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The effect of pressure on fast ion conductivity in glasses
Journal of Non-Crystalline Solids 65 (1984) 29-37 North-Holland, Amsterdam
29
THE EFFECT OF PRESSURE ON FAST ION CONDUCTIVITY IN GLASSES Martin J. R Y A N and Stuart I. S M E D L E Y Chemistry Department, Victoria University of Wellington, New Zealand
Received 13 April 1983 Revised manuscript received 27 September 1983
We have measured the effect of pressure and temperature on the ionic conductivity of glasses in the system BEO3/Li20(LiX)2 (X = F, C1, Br, I), where Xa~o3= 0.7 and the X/O ratio was varied while maintaining a constant Li + content of X (LiX)~ + Li20 0.3. All glasses exhibited a very small pressure coefficient of conductivity, i.e. AV----1cm~ mol-1 which decreased very slightly as the CI/O ratio increased or as Cl was replaced by Br at constant X/O ratio. The results were compared with the ELectrostatic Strain Energy theory; the results of this comparison lead to the conclusion that for Li + conduction the electrostatic term is dominant in determining the activation energy for conduction. m
I. Introduction Significant advances have been made in our knowledge and understanding of the p h e n o m e n o n of fast ion conductivity in solids. These advances have been made possible by the wide range of experimental techniques that have been applied to the study of fast ion conductors which apart from c o n d u c t a n c e studies include N M R , X-ray, infrared and R a m a n spectroscopy. One of the most studied glass conductors is the LiEO/B203 system [1], the structural and c o n d u c t a n c e properties of which were recently reviewed by Button et al. [2]. It suffices to say that LiEO/B203 glasses exhibit high conductivities such that the conductivity o increases and the activation energy E ~ decreases with increasing L i 2 0 content up to about 25 tool%, above which the trend becomes less significant. Levasseur et al. [3] have studied glass with the composition ( L i X ) 2 / L i 2 0 / B 2 0 3 (X = F, C1, Br, I) and have observed that E *~ decreases with increasing (LiC1)2 content up to - 30 mol%. However at a given composition E # decreased monotonically with increasing halogen ion size. Button et al. have systematically studied the conductivity of the system 0.63613203/0.364 [(LiC1)2 + Li20], where the total B203 and Li + content remain constant as the C I / O ratio is increased. They observed that E #, Tg and the density decreased with increasing C 1 / O ratio up to (LiC1)2/[(LiC1)2 + Li20] = 0.3, whereafter they remain approximately constant to a ratio of 0.5. Direct current transport measurement confirmed that the conductivity was ionic and due to the Li + ion. 0022-3093/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M.J. Ryan, S.I. Smedley / Effect of pressure on fast ion conductivity
30
Thus, because the properties of this system are well known at atmospheric pressure, we decided to make high pressure measurements on the ( L i X ) 2 / B z O 3 / L i 2 0 system over a small range of concentration but over the composition range X = F, C1, Br, I. It was our objective in making these studies to see if we could elucidate the effect of volume as well as temperature and composition on fast ion conductivity.
2. Theory The phenomenon of ionic conductivity in dense media is defined as the product of the concentration of mobile ions c , of charge eZ~ and m o b i l i t y / ~
o =Ecit~izie.
(1)
i
Several theories have been developed to account for the composition and temperature dependence of o. Most theories are concerned only with the temperature dependence of/~ and have their physical basis in the idea that thermal excitation of an ion will enable it to translate from one localized state or site to the next nearest. The energy required to excite the ion from its localized oscillatory state to a translational state is the activation energy E n, and is related to/~ through the random walk model equation to give
I.t
ziedZv . ~ exp(-E*~/RT),
(2)
where d is the ion j u m p distance, v the ion characteristic vibrational frequency [4]. The "Electrostatic Strain-Energy" model of Anderson and Stuart [5] is applicable to this study, and is based on the premise that the activation energy is comprised of two components; an electrostatic term E~ and a strain term E s. The electrostatic term is evaluated as the energy difference between the ion at its position of rest and at a position halfway between adjacent rest or equilibrium sites. This term may be written [5-7] Ee=
(
1-
1)
ZiZoe2 (
A 4 - - - ~ o~e
+ ro
R1)
,
(3)
where R is half the distance between equilibrium sites (R - 2.75J~ for silica [8]), r i and ro are the radii of the mobile ion and an oxygen ion respectively, ez i and ez o their charges, e0 the permittivity of free space and e~ the high frequency relative permittivity of the glass matrix. Anderson and Stuart stated that the product ( 1 - n - 1 ) A where .4 is the Madelung constant and n the exponent in the Born repulsion energy equation was approximately unity; it can be replaced by the fitting parameter P. The strain term E s arises from the elastic energy required to dilate a cavity from the "doorway" radius rd to r, the radius of the migrating ion. For a
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
31
spherical cavity this term is given by the Frenkel equation, modified to include
pAV work E~ = 8~rGrd(r
-
rd)2C 4- ~ ( r
2 - r3),
(4)
where G is the shear modulus of the glass matrix, p the hydrostatic pressure and C a correction factor (0 < C < 1) said to account for the difference between glass structure and that of a close packed liquid, for which the Frenkel equation was derived [6]. The values of ra and C may be derived from the x intercept and slope of a plot of the square root of activation energy for molecular diffusion in the glass matrix versus the radius of the diffusing species [7]. Values so calculated for fused silica are rd = 0.456 A and C = 0.363. However, from the work of Shelby on He diffusion in binary alkali silicate glasses [9-11] it may be seen that the composition dependence of activation energy changes at - 20 mol% alkali, i.e. above the composition region of phase separation. Extrapolating the high concentration trend back to 0 mol% alkali, and assuming an unchanged dependence of activation energy on radius of migrating species, values become rd = 0.306 h,, C = 0.541. Assuming that ZlR/R, Ara/r a change as ½AVm/Vm. o [61 with composition, where Vm is the molar volume of glass and Vm,o the molar volume of fused silica, and estimating G, eo~ from composition dependent literature values [5,6,12-14], eqs. (3) and (4) successfully predict the composition dependence of E ~ for conduction in binary Li:O ( r = 0.6 A), N a 2 0 ( r = 0.95 ,h), K 2 0 ( r = 1.33 A) silicate glasses of 20 mol% alkali and greater. As predicted, 1 < P < 2. Using the above approach we calculate that for Li 2° E s << E e, for N a 2 0 E s - E e and for K 2 0 E s >> E¢, as might be expected from the relative sizes of the ions. By substituting eqs. (3), (4) and (2) into eq. (1), and differentiating with respect to pressure, it is possible to estimate the activation volumes for conductance of alkali metal ions in a silicate glass matrix.
AV ~ = - R T ( 3 In o
T
ap IT t- -p]T
(5)
Now the second term on the right-hand side may be estimated by assuming that R in eq. (3) varies as A R / R = ]tip where fl is the isothermal compressibility [16] of the glass, and P is independent of pressure. The third term may be estimated by assuming that ra varies in a similar manner and that G is practically independent of pressure [15]. Using the appropriate value for these silicate glasses we find that E e depends on pressure to a negligible extent and that AV ~ arises largely from Es. The values calculated in this manner are AV~i.= 0.5 cm 3 mo1-1, AVr~A+----2.1 cm 3 mo1-1, AVe+= 5.9 cm 3 mo1-1. By comparison, the experimental results of Hamann [17] show that the activation volumes for alkali ion conduction in 0.25 L i 2 0 / 0 . 6 8 SiO2; 0.21 N a 2 0 / 0 . 7 2 SiO2; 0.20 K 2 0 / 0 . 7 6 SiO 2 are 1.0 cm 3 mo1-1, 3.5 cm 3 mo1-1, 6.0 cm 3 mol -l,
Md. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
32
respectively. Pressure dependence of the electrostatic term E e may explain the difference. As
AV#-
~-(r3-rd)
and rd < ½r, the correlation of observed AV * with the molar volume of the migrating ion [11] is explained. F r o m eq. (4), we calculate for Li ÷ that E s = 7 kJ mol-1 and since E ~ for Li conduction in these glasses is - 80 kJ tool-1 then Ee >> E s. Thus, on the basis of this model, it would be predicted that Li ÷ conducting oxide glasses of similar structure and Li ÷ concentration should have comparable activation energies. In accord with this prediction, the activation energies for 0.5 Li 20/0.5 Nb205 and 0.5 L i 2 0 / 0 . 5 Ta205 are 42 kJ m o l i and 38 kJ mo1-1 respectively [18], and for L i 2 0 / A I 2 0 3 , L i 2 0 / G a 2 0 3 and L i 2 0 / B i 2 0 3 are within 10% of each other over a mole fraction range of Li ÷ from 0.4 to 0.8 [7]. Although the model has not been tested against the data for this latter set of glasses, it does provide an explanation as to why their conductivities and activation energies are so very similar.
3. Experimental Reagent grade chemicals were used in the preparation of all glass samples. The lithium salts were dried under vacuum at 250°C for seven days and B203 was prepared by bubbling dry nitrogen through the liquid at l l 0 0 ° C . Glass samples were prepared by fusing appropriate amounts of the dried reagents at 1000°C for about five minutes in a Pt crucible. The melts were quenched between warm ( - 400°C) aluminium spacers, and upon glass formation this assembly was placed in an electric furnace at 400°C for 16 h. Glass samples prepared in this manner were disc shaped and had the approximate dimensions of 20 m m by 2 mm. Because the alkali halides are volatile at the fusion temperature, all glass samples were analysed for Li content by atomic absorption spectrometry, for C1, Br and I by precipitation titration; F was determined gravimetrically as CaF 2. The resulting compositions are shown in table 1. Densities were determined by the Archimedes buoyancy method in petroleum ether at 25.9°C. Glass transition temperatures Tg, and crystallization temperatures Tc were determined by differential thermal analysis. The heating rate was 1 0 ° / m i n and the transition temperatures were estimated to occur at the peak positions of the D T A peaks. After annealing, the glass discs were ground under kerosine with fine silicon carbide paper so that the faces were plane parallel. They were washed in trichlorethane and a coating of gold was vacuum deposited on each face. On one face a - 0.8 cm diameter circle was scribed through the gold, the outer
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
33
Table 1 Composition (mol%) Glass
FH
B203 Li 20 (LiX)2
65 69 23 27 12 4.5 Density (g cm -3)
CI l
CI H
CI m
65 29 5.8
2.28 2.17 2.18 Glass transition temperature Tg(°C) 474
510
475
Br I
68 22 10
Brll
67 26 7.0
2.11
2.32
463
Il
65 29 6.2 2.34
463
463
65 32 3.1 2.26 521
ring formed by this procedure was used as a guard electrode, the inner electrode as the conductivity electrode. The dimensions of the conductance cell so formed were measured with a travelling microscope and micrometer. The pressure vessel, its ancillary equipment and thermostat have been described elsewhere. Each glass sample was mounted in the pressure vessel and between three shielded wire conductors that passed through the pressure vessel head, where they were connected to the conductance bridge leads. Sample conductance and capacitance were measured at selected frequencies from 500 Hz to 20 Hz with a Wayne Kerr Bridge coupled to a Wayne Kerr source and detector models, B224, S121 and A321 respectively. Unfortunately the pressure intensifier failed at the beginning of this work and pressure was limited to 2 kbar, the temperature range was limited to a maximum of 250°C by the stability limits of the thermostating fluid, Dow Coming 550 Silicone Oil.
4. Results
Compositions, glass transition temperatures and densities of all the glass samples described in this work are displayed in table 1. For the chlorides the Tg values are slightly higher than those obtained by Button et al. at the same Table 2 Activation energies and volumes of conductivity in alkali borate glasses Glass
Fll
C11
Clll
Clll I
Br I
Bri!
II
E # (kJ m o l - 1) a) A# V (cm 2 m o l - 1) a)
72.2
68.1
60.0
60.0
55.8
57.3
75.5
1.6 1.2 1.4 1.6
1.5 1.4 1.9 1.3
1.5 1.7 1.5 1.5
0.9 1.1 1.2 1.3
1.0 1.0 1.0 1.1
1.1 1.1 1.3 1.8
1.5 1.7 1.6 1.55
1.45
1.53
1.55
1.1
1.0
1.33
1.55
12.2
12.0
373 423 473 523
K K K K
Average
ln(oo/S) (cm -1)
12.4
12.1
12.0
a) Error in A#V +0.1 cm 3 mol - l , and in E n +0.05 kJ mo1-1.
12.4
12.5
34
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
(LiC1)2/(LiX)2 ratio [2], this probably reflects the higher B203mole fraction in our glasses, our Tg values for the fluorides are comparable to those measured by Smedley and Angell. Resistance and capitance of each sample at a given temperature and pressure were converted to real Z', and imaginary Z", components of the complex impedance Z*. When plotted on a complex impedance plane diagram these components generally formed an arc, and in some cases two intersecting arcs. The value of Z ' at the point of intersection of these two arcs, or the point at which the extrapolation of a single arc
15 ~
2
3
4
5
6
T
8
I:11 e l I C111 C i l l I Br I Br11 Ill 10
0~-
-5
-10
0
1
2
3
103K T
Fig. 1. Effect of temperature on the conductivity of borate glasses. The inset code refers to compositions in table 1.
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
35
intersected the Z ' axis was taken as the true ohmic resistance R, of the sample. Conductivity was calculated from R and the cell dimensions. In all cases the glass conductivity decreased with increasing pressure, the activation volumes were independent of pressure within the limits of precision and are displayed in table 2. A significant portion of the quoted error of + 0.1 cm 3 mo1-1 for AV ~ arises from the extrapolation of the impedance data to estimate R, at any given temperature and pressure. The scatter in AV ~ values for any given glass samples most probably arises from the inadequacy of the thermostating system and small changes in temperature over the period of an experiment of approximately 4 h. Notwithstanding these limits on the reliability of the data, some trends in AV # as a function of composition are apparent. Glasses Cll, C1 m, Br~ have comparable compositions, XB2o3=0.68 and XtL~X)2+ Li20 = 0.32, and illustrate the effect on AV # of altering the X / O ratio at constant Li ÷ content. AV ~ decreases in going from sample CI~ to E l m , i.e. as the C I / O ratio is increased, AV ~ also decreases as CI is replaced by Br at constant X / O ratio. This latter trend is also observed of samples C1 m and Br H that have fixed XB~o3 = 0.65, and a fixed X / O ratio. Conductivities as a function of reciprocal temperature at constant pressure are displayed in fig. 1. They follow the characteristic Arrhenius temperature dependence that has been observed for most other fast ion conducting glasses where o = o o / r exp( - E = / R T ) .
Activation energies E #, and pre-exponential factors o0 at 1 atm. are shown in g¢ table 2. The following comparisons can be made; E = for sample CI~, Ecl0), is greater than that for sample Clm; similarly E a o ) > EB~(I), EcI(II) > EBrtii) , and In o 0 is independent of sample composition and pressure. These trends are qualitatively and quantitatively in agreement with those observed by other authors [2].
5. Discussion The most significant feature of the data in table 2 is that activation volumes are small and positive, and comparable to those observed for ionic conductivity in alkali silicate glasses and molten salts. It was shown in the introduction that in terms of the Anderson and Stuart model of ionic conductivity E e was almost independent of pressure and that AV ~* for alkali silicate glasses arose from the pressure dependence of the strain-energy term. Since the structural parameters of compressibility and distance terms for alkali silicate and borate glasses are similar, then the relationship E S << Ee must also apply to the lithium borate glasses. This inequality implies that Li + mobility in borate glasses is determined by the electrostatic energy term and therefore by P, r0 and e~. Previous work [2], as well as that presented here, shows that E ~ decreases as the C 1 / O ratio is
36
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
increased, i.e. as Li20 is replaced by (LiC1)2 whilst the mole fraction B203 is maintained. In terms of the above model the interpretation of these observations would be based on the change of P, ro, R and e~ with increasing (LiC1)2 content. For alkali silicate glasses e~ decreases with increasing amount of alkali oxide. In the absence of suitable high frequency data we would anticipate that since e~ is determined by the polarizability of the molecules comprising the glass; and since C1 is more polarizable than oxygen, there would be an increase in e~ as O is replaced by C1, and a concomitant decrease in E e. Furthermore, some Li ÷ sites will be adjacent to C1- ions and ro < rcj giving rise to a decrease in E e for those ions. Another contribution to the decrease in Ee will come from the diminution in the mean value of R as each O is replaced by 2C1 atoms. Button et al. [2] have preferred to ascribe the decrease in E ~ as the C1 ion content is increased to structural changes in the glass. They suggest that replacement of O by CI results in an increase of 3-fold at the expense of 4-fold coordinated boron and that these change result in either an increase in mobility, or in an increase in the concentration of charge carriers. In terms of the Anderson and Stuart model structural changes such as these would affect the j u m p distances and may also give rise to changes in the electrostatic force between charges. The diminution in AV ~ from glass CI~ to C1 m probably arises from the structural changes outlined above, which presumbly open up new pathways and enlarge others through the glass. The data in table 2 confirm the observations of Levasseur et al. [3] regarding the effect of the halogen ion on the activation energy where, at a given composition, E # decreases in the sequence X = F, C1, Br. In terms of the above discussion this could be attributed to the increasing polarizability from F to Br and the resulting increase in e~, and the increase in rx across the series of glasses. For a fixed composition AV ~ also decreases from C1 to Br (compare glasses CI~ and Br~, C1 n and Brn), and we attribute this fact to the generation of more open pathways as Br replaces C1.
6. Summary and conclusions Activation volumes for lithium ion conductivity in alkali borate glasses are small, in agreement with those observed for alkali silicate glasses. The Anderson and Stuart Electrostatic-Strain Energy model can be used to interpret these observations in terms of a negligibly small Strain Energy for ionic migration. Therefore the observed activation energy must arise principally from the electrostatic term. The diminution in activation energy as the C I / O ratio is increased is explained in terms of a probably increase in e~, as well as an increase in the distance terms in eq. (3). A similar explanation can be invoked to explain the decrease in E ~ across a series of glasses of constant composition as the halogen is replaced in the series F, C1, Br.
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
37
T o s u b t a n t i a t e t h e s e h y p o t h e s e s , o r o t h e r w i s e , we i n t e n d to c a r r y o u t m o r e p r e c i s e studies o n t h e s e glasses o v e r a w i d e r r a n g e o f p r e s s u r e a n d f r e q u e n c y to d e t e r m i n e t h e c o m p o s i t i o n , t e m p e r a t u r e a n d p r e s s u r e d e p e n d e n c e o f e~.
References [1] K. Otto, Phys. Chem. Glasses 7 (1966) 27. [2] D.P. Button, R.P. Tandon, H.L. Tuller and D.R. Uhlmann, J. Non-Crystalline Solids 42 (1980) 297. [3] A. Levasseur, J.C. Brethous, J.M. Reau and P. Hagenmuller, Mat. Res. Bull. 14 (1979) 121. [4] H.L. Tuller, D.P. Button and D.R. Uhlmann, J. Non-Crystalline Solids 40 (1980) 93. [5] O.L. Anderson and D.A. Stuart, J. Am. Ceram. Soc. 37 (1954) 573. [6] R.M. Hakim and D.R. Uhlmann, Phys. Chem. Glasses 12 (1971) 133. [7] R.H. Doremus, Glass Science (Wiley, New York, 1973). [8] R.H. Doremus, in Modern Aspects of the Vitreous State, Vol. 2, ed., J.D. Mackenzie (Butterworths, London, 1960). [9] J.E. Shelby, J. Am. Ceram. Soc. 56(5) (1973) 263. [10] J.E. Shelby, J. Am. Ceram. Soc. 57(6) (1974) 260. [11] J.E. Shelby, J. Appl. Phys. 48(4) (1977) 1497. [12] R.S. Shaw and D.R. Uhlmann, J. Non-Crystalline Solids 4 (1971) 237. [13] R.J. Charles, J. Am. Ceram. Soc. 46 (1963) 235. [14] H.E. Taylor, J. Soc. Glass Tech. 43 (1959) 124. [15] M.H. Manghnami, J. Am. Ceram. Soc. 55 (1972) 360. [16] C.E. Weir and L. Shartsis, J. Am. Ceram. Soc. 38 (1955) 299. [17] S.D. Hamann, Aust. J. Chem. 18 (1965) 1. [18] A.M. Glass, K. Nassau and T.J. Negran, J. Appl. Phys. 49 (1978) 4808.
29
THE EFFECT OF PRESSURE ON FAST ION CONDUCTIVITY IN GLASSES Martin J. R Y A N and Stuart I. S M E D L E Y Chemistry Department, Victoria University of Wellington, New Zealand
Received 13 April 1983 Revised manuscript received 27 September 1983
We have measured the effect of pressure and temperature on the ionic conductivity of glasses in the system BEO3/Li20(LiX)2 (X = F, C1, Br, I), where Xa~o3= 0.7 and the X/O ratio was varied while maintaining a constant Li + content of X (LiX)~ + Li20 0.3. All glasses exhibited a very small pressure coefficient of conductivity, i.e. AV----1cm~ mol-1 which decreased very slightly as the CI/O ratio increased or as Cl was replaced by Br at constant X/O ratio. The results were compared with the ELectrostatic Strain Energy theory; the results of this comparison lead to the conclusion that for Li + conduction the electrostatic term is dominant in determining the activation energy for conduction. m
I. Introduction Significant advances have been made in our knowledge and understanding of the p h e n o m e n o n of fast ion conductivity in solids. These advances have been made possible by the wide range of experimental techniques that have been applied to the study of fast ion conductors which apart from c o n d u c t a n c e studies include N M R , X-ray, infrared and R a m a n spectroscopy. One of the most studied glass conductors is the LiEO/B203 system [1], the structural and c o n d u c t a n c e properties of which were recently reviewed by Button et al. [2]. It suffices to say that LiEO/B203 glasses exhibit high conductivities such that the conductivity o increases and the activation energy E ~ decreases with increasing L i 2 0 content up to about 25 tool%, above which the trend becomes less significant. Levasseur et al. [3] have studied glass with the composition ( L i X ) 2 / L i 2 0 / B 2 0 3 (X = F, C1, Br, I) and have observed that E *~ decreases with increasing (LiC1)2 content up to - 30 mol%. However at a given composition E # decreased monotonically with increasing halogen ion size. Button et al. have systematically studied the conductivity of the system 0.63613203/0.364 [(LiC1)2 + Li20], where the total B203 and Li + content remain constant as the C I / O ratio is increased. They observed that E #, Tg and the density decreased with increasing C 1 / O ratio up to (LiC1)2/[(LiC1)2 + Li20] = 0.3, whereafter they remain approximately constant to a ratio of 0.5. Direct current transport measurement confirmed that the conductivity was ionic and due to the Li + ion. 0022-3093/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M.J. Ryan, S.I. Smedley / Effect of pressure on fast ion conductivity
30
Thus, because the properties of this system are well known at atmospheric pressure, we decided to make high pressure measurements on the ( L i X ) 2 / B z O 3 / L i 2 0 system over a small range of concentration but over the composition range X = F, C1, Br, I. It was our objective in making these studies to see if we could elucidate the effect of volume as well as temperature and composition on fast ion conductivity.
2. Theory The phenomenon of ionic conductivity in dense media is defined as the product of the concentration of mobile ions c , of charge eZ~ and m o b i l i t y / ~
o =Ecit~izie.
(1)
i
Several theories have been developed to account for the composition and temperature dependence of o. Most theories are concerned only with the temperature dependence of/~ and have their physical basis in the idea that thermal excitation of an ion will enable it to translate from one localized state or site to the next nearest. The energy required to excite the ion from its localized oscillatory state to a translational state is the activation energy E n, and is related to/~ through the random walk model equation to give
I.t
ziedZv . ~ exp(-E*~/RT),
(2)
where d is the ion j u m p distance, v the ion characteristic vibrational frequency [4]. The "Electrostatic Strain-Energy" model of Anderson and Stuart [5] is applicable to this study, and is based on the premise that the activation energy is comprised of two components; an electrostatic term E~ and a strain term E s. The electrostatic term is evaluated as the energy difference between the ion at its position of rest and at a position halfway between adjacent rest or equilibrium sites. This term may be written [5-7] Ee=
(
1-
1)
ZiZoe2 (
A 4 - - - ~ o~e
+ ro
R1)
,
(3)
where R is half the distance between equilibrium sites (R - 2.75J~ for silica [8]), r i and ro are the radii of the mobile ion and an oxygen ion respectively, ez i and ez o their charges, e0 the permittivity of free space and e~ the high frequency relative permittivity of the glass matrix. Anderson and Stuart stated that the product ( 1 - n - 1 ) A where .4 is the Madelung constant and n the exponent in the Born repulsion energy equation was approximately unity; it can be replaced by the fitting parameter P. The strain term E s arises from the elastic energy required to dilate a cavity from the "doorway" radius rd to r, the radius of the migrating ion. For a
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
31
spherical cavity this term is given by the Frenkel equation, modified to include
pAV work E~ = 8~rGrd(r
-
rd)2C 4- ~ ( r
2 - r3),
(4)
where G is the shear modulus of the glass matrix, p the hydrostatic pressure and C a correction factor (0 < C < 1) said to account for the difference between glass structure and that of a close packed liquid, for which the Frenkel equation was derived [6]. The values of ra and C may be derived from the x intercept and slope of a plot of the square root of activation energy for molecular diffusion in the glass matrix versus the radius of the diffusing species [7]. Values so calculated for fused silica are rd = 0.456 A and C = 0.363. However, from the work of Shelby on He diffusion in binary alkali silicate glasses [9-11] it may be seen that the composition dependence of activation energy changes at - 20 mol% alkali, i.e. above the composition region of phase separation. Extrapolating the high concentration trend back to 0 mol% alkali, and assuming an unchanged dependence of activation energy on radius of migrating species, values become rd = 0.306 h,, C = 0.541. Assuming that ZlR/R, Ara/r a change as ½AVm/Vm. o [61 with composition, where Vm is the molar volume of glass and Vm,o the molar volume of fused silica, and estimating G, eo~ from composition dependent literature values [5,6,12-14], eqs. (3) and (4) successfully predict the composition dependence of E ~ for conduction in binary Li:O ( r = 0.6 A), N a 2 0 ( r = 0.95 ,h), K 2 0 ( r = 1.33 A) silicate glasses of 20 mol% alkali and greater. As predicted, 1 < P < 2. Using the above approach we calculate that for Li 2° E s << E e, for N a 2 0 E s - E e and for K 2 0 E s >> E¢, as might be expected from the relative sizes of the ions. By substituting eqs. (3), (4) and (2) into eq. (1), and differentiating with respect to pressure, it is possible to estimate the activation volumes for conductance of alkali metal ions in a silicate glass matrix.
AV ~ = - R T ( 3 In o
T
ap IT t- -p]T
(5)
Now the second term on the right-hand side may be estimated by assuming that R in eq. (3) varies as A R / R = ]tip where fl is the isothermal compressibility [16] of the glass, and P is independent of pressure. The third term may be estimated by assuming that ra varies in a similar manner and that G is practically independent of pressure [15]. Using the appropriate value for these silicate glasses we find that E e depends on pressure to a negligible extent and that AV ~ arises largely from Es. The values calculated in this manner are AV~i.= 0.5 cm 3 mo1-1, AVr~A+----2.1 cm 3 mo1-1, AVe+= 5.9 cm 3 mo1-1. By comparison, the experimental results of Hamann [17] show that the activation volumes for alkali ion conduction in 0.25 L i 2 0 / 0 . 6 8 SiO2; 0.21 N a 2 0 / 0 . 7 2 SiO2; 0.20 K 2 0 / 0 . 7 6 SiO 2 are 1.0 cm 3 mo1-1, 3.5 cm 3 mo1-1, 6.0 cm 3 mol -l,
Md. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
32
respectively. Pressure dependence of the electrostatic term E e may explain the difference. As
AV#-
~-(r3-rd)
and rd < ½r, the correlation of observed AV * with the molar volume of the migrating ion [11] is explained. F r o m eq. (4), we calculate for Li ÷ that E s = 7 kJ mol-1 and since E ~ for Li conduction in these glasses is - 80 kJ tool-1 then Ee >> E s. Thus, on the basis of this model, it would be predicted that Li ÷ conducting oxide glasses of similar structure and Li ÷ concentration should have comparable activation energies. In accord with this prediction, the activation energies for 0.5 Li 20/0.5 Nb205 and 0.5 L i 2 0 / 0 . 5 Ta205 are 42 kJ m o l i and 38 kJ mo1-1 respectively [18], and for L i 2 0 / A I 2 0 3 , L i 2 0 / G a 2 0 3 and L i 2 0 / B i 2 0 3 are within 10% of each other over a mole fraction range of Li ÷ from 0.4 to 0.8 [7]. Although the model has not been tested against the data for this latter set of glasses, it does provide an explanation as to why their conductivities and activation energies are so very similar.
3. Experimental Reagent grade chemicals were used in the preparation of all glass samples. The lithium salts were dried under vacuum at 250°C for seven days and B203 was prepared by bubbling dry nitrogen through the liquid at l l 0 0 ° C . Glass samples were prepared by fusing appropriate amounts of the dried reagents at 1000°C for about five minutes in a Pt crucible. The melts were quenched between warm ( - 400°C) aluminium spacers, and upon glass formation this assembly was placed in an electric furnace at 400°C for 16 h. Glass samples prepared in this manner were disc shaped and had the approximate dimensions of 20 m m by 2 mm. Because the alkali halides are volatile at the fusion temperature, all glass samples were analysed for Li content by atomic absorption spectrometry, for C1, Br and I by precipitation titration; F was determined gravimetrically as CaF 2. The resulting compositions are shown in table 1. Densities were determined by the Archimedes buoyancy method in petroleum ether at 25.9°C. Glass transition temperatures Tg, and crystallization temperatures Tc were determined by differential thermal analysis. The heating rate was 1 0 ° / m i n and the transition temperatures were estimated to occur at the peak positions of the D T A peaks. After annealing, the glass discs were ground under kerosine with fine silicon carbide paper so that the faces were plane parallel. They were washed in trichlorethane and a coating of gold was vacuum deposited on each face. On one face a - 0.8 cm diameter circle was scribed through the gold, the outer
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
33
Table 1 Composition (mol%) Glass
FH
B203 Li 20 (LiX)2
65 69 23 27 12 4.5 Density (g cm -3)
CI l
CI H
CI m
65 29 5.8
2.28 2.17 2.18 Glass transition temperature Tg(°C) 474
510
475
Br I
68 22 10
Brll
67 26 7.0
2.11
2.32
463
Il
65 29 6.2 2.34
463
463
65 32 3.1 2.26 521
ring formed by this procedure was used as a guard electrode, the inner electrode as the conductivity electrode. The dimensions of the conductance cell so formed were measured with a travelling microscope and micrometer. The pressure vessel, its ancillary equipment and thermostat have been described elsewhere. Each glass sample was mounted in the pressure vessel and between three shielded wire conductors that passed through the pressure vessel head, where they were connected to the conductance bridge leads. Sample conductance and capacitance were measured at selected frequencies from 500 Hz to 20 Hz with a Wayne Kerr Bridge coupled to a Wayne Kerr source and detector models, B224, S121 and A321 respectively. Unfortunately the pressure intensifier failed at the beginning of this work and pressure was limited to 2 kbar, the temperature range was limited to a maximum of 250°C by the stability limits of the thermostating fluid, Dow Coming 550 Silicone Oil.
4. Results
Compositions, glass transition temperatures and densities of all the glass samples described in this work are displayed in table 1. For the chlorides the Tg values are slightly higher than those obtained by Button et al. at the same Table 2 Activation energies and volumes of conductivity in alkali borate glasses Glass
Fll
C11
Clll
Clll I
Br I
Bri!
II
E # (kJ m o l - 1) a) A# V (cm 2 m o l - 1) a)
72.2
68.1
60.0
60.0
55.8
57.3
75.5
1.6 1.2 1.4 1.6
1.5 1.4 1.9 1.3
1.5 1.7 1.5 1.5
0.9 1.1 1.2 1.3
1.0 1.0 1.0 1.1
1.1 1.1 1.3 1.8
1.5 1.7 1.6 1.55
1.45
1.53
1.55
1.1
1.0
1.33
1.55
12.2
12.0
373 423 473 523
K K K K
Average
ln(oo/S) (cm -1)
12.4
12.1
12.0
a) Error in A#V +0.1 cm 3 mol - l , and in E n +0.05 kJ mo1-1.
12.4
12.5
34
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
(LiC1)2/(LiX)2 ratio [2], this probably reflects the higher B203mole fraction in our glasses, our Tg values for the fluorides are comparable to those measured by Smedley and Angell. Resistance and capitance of each sample at a given temperature and pressure were converted to real Z', and imaginary Z", components of the complex impedance Z*. When plotted on a complex impedance plane diagram these components generally formed an arc, and in some cases two intersecting arcs. The value of Z ' at the point of intersection of these two arcs, or the point at which the extrapolation of a single arc
15 ~
2
3
4
5
6
T
8
I:11 e l I C111 C i l l I Br I Br11 Ill 10
0~-
-5
-10
0
1
2
3
103K T
Fig. 1. Effect of temperature on the conductivity of borate glasses. The inset code refers to compositions in table 1.
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
35
intersected the Z ' axis was taken as the true ohmic resistance R, of the sample. Conductivity was calculated from R and the cell dimensions. In all cases the glass conductivity decreased with increasing pressure, the activation volumes were independent of pressure within the limits of precision and are displayed in table 2. A significant portion of the quoted error of + 0.1 cm 3 mo1-1 for AV ~ arises from the extrapolation of the impedance data to estimate R, at any given temperature and pressure. The scatter in AV ~ values for any given glass samples most probably arises from the inadequacy of the thermostating system and small changes in temperature over the period of an experiment of approximately 4 h. Notwithstanding these limits on the reliability of the data, some trends in AV # as a function of composition are apparent. Glasses Cll, C1 m, Br~ have comparable compositions, XB2o3=0.68 and XtL~X)2+ Li20 = 0.32, and illustrate the effect on AV # of altering the X / O ratio at constant Li ÷ content. AV ~ decreases in going from sample CI~ to E l m , i.e. as the C I / O ratio is increased, AV ~ also decreases as CI is replaced by Br at constant X / O ratio. This latter trend is also observed of samples C1 m and Br H that have fixed XB~o3 = 0.65, and a fixed X / O ratio. Conductivities as a function of reciprocal temperature at constant pressure are displayed in fig. 1. They follow the characteristic Arrhenius temperature dependence that has been observed for most other fast ion conducting glasses where o = o o / r exp( - E = / R T ) .
Activation energies E #, and pre-exponential factors o0 at 1 atm. are shown in g¢ table 2. The following comparisons can be made; E = for sample CI~, Ecl0), is greater than that for sample Clm; similarly E a o ) > EB~(I), EcI(II) > EBrtii) , and In o 0 is independent of sample composition and pressure. These trends are qualitatively and quantitatively in agreement with those observed by other authors [2].
5. Discussion The most significant feature of the data in table 2 is that activation volumes are small and positive, and comparable to those observed for ionic conductivity in alkali silicate glasses and molten salts. It was shown in the introduction that in terms of the Anderson and Stuart model of ionic conductivity E e was almost independent of pressure and that AV ~* for alkali silicate glasses arose from the pressure dependence of the strain-energy term. Since the structural parameters of compressibility and distance terms for alkali silicate and borate glasses are similar, then the relationship E S << Ee must also apply to the lithium borate glasses. This inequality implies that Li + mobility in borate glasses is determined by the electrostatic energy term and therefore by P, r0 and e~. Previous work [2], as well as that presented here, shows that E ~ decreases as the C 1 / O ratio is
36
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
increased, i.e. as Li20 is replaced by (LiC1)2 whilst the mole fraction B203 is maintained. In terms of the above model the interpretation of these observations would be based on the change of P, ro, R and e~ with increasing (LiC1)2 content. For alkali silicate glasses e~ decreases with increasing amount of alkali oxide. In the absence of suitable high frequency data we would anticipate that since e~ is determined by the polarizability of the molecules comprising the glass; and since C1 is more polarizable than oxygen, there would be an increase in e~ as O is replaced by C1, and a concomitant decrease in E e. Furthermore, some Li ÷ sites will be adjacent to C1- ions and ro < rcj giving rise to a decrease in E e for those ions. Another contribution to the decrease in Ee will come from the diminution in the mean value of R as each O is replaced by 2C1 atoms. Button et al. [2] have preferred to ascribe the decrease in E ~ as the C1 ion content is increased to structural changes in the glass. They suggest that replacement of O by CI results in an increase of 3-fold at the expense of 4-fold coordinated boron and that these change result in either an increase in mobility, or in an increase in the concentration of charge carriers. In terms of the Anderson and Stuart model structural changes such as these would affect the j u m p distances and may also give rise to changes in the electrostatic force between charges. The diminution in AV ~ from glass CI~ to C1 m probably arises from the structural changes outlined above, which presumbly open up new pathways and enlarge others through the glass. The data in table 2 confirm the observations of Levasseur et al. [3] regarding the effect of the halogen ion on the activation energy where, at a given composition, E # decreases in the sequence X = F, C1, Br. In terms of the above discussion this could be attributed to the increasing polarizability from F to Br and the resulting increase in e~, and the increase in rx across the series of glasses. For a fixed composition AV ~ also decreases from C1 to Br (compare glasses CI~ and Br~, C1 n and Brn), and we attribute this fact to the generation of more open pathways as Br replaces C1.
6. Summary and conclusions Activation volumes for lithium ion conductivity in alkali borate glasses are small, in agreement with those observed for alkali silicate glasses. The Anderson and Stuart Electrostatic-Strain Energy model can be used to interpret these observations in terms of a negligibly small Strain Energy for ionic migration. Therefore the observed activation energy must arise principally from the electrostatic term. The diminution in activation energy as the C I / O ratio is increased is explained in terms of a probably increase in e~, as well as an increase in the distance terms in eq. (3). A similar explanation can be invoked to explain the decrease in E ~ across a series of glasses of constant composition as the halogen is replaced in the series F, C1, Br.
M.J. Ryan, S.L Smedley / Effect of pressure on fast ion conductivity
37
T o s u b t a n t i a t e t h e s e h y p o t h e s e s , o r o t h e r w i s e , we i n t e n d to c a r r y o u t m o r e p r e c i s e studies o n t h e s e glasses o v e r a w i d e r r a n g e o f p r e s s u r e a n d f r e q u e n c y to d e t e r m i n e t h e c o m p o s i t i o n , t e m p e r a t u r e a n d p r e s s u r e d e p e n d e n c e o f e~.
References [1] K. Otto, Phys. Chem. Glasses 7 (1966) 27. [2] D.P. Button, R.P. Tandon, H.L. Tuller and D.R. Uhlmann, J. Non-Crystalline Solids 42 (1980) 297. [3] A. Levasseur, J.C. Brethous, J.M. Reau and P. Hagenmuller, Mat. Res. Bull. 14 (1979) 121. [4] H.L. Tuller, D.P. Button and D.R. Uhlmann, J. Non-Crystalline Solids 40 (1980) 93. [5] O.L. Anderson and D.A. Stuart, J. Am. Ceram. Soc. 37 (1954) 573. [6] R.M. Hakim and D.R. Uhlmann, Phys. Chem. Glasses 12 (1971) 133. [7] R.H. Doremus, Glass Science (Wiley, New York, 1973). [8] R.H. Doremus, in Modern Aspects of the Vitreous State, Vol. 2, ed., J.D. Mackenzie (Butterworths, London, 1960). [9] J.E. Shelby, J. Am. Ceram. Soc. 56(5) (1973) 263. [10] J.E. Shelby, J. Am. Ceram. Soc. 57(6) (1974) 260. [11] J.E. Shelby, J. Appl. Phys. 48(4) (1977) 1497. [12] R.S. Shaw and D.R. Uhlmann, J. Non-Crystalline Solids 4 (1971) 237. [13] R.J. Charles, J. Am. Ceram. Soc. 46 (1963) 235. [14] H.E. Taylor, J. Soc. Glass Tech. 43 (1959) 124. [15] M.H. Manghnami, J. Am. Ceram. Soc. 55 (1972) 360. [16] C.E. Weir and L. Shartsis, J. Am. Ceram. Soc. 38 (1955) 299. [17] S.D. Hamann, Aust. J. Chem. 18 (1965) 1. [18] A.M. Glass, K. Nassau and T.J. Negran, J. Appl. Phys. 49 (1978) 4808.