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Effect of mixing mobile ions in glasses on transport processes
324
Journal of Non-Crystalline Solids 123 (1990) 324-327 North-Holland
EFFECT OF MIXING MOBILE IONS IN GLASSES ON TRANSPORT PROCESSES Z. BOKSAY Institute for General and Inorganic Chemistry, E6toi~s LorAnd Unioersity, Budapest, Hungary
A computer simulation for the ionic movement in glass has been developed in order to explain the decrease of conductivity caused by mixing alkali ions. An account of the composition dependence of the transference number of alkali ions involved is also included. According to the model used, a vacancy in the glass migrates in a three-dimensional path consisting of junctions, terminal and internal sites for alkali ions. It is assumed that the jump of a larger alkali ion into an empty site vacated by a smaller ion is impeded. The consequences of the replacement of alkali ions by protons are also discussed.
1. InWoduction Glasses containing two types of alkali ions in a commensurable amount exhibit conductivities lower by several orders of magnitude than do glasses with one type of alkali ion. The transference number of ions involved (the ratio of the charge transported by the given ion to the total transported charge) varies monotonically with the composition, usually according to an S-shaped curve. In figs. I(A) and 2(A) the logarithm of the conductivity and the transference number, respectively, of the larger ion are plotted versus the mole fraction of the larger ion for Li-Na, L i - K and N a - K glasses [3,5]. The decrease of conductivity is a rather astonishing effect not quite free of mystery. From 1925 when the effect was discovered [1], many efforts have been made to solve the problem; however, no explanation proposed has been generally accepted so far. An attempt to elucidate the foundation of this phenomenon will be developed using a computer simulation.
(1) In the glasses considered charge is carried by alkali ions located in the holes of the random silicate network. (2) The alkali ions are coordinated by more than one non-bridging oxygen ion; the rest of the coordination sphere consists of bridging oxygen atoms. The total coordination number depends on the diameter of an alkali ion present in the posi-
-4q
A
B
-6
-7 0.5
2. Assumptions In order to keep the extent of this paper within reasonable limits, the structural background will only be outlined in a set of postulates forming a conduction model of glass.
2
1.
0
.0
Mole f r a c t i o n of the larger ion Fig. 1. Comparison of conductivity measured and vacancy diffnsivity calculated. (A) The conductivity measured at 300 ° C, of mixed alkali glasses containing 16 mol% M20, 8 mol% MgO, 8 mol~ BaO and 68 mol~ SiO2. (B) The relative diffusivity of the vacancy calculated with parameters specified in the text.
0022-3093/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
Z. Boksay / Effect of mixing mobile ions in glasses on transport processes
1.0
0.5 2 ~
0
0.5 1.0 0 0.5 Mole fraceion of the larger ion
1.0
Fig. 2. Similarity of curves for the transference numbers of alkali ions, (A) measured in mixed alkali glasses at 270 o C, and (B) calculated according to the model, respectively. tion when the glass structure was frozen. The coordination number for a site remains unchanged even when the original alkali ion is replaced by another one. (3) The number of sites able to accommodate alkali ions is somewhat higher than that of alkali ions. The number of the surplus sites, i.e. the vacancies, is constant in a wide temperature range. (4) The ions and the vacancies move in a threedimensional network. An empty site can be occupied by a neighbouring alkali ion only from one, two or three directions depending on the local conditions. These directions are associated with a relatively low potential barrier the height of which is identical with the activation energy of the electrical conduction. (5) Most sites are only linked with two others through such a barrier and so form internal members of a chain of sites. In the network there are also junctions with three exits. Some chains of sites form impasses with a junction and a terminal site at the ends. Figure 3 shows a two-dimensional representation of the network of ionic movement.
325
(6) In mixed alkali glasses, the j u m p of the larger ion into the vacated site of a smaller one requires an elevated energy. The relevant potential barrier should be either overcome or avoided in a more or less long roundabout way by a migrating vacancy. Both alternatives diminish the electric conduction, diffusion of ions and other ionic transport processes. (7) The impeded movement of the vacancy in the network is assumed to be decisively responsible for the mixed alkali effect. Other influences, e.g. that of the density anomaly, are less significant ones and are not considered here.
3. Model simulation The above items represent the final stage of the development of a model elaborated from 1954 until 1974 [2-6]. In order to confirm this conduction model, a computer simulation was performed. The program used is regarded as a first approximation and, obviously, has incorporated some simplifications as follows. (1) The network for the migrating vacancy is gradually generated by computer in the nearby region of the actual location of the vacancy. When a new chain of sites is generated an old one is cancelled in the computer memory. (2) The chains of sites consist of linked elementary distances. All j u m p distances are equal and parallel with one of the Cartesian coordinate axes. The total length of each chain is constant (see fig. 4). (3) The larger and the smaller alkali ions are statistically distributed in junctions, chain sites and terminal sites..
Termin~!c ~mpass
~tion
~ ' ~ ' ~ ' Vacancy Fig. 3. Junctions, terminal and internal sites in the network for the migration of the vacancy and alkali ions.
Fig. 4. Two-dimensional representation of the network of ionic movement according to the model applied.
326
Z. Boksay / Effect of mixing mobile ions in glasses on transport processes
(4) The alkali ion concentration is assumed to be constant in the whole glass series. (5) The time between two subsequent jumps is taken as constant; the number of displacements of a vacancy measures the time. (6) In the calculation of the probability of alternative jumps, the weighting factor for the jump of the larger ion into a smaller site is less than the common value attributed to the rest of the jumps of any kind. (7) The random walk of the vacancy always starts from the centre of the coordinate system. After certain steps the square of the distance between the starting point and the final position of the vacancy is registered. The mean value of the square of the distance for a relatively large number of vacancy runs is calculated. (8) The ratio of the square of the distance to the number of steps is used as a relative diffusivity of the vacancy. (9) If the vacancy has succeeded in moving from one junction into another through a chain of sites, the displacements of the ions involved are considered in the calculation of the transference number. If the vacancy leaves the chain where it entered, the displacements in the chain are cancelled. (10) The parameters chosen are as follows. The ratio of the weighting factor for the simple and hindered jumps is 20:1. The length of the chains is 10 in distances of elementary jumps as units. One fifth of the chains forms impasse. The random walk for a given mole fraction was simulated in five series with 20 runs in each. For the relative diffusivity of the vacancy, the standard deviation of the 5 series is as high as 22%, on the average. The transference number of the larger ion was calculated with an accuracy of +0.002. The logarithm of the relative diffusivity was plotted against the composition in fig. I(B). Since the diffusivity D of the vacancy is proportional to the ionic conductivity ~¢, according to the Nernst-Einstein equation lg D and lg x have curves of the same shape. Thus the composition dependence of the logarithm of calculated DreI can be compared with that of the measured conductivity. The characteristic similarity of the curves, manifested in the deep minima, indicates the use-
+m z
~
o
~
0
1
2
Depth/pm Fig. 5. Relative sodium ion concentration after 14 days leaching at 40 o C, in the surface layer of a glass containing 22% Na20, 6% CaO and 68% SiO2 (wt%). fulness of the hindered migration model. While the conductivity goes through a minimum, the related transference number obtained varies according to an S-shaped curve similar to the experimental ones (fig. 2). Thus the model correctly describes the experiences in two respects: it describes the compositional dependence of both the conductivity and the transference number. By the variation of the parameters a close agreement between the calculated results and the experimental data is expected. The model can be applied to relaxation phenomena as well. Replacement of an alkali ion by a proton may lead to high resistance [8]. One may think that H + ions behave like alkali ions, causing a mixed alkali effect, but this interpretation is wrong. Mixing of alkali ions with protons takes place when the glass is leached. Owing to the leaching a thin layer is formed on the glass surface in which the alkali
-5
f
tx0
10
0
2
Depth/pm Fig. 6. Variation of the conductivity vdth the depth in the
surface layer of the leached sodium calcium silicate glass [9].
327
Z. Boksay / Effect of mixing mobile ions in glasses on transport processes
concentration increases with the depth and tends to a constant value while the proton concentration decreases and tends to zero. Figure 5 shows the relative sodium ion concentration of a sodium calcium silicate glass. (The alkali content in the outer part of the surface layer is supposed to be blocked in the structure and not to take part in the conduction.) The proton in the surface layer is usually bound to a silicate anionic group and forms a silanol group. Its movement reqires the presence o f a nearby water molecule or an anionic group. Obviously, the latter is associated with metal ions, mostly with alkali ions. If the proton movements are shown by bent arrows, we may write I
[
--si-\ ~ O--H O /\ H
--si-\ O--H H
-5
/
-lo
i
0
0.5
1.0
Mole f r a c t i o n of Na Fig. 7. Conductivity in the inner part of the interface layer of the leached glass plotted against the mole fraction of the sodium ion.
I
--si-/ ~ a O-"~ Na ÷
+
- O\
--Si--. Where the mobile ion concentration is very low, the contribution of the alkali ion to the conductivity should be small. However, the low alkali concentration involved a low concentration of anionic groups. Thus, if the penetrating water molecules have not reached this part of glass the protons cannot move easily because the lack of proton acceptors. The deep conductivity minimum in fig. 6 is explained by these circumstances. Beyond the minimum in a wide range the conductivity inreases with increasing alkali concentration (cf. figs. 5 and 6). This part of the surface layer contains the alkali ions and the protons in commensurable amounts and is supposed to be almost free of water molecules. It should be taken into account that when an alkali ion is replaced by a proton the volume of the alkali ion site will be empty because the substituting proton is immersed into the electron cloud of an anionic group. Such empty sites serve as additional vacancies for the migrating alkali ions. On the other hand, the non-protonated anionic groups play the role of the proton acceptors required by the proton movement. Beneath a certain depth this advantageous
condition for the electrical conduction vanishes and finally the conductivity takes the value of the bulk as it is shown by fig._6. From fig. 7 it is evident that the effect of mixing the alkali ion with the proton is unlike the well known mixed alkali effect: the shapes of the curves in figs. I(A) and 7 are very different. However, when the resis"tance is measured perpendicular to the glass surface [7], which i~Lthe u s u a l case, the contribution of the high resistivity layer is predominant. Therefore after leaching a glass an increase in the resistance is observed. The author is indebted to Dr J. Rohonczy for the computer program.
References
[1] [2] [3] [4] [5] [6] [7] [8] [9]
G. Gehlhoff and M. Thomas, Z. Techn. Phys. 6 (1925) 544. B. Lengyel and Z. Boksay, Z. Phys. Chem. 203 (1954) 93. B. Lengyel and Z. Boksay, Z. Phys. Chem. 204 (1955) 154. B. Lengyel and Z. Boksay, Z. Phys. Chem. 223 (1963) 49. B. Lengyel, Z. Boksay and S. Dobos, Z. Phys. Chem. 223 (1963) 49. B. Lengyel and Z. Boksay, Z. Phys. Chem. 241 (1969) 36. Z. Boksay and B. Lengyel, J. Non-Cryst. Solids 14 (1974) 79. A. Wikby, J. Electroanal Chem. 18 (1968) 363. Z. Boksay, Wiss. Z. Friedrich-Schiller-Univ. Jena Math. -Nat. R. 28 (1979) 477.
Journal of Non-Crystalline Solids 123 (1990) 324-327 North-Holland
EFFECT OF MIXING MOBILE IONS IN GLASSES ON TRANSPORT PROCESSES Z. BOKSAY Institute for General and Inorganic Chemistry, E6toi~s LorAnd Unioersity, Budapest, Hungary
A computer simulation for the ionic movement in glass has been developed in order to explain the decrease of conductivity caused by mixing alkali ions. An account of the composition dependence of the transference number of alkali ions involved is also included. According to the model used, a vacancy in the glass migrates in a three-dimensional path consisting of junctions, terminal and internal sites for alkali ions. It is assumed that the jump of a larger alkali ion into an empty site vacated by a smaller ion is impeded. The consequences of the replacement of alkali ions by protons are also discussed.
1. InWoduction Glasses containing two types of alkali ions in a commensurable amount exhibit conductivities lower by several orders of magnitude than do glasses with one type of alkali ion. The transference number of ions involved (the ratio of the charge transported by the given ion to the total transported charge) varies monotonically with the composition, usually according to an S-shaped curve. In figs. I(A) and 2(A) the logarithm of the conductivity and the transference number, respectively, of the larger ion are plotted versus the mole fraction of the larger ion for Li-Na, L i - K and N a - K glasses [3,5]. The decrease of conductivity is a rather astonishing effect not quite free of mystery. From 1925 when the effect was discovered [1], many efforts have been made to solve the problem; however, no explanation proposed has been generally accepted so far. An attempt to elucidate the foundation of this phenomenon will be developed using a computer simulation.
(1) In the glasses considered charge is carried by alkali ions located in the holes of the random silicate network. (2) The alkali ions are coordinated by more than one non-bridging oxygen ion; the rest of the coordination sphere consists of bridging oxygen atoms. The total coordination number depends on the diameter of an alkali ion present in the posi-
-4q
A
B
-6
-7 0.5
2. Assumptions In order to keep the extent of this paper within reasonable limits, the structural background will only be outlined in a set of postulates forming a conduction model of glass.
2
1.
0
.0
Mole f r a c t i o n of the larger ion Fig. 1. Comparison of conductivity measured and vacancy diffnsivity calculated. (A) The conductivity measured at 300 ° C, of mixed alkali glasses containing 16 mol% M20, 8 mol% MgO, 8 mol~ BaO and 68 mol~ SiO2. (B) The relative diffusivity of the vacancy calculated with parameters specified in the text.
0022-3093/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
Z. Boksay / Effect of mixing mobile ions in glasses on transport processes
1.0
0.5 2 ~
0
0.5 1.0 0 0.5 Mole fraceion of the larger ion
1.0
Fig. 2. Similarity of curves for the transference numbers of alkali ions, (A) measured in mixed alkali glasses at 270 o C, and (B) calculated according to the model, respectively. tion when the glass structure was frozen. The coordination number for a site remains unchanged even when the original alkali ion is replaced by another one. (3) The number of sites able to accommodate alkali ions is somewhat higher than that of alkali ions. The number of the surplus sites, i.e. the vacancies, is constant in a wide temperature range. (4) The ions and the vacancies move in a threedimensional network. An empty site can be occupied by a neighbouring alkali ion only from one, two or three directions depending on the local conditions. These directions are associated with a relatively low potential barrier the height of which is identical with the activation energy of the electrical conduction. (5) Most sites are only linked with two others through such a barrier and so form internal members of a chain of sites. In the network there are also junctions with three exits. Some chains of sites form impasses with a junction and a terminal site at the ends. Figure 3 shows a two-dimensional representation of the network of ionic movement.
325
(6) In mixed alkali glasses, the j u m p of the larger ion into the vacated site of a smaller one requires an elevated energy. The relevant potential barrier should be either overcome or avoided in a more or less long roundabout way by a migrating vacancy. Both alternatives diminish the electric conduction, diffusion of ions and other ionic transport processes. (7) The impeded movement of the vacancy in the network is assumed to be decisively responsible for the mixed alkali effect. Other influences, e.g. that of the density anomaly, are less significant ones and are not considered here.
3. Model simulation The above items represent the final stage of the development of a model elaborated from 1954 until 1974 [2-6]. In order to confirm this conduction model, a computer simulation was performed. The program used is regarded as a first approximation and, obviously, has incorporated some simplifications as follows. (1) The network for the migrating vacancy is gradually generated by computer in the nearby region of the actual location of the vacancy. When a new chain of sites is generated an old one is cancelled in the computer memory. (2) The chains of sites consist of linked elementary distances. All j u m p distances are equal and parallel with one of the Cartesian coordinate axes. The total length of each chain is constant (see fig. 4). (3) The larger and the smaller alkali ions are statistically distributed in junctions, chain sites and terminal sites..
Termin~!c ~mpass
~tion
~ ' ~ ' ~ ' Vacancy Fig. 3. Junctions, terminal and internal sites in the network for the migration of the vacancy and alkali ions.
Fig. 4. Two-dimensional representation of the network of ionic movement according to the model applied.
326
Z. Boksay / Effect of mixing mobile ions in glasses on transport processes
(4) The alkali ion concentration is assumed to be constant in the whole glass series. (5) The time between two subsequent jumps is taken as constant; the number of displacements of a vacancy measures the time. (6) In the calculation of the probability of alternative jumps, the weighting factor for the jump of the larger ion into a smaller site is less than the common value attributed to the rest of the jumps of any kind. (7) The random walk of the vacancy always starts from the centre of the coordinate system. After certain steps the square of the distance between the starting point and the final position of the vacancy is registered. The mean value of the square of the distance for a relatively large number of vacancy runs is calculated. (8) The ratio of the square of the distance to the number of steps is used as a relative diffusivity of the vacancy. (9) If the vacancy has succeeded in moving from one junction into another through a chain of sites, the displacements of the ions involved are considered in the calculation of the transference number. If the vacancy leaves the chain where it entered, the displacements in the chain are cancelled. (10) The parameters chosen are as follows. The ratio of the weighting factor for the simple and hindered jumps is 20:1. The length of the chains is 10 in distances of elementary jumps as units. One fifth of the chains forms impasse. The random walk for a given mole fraction was simulated in five series with 20 runs in each. For the relative diffusivity of the vacancy, the standard deviation of the 5 series is as high as 22%, on the average. The transference number of the larger ion was calculated with an accuracy of +0.002. The logarithm of the relative diffusivity was plotted against the composition in fig. I(B). Since the diffusivity D of the vacancy is proportional to the ionic conductivity ~¢, according to the Nernst-Einstein equation lg D and lg x have curves of the same shape. Thus the composition dependence of the logarithm of calculated DreI can be compared with that of the measured conductivity. The characteristic similarity of the curves, manifested in the deep minima, indicates the use-
+m z
~
o
~
0
1
2
Depth/pm Fig. 5. Relative sodium ion concentration after 14 days leaching at 40 o C, in the surface layer of a glass containing 22% Na20, 6% CaO and 68% SiO2 (wt%). fulness of the hindered migration model. While the conductivity goes through a minimum, the related transference number obtained varies according to an S-shaped curve similar to the experimental ones (fig. 2). Thus the model correctly describes the experiences in two respects: it describes the compositional dependence of both the conductivity and the transference number. By the variation of the parameters a close agreement between the calculated results and the experimental data is expected. The model can be applied to relaxation phenomena as well. Replacement of an alkali ion by a proton may lead to high resistance [8]. One may think that H + ions behave like alkali ions, causing a mixed alkali effect, but this interpretation is wrong. Mixing of alkali ions with protons takes place when the glass is leached. Owing to the leaching a thin layer is formed on the glass surface in which the alkali
-5
f
tx0
10
0
2
Depth/pm Fig. 6. Variation of the conductivity vdth the depth in the
surface layer of the leached sodium calcium silicate glass [9].
327
Z. Boksay / Effect of mixing mobile ions in glasses on transport processes
concentration increases with the depth and tends to a constant value while the proton concentration decreases and tends to zero. Figure 5 shows the relative sodium ion concentration of a sodium calcium silicate glass. (The alkali content in the outer part of the surface layer is supposed to be blocked in the structure and not to take part in the conduction.) The proton in the surface layer is usually bound to a silicate anionic group and forms a silanol group. Its movement reqires the presence o f a nearby water molecule or an anionic group. Obviously, the latter is associated with metal ions, mostly with alkali ions. If the proton movements are shown by bent arrows, we may write I
[
--si-\ ~ O--H O /\ H
--si-\ O--H H
-5
/
-lo
i
0
0.5
1.0
Mole f r a c t i o n of Na Fig. 7. Conductivity in the inner part of the interface layer of the leached glass plotted against the mole fraction of the sodium ion.
I
--si-/ ~ a O-"~ Na ÷
+
- O\
--Si--. Where the mobile ion concentration is very low, the contribution of the alkali ion to the conductivity should be small. However, the low alkali concentration involved a low concentration of anionic groups. Thus, if the penetrating water molecules have not reached this part of glass the protons cannot move easily because the lack of proton acceptors. The deep conductivity minimum in fig. 6 is explained by these circumstances. Beyond the minimum in a wide range the conductivity inreases with increasing alkali concentration (cf. figs. 5 and 6). This part of the surface layer contains the alkali ions and the protons in commensurable amounts and is supposed to be almost free of water molecules. It should be taken into account that when an alkali ion is replaced by a proton the volume of the alkali ion site will be empty because the substituting proton is immersed into the electron cloud of an anionic group. Such empty sites serve as additional vacancies for the migrating alkali ions. On the other hand, the non-protonated anionic groups play the role of the proton acceptors required by the proton movement. Beneath a certain depth this advantageous
condition for the electrical conduction vanishes and finally the conductivity takes the value of the bulk as it is shown by fig._6. From fig. 7 it is evident that the effect of mixing the alkali ion with the proton is unlike the well known mixed alkali effect: the shapes of the curves in figs. I(A) and 7 are very different. However, when the resis"tance is measured perpendicular to the glass surface [7], which i~Lthe u s u a l case, the contribution of the high resistivity layer is predominant. Therefore after leaching a glass an increase in the resistance is observed. The author is indebted to Dr J. Rohonczy for the computer program.
References
[1] [2] [3] [4] [5] [6] [7] [8] [9]
G. Gehlhoff and M. Thomas, Z. Techn. Phys. 6 (1925) 544. B. Lengyel and Z. Boksay, Z. Phys. Chem. 203 (1954) 93. B. Lengyel and Z. Boksay, Z. Phys. Chem. 204 (1955) 154. B. Lengyel and Z. Boksay, Z. Phys. Chem. 223 (1963) 49. B. Lengyel, Z. Boksay and S. Dobos, Z. Phys. Chem. 223 (1963) 49. B. Lengyel and Z. Boksay, Z. Phys. Chem. 241 (1969) 36. Z. Boksay and B. Lengyel, J. Non-Cryst. Solids 14 (1974) 79. A. Wikby, J. Electroanal Chem. 18 (1968) 363. Z. Boksay, Wiss. Z. Friedrich-Schiller-Univ. Jena Math. -Nat. R. 28 (1979) 477.