due to the electric field, thus reducing the flux of the M component from anode to cathode. This contrasts with the situation for neutral ion pairs, MX, for which there can be no migratory flux. (iv) In the case of a fully dissociated ideal electrolyte, ueff is independent of the applied potential only for A V S 10 mV." In contrast, we have shown by computer simulation for systems containing M+, X- and MX that if the concentration of mobile ion pairs is high, oeff may remain independent of the applied voltage up to much higher limits, exceeding 10 V in some cases2' In consequence, measurements of aefi as a function A V may be useful as a test for ion association in polymer electrolytes.In table 1 results are summarised for a series of measurernent~~~ where the applied potential has been corrected for electrode effects. aeff clearly remains constant until values well in excess of 10mV, and it may be concluded that this result gives useful additional evidence for ion association in this system, complementing that from spectroscopic measurements. Conclusions Where simple ion-pair formation is the only form of ion association in solid polymer electrolytes, the Hittorf, concentration cell and force-field methods yield individual ion-transport numbers since these techniques are not affected by neutral species whether these are mobile or not.Even for such simple association, however, radio-tracer, pulsed field gradient n.m.r., Cottrell and electrochemical polarisation methods do not yield transport numbers when the measurements are affected by contributions from mobile ion pairs. Full analysis requires knowledge of both the association constant and the mobilities of ions and ion pairs.On the other hand, if the ion pairs are immobile, these techniques do permit the transport number of the ions to be evaluated. Where there is54 Transport in Polymer Electrolytes higher association (triples and above) none of the methods yield ionic transport numbers if any of the more highly charged species are mobile. Despite this, it is probable that a combination of techniques from classes I-IV will lead to a clearer insight into the nature of transport in polymer electrolytes, since the different techniques provide different and possibly complementary information. Study of d.c. polarisation of cells of the form M( s) Ipolymer-MX( s) 1 M( s) leads to evaluation of the term ueff which is a useful parameter for determining the practical merit of a polymer electrolyte in devices such as power sources.We thank the S.E.R.C. for financial support, and Mr M. T. Hardgrave for making available some of the results of his electrochemical measurements. References 1 M. Spiro, in Physical Methods ofchemistry, ed. A. Weissberger and B. W. Rossiter ( Wiley-Interscience, New York, 1971), vol. 1, part I 1 A, chapt. IV. 2 S. Schantz, J. Sandahl, L. Borjesson, L. M. Torell and J. R. Stevens, Solid State Ionics, 1988,28-30, 1047. 3 L. M. Torell and S. Schantz, in Polymer Electrolyte Reviews I I , ed. J. R. MacCallum and C. A. Vincent, 4 J. R. MacCallum, A. S. Tomlin and C. A. Vincent, Europ. Polym. J. 1986, 2, 787. 5 C. A. Vincent, Prog. Solid State Chem., 1987, 17, 145. 6 M. Leveque, J. F. Le Noest, A. Gandini and H. Cheradame, Marcromol. Chem. Rapid Commun., 1983, 7 A. Bouridah, F. Dalard, D. Deroo and M. B. Armand, Solid State Ionics, 1986, 18-19, 287. 8 A. Bourdiah, F. Dalard and M. B. Armand, Solid State lonics, 1988, 28-30, 950. 9 A. V. Chadwick, J. H. Strange and M. K. Worboys, Solid State lonics, 1983, 9-10, 1155. (Elsevier, London, 1989), chap. 1. 4, 497. 10 C. Bridges, A. V. Chadwick and M. R. Worboys, Br. Poly. J. 1988, 20, 207. 11 A. A. Al-Mudaris and A. V. Chadwick, Br. Polym. J., 1988, 20, 213. 12 S. I. Smedley, in The Interpretation of Ionic Conductivity in Liquids (Plenum Press, New York, 1980). 13 M. Minier, C. Berthier and W. Gorecki, J. Phys. (Paris), 1984, 45, 739. 14 S. Bhattacharja, S. W. Smoot and D. H. Whitmore, Solid Stare lonics, 1986, 18-19, 306. 15 S. E. Lindsey, D. H. Whitmore, W. P. Halperin and J. M. Torkelson, Polym. Preprints, 1989, 30, 442. 16 M. McLin and C. A. Angell, Polym. Preprints, 1989, 30, 439. 17 P. G. Bruce and C. A. Vincent, J. Electroanal. Chem., 1987, 225, 1. 18 P. M. Blonsky, D. F. Shriver, P. Austin and H. R Allcock, Solid Stare lonics, 1986, 18-19, 258. 19 P. R. Sprrensen and T. Jacobsen, Electrochim. Acta, 1982, 27, 1671. 20 P. G. Bruce, J. Evans and C. A. Vincent, Polymer, 1987, 28, 2324. 21 P. G. Bruce, M. T. Hardgrave and C. A. Vincent, J. Electroanal. Chem., in press. 22 M. D. Ingram, personnel communication. 23 G. G. Cameron, J . L. Harvie and M. D. Ingram, Solid State Ionics, in press. 24 M. T. Hardgrave, Ph.D. Thesis, (University of St Andrews; 0000). 25 P. G. Bruce, M. T. Hardgrave and C. A. Vincent, to be published. 26 H. Cheradame, personal communication. Paper 9/02349F; Received 5th June, 1989