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Self-diffusivity of polyvalent ions in silicate liquids
Journal of Non-Crystalline Solids 253 (1999) 76±83
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Self-diusivity of polyvalent ions in silicate liquids Olaf Clauûen, Sandra Gerlach, Christian R ussel * Otto-Schott-Institut, Friedrich-Schiller-Universit at Jena, Fraunhoferstraûe 6, 07743 Jena, Germany
Abstract Self-diusion coecients of tin, lead, arsenic, antimony, bismuth, vanadium, chromium, manganese, iron, nickel, cobalt, zinc, cadmium, copper and cerium were measured as a function of temperature in a borosilicate glass liquid. Self-diusion coecients of iron were also measured in some alkali±magnesia±silicate, alkali±lime±silicate and alkali± lime±alumosilicate liquids. The method used was the square-wave voltammetry, a fast potentiostatic pulse method. In the temperature range of 700±1500°C, a linear correlation between log D and 1/T in which D is the diusivity and T is sample temperature is observed for all polyvalent elements and compositions studied. The self-diusion coecients of Cu are up to six orders of magnitude larger than those of Cr6 . Self-diusion coecients of iron in liquids possessing larger alkali content and hence smaller viscosity were larger. At all temperatures studied, the self-diusion coecients of iron decrease from lithium to cesium. However, if liquids of constant viscosity are compared, the self-diusion coef®cients of iron decrease with decreasing size of the alkali cation, increasing alkali concentration and decreasing alumina content. This decrease is explained by a structural model for the incorporation of iron. Ó 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Self-diusion coecients of glasses and glass liquids have been measured by tracer methods [1± 11]. Here, besides the main components, also the diusion of some polyvalent compounds, such as iron has been studied [8±11]. Redox properties of polyvalent ions and the kinetics of redox equilibration have an eect on the physical properties of glass liquids and the ®nal glass product. The thermodynamics of redox equilibria may be studied by either equilibration experiments or electrochemical methods [12±22]. The latter is enabled by the electric conductivity of network modi®er containing glass liquids at temperatures greater than the glass
* Corresponding author. Tel.: +49-3641 948 501; fax: +493641 948 502; e-mail: [email protected]
transition temperature, Tg . However, voltammetric methods also enable the determination of self-diffusion coecients of polyvalent elements at high temperatures [12,13,18±27]. Electrochemical methods provide a possibility to determine the diusivity of components which can be reduced or oxidized in a certain potential range and at temperatures >Tg . Self-diusion coecients of polyvalent ions play a role during kinetically determined processes of equilibrating and cooling of glass melts. Other properties such as nucleation and crystallization as well as the corrosion of metals in glass liquids may also be controlled by the diusion of polyvalent elements. The Stokes±Einstein equation provides a theoretical dependence of the self-diusion coecient upon viscosity. Self-diusion coecients of various polyvalent elements measured in the same liquid at the same temperature, dier by up to six orders of magnitude [22,24,27,28].
0022-3093/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 3 4 5 - 2
O. Clauûen et al. / Journal of Non-Crystalline Solids 253 (1999) 76±83
It should be mentioned that the quantitative determination of polyvalent elements [18,25,26] can also be carried out directly in the glass liquid using electrochemical methods.
Electron transfer reactions can be studied using voltammetric methods. In the case of polyvalent elements, the electrode reaction can be described as follows: 1
The voltammetric method used in our experiments is the square-wave voltammetry, a fast pulse method. The applied potential is a staircase ramp, upon which is superimposed a rectangular wave of comparably short pulse time (1 6 s 6 500 ms) and high voltage amplitude (50 6 DE 6 250 mV). The current is measured at the end of every half wave and then dierentiated [29,30]. If the reactions of polyvalent elements during voltammetric measurements are controlled by diusion, the peak currents, DIP , of the current as a function of the potential obtained are proportional to the total concentration of the polyvalent element, c0 , the number of the electrons transferred, n, and the self-diusion coecient, D [29,30]. The peak currents are given by Eq. (2): DIP 0:31 pÿ1=2 Rÿ1 T ÿ1 A c0 D1=2 n2 F 2 DE sÿ1=2 ; 2 where A is the area of the working electrode and c0 the total concentration of the polyvalent element. The temperature dependence of the self-diusion coecients is usually described using Arrhenius equation. D D0 exp ÿED =R T
kT : 6p g r
4
3. Experimental
2. Theory
A xn n eÿ ¢ Ax :
D
77
3
with ED is the activation energy of the diusion process. In aqueous or organic solutions, the self-diusion coecient is dependent on the viscosity of the solution, g, and the radius of the diusing species, r, according to Stokes±Einstein equation [31]
The measurements were carried out in a resistance heated furnace. In its middle, a platinum crucible containing the glass liquid was located. From the top, three electrodes were dipped into the liquid: a platinum wire (diameter: 1 mm) as the working electrode, a platinum plate (size: about 2 cm2 ) as the counter electrode and a zirconia probe ¯ushed with air as the reference electrode. All potentials mentioned in this paper are referenced to the zirconia probe. Electronics were constructed, the main part being a potentiostat in our laboratory. It is connected to a computer via digital/ analogue and analogue/digital converters so that any potential-time dependence can be supplied. The computer also records the current-potential curve. The experimental equipment and the procedure applied were described in Refs. [12,18]. The glass liquids studied were a borosilicate composition(glassA:57SiO2 á 12.2B2 O3 á 16.8Na2 O á 11.1Li2 O á 2.9MgO á 0.2ZrO2 á 0.1La2 O3 ), various alkali±magnesia±silicate compositions (glass B: 74SiO2 á 16Li2 O á 10MgO; glass series C: (90ÿx)SiO2 á xNa2 O á 10MgO; glass series D: (90ÿx)SiO2 á xK2 O á 10MgO with x 10, 15, 20 and 25; and glass E: 74SiO2 á 16Cs2 O á 10MgO) as well as dierent alkali±lime±silicate compositions (glass series F: 74SiO2 á (26ÿx)Na2 O á 10CaO, with x 0, 5, 10, 15; glass series G: (90ÿx)SiO2 á xNa2 O á 10CaO; and glass series H: (90ÿx)SiO2 á xK2 O á 10CaO). All glass melts were doped with 0.25±0.5 mol% of the oxide of the polyvalent element, in the case of iron always with 0.2 mol% Fe2 O3 . The square-wave voltammograms were deconvoluted using a procedure described in Ref. [26]. From the experimentally recorded curves, a current±potential curve also experimentally recorded in a glass liquid without any polyvalent elements was subtracted. The resulting curve then was approximated by theoretical current±potential dependences using a least-square ®t.
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Viscosities were measured using a rotational viscometer (Bahr VIS 403). A Pt/Rh crucible and Pt/Rh rotor were used. 4. Results Fig. 1 (curve 1) presents a voltammogram of a chromium doped borosilicate glass liquid (Sample A) recorded at 900°C. One peak is observed at a potential of ÿ45 mV and another one, after subtracting the matrix current (curve 2) can be seen in curve 3 at ÿ865 mV. The ®rst peak is caused by the reduction of Cr6 to Cr3 while the second one is attributed to the reduction of Cr3 to Cr2 [21]: 3eÿ
eÿ
Cr6 ¢ Cr3 ¢ Cr2 :
5
Fig. 2 shows self-diusion coecients calculated from the peak currents of voltammograms recorded in the Cr2 O3 -doped sample (see Fig. 1) as a function of temperature. A linear correlation between log D and 1/T is observed. Curve 2 is attributed to the reduction of Cr6 to Cr3 , (correlation coecient R > 0.99) while curve 1 to that of Cr3 to Cr2 (correlation coecient R > 0.99). The self-diusion coecients shown in curve 1 are larger than those in curve 2. Although self diusion coecients are mean values (e.g., Cr6 and Cr3 ), in the following, only the oxidized species will be mentioned.
Fig. 1. Square-wave voltammogram of a borosilicate glass melt doped with 0.5 mol% Cr2 O3 (curve 1) at 900°C. Curve 2: matrix current; curve 3: curve 1 ÿ curve 2.
Fig. 2. Voltammetrically determined self-diusion coecients in the borosilicate glass melt as a function of temperature. Curve 1: Cr3 ; curve 2: Cr6 .
Samples doped with other polyvalent elements also have resolved peaks from which peak currents, self-diusion coecients were calculated. Fig. 3 summarizes self-diusion coecients of various multivalent elements in melt composition
Fig. 3. Voltammetrically determined self-diusion coecients of various polyvalent elements in the borosilicate glass melt as a function of temperature: 1: Cu2 , 2: Fe3 , 3: Cu , 4: As5 , 5: V4 , 6: Ni2 , 7: As3 , 8: Cd2 , 9: Pb2 , 10: Co2 , 11: Mn2 , 12: Zn2 , 13: V5 , 14: Ce4 , 15: Cr3 , 16: Sn2 , 17: Sb5 , 18: Bi3 , 19: Sb3 , 20: Cr6 .
O. Clauûen et al. / Journal of Non-Crystalline Solids 253 (1999) 76±83
A and the temperature range of 700±1100°C (including datas from Refs. [24,28]). They all have a temperature dependence which can be ®tted to Eq. (3) and are illustrated by means of their regression lines (all correlation coecients are in the range of 0.98±0.99). The self-diusion coecients in Fig. 3 and their temperature dependences dier. The largest self-diusion coecients were measured for iron and copper and were in the range of 1:8 10ÿ10 ±2:8 10ÿ10 m2 /s at 1100°C. At the same temperature the smallest self-diusion coef®cient measured was that of Cr6 , more than two orders of magnitude smaller (1:2 10ÿ12 m2 /s) than that of iron. For all polyvalent elements investigated, the self-diusion coecients decrease by two to three orders of magnitude while decreasing the temperature from 1100°C to 700°C. At 800°C they are in the range of 1:1 10ÿ14 ± 6:2 10ÿ12 m2 /s. Fig. 4 shows the self-diusion coecients of Fe3 at 1300°C in dierent soda±lime±silica glass melts (74SiO2 , (26ÿx)Na2 O, xCaO; x 0, 5, 10, 15) (series F) as a function of the Na2 O-content. It can be observed that an increasing Na2 O-content leads to larger self-diusion coecients. Fig. 5 shows the viscosities of dierent soda± lime±silica samples of series F as a function of the Na2O-content also at a temperature of 1300°C. We observe that an increasing Na2 O-content leads
Fig. 4. Self-diusion coecients of iron as a function of the Na2 O-content for soda±lime±silica glass melts (74SiO2 , (26 ÿ x)Na2 O, xCaO; x 0, 5, 10, 15) doped with 0.2 mol% Fe2 O3 at 1300°C.
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Fig. 5. Viscosities of the four soda±lime±silica glass melts (see Fig. 4) at 1300°C.
to a smaller viscosity. A plot of the self-diusion coecients versus the reciprocal viscosity is shown in Fig. 6. Within this glass system, the self-diusion coecients increase with decreasing viscosity and a linear dependence of the self-diusion coef®cient on the reciprocal viscosity is observed (The full line is a regression line, R 0.98). Table 1 shows the self-diusion coecients of iron in dierent alkali±magnesia±silica samples as a function of the alkali content (1300°C). The selfdiusion coecients of iron increase with increasing R2 O-content. For the Na2 O-containing samples (series C), the self-diusion coecients
Fig. 6. Self-diusion coecients of iron as a function of the viscosity for soda±lime±silica glass melts (see Figs. 4 and 5) at various temperatures. The mol% compositions are given in the box: [Na2 O]±[CaO].
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Table 1 Self-diusion coecients of iron in dierent alkali-magnesiasilica samples at 1300°C and at a viscosity of 3.16 ´ 103 dPa s (error 15%) Alkali content
D at 1300°C (in m2 /s)
D at g 3:16 103 dPas (in m2 /s)
16Cs2 O 10K2 O 15K2 O 20K2 O 25K2 O 10Na2 O 15Na2 O 20Na2 O 25Na2 O
1.62 ´ 10ÿ11 0.78 ´ 10ÿ11 1.70 ´ 10ÿ11 5.37 ´ 10ÿ11 6.46 ´ 10ÿ11 1.62 ´ 10ÿ11 4.37 ´ 10ÿ11 24.0 ´ 10ÿ11 39.8 ´ 10ÿ11
6.79 ´ 10ÿ11 3.72 ´ 10ÿ11 1.62 ´ 10ÿ11 1.07 ´ 10ÿ11 0.37 ´ 10ÿ11 14.1 ´ 10ÿ12 4.0 ´ 10ÿ12 3.31 ´ 10ÿ12 1.62 ´ 10ÿ12
samples) for a temperature of 1300°C. Within the series, from Li to K, the self-diusion coecients are decreasing. The self-diusion coecients of iron in the magnesia-containing samples are nearly the same for K and Cs, however, we note that the concentration of the Cs2 O is 1 mol% larger than the concentration of the K2 O. The self-diusion coecient in the soda±lime±silicate sample is larger than in the soda±magnesia±silicate sample, while the self-diusion coecients in the K2 Ocontaining samples are the same within the error limits. 5. Discussion
dier by about a factor of 25 and for the K2 Ocontaining samples (series D) by about a factor of 8. The self-diusion coecients of iron in the Na2 O-containing samples are generally larger than those in the K2 O-containing samples in the temperature range studied. Fig. 7 presents self-diusion coecients of iron in four alkali±magnesia±silicate samples ((90ÿx)SiO2 , xR2 O, 10MgO) (Samples B±E) and two alkali±lime±silicate samples ((90ÿx)SiO2 , xR2 O, 10CaO) (Samples G and H) containing dierent types of alkali, however, all possessing a similar R2 O-content (16Li2 O, 15Na2 O, 15K2 O, 16Cs2 O for the alkali±magnesia±silicate samples and 16Na2 O, 16K2 O for the alkali±lime±silicate
Fig. 7. Self-diusion coecients of iron as a function of the type of alkali ions at a temperature of 1300°C; d alkali±magnesia±silica glass melts (16Li2 O, 15Na2 O, 15K2 O, 16Cs2 O), n alkali-lime-silica glass melts (16Na2 O, 16K2 O).
All self-diusion coecients measured could be ®tted with the Arrhenius equations with correlation coecients R > 0.98. Hence within the temperature range studied, the activation energy, ED , is constant and it can be assumed that the diusion mechanism does not change with temperature for any of the polyvalent elements in our samples. Fig. 3 illustrates that the ion possessing the largest charge (Cr6 ) has the smallest selfdiusion coecient while ions with the smallest charge, such as copper or iron have the largest self-diusion coecients. A similar dependence on charge has already been reported for soda± lime±silicate samples with composition of 74SiO2 á 16Na2 O á 10CaO in Ref. [22]. In that liquid, the self-diusion coecients dier by more than six orders of magnitude at 800°C. By analogy to Fig. 3, also in the soda±lime±silicate samples usually ions of smaller charge possess larger self-diusion coecients than those of larger charge. In the samples a strict correlation is not observed. The polyvalent cations have larger dierences in their self-diusion coecients at lower temperatures than at higher temperatures. As already observed in the case of a soda±lime± silica sample, a correlation of the self-diusion coecients with the atomic radius, the electronegativity or the ®eld strength is not observed. Fig. 8 shows a plot of the logarithm of the selfdiusion coecients measured in the borosilicate sample against those obtained in the soda±lime± silicate sample (74SiO2 , 16Na2 O, 10CaO from
O. Clauûen et al. / Journal of Non-Crystalline Solids 253 (1999) 76±83
Fig. 8. Self-diusion coecients of various polyvalent elements, measured in the borosilicate glass melt and a soda±lime± silica glass melt at 1000°C. Full line: linear regression, dashed line: equal diusion coecients in both glass melts.
Ref. [22]) both at a temperature of 1000°C. In both cases, Cr6 , has the smallest self-diusion coecient. In the borosilicate sample this coecient is around two orders of magnitude larger. In the case of Cd2 , the dierence is smallest. The full line in Fig. 4 is a regression line, deviations from it are less than 0.5 in the logarithm scale which corresponds to a factor of 3. Hence the deviation from the full line is clearly larger than the error limit. A possible reason for the larger self-diusion coecients is the smaller viscosity of the borosilicate sample in comparison to that of the soda±lime± silica sample. Stokes±Einstein equation (see Eq. (4)) predicts a correlation between the self-diusion coecient and the reciprocal ionic radius in media possessing the same viscosity. Since the self-diusion coecients dier by several orders of magnitude in the samples of the same composition (except for the polyvalent dopant) at the same temperature, the dierences in the self-diusion coecients are not an eect of dierences in the ionic radii. Thus, Stokes±Einstein equation is not ful®lled. As shown in Figs. 4, 6 and 8, the self-diffusion coecient of a polyvalent ion is usually larger in a glass melt with smaller viscosity. As shown for iron in samples of the series F (see Fig. 6) within this composition range, a linear function can be ®t to the data of the self-diusion coe-
81
cient on the reciprocal viscosity. It should be noted that the diusivity of other components of a glass liquid, such as silicon or oxygen follows the Stokes±Einstein equation [32,33]. By contrast, diusion coecients of network modi®ers have been reported to be some orders of magnitude larger [34]. As illustrated in Fig. 7, self-diusion coecients of iron are larger for alkali±lime±and alkali± magnesia±silicate samples with smaller alkali cations, such as Li2 O-and Na2 O-containing ones. However, these samples also have smaller viscosities. Hence, also in these cases, the self-diusion coecients are larger in samples with smaller viscosities. As illustrated for the case of the glass series F in Fig. 6, there is a correlation of D and 1/g for each system studied. However, there are deviations from a strict correlation if dierent glass systems are compared. This deviation can be illustrated with the aid of Table 1, column 3 (including data from Refs. [28,35,36]) as follows: At a constant viscosity of g 3:16 103 dPas (i.e. the temperature is not constant), a dierent eect of the self-diusion coecients of iron upon the glass composition is observed. With increasing alkali content, the self-diusion coecients decrease. With increasing size of the alkali cations, the selfdiusion coecients increase. With increasing alumina content, the self-diusion coecients also increase (see Ref. [36]). It should be noted that at constant viscosity the iron self-diusion coecient observed is around 40 times larger than the smallest one. In former studies [20,36], it has been concluded that Fe3 in silicate glasses mainly occurs in a distorted fourfold coordination tedrahedra, formally possessing a negative charge. Two oxygens are coordinated with an alkali ion for charge compensation and hence the resulting symmetry is C2V already known from electron paramagnetic resonance spectra (EPR) [37]. This complex is more eectively stabilized by smaller alkali cations such as Li since the Coulomb forces are larger. In the present paper as well as in Ref. [35], it was observed that iron has larger self-diusion coef®cients if larger alkali cations are present in liquids of the same viscosity, i.e. the same mobility of the network. This eect should be caused by
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the better stabilization of the Fe3 -tetrahedra in glass networks containing smaller alkali cations. If larger cations are present, the C2V -complex is less stabilized (see Refs. [35,37]). Otherwise, an increase in the alkali content causes a better stabilization. This dierence explains the decrease in the self-diusion coecients in liquids of the same viscosity containing larger quantities of alkali cations. Another interesting result recently reported in Ref. [36] is that an increase in the Al2 O3 -content leads to larger iron self-diusion coecients in liquids possessing the same viscosity. In glasses with an Al2 O3 -content smaller than the sum of the alkali and the earth alkali content, aluminum is mainly incorporated in fourfold coordination [38]. By analogy to Fe3 , this complex possesses a formal negative charge compensated by an alkali cation. These alkali cations are located near this tetrahedral complex and no longer contribute to the stabilization of the C2V -complex of Fe3 . Thus larger Al2 O3 -contents have the same eect as smaller alkali contents. In both cases, the incorporation of Fe3 into the glass network decreases and therefore, the self-diusion coecients of iron in liquids with the same viscosity increases. 6. Conclusions The self-diusion coecients of various polyvalent elements dier by some order of magnitudes in the same basic glass systems at the same temperature. The Stokes±Einstein equation is not valid for the ionic radius of the diusing species. Generally, in liquids possessing smaller viscosities, the self-diusion coecients are larger. The dependence of the self-diusion coecients of iron upon the reciprocal viscosity of a glass liquid is ®t to the Stokes±Einstein equation within any glass system. If liquids of constant viscosity are compared, the self-diusion coecient of iron decreases with increasing alkali concentration, decreasing size of the alkali cation and decreasing alumina content. This is consistent with the stabilization of tetrahedrally coordinated iron by increasing alkali concentrations and decreasing size of the alkali cation as well as by a deterio-
rating stabilization with increasing alumina content. Acknowledgements Thanks are due to the Deutsche Forschungsgemeinschaft (DFG), Bonn Bad Godesberg, for ®nancial support. References [1] H.A. Schaeer, J. Mecha, J. Steinmann, J. Am. Ceram. Soc. 62 (1979) 343. [2] G.H. Frischat, Ionic Diusion of Oxide Glasses, Trans. Tech, Aedermannsdorf, Switzerland, 1975. [3] G.H. Frischat, H.-J. Oel, Glastechn. Ber. 39 (1966) 524. [4] G.H. Frischat, Glastech. Ber. 44 (1971) 93. [5] R. Lindner, W. Hassenteufel, Y. Kotera, H. Matzke, Z. Phys. Chem. NF 23 (1960) 408. [6] R.H. Doremus, Glass Science, Wiley, New York, 1994, p. 276. [7] H.A. Schaeer, Phys. Status Solidi A 22 (1974) 281. [8] W. K ohler, G.H. Frischat, Phys. Chem. Glasses 19 (1978) 103. [9] T. Das, A.S. Sanyal, J. Mukerji, Phys. Chem. Glasses 35 (1994) 180. [10] T. Das, A.S. Sanyal, J. Mukerji, Phys. Chem. Glasses 35 (1994) 191. [11] T. Das, A.S. Sanyal, J. Mukerji, Phys. Chem. Glasses 35 (1994) 198. [12] E. Freude, C. R ussel, Glastech. Ber. 60 (1987) 202. [13] C. R ussel, G. Sprachmann, J. Non-Cryst. Solids 127 (1991) 197. [14] K. Takahasi, Y. Miura, J. Non-Cryst. Solids 95 & 96 (1987) 119. [15] J.-Y. Tilquin, J. Glibert, P. Claes, Electrochim. Acta 38 (1993) 479. [16] M. Shibata, M. Oakawa, T. Yokokawa, J. Non-Cryst. Solids 190 (1995) 226. [17] A. Sasahira, T. Yokokawa, Electrochim. Acta 30 (1985) 441. [18] G. Montel, C. R ussel, E. Freude, Glastech. Ber. 61 (1988) 59. [19] C. R ussel, Phys. Chem. Glasses 32 (1991) 138. [20] C. R ussel, Glastech. Ber. 66 (1993) 93. [21] O. Clauûen, C. R ussel, Ber. Bunsenges. Phys. Chem. 100 (1996) 1475. [22] C. R ussel, J. Non-Cryst. Solids 134 (1991) 169. [23] R. Pascova, C. R ussel, J. Non-Cryst. Solids 208 (1997) 237. [24] O. Clauûen, C. R ussel, J. Non-Cryst. Solids 215 (1997) 68. [25] H. M uller-Simon, K.W. Mergler, Glastech. Ber. Glass Sci. Technol. 68 (1995) 273.
O. Clauûen et al. / Journal of Non-Crystalline Solids 253 (1999) 76±83 [26] O. Clauûen, C. R ussel, Glastech. Ber. Glass Sci. Technol. 69 (1996) 95. [27] O. Clauûen, C. R ussel, Phys. Chem. Glasses 39 (1998) 200. [28] O. Clauûen, C. R ussel, Solid State Ionics 105 (1998) 289. [29] G.C. Barker, Anal. Chim. Acta 18 (1958) 118. [30] J.G. Osteryoung, J.J. O'Dea, in: A.J. Bard, Electroanalytical Chemistry, 14th ed., Dekker, New York, Basel, 1986. [31] P.W. Atkins, Physikalische Chemie, Verlag Chemie, Weinheim, 1990, pp. 692. [32] R. Terai, Y. Dishi, Glastechn. Ber. 50 (1977) 68.
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[33] H.A. Schaeer, Phys. Status Solidi A 22 (1974) 281. [34] H.A. Schaeer, J. Mecha, J. Steinmann, J. Am. Ceram. Soc. 62 (1979) 343. [35] S. Gerlach, O. Clauûen, C. R ussel, J. Non-Cryst. Solids 226 (1998) 11. [36] S. Gerlach, O. Clauûen, C. R ussel, J. Non-Cryst. Solids 240 (1998) 110. [37] C. R ussel, Glastech. Ber. 66 (1993) 68. [38] H. Scholze, Glas-Natur, Struktur und Eigenschaften, Springer, Berlin, 1988, p. 158.
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Self-diusivity of polyvalent ions in silicate liquids Olaf Clauûen, Sandra Gerlach, Christian R ussel * Otto-Schott-Institut, Friedrich-Schiller-Universit at Jena, Fraunhoferstraûe 6, 07743 Jena, Germany
Abstract Self-diusion coecients of tin, lead, arsenic, antimony, bismuth, vanadium, chromium, manganese, iron, nickel, cobalt, zinc, cadmium, copper and cerium were measured as a function of temperature in a borosilicate glass liquid. Self-diusion coecients of iron were also measured in some alkali±magnesia±silicate, alkali±lime±silicate and alkali± lime±alumosilicate liquids. The method used was the square-wave voltammetry, a fast potentiostatic pulse method. In the temperature range of 700±1500°C, a linear correlation between log D and 1/T in which D is the diusivity and T is sample temperature is observed for all polyvalent elements and compositions studied. The self-diusion coecients of Cu are up to six orders of magnitude larger than those of Cr6 . Self-diusion coecients of iron in liquids possessing larger alkali content and hence smaller viscosity were larger. At all temperatures studied, the self-diusion coecients of iron decrease from lithium to cesium. However, if liquids of constant viscosity are compared, the self-diusion coef®cients of iron decrease with decreasing size of the alkali cation, increasing alkali concentration and decreasing alumina content. This decrease is explained by a structural model for the incorporation of iron. Ó 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Self-diusion coecients of glasses and glass liquids have been measured by tracer methods [1± 11]. Here, besides the main components, also the diusion of some polyvalent compounds, such as iron has been studied [8±11]. Redox properties of polyvalent ions and the kinetics of redox equilibration have an eect on the physical properties of glass liquids and the ®nal glass product. The thermodynamics of redox equilibria may be studied by either equilibration experiments or electrochemical methods [12±22]. The latter is enabled by the electric conductivity of network modi®er containing glass liquids at temperatures greater than the glass
* Corresponding author. Tel.: +49-3641 948 501; fax: +493641 948 502; e-mail: [email protected]
transition temperature, Tg . However, voltammetric methods also enable the determination of self-diffusion coecients of polyvalent elements at high temperatures [12,13,18±27]. Electrochemical methods provide a possibility to determine the diusivity of components which can be reduced or oxidized in a certain potential range and at temperatures >Tg . Self-diusion coecients of polyvalent ions play a role during kinetically determined processes of equilibrating and cooling of glass melts. Other properties such as nucleation and crystallization as well as the corrosion of metals in glass liquids may also be controlled by the diusion of polyvalent elements. The Stokes±Einstein equation provides a theoretical dependence of the self-diusion coecient upon viscosity. Self-diusion coecients of various polyvalent elements measured in the same liquid at the same temperature, dier by up to six orders of magnitude [22,24,27,28].
0022-3093/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 3 4 5 - 2
O. Clauûen et al. / Journal of Non-Crystalline Solids 253 (1999) 76±83
It should be mentioned that the quantitative determination of polyvalent elements [18,25,26] can also be carried out directly in the glass liquid using electrochemical methods.
Electron transfer reactions can be studied using voltammetric methods. In the case of polyvalent elements, the electrode reaction can be described as follows: 1
The voltammetric method used in our experiments is the square-wave voltammetry, a fast pulse method. The applied potential is a staircase ramp, upon which is superimposed a rectangular wave of comparably short pulse time (1 6 s 6 500 ms) and high voltage amplitude (50 6 DE 6 250 mV). The current is measured at the end of every half wave and then dierentiated [29,30]. If the reactions of polyvalent elements during voltammetric measurements are controlled by diusion, the peak currents, DIP , of the current as a function of the potential obtained are proportional to the total concentration of the polyvalent element, c0 , the number of the electrons transferred, n, and the self-diusion coecient, D [29,30]. The peak currents are given by Eq. (2): DIP 0:31 pÿ1=2 Rÿ1 T ÿ1 A c0 D1=2 n2 F 2 DE sÿ1=2 ; 2 where A is the area of the working electrode and c0 the total concentration of the polyvalent element. The temperature dependence of the self-diusion coecients is usually described using Arrhenius equation. D D0 exp ÿED =R T
kT : 6p g r
4
3. Experimental
2. Theory
A xn n eÿ ¢ Ax :
D
77
3
with ED is the activation energy of the diusion process. In aqueous or organic solutions, the self-diusion coecient is dependent on the viscosity of the solution, g, and the radius of the diusing species, r, according to Stokes±Einstein equation [31]
The measurements were carried out in a resistance heated furnace. In its middle, a platinum crucible containing the glass liquid was located. From the top, three electrodes were dipped into the liquid: a platinum wire (diameter: 1 mm) as the working electrode, a platinum plate (size: about 2 cm2 ) as the counter electrode and a zirconia probe ¯ushed with air as the reference electrode. All potentials mentioned in this paper are referenced to the zirconia probe. Electronics were constructed, the main part being a potentiostat in our laboratory. It is connected to a computer via digital/ analogue and analogue/digital converters so that any potential-time dependence can be supplied. The computer also records the current-potential curve. The experimental equipment and the procedure applied were described in Refs. [12,18]. The glass liquids studied were a borosilicate composition(glassA:57SiO2 á 12.2B2 O3 á 16.8Na2 O á 11.1Li2 O á 2.9MgO á 0.2ZrO2 á 0.1La2 O3 ), various alkali±magnesia±silicate compositions (glass B: 74SiO2 á 16Li2 O á 10MgO; glass series C: (90ÿx)SiO2 á xNa2 O á 10MgO; glass series D: (90ÿx)SiO2 á xK2 O á 10MgO with x 10, 15, 20 and 25; and glass E: 74SiO2 á 16Cs2 O á 10MgO) as well as dierent alkali±lime±silicate compositions (glass series F: 74SiO2 á (26ÿx)Na2 O á 10CaO, with x 0, 5, 10, 15; glass series G: (90ÿx)SiO2 á xNa2 O á 10CaO; and glass series H: (90ÿx)SiO2 á xK2 O á 10CaO). All glass melts were doped with 0.25±0.5 mol% of the oxide of the polyvalent element, in the case of iron always with 0.2 mol% Fe2 O3 . The square-wave voltammograms were deconvoluted using a procedure described in Ref. [26]. From the experimentally recorded curves, a current±potential curve also experimentally recorded in a glass liquid without any polyvalent elements was subtracted. The resulting curve then was approximated by theoretical current±potential dependences using a least-square ®t.
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Viscosities were measured using a rotational viscometer (Bahr VIS 403). A Pt/Rh crucible and Pt/Rh rotor were used. 4. Results Fig. 1 (curve 1) presents a voltammogram of a chromium doped borosilicate glass liquid (Sample A) recorded at 900°C. One peak is observed at a potential of ÿ45 mV and another one, after subtracting the matrix current (curve 2) can be seen in curve 3 at ÿ865 mV. The ®rst peak is caused by the reduction of Cr6 to Cr3 while the second one is attributed to the reduction of Cr3 to Cr2 [21]: 3eÿ
eÿ
Cr6 ¢ Cr3 ¢ Cr2 :
5
Fig. 2 shows self-diusion coecients calculated from the peak currents of voltammograms recorded in the Cr2 O3 -doped sample (see Fig. 1) as a function of temperature. A linear correlation between log D and 1/T is observed. Curve 2 is attributed to the reduction of Cr6 to Cr3 , (correlation coecient R > 0.99) while curve 1 to that of Cr3 to Cr2 (correlation coecient R > 0.99). The self-diusion coecients shown in curve 1 are larger than those in curve 2. Although self diusion coecients are mean values (e.g., Cr6 and Cr3 ), in the following, only the oxidized species will be mentioned.
Fig. 1. Square-wave voltammogram of a borosilicate glass melt doped with 0.5 mol% Cr2 O3 (curve 1) at 900°C. Curve 2: matrix current; curve 3: curve 1 ÿ curve 2.
Fig. 2. Voltammetrically determined self-diusion coecients in the borosilicate glass melt as a function of temperature. Curve 1: Cr3 ; curve 2: Cr6 .
Samples doped with other polyvalent elements also have resolved peaks from which peak currents, self-diusion coecients were calculated. Fig. 3 summarizes self-diusion coecients of various multivalent elements in melt composition
Fig. 3. Voltammetrically determined self-diusion coecients of various polyvalent elements in the borosilicate glass melt as a function of temperature: 1: Cu2 , 2: Fe3 , 3: Cu , 4: As5 , 5: V4 , 6: Ni2 , 7: As3 , 8: Cd2 , 9: Pb2 , 10: Co2 , 11: Mn2 , 12: Zn2 , 13: V5 , 14: Ce4 , 15: Cr3 , 16: Sn2 , 17: Sb5 , 18: Bi3 , 19: Sb3 , 20: Cr6 .
O. Clauûen et al. / Journal of Non-Crystalline Solids 253 (1999) 76±83
A and the temperature range of 700±1100°C (including datas from Refs. [24,28]). They all have a temperature dependence which can be ®tted to Eq. (3) and are illustrated by means of their regression lines (all correlation coecients are in the range of 0.98±0.99). The self-diusion coecients in Fig. 3 and their temperature dependences dier. The largest self-diusion coecients were measured for iron and copper and were in the range of 1:8 10ÿ10 ±2:8 10ÿ10 m2 /s at 1100°C. At the same temperature the smallest self-diusion coef®cient measured was that of Cr6 , more than two orders of magnitude smaller (1:2 10ÿ12 m2 /s) than that of iron. For all polyvalent elements investigated, the self-diusion coecients decrease by two to three orders of magnitude while decreasing the temperature from 1100°C to 700°C. At 800°C they are in the range of 1:1 10ÿ14 ± 6:2 10ÿ12 m2 /s. Fig. 4 shows the self-diusion coecients of Fe3 at 1300°C in dierent soda±lime±silica glass melts (74SiO2 , (26ÿx)Na2 O, xCaO; x 0, 5, 10, 15) (series F) as a function of the Na2 O-content. It can be observed that an increasing Na2 O-content leads to larger self-diusion coecients. Fig. 5 shows the viscosities of dierent soda± lime±silica samples of series F as a function of the Na2O-content also at a temperature of 1300°C. We observe that an increasing Na2 O-content leads
Fig. 4. Self-diusion coecients of iron as a function of the Na2 O-content for soda±lime±silica glass melts (74SiO2 , (26 ÿ x)Na2 O, xCaO; x 0, 5, 10, 15) doped with 0.2 mol% Fe2 O3 at 1300°C.
79
Fig. 5. Viscosities of the four soda±lime±silica glass melts (see Fig. 4) at 1300°C.
to a smaller viscosity. A plot of the self-diusion coecients versus the reciprocal viscosity is shown in Fig. 6. Within this glass system, the self-diusion coecients increase with decreasing viscosity and a linear dependence of the self-diusion coef®cient on the reciprocal viscosity is observed (The full line is a regression line, R 0.98). Table 1 shows the self-diusion coecients of iron in dierent alkali±magnesia±silica samples as a function of the alkali content (1300°C). The selfdiusion coecients of iron increase with increasing R2 O-content. For the Na2 O-containing samples (series C), the self-diusion coecients
Fig. 6. Self-diusion coecients of iron as a function of the viscosity for soda±lime±silica glass melts (see Figs. 4 and 5) at various temperatures. The mol% compositions are given in the box: [Na2 O]±[CaO].
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Table 1 Self-diusion coecients of iron in dierent alkali-magnesiasilica samples at 1300°C and at a viscosity of 3.16 ´ 103 dPa s (error 15%) Alkali content
D at 1300°C (in m2 /s)
D at g 3:16 103 dPas (in m2 /s)
16Cs2 O 10K2 O 15K2 O 20K2 O 25K2 O 10Na2 O 15Na2 O 20Na2 O 25Na2 O
1.62 ´ 10ÿ11 0.78 ´ 10ÿ11 1.70 ´ 10ÿ11 5.37 ´ 10ÿ11 6.46 ´ 10ÿ11 1.62 ´ 10ÿ11 4.37 ´ 10ÿ11 24.0 ´ 10ÿ11 39.8 ´ 10ÿ11
6.79 ´ 10ÿ11 3.72 ´ 10ÿ11 1.62 ´ 10ÿ11 1.07 ´ 10ÿ11 0.37 ´ 10ÿ11 14.1 ´ 10ÿ12 4.0 ´ 10ÿ12 3.31 ´ 10ÿ12 1.62 ´ 10ÿ12
samples) for a temperature of 1300°C. Within the series, from Li to K, the self-diusion coecients are decreasing. The self-diusion coecients of iron in the magnesia-containing samples are nearly the same for K and Cs, however, we note that the concentration of the Cs2 O is 1 mol% larger than the concentration of the K2 O. The self-diusion coecient in the soda±lime±silicate sample is larger than in the soda±magnesia±silicate sample, while the self-diusion coecients in the K2 Ocontaining samples are the same within the error limits. 5. Discussion
dier by about a factor of 25 and for the K2 Ocontaining samples (series D) by about a factor of 8. The self-diusion coecients of iron in the Na2 O-containing samples are generally larger than those in the K2 O-containing samples in the temperature range studied. Fig. 7 presents self-diusion coecients of iron in four alkali±magnesia±silicate samples ((90ÿx)SiO2 , xR2 O, 10MgO) (Samples B±E) and two alkali±lime±silicate samples ((90ÿx)SiO2 , xR2 O, 10CaO) (Samples G and H) containing dierent types of alkali, however, all possessing a similar R2 O-content (16Li2 O, 15Na2 O, 15K2 O, 16Cs2 O for the alkali±magnesia±silicate samples and 16Na2 O, 16K2 O for the alkali±lime±silicate
Fig. 7. Self-diusion coecients of iron as a function of the type of alkali ions at a temperature of 1300°C; d alkali±magnesia±silica glass melts (16Li2 O, 15Na2 O, 15K2 O, 16Cs2 O), n alkali-lime-silica glass melts (16Na2 O, 16K2 O).
All self-diusion coecients measured could be ®tted with the Arrhenius equations with correlation coecients R > 0.98. Hence within the temperature range studied, the activation energy, ED , is constant and it can be assumed that the diusion mechanism does not change with temperature for any of the polyvalent elements in our samples. Fig. 3 illustrates that the ion possessing the largest charge (Cr6 ) has the smallest selfdiusion coecient while ions with the smallest charge, such as copper or iron have the largest self-diusion coecients. A similar dependence on charge has already been reported for soda± lime±silicate samples with composition of 74SiO2 á 16Na2 O á 10CaO in Ref. [22]. In that liquid, the self-diusion coecients dier by more than six orders of magnitude at 800°C. By analogy to Fig. 3, also in the soda±lime±silicate samples usually ions of smaller charge possess larger self-diusion coecients than those of larger charge. In the samples a strict correlation is not observed. The polyvalent cations have larger dierences in their self-diusion coecients at lower temperatures than at higher temperatures. As already observed in the case of a soda±lime± silica sample, a correlation of the self-diusion coecients with the atomic radius, the electronegativity or the ®eld strength is not observed. Fig. 8 shows a plot of the logarithm of the selfdiusion coecients measured in the borosilicate sample against those obtained in the soda±lime± silicate sample (74SiO2 , 16Na2 O, 10CaO from
O. Clauûen et al. / Journal of Non-Crystalline Solids 253 (1999) 76±83
Fig. 8. Self-diusion coecients of various polyvalent elements, measured in the borosilicate glass melt and a soda±lime± silica glass melt at 1000°C. Full line: linear regression, dashed line: equal diusion coecients in both glass melts.
Ref. [22]) both at a temperature of 1000°C. In both cases, Cr6 , has the smallest self-diusion coecient. In the borosilicate sample this coecient is around two orders of magnitude larger. In the case of Cd2 , the dierence is smallest. The full line in Fig. 4 is a regression line, deviations from it are less than 0.5 in the logarithm scale which corresponds to a factor of 3. Hence the deviation from the full line is clearly larger than the error limit. A possible reason for the larger self-diusion coecients is the smaller viscosity of the borosilicate sample in comparison to that of the soda±lime± silica sample. Stokes±Einstein equation (see Eq. (4)) predicts a correlation between the self-diusion coecient and the reciprocal ionic radius in media possessing the same viscosity. Since the self-diusion coecients dier by several orders of magnitude in the samples of the same composition (except for the polyvalent dopant) at the same temperature, the dierences in the self-diusion coecients are not an eect of dierences in the ionic radii. Thus, Stokes±Einstein equation is not ful®lled. As shown in Figs. 4, 6 and 8, the self-diffusion coecient of a polyvalent ion is usually larger in a glass melt with smaller viscosity. As shown for iron in samples of the series F (see Fig. 6) within this composition range, a linear function can be ®t to the data of the self-diusion coe-
81
cient on the reciprocal viscosity. It should be noted that the diusivity of other components of a glass liquid, such as silicon or oxygen follows the Stokes±Einstein equation [32,33]. By contrast, diusion coecients of network modi®ers have been reported to be some orders of magnitude larger [34]. As illustrated in Fig. 7, self-diusion coecients of iron are larger for alkali±lime±and alkali± magnesia±silicate samples with smaller alkali cations, such as Li2 O-and Na2 O-containing ones. However, these samples also have smaller viscosities. Hence, also in these cases, the self-diusion coecients are larger in samples with smaller viscosities. As illustrated for the case of the glass series F in Fig. 6, there is a correlation of D and 1/g for each system studied. However, there are deviations from a strict correlation if dierent glass systems are compared. This deviation can be illustrated with the aid of Table 1, column 3 (including data from Refs. [28,35,36]) as follows: At a constant viscosity of g 3:16 103 dPas (i.e. the temperature is not constant), a dierent eect of the self-diusion coecients of iron upon the glass composition is observed. With increasing alkali content, the self-diusion coecients decrease. With increasing size of the alkali cations, the selfdiusion coecients increase. With increasing alumina content, the self-diusion coecients also increase (see Ref. [36]). It should be noted that at constant viscosity the iron self-diusion coecient observed is around 40 times larger than the smallest one. In former studies [20,36], it has been concluded that Fe3 in silicate glasses mainly occurs in a distorted fourfold coordination tedrahedra, formally possessing a negative charge. Two oxygens are coordinated with an alkali ion for charge compensation and hence the resulting symmetry is C2V already known from electron paramagnetic resonance spectra (EPR) [37]. This complex is more eectively stabilized by smaller alkali cations such as Li since the Coulomb forces are larger. In the present paper as well as in Ref. [35], it was observed that iron has larger self-diusion coef®cients if larger alkali cations are present in liquids of the same viscosity, i.e. the same mobility of the network. This eect should be caused by
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the better stabilization of the Fe3 -tetrahedra in glass networks containing smaller alkali cations. If larger cations are present, the C2V -complex is less stabilized (see Refs. [35,37]). Otherwise, an increase in the alkali content causes a better stabilization. This dierence explains the decrease in the self-diusion coecients in liquids of the same viscosity containing larger quantities of alkali cations. Another interesting result recently reported in Ref. [36] is that an increase in the Al2 O3 -content leads to larger iron self-diusion coecients in liquids possessing the same viscosity. In glasses with an Al2 O3 -content smaller than the sum of the alkali and the earth alkali content, aluminum is mainly incorporated in fourfold coordination [38]. By analogy to Fe3 , this complex possesses a formal negative charge compensated by an alkali cation. These alkali cations are located near this tetrahedral complex and no longer contribute to the stabilization of the C2V -complex of Fe3 . Thus larger Al2 O3 -contents have the same eect as smaller alkali contents. In both cases, the incorporation of Fe3 into the glass network decreases and therefore, the self-diusion coecients of iron in liquids with the same viscosity increases. 6. Conclusions The self-diusion coecients of various polyvalent elements dier by some order of magnitudes in the same basic glass systems at the same temperature. The Stokes±Einstein equation is not valid for the ionic radius of the diusing species. Generally, in liquids possessing smaller viscosities, the self-diusion coecients are larger. The dependence of the self-diusion coecients of iron upon the reciprocal viscosity of a glass liquid is ®t to the Stokes±Einstein equation within any glass system. If liquids of constant viscosity are compared, the self-diusion coecient of iron decreases with increasing alkali concentration, decreasing size of the alkali cation and decreasing alumina content. This is consistent with the stabilization of tetrahedrally coordinated iron by increasing alkali concentrations and decreasing size of the alkali cation as well as by a deterio-
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