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VOLUME 97, NUMBER 35, SEPTEMBER 2, 1993
LETTERS Preferential Paths in Alkali Ion Migration and the Mixed Alkali Effect in Silicate Glassed S. Balasubramanian and K. J. Rao' Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, India Received: April 27, 1993; In Final Form: June 21, 1993'
The mixed alkali effect (MAE) in glasses has been investigated using molecular dynamics simulation. For the first time, it has been establishedusing Van Hove type correlation functions that alkali ions exhibit site preferences during migration. The occurrence of conductivity minimum has been demonstrated, and reasons for suppression of MAE a t high temperatures are indicated.
Introduction
Methodology
The mixed alkali effect (MAE) is the manifestation of pronounced nonlinearity in physical properties of glasses when one alkali is substituted for the other.'" Transport properties such as conductivity, viscosity, etc., exhibit such nonlinearities. Static properties such as density are generally insensitive to such substitution. For example, in a xLizO( 1 - x)Kz0-2SiOz glass, the conductivity dips by 5 orders of magnitude below linearly interpolatedvalueswhen lithium ions are substituted by potassium ions.2 It is observed experimentally that the self-diffusion coefficientsof the alkali ions decrease markedly in the interalkali composition^.^ However, the mixed alkali behavior in conductivity tends to disappear at higher temperatures? There has been a renewed interest toward understanding the origin of MAE recently.9-15 Bunde et al.10-14have approached MAE on the basis of memory of occupancy of the vacancies. Thus, Li+/K+ ion vacancies retain memory of their original occupants for a typical time within which period only an ion of the same type is allowed tooccupy thevacancy.14 In the context of conductivity,it amounts tothepresenceofa pathmemory. This feature has beenidentified in the transport behavior of mixed cation glasses.I6 In this Letter, we provide microscopic evidence for the existence of preferred pathways for the migration of two types of alkali ions by probing their dynamics in a 35(xLiZO(1 -x)K20).65Si02 glass using the molecular dynamics method.
Molecular dynamics (MD) simulation was performed on a system of 960 atoms (224 alkali, 208 silicon, and 528 oxygen atoms) in the microcanonical ensemble (NEV). The initial configuration was chosen randomly with a minimum interatomic separation of 1.8 A to avoid unphysical contacts. The BornMayer-Huggins potential with the same parameters as those used by Tesar and Varshneya" was employed. The dynamical equations of motion were integrated using Verlet algorithm with a time step of 1 fs. The Ewald summation procedure was employed to evaluate the coulombic interactions.1* The system was first equilibrated at 5000 K for 40 ps and then quenched in stages (quench rate of 0.2 K/fs) to 3000 and 1000 K. Similar equilibration times (40 ps) were employed at each temperature. At 1000K, static and dynamical property averages were collected for 90 ps. Various properties including pair correlation functions, mean-square displacements (MSD), velocity autocorrelation functions (VAF), and time-dependent pair correlation functions were evaluated. In all, five compositions were studied. They wereC1 ( x = l.O), C2 (x = 0.741), C3 (x = O S ) , C4 (x = 0.259), and C5 (x = 0.0).
Contribution No. 926. *Abstract published in Advance ACS Abstracts, August 15, 1993,
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Results and Discussion At 1000 K, we have observed that the skeleton of the silicate network consistingof siliconand oxygen atoms is essentially static since the long time slope of the,MSD curves of silicon and oxygen atoms are nearly zero. Indeed, 1000 K is well below the glass 0 1993 American Chemical Society
8836 The Journal of Physical Chemistry, Vol. 97, No. 35, 1993
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J 0.2
0.4
0.6
0.8
1
K/(K+Li)
Figure 1. (a) Alkalidxygen pair distribution function in composition C3. (b) Variation of self-diffusion coefficient of Li+(X) and K+(0)ions
with mole fractionof potassium. Inset: variation of electricalconductivity with mole fraction of potassium. Lines are only indicativeof the trends. Note the Occurrence of the mixed alkali effect. transition temperature of the computer-simulatedsilicateglasses1g (typically around 2000 K) and is thus considered as a glassy region in our work. We have also observed strong spatial correlations among Li-Li, K-K, and Li-K pairs (see later) in their respective pair correlation functions. Figure l a shows the Li-O and K-O pair correlation functions in composition C3. The peak positions areat 2.05 and 2.7A, respectively. Thesedistances are unchanged from the values in single alkali silicate glasses as observed in EXAFS measurements.20Areas under the first peak give a coordination number of 3.8 for Li and 7.6 for K. This difference in the number of oxygens surrounding the alkali ions can be attributed to differences in the sizes of K+ ( r = 1.33 A) and Li+ ( r = 0.67 A) ions. The power spectrum of lithium ions (Fourier transform of the VAF of Li+ ions) exhibits a peak at 380 cm-1 while that of potassium exhibits a peak at 120 cm-l. These correspond to the cage vibration frequencies of the alkali ions as observed in far-infrared experiments on these glasses.21.22 The half-widths at half-maximum (hwhm) values are 200 and 100 cm-1 for Li+ and K+ ions, respectively, and they remain essentially unaltered in the mixed alkali compositions. The constancy of both vibrational frequencies and the hwhm values are a strong indication of the absence of any unusual interalkali interactions related to vibrational properties.23 Composition independenceof cage vibration frequencies is in good agreement with experiments.24 The effect of the addition of alkali oxide is generally visualized as breaking the network and creating nonbridging oxygens (NBO) .25 In our simulations, NBOs are identified using a distance criterion. An oxygen ion is counted as a NBO if its distance to one of the silicon neighbors is greater than 2.4 A. We observe that nearly 26% of the total number of oxygens are NBOs in the pure Li composition C1, while 35% of the oxygens are NBOs in the pure potassium CS composition. The latter is in good agreement with experimental results. The lower value of 26% of NBO in composition C1 is an indication of the inadequacy of a distance criterion when cation radius is small. The coordination sphere of the small Li+ ion consists of fewer oxygen atoms than that of the K+ ion. The self-diffusion coefficients of the alkali ions have been calculated from the MSD data for all compositionsand are presented in Figure 1b. Electrical conductivity has been calculated using the Nernst-Einstein
I
r (A)
Figure 2. Time-dependent pair correlation function Gi(r,t) for (a) Li+ ion migration to Li+vacancy site in composition C1 and (b) same as (a), but shown for r I 5.0 A. Time in picoseconds: 0, (-)0.2, (- -)l,
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(-e)
(- -)20, (- -)50.
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and the variation of the same is shown as an inset to Figure lb. MAE is clearly evident in Figure 1b (inset) although its magnitude is somewhat lower than those reported in experiments. Both static and dynamical properties of mixed alkali silicate glasses are satisfactorily simulated in the present work. In order to gain further insight to the ionic motion, we have probed the microscopic dynamics of alkali cations. Toward this purpose, we have used the time-dependent pair correlation function27 which is defined as
Gf(r,t) = Vn,(r,t)/NiN,4d d r where nj(r,t) is the number of particles of type j situated at a distance between r and r + dr at time t from the position which was occupied by a given atom of type i at t = 0. Vis the system volume, and Ni and N,represent the number of particles of type i and j . n,(r,t) is a time and ensemble averaged quantity. Gz(r,t) probes the temporal evolution of spatial correlations. It is closely related to the Van Hove correlation function26 and is the Fourier transform of the dynamic structure factor which can be obtained from inelastic neutron scattering experiments. At time t = 0, Gj(r,t) reduces to the familiar pair correlation function, g-’(r). We now define a K+/Li+ vacancy site for convenience as the position occupied by a K+/Li+ ion at t = 0. Such a Li+ or K+ ion occupying a site at t = 0 quite generally moves out after a time rendering the site vacant. The site which is referred to hereafter as a “vacancy” is then ready for reoccupation by any other ion. In particular, we have employed G$(r,t)to study the nature of the migration of one type of alkali ion (Li+ or K+) to a vacancy site of its own type (Li+ or K+) or of the complementary type (K+ or Li+). Figure 2a shows the Gj(r,t)curves at five different intervals for the migration of Li+ ions with respect to a Li+ vacancy site (j represents Li+ ion, i the Li+ vacancy) in composition C1. The region of G j ( r , f )up to r = 5.0 A is separately shown in Figure 2b to enhance clarity in this region. A striking feature of Figure
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The Journal of Physical Chemistry, Vol. 97,No. 35, 1993 8837
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0
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25 (c)
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LI-LI ( 0 1 )
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LI-K ( 0 9 )
20
60
40 time (ps)
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1 -
! 60
Figure 4. Number of alkali ions occupying (a) a Li+ vacancy site and (b) a K+ vacancy site (see text for details).
Figure 3. Time-dependent pair correlation function GY(r,t) in composition C3 for (a) Li+ ion migration to Li+vacancy, (b) K+ ion migration to Li+ ion vacancy, (c) Li+ ion migration to K+ vacancy, and (d) K+ ion migration to K+ vacancy. Time in picoseconds: 0, (-)0.2, (- - -)l, (- - -)20,(- -)50. (.e.)
2a or of Figure 2b is the gradual buildup of Gy(r,t) close to the origin (0.1 A) as a function of time with a simultaneous drop in the intensity of the curve at 2.8 A. It is evident that it involves at least two steps. One, the reference ion has moved out of its own site creating a vacancy, and two, any of the other ions has moved into this reference site. During this time some of the ions contributing to the first peak in Gy(r,t)have also moved out. The 0.1-A position is very close to the reference site, and we treat it as the reference site itself because the 0.1-A shift partly reflects structural relaxation of the local environment. In Figure 3a similiar Gy(r,t) has been presented for Li+-Li+ correlation in a mixed alkali composition,C3. Again, weobserve that the temporal behavior of the peaks at 2.8 and 0.1 A suggests that Li+ ion migration is almost identical to that in Figure 2b. The inference is clear; K+ ions do not influence Li+ ion migration to a Li+ vacancy. Further, while the peak at 0.1 A grows in intensity, there is no intensity buildup in the region between the 0.1-A peak and the 2.8-A peak. It is a definite indication of the presence of spatially well-defined alkali sites in the glassy matrix. The most important finding of this work is contained in Figure 3b where G$(r,t) of K+ ions with reference to a Li+ site is presented. For the same time interval of 50 ps, we note that there is hardly any buildup of intensity at 0.1 A (or, for that matter, up to a distanceof 2.0 A), although thereis a clear loss of intensity of the peak at 3.025 A. This is in sharp contrast to the Gf (r,t) of the Li+ ion in Figure 3a. The inference from Figure 3b is that K+ ions do not prefer migration to a Li+ vacancy. The nonpreferenceof K+ ion to migrate to a Li+vacancy would appear to be due to the smaller size of the Li+ vacancy compared to the size of K+ ions. We have therefore examined Gj(r,t) for the complementary migration, namely, Li+ ions to K+ vacancies (Figure 3c). Indeed, the behavior exhibited in Figure 3c is very similar to that in Figure 3b. In Figure 3d we present Gy(r,t)for K+ ion migration to K+ vacancy site. The behavior is rather close to that in Figure 3a. Therefore, Li+/K+ions do not migrate
to K+/Li+vacancies while they migrate preferentially to Li+/K+ ~acuncies.1~J4 Site preferences exhibited by the alkali ions could be due to unfavorable potential energies in unlike ion sites. We have therefore evaluated in composition C3 the various site energies which are as follows: Li+ ion in Li+ site
-8.18 f 0.56 eV/atom
Li+ ion in K+ site
-5.18 f 0.52 eV/atom
K+ ion in K+ site
-4.41 f 0.36 eV/atom
K+ ion in Li+ site
-3.08 f 0.52 eV/atom
These energies were averaged over several sites. Li+ ion in Li+ site and K+ ion in K+ site are normal occupancies while Li+ ion in K+ site and K+ ion in Li+ site are not. Therefore, energies for the latter situations were obtained by artificially relabeling a single K+ ion as Li+ or Li+ ion as K+. Energies were averaged over several such single site substitutions. It is seen above that an alkali ion experiences a higher potential energy in an unlike ion vacancy site than in a like ion vacancy site. These results agree well with those of Uchino et al., who calculated cation mismatch energies using ab initio Hartree-Fock calculations on mixed alkali disilicate clusters.29 It should also be noted that the thermal energy at 1000 K is only 0.086 eV/atom and is not high enough to suppress the site preferences. It is also interesting to note that the growth of the 0.1-A peak occurs at the expense of the first-neighborpeakcorrespondingto like ion correlationduring the typical simulation time of 50 ps. This is an indication that the migration into the vacancy is largely from nearest-neighbor distances. This is further confirmed in Figure 2a where the second peak remains unaffected during this period. The same behavior is observed for the occupation of K+ ion vacancy also. We have calculated the number of ions corresponding to the area under the new peak. This is plotted as a function of time for Li-Li (Cl), Li-Li (C3), and Li-K (C3) in Figure 4a and for K-K (CS), K-K (C3), and K-Li (C3) in Figure 4b. The compositionsare indicated in parentheses. It is evident from the figures that the number of ions appearing at the vacancy and its
8838 The Journal of Physical Chemistry, Vol. 97, No. 35, 1993 variation with time (the slope) are larger in the Li-Li (C3) case than in the Li-K (C3) case. These results conclusively indicate that the migration of an ion to a vacancy of unlike ion is not as facile as that to a like ion vacancy. We are therefore led to conclude that thevacancy retains memoryof the occupant ion.1*J4 The implications of the curves in Figure 4b are identical. Preferential jumps into vacancy sites are tantamount to preferred path migration. We have so far examined in detail the microscopic dynamics associated with the MAE and have established a preferred path migration of alkali ions. We wish to speculate the reason for this behavior as follows: We have seen that a lithium ion is typically surrounded by four oxygen atoms, whereas a potassium ion is surrounded by about eight oxygen atoms. These coordinations are dictated by their sizes and interaction potentials. Thus, when a lithium site becomes vacant, the structure around the site and hence the number of oxygen atom neighbors remain unaffected. Well below the glass transition temperature where MAE is pronounced, the ‘rigid’ network to which oxygen atoms belong takes a long time to relax. Within this period, a neighboring lithium can migrate into this vacancy, whereas a potassium ion cannot as the local potential is quite unfavorable for its occupation, as illustrated earlier. It should also be noted that though a K+ vacancy site is large enough to hold a Li+ ion migrant, the probability for such an occupation is low as the potential experienced by the lithium ion in this vacancy site is higher than the energy it experiences in its own site. At higher temperatures the anion matrix relaxations become easier (shorter relaxation times). The oxygen ions rapidly rearrange in order to accommodate either of the cations at the vacancy site and with appropriate site energies. Thus, conductivity tends to exhibit linear variations and the magnitude of MAE diminishes. Conclusions The important result of this work is the identification of preferred site migration which determines the migration paths for alkali ions in mixed alkali silicate glasses. When account is taken of this feature, the calculated conductivities exhibit MAE. A decrease in the magnitude of MAE a t high temperatures can be attributed to the facile relaxation of anions. The possibility of special interalkali ion interactionsz1is not favored since neither the alkali ion vibrational frequencies nor their widths vary as a function of composition.
Letters Acknowledgment. The authors are thankful to the Commission of the European Communities for the award of a fellowship to S.B. and to the Jawaharlal Nehru Centre for Advanced Scientific Research for computational facility. References and Notes (1) Isard, J. 0. J. Non-Cryst. Solids 1969, I , 235. (2) Day, D. E. J. Non-Cryst. Solids 1976, 21, 343. (3) Ingram, M. D. Phys. Chem. Glasses 1987, 28,215. (4) Rao, K. J.; Sundar, H. G. K. Phys. Chem. Glosses 1980, 21, 216. (5) Selvaraj, U.; Rao, K. J. Spectrahim. Acta 1984, IOA, 1081. (6) Damodaran, K. V.; Rao, K. J. Phys. Chem. Glumes 1989,30, 130. (7) Varshneya, A. K. J. Am. Cerum. Sa. 1974,57, 37. (8) Moynihan, C. T.; Saad, N. S.; Tran, D. C.; Lesikar, A. V. J. Am. Cerum. Soc. 1980, 63, 458. (9) Huang, C.; Cormack, A. N. J. Muter. Chem. 1992, 2, 281. (10) Maass, P.; Peterson, J.; Bunde, A.; Dieterich, W.; Roman, H. E. Phys. Rev. Lezt. 1991, 66, 52. (1 1) Bunde, A.; Ingram, M. D.; Maass, P.; Ngai, K. L. J. Phys. A: Murh. Gen. 1991, 24, L881. (12) Bunde, A.; Ingram, M. D.; Maass, P.; Ngai, K. L. J. Non-Crysr. Solids 1991, 131, 1109. (13) Harder, H.; Bunde,A.; Dieterich, W. J. Chem.Phys. 1986,85,4123. (14) Maass, P.; Bunde, A,; Ingram, M. D. Phys. Rev. Lett. 1992,68,3064. (15) Vessal, B.; Greaves, G. N.; Marten, P. T.; Chadwick, A. V.; Mole, R.; Houde-Walter, S. Nuture 1992, 356, 504. (16) Ananthraj, S.; Rao, K. J. Phys. Chem. Glasses, communicated. (17) Tesar, A. A.; Varshneya, A. K. J. Chem. Phys. 1987,87,2986. (18) Allen, M. P.; Tildesley, D. J. Computer Simulution of Liquids; Clarendon Press: Oxford, 1987; p 156. (19) Soules, T. F. J. Non-Cryst. Solids 1982, 49, 29. (20) Greaves,G. N.Philos.Mug.1989,B60,793. Greaves, G. N.;Catlow,
C. R. A.; Vasal, B.; Charnock, J.; Henderson, C. M.B.;Zhu, R.; Qiao, S.; Wang, Y.;Gurman, S. J.; Houde-Walter, S . Int. Con$ Proc. Ser. 1990,111,
411. (21) Exarhos, G. J.; Miller, P. J.; Risen, Jr., W. M. SolidSzute Commun. 1975, 17, 29. (22) Exarhos, G. J.; Risen, Jr., W. M. Solid Store Commun. 1972, 11, 755. (23) Hendrichon, J. R.; Bray, P. J. Phys. Chem. Glasses 1972, 13, 43. (24) Rouse, Jr., G. B.; Miller, P. J.; Risen, Jr., W. M. J.Non-Crysz.Solids 1978, 28, 193. (25) Faman, I.; Grandinetti, P. J.; Baltisberger, J. H.; Stebbins, J. F.; Werner, U.; Eastman, M. A.; Pines, A. Nuture 1992, 358, 31. (26) Atkins, P. W. Physicul Chemistry;Oxford University Press: Oxford, 1987; p 675. (27) Rahman, A. Phys. Reu. 1964,136, A405. (28) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids;Academic Press: London, 1986; p 216. (29) Uchino, T.; S i b , T.; Ogata, Y.;Iwasaki, M. J. Non-Cryst. Solids 1992, 146, 26.
