J. Phys. Chem. 1996, 100, 1181-1188
1181
Probe Ion Diffusivity Measurements in Salt-in-Polymer Electrolytes: Stokes Radii and the Transport Number Problem M. G. McLin† and C. A. Angell* Department of Chemistry, Arizona State UniVersity, Tempe, Arizona 85287 ReceiVed: July 14, 1995; In Final Form: October 30, 1995X
In order to explore the factors which distinguish ionic motions in salt-in-polymer electrolytes from those in molecular and aqueous solvents, self-diffusivity measurements of cations and anions have been made utilizing the electrochemical technique of chronoamperometry. Stokes’ law radii have been calculated using both the macroscopic and microscopic solvent viscosities and found to differ greatly from the crystal radii. The friction acting on the small cation Ag+ is found to be more than a factor of 20 greater than calculated from the microscopic viscosity. Nernst-Einstein conductivities have been calculated for the probe ion species and are found to be much greater than the measured host conductivity on the supposition of comparable host ion diffusivities. The excess conductivity correlates with the ion pairing propensity of the host anions, triflate and perchlorate. Correcting the measured probe diffusivities for ion pair contributions, our data imply host cation transport numbers as low as 0.1.
Introduction There are few problems in the physical chemistry of ionic solutions which have not been directly impacted by Harold Friedman’s incisive thinking. One which has received less attention than most is the problem of ionic motion in solvents which are polymeric in nature. Mechanically, such systems are rubbery solids when the molecular weight is sufficiently high. The importance of such systems has come into focus with the growing interest in solid state electrochemical devices, and the study of salt-in-polymeric-solvent (“polymer electrolyte”) phenomena is now a significant field of research.1 Improving the ambient temperature conductivity of such electrolyte systems in order that they may serve as components in advanced high voltage solid state battery systems has become one of the important challenges in materials chemistry.1,2 Unfortunately, polymer electrolytes, like nonaqueous solutions in general, conduct very poorly relative to aqueous systems of the same concentration.3 This is because the ions, and in particular the cations, encounter a much higher frictional resistance to their motions than expected from the StokesEinstein relation.4,5 The higher friction on the cation leads to low cation transport numbers.6-8 This constitutes another important problem obstructing technological utilization of the salt-in-polymer type of electrolyte because it leads to polarization during discharge of metallic anode cells.2 To study these problems, we have initiated a study of the motion of individual ions of different sizes and charges in polymer electrolytes, and in this paper we report initial results. A first attempt to interpret the latter observation would follow the line of argument used to explain the failure of Stokes’ law for Li+ (and also doubly charged cations) in aqueous solutions; viz., the cation can only move when a part of its environment (usually conceived of as a hydration shell) also moves, i.e., that cations must drag the polymer chains as they move. However, the low transport number of the cation seems not to depend strongly on the polymer molecular weight,7 so chain dragging * To whom correspondence should be addressed. † Research contained in this paper was performed in partial fulfillment of the Ph.D. requirements in Physical Chemistry at Purdue University. M.G.L. is presently at Arthur D. Little Inc., Acorn Park, Cambridge, MA 02140-2390. X Abstract published in AdVance ACS Abstracts, December 15, 1995.
0022-3654/96/20100-1181$12.00/0
is not supported. In any case a given polymer chain will generally be ligated to many individual cations so no single cation could drag the chain independently. Therefore, the interpretation of the cation behavior in such systems is more profitably made in terms of a more generalized concept of friction, in which the deviation from Stokes’ law is taken as a measure of the reduced probability that an ion can move relative to its environment on the same time scale as the components of the environment (the chain “beads”) can more relative to each other. Torell and Angell4 attempted to give this line of thought a phenomenological basis by comparing directly the time scale for the structural relaxation of the polymer electrolyte solution (taken as the longitudinal relaxation time measured by Brillouin scattering for short relaxation times) with the time scale for electrical relaxation. They obtained the decoupling ratio Rτ, the ratio of these two time scales,9 and found that in contrast to concentrated aqueous solutions in which the two time scales are very similar9 and to superionic glasses10 in which the conductivity relaxation is many orders of magnitude faster (i.e., Rτ . 1), the polymer electrolytes responded to electric stresses much more slowly than they respond to mechanical stress, i.e., Rτ , 1. Although the use of the Brillouin scattering-based longitudinal relaxation time as a basis for the mechanical relaxation time scale in polymer electrolytes has been criticized,11 it has the advantage of giving results for PPO which coincide with those from 13C NMR, impulsive light scattering, and inelastic neutron scattering estimates of the primary structural (or local viscosity) relaxation,12,13 and the authors can see no more appropriate choice. In a recent extension of these latter studies, the present authors5 demonstrated that the unfavorable relaxation time ratio Rτ reported by Torell and Angell at relatively low salt concentrations, M:O ∼ 1:10, was changing in a favorable direction with increasing salt concentration despite the evidence14 that the fraction of salt present as ion pairs is also increasing with concentration. This implies that the large and unfavorable friction on the ions is decreasing with increasing salt concentration more rapidly than the free ion concentration is decreasing, © 1996 American Chemical Society
1182 J. Phys. Chem., Vol. 100, No. 4, 1996 implying the possibility of cation-dominated conductivity at sufficiently high salt concentrations. To deepen our understanding of the factors which determine these subtleties of ion dynamics in polymer electrolytes, it seems clear that measurements which are sensitive to the movement of the individual ionic species must be introduced. The most fundamental transport measurement which can be made is that of the self-diffusion coefficient of an individual chemical species, Di. Di can be described in linear response theory by a single particle correlation function15 and furthermore can be measured in the laboratory in the absence of any driving force other than that of simple probability. Unfortunately, the selfdiffusion coefficient is very difficult and tedious to measure by the classical radiotracer techniques16 and can only be measured accurately by the simpler NMR spin-echo method17 when diffusivities are higher than those encountered in most polymer electrolytes. While the latter problem can be partly overcome by using the pulsed field gradient method,18 few workers have had access to the necessary equipment. Despite the difficulties, some measurements have been made. Chadwick and co-workers16 reported tracer diffusion measurements on Na+- and SCN--containing entities in PEO polymer electrolytes at the 1:8 concentration, and the results showed that cations indeed move at least as slowly as anions in polymersalt systems. Furthermore, any difference in free ion diffusivities must be greater than these D values suggested because the conductivities calculated from their results, using the NernstEinstein equation, implied that much of the observed diffusion was due to ion pairs. Due to their difficulty, such measurements are unlikely to be performed on a broad enough range of systems and concentrations to provide the empirical information we believe is needed to resolve the relative ion mobility problem satisfactorily. An important study of diffusivity of Li+ and triflate (CF3SO3-) anions was later made using the pulsed field gradient technique by Whitmore and co-workers.19 They succeeded in measuring diffusivities as low as 1 × 10-7cm2 s and established that the cation is the more slowly diffusing species. The difference in diffusivity was not large, and it was again clear that an important part of the diffusive displacements occurred by ion pair displacements. Unfortunately, no concentration dependences were reported. Like the earlier study of Chadwick et al., this study was executed on the technologically relevant “solid” (i.e., rubbery) electrolyte system PEO + salt, so nothing was learned about ionic diffusion in relation to solvent viscosity. This latter limitation was lifted by the more recent studies of Bridges et al.20 and Boden et al.21 While this latter technique is much more practical for wide ranging studies than the radiotracer technique (and will no doubt be widely applied in the future), we believe there is a need for methods in which the diffusivity of a number of different species in the same solvent can be made with minimum expenditure of equipment or investigator time. For such purposes the electrochemical techniques, which depend on establishing a diffusion-controlled process at an electrode, are ideally suited. Any species which can participate in an electrochemical oxidation or reduction process at an electrode at a potential within the solvent electrochemical “window” can be used as a probe of diffusion kinetics. In the case of LiClO4 and NaCF3SO3 solutions in PPO(4000), the electrochemical window is such that a wide range of cationic, anionic, and molecular probes could be used, for example Cu+, Ag+, Tl+, Cd2+, Zn2+, In3+, I-, Fe(CN)64- as well as a variety of organic cations and also oxidizable or reducible molecular species. The present study will focus on just three of these possibilities,
McLin and Angell viz., Ag+, Tl+, and I-. Ag+ has about the same crystal radius as Na+, Tl+ is similar in radius to the weakly coordinating K+ cation, and I- is not much different in size from the quasispherical ClO4- anion. We have chosen to study the diffusivity of these three species in a variety of solutions of different salt concentrations in PPO(4000), focusing on the two solutes LiClO4 and NaCF3SO3 for which so much information is now available.5, 13, 14, 22-24 It is important to note at the outset that, for the conditions of the electrochemical experiment in which the diffusing species is very dilute, the measured diffusivity differs negligibly from the self-diffusivity of the electroactive species. This was argued theoretically by Laity and McIntyre25 and verified experimentally to (3% for the case of Ag+ diffusing in the ionic liquid solvent 3Ca(NO3)2‚2KNO, by Welch and Angell.26 The reason is basically that the thermodynamic correction term in the Darken equation27 relating self-diffusion to diffusion in the presence of a concentration gradient,
Di,self[1 + Ni d ln γi/dNi]
(1)
(where γi is the activity coefficient of species i and Ni its concentration) is essentially unity in the very dilute probe species case. Experimental Section Solutions of LiClO4 and NaCF3SO3 in PPO(4000) were prepared using reagent grade materials and careful drying procedures fully described in earlier papers.5,22 The compositions, in units of (moles of salt):(moles of polymer ether oxygens), i.e., polymer repeat units, ranged from 1:8 to 1:40. This corresponds to n values of 8 to 40 in the notation LiX‚P(PO)n. The only distinction in the present study is that samples of the salt-in-polymer solutions were doped with an electroactive probe ion at a concentration of 0.02 M ((0.01 M for Tl+). This is a higher concentration of probe ion than is normally used29 but was necessitated by the low mobility; a minimum current is needed for acceptable precision, and the current is proportional to the electroactive ion concentration. Our ratio of probe ion to host is, however, still more than adequate to satisfy the probe diffusion current control requirement. The probe ions Ag+, Tl+, and I- were added to separate solvent samples in the forms AgCF3SO3, TINO3, and LiI, respectively. The TlNO3 is barely soluble in the polymer solution, because the Tl+ ion ligates only weakly and the dielectric constant of the solvent is too low ( ) 3.5) to provide an electrical driving force to solution. Therefore, only very dilute solutions could be studied. Consequently, the results for thallium diffusion are in principle less precise than for the other species. The diffusion coefficients of these electroactive probe ions were determined by chronoamperometry using an interfaced PAR Model 273 potentiostat/galvanostat and a standard macroelectrode cell arrangement. The samples were placed in a 25 mm diameter Pyrex vial with a fitted Teflon cap through which the electrodes, a thermocouple, and N2 gas purging tube were placed. A planar Pt electrode purchased from BioAnalytical Systems was used as the working electrode, a Pt wire was used as the auxiliary electrode, and a bar Ag wire was used as a quasi-reference electrode. Temperature regulation of the sample to within (0.1 °C between ∼70 and 110 °C was achieved using a circulating bath and a copper tube thermostat. The latter consisted of a copper tube of diameter ∼1 in. with a solid bottom around which copper tubing was wrapped and soldered in place. The exterior of this device was wrapped in fiberglass tape to provide insulation,
Probe Ion Diffusivity in Salt-in-Polymer Electrolytes
J. Phys. Chem., Vol. 100, No. 4, 1996 1183
Figure 1. Cyclic voltammogram of Ag+-doped 1:30 LiClO4-PPG(4000). Figure 3. Arrhenius representation of silver probe ion diffusivities in LiClO4-PPG(4000) from 1:40-1:10 (M:O).
Figure 2. i(t) vs t-1/2 for reduction of Ag+ doped in 1:30 LiClO4PPG(4000) at 83.0 °C.
and ethylene glycol circulated through the tubing to establish the temperature of the fluid reservoir of the Lauda bath. Chronoamperometry is a controlled potential technique in which the potential of an electrode in the solution containing an electroactive species is stepped, at t ) 0, to a preselected value such that the concentration of the electroactive species at the electrode surface drops suddenly from the bulk concentration to zero. The current flowing through this electrode is then monitored as a function of time. Because only the one species can be deposited at the chosen potential, a concentration profile develops as time elapses and the current becomes diffusionlimited. The steps are chosen to minimize RC charging after the potential step while ensuring that the final potential exceeds the half-reaction potential sufficiently to result in the probe ions being completely electrolyzed at the electrode surface. The evolution of the current, i(t), with time for a diffusioncontrolled electrode reaction is given by the Cottrell equation29
i(t) ) nFAD1/2C*/(π1/2t1/2)
(2)
where n is the number of electrons being transferred in the electrode reaction, A is the area of the electrode in cm2, F is
Faraday’s constant, D is the diffusion coefficient of the electroactive species in cm2/s, C* is the bulk concentration of the electroactive species in mol/cm3, and t is time in seconds. The diffusion coefficient of the electroactive species can be determined from the slope of the plot. The concentration being initially very small, the diffusivity measured differs negligibly from the self-diffusion coefficient of the species, as emphasized earlier. Before chronoamperometry could be performed on these samples, the uncompensated resistance arising from the different positions of the reference and working electrodes relative to the auxiliary electrode28 had to be removed. From 75% to 90% of the uncompensated resistance can be electronically removed from the cell circuit using the positive feedback IR compensation feature of the Model 273 potentiostat/galvanostat. It was necessary to perform this task at each temperature since the specific resistivity of the electrolyte, and therefore the uncompensated resistance, decreased with increasing temperature. In order to determine the appropriate potentials for the potential step, cyclic voltammograms were also performed at each temperature. The cyclic voltammogram of Ag+-doped 1:30 LiClO4-PPO(4000) samples is shown in Figure 1. The deposition of the iodide ion at an anode occurs not by iodine generation directly but by oxidation to I3-. The oxidation halfreaction for I- to I3- occurs on the edge of the electrochemical window of the supporting electrolyte, so the diffusion coefficients for this species have a greater uncertainty than those of Ag+ though the reproducibility was good. The method was tested by reproducing the well-known28 diffusivity of Fe(CN)63-. A typical plot of i(t) vs t-1/2 is shown in Figure 2. The initial part of this plot is not linear due to RC charging of the residual uncompensated resistance and double layer capacitance. Thereafter the plot becomes linear, indicating the electrode reaction is diffusion-controlled, and the diffusion coefficient for the electroactive probe ion can then be evaluated as described above. The current decay was monitored for 25 s without any noticeable distortions from convection due to density gradients or mechanical vibrations. Results In Figures 3 and 4 silver and iodide probe ion diffusivity data in LiClO4 + PPG(4000) salt-in-polymer solutions of
1184 J. Phys. Chem., Vol. 100, No. 4, 1996
McLin and Angell
Figure 6. Arrhenius representation of silver, iodide, and thallium probe ion diffusivities in 1:40 LiClO4-PPG(4000).
Figure 4. Arrhenius representation of iodide probe ion diffusivities in LiClO4-PPG(4000) from 1:40-1:10 (M:O).
Figure 7. Arrhenius representation of iodide probe ion diffusivities in NaCF3SO3-PPG(4000) solutions.
indistinguishable from the iodide probe at this concentration). Any error due to undissolved Tl would cause an underestimate of the Tl diffusivity. The effect of change of host solution salt concentration is shown in Figure 7 for the case of iodide diffusion in sodium triflate solutions. Figure 5. Comparison of silver and iodide probe ion diffusivities in 1:30 LiClO4-PPG(4000) and 1:30 NaCF3SO3-PPG(4000) in the Arrhenius representation.
different compositions are shown. The diffusivities of both probe ions decrease with increasing host electrolyte concentration and, unlike the conductivities of the undoped solutions, exhibit Arrhenius temperature dependences within our experimental uncertainties. In Figure 5, a comparison between the diffusivities of the silver and iodide probe ions in 1:30 NaCF3SO3 and 1:30 Li ClO4 is shown. Both probe ions species have greater diffusivities in the 1:30 NaCF3SO3 solutions than in the 1:30 LiClO4 solution. This qualitative trendsgreater diffusivities for both probes in the NaCF3SO3 solutions at a given compositionswas also observed at the other stoichiometries examined. A comparison of the Tl+, Ag+, and I- probe ions in the most dilute solution, 1:40 LiClO4 (the only one in which enough Tl for measurement would stay in solution), is shown in Figure 6. Figure 6 also contains the product σT for the host solution, to be discussed below. The weakly solvated Tl+ is observed to be more diffusive than the smaller silver probe (which is
Discussion Although the diffusivity measurements reported above do not give direct information on the motion of the host cations or anions, they do give important insights into the behavior of individual chemical species in these solutions. Furthermore, at the compositions 1:30 at the dilute ends of the systems studied, the Ag and I probe ion data prove surprisingly predictive of the host properties and can therefore plausibly be used to discuss the contentious issue of host solution transference numbers. The first observation to be made is that, except for 1:40 LiClO4 solutions in which iodide diffusion is anomalously slow (see below), the small Ag+ cation diffuses less rapidly than the larger I- anion. Since the ion-pairing phenomena known to be present will only serve to reduce the difference between the measured diffusivities, it is clear that the probe diffusion measurements have established directly a qualitative breakdown of the Stokes-Einstein equation for ionic motion in these solutions. The equation is
Di ) kT/(6πηri)
(3)
where η is the viscosity, ri is the radius of species i, and the
Probe Ion Diffusivity in Salt-in-Polymer Electrolytes factor 6 is appropriate for pure stick conditions. The breakdown is consistent with that implied by earlier self-diffusion measurements in PEO solutions made by radiotracer16 and pulsed field gradient NMR spin echo18,19,21 techniques and with previous transport number measurements,6-8,30,31 all of which imply that the cation is the less mobile ionic species. The anomalously small cation mobility and the extent of ion pairing, both of which are deleterious to performance in device applications, are the two key features of the salt-in-polymer solvent electrolytes. Our discussion of the present diffusivity findings will be undertaken in three stages. First, it will be noted how the difference in the diffusivity and conductivity temperature dependence is consistent with our earlier discussion of conductance/viscosity relations in terms of ion pairing.22 Then the effect of viscosity on the diffusivity data will be examined via the Stokes-Einstein equation to gain insight into the nature of the forces which impede the ionic motion. Finally, all observations will be integrated to provide a basis for extracting the most probable diffusion coefficients for the free ions from which the relative ion mobilities, in a form analogous to the transference numbers for the host cations, may be calculated. By making the trial assumption that the probe ions mimic the host cations, a total conductivity can then be calculated via the Nernst-Einstein equation which can be compared with the observed value. From the quality of the agreement, a judgment on the initial assumption can be made which will then indicate how closely the relative mobility fraction calculated can be taken to represent the actual host ion transport numbers. The presence of ion pairing, which diminishes the high temperature conductivity,22 can be seen directly from comparison of the diffusivity Arrhenius plots for the probe species with those for bulk conductivity (see Figure 6). The evidence is more direct than that obtained from the viscosity-conductivity comparisons discussed earlier.22 Whereas the conductivity Arrhenius plot is strongly curved at high temperatures, the diffusivity plots maintain an almost constant slope with an activation energy which is close to that of the conductivity in the lower part of the common temperature range and essentially the same as the average value for the viscosity in the range of the measurements. Since ion-paired species contribute to the observed diffusivity but not to the conductivity, the obvious interpretation of the difference in temperature dependence of conductivity and diffusivity must invoke the same loss of conducting species into the ion pair population that was earlier deduced from the viscosity/conductivity relations22 (and connected quantitatively to the Raman spectral results of Torell and co-workers14,23). This loss is progressively more important the higher the temperature.14,23 The above observation makes clear that, in order to compare the mobilities of free cation and anion species in polymer electrolytes using our self-diffusivity data, a correction for the ion pair contribution must be made, and this will be addressed in the last part of this discussion. Stokes Radii and Ion-Solvent Friction. With self-diffusion data in hand, the applicability of the Stokes-Einstein equation to the diffusion of ions in polymer electrolytes can be tested. If, in the solutions of this study, the friction on the ions is much less than the macroscopic viscosity (as is most obviously the case in the conducting solids obtained when high MW PPO solvents are used), then the viscosity calculated from the Stokes-Einstein equation using the measured Di should be found much less than the directly measured value. Alternatively, if the measured viscosity is used in eq 3, then the inappropriateness of this viscosity for assessing the microscopic friction acting on the ions should be indicated by a difference between the
J. Phys. Chem., Vol. 100, No. 4, 1996 1185
Figure 8. Stokes radii for silver and iodide probe ions in 1:30 NaCF3SO3. The dashed lines are the values of the crystal radii for the iodide anion (top) and the silver cation (bottom). Note that points on the Ag crystal line are for I-, not Ag+.
calculated Stokes radius and the crystal radius. The data have been treated using the latter approach, and the results, while fully consistent with the fractional decoupling ratios for the host solutions derived in the earlier papers,4,5 are somewhat surprising. In Figure 8, the Stokes radii for pure stick conditions calculated for the silver and iodine species in 1:30 NaCF3SO3 solutions using eq 3 are shown over the temperature range of the diffusivity measurements. These radii are based on the raw diffusivity data (uncorrected for any ion pair effects) and the macroscopic viscosity data. The Pauling crystal radii32 are marked as horizontal dashed lines. Results for the perchlorate solutions are qualitatively similar with Stokes radii 30-50% larger and also more temperature dependent than in the triflate solutions. Fortuitously, in the dilute 1:30 triflate cases (Figure 8), the I- Stokes radii values fall approximately where the smaller Ag+ values would be expected if the Stokes-Einstein equation applied, and vice versa. More surprising than this inversion, however, are the roughly correct (within a factor of 2) values obtained since we have used the macroscopic, not the microscopic, viscosities in calculating the Stokes radii. This is a matter peculiar to polymer electrolytes which we now discuss. We consider the I- values first. The small Stokes radius relative to the crystal value means that the friction acting on the ion is less than expected from the macroscopic viscosity. This is qualitatively what is expected from the “polymer effect” on the viscosity and shows that the I- ion, which we do not expect to be much affected by solvation effects, is responding to a smaller viscosity than that measured by the rotating cylinder22 or other macroscopic probes used to measure the viscosity. The iodide ion feels a “local” viscosity which reflects the motion of its immediate surroundings (which can be measured by Brillouin light scattering, impulsive light scattering, and 13C NMR correlation techniques12,13) rather than that of the slower whole chain motion of the solvent. In this sense it is decoupled from the macroscopic viscosity as in superionic glasses. However, in the latter cases and in all ionic glassforming nonpolymeric liquids, there is no distinction between macroscopic and microscopic viscosities. The microscopic viscosity, which is associated with the mobility of rearrangeable
1186 J. Phys. Chem., Vol. 100, No. 4, 1996
Figure 9. Stokes radii for silver probe ions in LiClO4-PPG(4000) from 1:40-1:10 (M:O) versus temperature. Stokes radii for thallium probe ion in 1:40 LiClO4 PPG(4000) versus temperature are also included.
units of the structure, is the dynamic quantity involved in the glass transition, so all nonpolymeric liquids exhibit glass transitions when the viscosity reaches a roughly common value, 1013 P. It is the decoupling of microscopic from macroscopic viscosity which makes possible the rubbery or at least “plastic” behavior of polymer electrolytes and polymers in general. In superionic glasses the conducting modes are decoupled from the microscopic viscosity, this of course being necessary for high conductivity in the glassy state to be manifested. Quantitatively, the difference in crystal and Stokes-Einstein radii is smaller than the factor of ∼10 expected from the polymer molecular weight contribution to the prefactor in the viscosity of PPO(4000)33 or from the solvent structural relaxation time probed by the above-mentioned techniques. This difference probably reflects the coupling of the anion-to-cation motion implied by the extensive ion pairing in these systems14 which would require anions to feel some of the friction acting on the cations. Most significant is the very large friction acting on the Ag+ ion, implied by the observation in Figure 8 that its Stokes radius is larger than the crystal radius notwithstanding the inappropriate use of the macroscopic viscosity in the calculation. The friction in this case is some 20 times greater than expected from the microscopic viscosity. This excess friction is to be correlated directly with the very small (subunity) decoupling ratios found4,5 in the dilute solutions since the decoupling ratios were calculated using the microscopic structural relaxation time. The findings for the case of Tl are in between as might be expected. In this case the Tl+ solvates the ether oxygens only weakly which is why, when of PPO is only 3.5, the TlNO3 barely dissolves in the polymer + salt solution. Still the interaction is sufficient that the species feels roughly twice the friction felt by I- (Figure 6). Due to fortuitous cancellation of polymer effect and weak solvation effect, the Stokes radius in this case turns out to be the same as the crystal radius. The trends with composition are summarized in Figures 9 and 10. Figure 9 shows that the Stokes radii for Ag, in contrast to those for I seen in Figure 10, are independent of composition within the experimental scatter and show no clear trend with temperature. This implies either that the Ag as a probe species is not reflecting the factors causing the increase in decoupling
McLin and Angell
Figure 10. Stokes radii for silver probe ions in LiClO4-PPG(4000) from 1:40-1:10 (M:O) versus temperature.
ratio Rτ with concentration seen in ref 5 or that these factors do not involve the cations. We think the former is most likely and that the Ag+ is not a good probe for the majority cation except in the dilute range (1:30) for a particular reason which is related to its small concentration, namely, that it selectively solvates the PPO chain ends. As a basis for this idea, we refer to the extraordinary viscosity behavior in the NaClO4-PPO(4000), at the salt content 1:30, being reported elsewhere.34 In this system the viscosity exceeded 1013 P at a temperature far above the calorimetric Tg, implying high MW polymer (or chain cross-linking) behavior. The interpretation given this viscosity/Tg anomaly was34 that, at this composition, the sodium ion concentration was optimized to end-link the hydroxyl-terminated short 4000 MW chains into self-entangling macromolecular chains. Now, in the present measurements the Ag ion concentration is always smaller than 1:30. Since rAg+ ≈ rNa+, the “hard and soft acid-base” argument35 predicts that “soft” Ag+ preferentially seeks the OH ligands rather than the “harder” ether ligands, and it may be inferred that Ag+ will be found almost exclusively in OH coordination at all compositions. In this case its behavior will not reflect the changing relation between chain end and ether ligand populations which determine the majority of the cation species’ behavior with increasing concentration. The conclusion to be drawn is that the Ag+ may serve as reasonable probes for majority cation species behavior at the composition 1:30 but not at higher concentrations. For successful probe studies at other compositions, solvents with methoxy-terminated chains should be used. In the meantime we will assume that the data taken at the M:O ratio 1:30 is representative of host salt behavior and take the analysis to the next stage. To test the ability of the probe ion data to model the 1:30 NaCF3SO3 and LiClO4 solutions, trial Nernst-Einstein conductivities have been calculated using eq 4
σ(NE) ) nF2(D+ + D-)/RT
(4)
where n is the concentration of host ions, R is the ideal gas constant, F is the Faraday constant, T is the temperature, and D+ and D- are the self-diffusion coefficients of the probe species Ag and I. The use of this equation assumes that the electrolyte
Probe Ion Diffusivity in Salt-in-Polymer Electrolytes
J. Phys. Chem., Vol. 100, No. 4, 1996 1187 transport numbers to represent those of the Na+ and CF3SO3ions of the host electrolyte. Estimates of the concentration of the ion pairs in the solutions at different temperatures were those used in the Walden plots comparisons presented earlier (based on the Raman data of Kakihani et al.14). Since our diffusion coefficient measurements were usually made at temperatures exceeding those for which the Raman estimates of ion pair concentrations14 are available, the values were extrapolated using the functional form for the temperature dependence given by Kakihana et al.23 Diffusivities for remaining “free” cations and anions were then calculated according to
Dfree ) ((Dtotal - RDion pair)/1 - R)
Figure 11. Comparison of measured and Nernst-Einstein probe conductivities for 1:30 LiClO4-PPG(4000).
where R is the fraction of the total Ag in ion pairs, and Dion pair is obtained as follows. In the first calculations Dion pair was assumed to be the average of the measured diffusivities of the silver and iodide probes. In the second calculation Dion pair was assumed to be equal to the measured iodide diffusivity. These yielded two sets of free ion diffusivities which were used to calculate Nernst-Einstein conductivities, which are compared with the experimentally measured values in Figures 12 and 13. Considering the improbability of the assumption that Ag+ and I- can accurately model Li+, ClO4-, Na+, and CF3SO3-, the calculated conductivities agree rather well with the measured values, falling on either side of them depending on the ion pair diffusivity assumption in the case of the triflate solutions where the ion pairing is most pronounced. Transference Numbers. Encouraged by the latter results, we now calculate cation transference numbers, t+, from free ion diffusivities according to
t+ ) D+free/(D+free + D-free)
Figure 12. Comparison of measured and Nernst-Einstein probe conductivities for 1:30 NaCF3SO3-PPG(4000).
is fully dissociated, that the cation-anion friction can be neglected, and that the diffusion coefficients for silver and iodide represent those of the free cation and anions of the majority species. The results are compared with the experimental conductivities in Figures 11 and 12. In each case the calculated conductivity exceeds the measured value, and for the triflate solutions the gap is much larger. This is in accord with previous studies of salt-in-polymer solutions20,21 and with the Raman spectral evidence14 that the 1:30 NaCF3SO3 solution is significantly more associated than the 1:30 LiClO4 solution. The result here implies a comparable Ag+CF3SO3 ion pairing. Therefore, the probe ion data for the two 1:30 solutions must be corrected to remove the ion pair contributions. We try two alternative calculable choices for the diffusion coefficient of the ion pair and use the resultant Di values for the free ions to calculate “probe-based” Nernst-Einstein conductivities for comparison with the measured values. A satisfactory comparison will then justify the use of the free ion diffusivities to obtain
(5)
(6)
where the transference numbers calculated from the raw diffusion data are, for each solution, near 0.4 and increasing slowly with increasing temperature; the values from eq 6 are around 0.3 for the LiClO4 solution and 0.2-0.1 for the triflate solution depending on the correction assumed. In the latter case the temperature dependence either disappeared (when t+ ∼ 0.2) or reversed. In the only case in which the transference numbers of similar solutions (LiClO4 and LiCF3SO3 at 1:24 in a PEObased network) have been directly measured, Cheradame and co-workers30,31 reported that t+ was 0.2 within experimental error for each solution. Treating these values as acceptably consistent, we may finally observe that transport numbers typical for the cation in these media must increase somewhat with increasing concentration since at the low concentrations examined by Ingram and coworkers8,9 values of t+ ∼ 0.07 were obtained by a reliable method. This is consistent with the trend in the decoupling ratios found by the authors at higher concentrations5 and the more recent finding that decoupling ratios exceeding unity may be realized as the salt-in-polymer domain is transcended in lowmelting salt + polymer systems.36 It is in the ensuing polymerin-salt domain that the cation transport numbers can approach unity as in superionic glasses, while rubbery elasticity as in polymer electrolytes is maintained.36,37 Concluding Remarks. It is being observed in current neutron scattering and computer simulation studies of diffusion in viscous liquids38-40 that the onset of diffusion with increasing temperature is preceded by a temperature regime in which the motions of the particles become highly anharmonic. While this has been studied mainly for simple liquids and some coupled systems, it has been observed in one case39 that the motion of
1188 J. Phys. Chem., Vol. 100, No. 4, 1996 a subset of the particles in a system can become highly anharmonic at much lower temperatures than for the majority and that this phenomenon is then followed by onset of onespecies diffusion (implying single ion conductivity). An object of polymer design for salt-in-polymer systems should therefore be the production of polymer structures in which harmonic solvation cages are not possible due to the size or chemical nature of successive ligating groups in the chain. It would then seem reasonable to test the efficacy of such polymer design by use of the Ag+ probe chronoamperametry and the comparison of derived Stokes radii with crystal values as described in this paper. Acknowledgment. This work was supported by the ONR in its early stages and completed under the auspices of DOE, Grant DE-F902-89ER 4535398. References and Notes (1) Ratner, M. A.; Shriver, D. F. Chem. ReV. 1988, 88, 109. (2) Steele, B. C. H. Mater. Sci., Eng. B. Solid State 1992, 13, 79. (3) (a) Friedman, H. L. Ionic Solution Theory; Interscience: New York, 1962; Annu. ReV. Phys. Chem. 1981, 32, 179. (b) Barthel, J.; Gores, H. J.; Schmeer, G. Ber. Bunsenges. Phys. Chem. 1979, 83, 911. (4) Torell, L. M.; Angell, C. A. Br. Polym. J. 1988, 20, 173. (5) McLin, M. G.; Angell, C. A. Solid State Ionics 1992, 53-56, 1027. (6) Cameron, G. G.; Harvie, J. L.; Ingram, M. D.; Sorrie, G. A. Br. Polym. J. 1988, 20, 199. (7) Cameron, G. G.; Ingram, M. D.; Harvie, J. L. Faraday Discuss. Chem. Soc. 1989, 88, 55. (8) Bruce, P. G.; Hardgrave, M. T.; Vincent, C. A. Solid State Ionics 1992, 53-56, 1087. (9) (a) Moynihan, C. T.; Balitactac, N.; Boone, L.; Litovitz, T. A. J. Chem. Phys. 1971, 55, 3013. (b) Ambrus, J. H.; Moynihan, C. T.; Macedo, P. B. J. Phys. Chem. 1972, 76, 3287. (10) Angell, C. A. Solid State Ionics 1983, 9&10, 3. (11) Mendolia, M. S.; Farrington, G. C. Chem. Mat. 1993, 5, 174. (12) Angell, C. A.; Monnerie, L.; Torell, L. M. Symp. Mat. Res. Symp. Soc. 1991, 215, 3. (13) Duggal, A. R.; Nelson, K. E. J. Chem. Phys. 1991, 94, 7677. (14) (a) Schantz, S.; Torell, L. M.; Stevens, J. R. J. Appl. Phys. 1988, 64, 2038. (b) Kakihana, M.; Schantz, S.; Torell, L. M. J. Chem. Phys. 1990, 92, 6271.
McLin and Angell (15) Zwanzig, R. Ann. ReV. Phys. Chem. 1965, 16, 67. (16) (a) Chadwick, A. V.; Strange, J. H.; Worboys, M. R. Solid State Ionics 1983, 9-10, 1155. (b) Chadwick, A. V.; Worboys, M. R. In Polymer Electrolyte ReViews I; MacCallum, J. R., Vincent, C. A., Eds.; Elsevier: London, 1987. (17) Hahn, E. L. Phys. ReV. 1950, 80, 580. (18) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288. (19) Bhattacharja, S.; Smoot, S. W.; Whitmore, D. H. Solid State Ionics 1986, 18-19, 306. (20) Bridges, C.; Chadwick, A.; Worboys, M. R. Brit. Polymer J. 1988, 20, 207. (21) Boden, N.; Leng, S. A.; Ward, I. M. Solid State Ionics 1991, 45, 26. (22) McLin, M. G.; Angell, C. A. J. Phys. Chem. 1991, 95, 9464. (23) Kakihana, M.; Schantz, S.; Mellander, B.-E.; Torell, L. M. In Proceedings of the 2nd International Symposium on Polymer Electrolytes; Scrosati, B., Ed.; Elsevier: London, 1990; p 23. (24) Sandahl, J.; Schantz, S.; Borjesson, L.; Torell, L. M.; Stevens, J. R. J. Chem. Phys. 1989, 91, 655. (25) Laity, R. W.; McIntyre, J. D. E. J. Am. Chem. Soc. 1965, 87, 3806. (26) Welch, B. J.; Angell, C. A. Aust. J. Chem. 1972, 25, 1613. (27) Darken, J. R. Trans. Am. Inst. Mech. Engr. 1948, 175, 84. (28) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; Wiley: New York, 1980; p 569. (29) Cottrell, F. G. Z. Phys. Chem. 1902, 42, 385. (30) Leveque, M.; LeNest, J. F.; Gandini, A.; Cheradame, H. Makrolmol. Chem. Rapid Commun. 1983, 4, 497. (31) Leveque, M.; LeNest, J. F.; Gandini, A.; and Cheradame, H.; J. Power Sources 1985, 14, 27. (32) Cotton, F. A.; Wilkinson, G. AdVanced Inorganic Chemistry; Wiley: New York, 1980; p 14. (33) Wang, C. H.; Fytas, G.; Lilge, D.; Dorfmu¨ller, Th. Macromolecules 1981, 14, 1363. (34) McLin, M. G.; Angell, C. A. Polymer, in press. (35) Pearson, R. G. J. Am. Chem. Soc. 1963, 85, 3533. (36) Angell, C. A.; Liu, C.; Sanchez, E. Nature 1993, 362, 137. (37) Angell, C. A.; Fan, J.; Liu, C.; Sanchez, E.; Xu, K. Solid State Ionics 1994, 69, 343. (38) Zorn, R.; Arbe, A.; Colmenero, J.; Frick, B.; Richter, D.; Buchenau, U. Phys. ReV. E, in press. (39) Angell, C. A. Science 1995, 267, 1924. (40) Shao, J.; Angell, C. A. In Diffusion in Amorphous Materials; Jain, H., Gupta, D., Eds.; The Minerals, Metals and Materials Society: 1994; pp 1-16.
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