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Evaluation of the number density of nuclei in Li2O·2SiO2 glass by DTA method
Journal of Non-Crystalline Solids 290 (2001) 64±72
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Evaluation of the number density of nuclei in Li2O 2SiO2 glass by DTA method Takashi Wakasugi *, Takuya Kadoguchi, Rikuo Ota Department of Chemistry and Materials Technology, Kyoto Institute of Technology Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan Received 4 December 2000
Abstract DTA measurements were performed to obtain the crystallization peak temperature, TC , for Li2 O 2SiO2 glass heattreated for nucleation. The relationship between TC and the number density of nuclei was obtained. Calculation of TC was performed by simulating the crystallization process, and the eects of the size and the number density of nuclei formed during DTA measurement on TC was evaluated. The calculated relationship between the number density of nuclei and TC agreed with the experimental one, and it was shown that the eects of the size and the number density of nuclei formed during DTA measurements is negligible in the evaluation of the number density of nuclei by TC . The steady state nucleation rate for the Li2 O 2SiO2 glass evaluated by DTA agreed well with the literature. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Since many glasses lose their desirable properties when they crystallize, the stability of glasses under usage conditions is very important to the application of glass materials. On the other hand, the properties of crystallized glass-ceramics depend on their crystallization process. The crystallization process is composed of nucleation and crystal growth, and the investigation of the nucleation behavior of glasses provides an important component to the production of glass materials including crystallized glass-ceramics. For the investigation of nucleation, it is necessary to count the number of nuclei in the glass.
* Corresponding author. Tel.: +81-75 724 7575; fax: +81-75 724 7580. E-mail address: [email protected] (T. Wakasugi).
Since the size of nuclei in glasses heat-treated for nucleation is too small for a direct microscopic observation, the glasses are heat treated again at higher temperature for a short time to make the nuclei grow to a detectable size. This time-consuming technique has been commonly used for the investigation of nucleation. Recently a new technique using dierential thermal analysis (DTA) has been proposed. Marrotta et al. [1] measured crystallization peak temperature by DTA and used the inverse function as a measure of the number density of nuclei, N. Ray and Day [2] measured the crystallization peak temperature, the height of the crystallization peak and the width of the crystallization peak at half maximum, and stated that it was preferable to use the height of crystallization peak as a measure of the number density of nuclei in a glass. This technique is a convenient way to know the variation of the relative number density of nuclei and to
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 7 1 8 - 9
T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
®nd the temperature at which nucleation becomes maximum. However, the absolute value of the number density cannot be obtained because the variation of crystal growth rate with temperature, which is necessary to analyze the crystallization process during DTA measurement, is not available. The latest report by Ray et al. [3] proposed a unique method to determine the rate of nucleation and crystal growth in glasses. Prior to the present study, the authors succeeded in evaluating the temperature dependence of nucleation rate semi-quantitatively by studying the relationship between crystallization peak temperature TC and heat treatment time for nucleation [4,5]. Also the relative value of the number density of nuclei was evaluated from TC by estimating the temperature dependence of crystal growth rate [6]. For further investigation of nucleation behavior, more quantitative analysis is required to determine the relationship between the number density of nuclei and TC . For this purpose, Li2 O 2SiO2 glass was selected in this study because various data necessary for the analysis of nucleation behavior are available for this glass [7±11]. This glass was heat treated under various conditions and TC was measured by DTA. The relationship between the number density of nuclei and TC was obtained and the variation of TC with heat treatment was discussed.
65
reduce the eect of surface nucleation, bulk samples were used on DTA runs and the surface of samples was ground before the DTA measurements. The release of the heat of crystallization makes the heating rate larger and causes TC shift to higher temperatures. TC tended to increase as the sample weight increased because of the increase of the heat of crystallization relative to the system. Generally, the sample with a large number of nuclei showed a large crystallization peak, which made the heating rate increase. Therefore, the weight of sample was adjusted in the range 10±20 mg depending on its nucleation conditions, so that the reproducibility of temperature in TC was 2°C.
3. Result 3.1. Dependence of TC on heat treatment temperature Fig. 1 shows DTA curves for Li2 O 2SiO2 glasses heat-treated for 20 h at the temperatures between 420°C and 540°C. After the heat treatment for 20 h at 560°C, the glass became opaque slightly and its crystallization peak was smaller than others. There are two possibilities for this
2. Experimental Li2 O 2SiO2 glass was prepared from reagent grade of Li2 CO3 and SiO2 . Well mixed batches were melted in a Pt crucible in an electric furnace at 1300°C for 1 h. The melts were stirred several times by moving the crucible to obtain homogeneous glasses and quenched by pressing the melt between steel plates to a thickness of about 1 mm. The glass was heat treated at temperatures between 400°C and 560°C and was cut to be about 2 mm square for DTA measurement to obtain TC . Though the shape of the samples was not the same for all samples, the dierence did not cause a measurable eect on TC . DTA measurements were conducted with a heating rate of 10°C min 1 using a Pt crucible (5 mm£ 5 mm). In order to
Fig. 1. DTA curves of Li2 O 2SiO2 glass heat-treated at various temperature for 20 h.
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T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
Fig. 2. Variation of TC for Li2 O 2SiO2 glass heat-treated for 20 h with heat-treatment temperature.
observation: crystallization or phase separation. In this case, a partial crystallization during the heat treatment would proceed because the glass composition was out of the immiscibility region [12]. Fig. 2 shows the variation of TC with heat treatment temperature for Li2 O 2SiO2 glass heat treated for 20 h. The sample heat-treated at 560°C was partially crystallized before the DTA measurement. As discussed later, partially crystallized samples tend to show low TC . TC decreased with the increase of heat treatment temperature and showed a minimum at 460°C and increased after that. According to Marotta [1], this means the number density of nuclei formed for 20 h becomes maximum at 460°C. Many investigators reported the temperature with maximum nucleation rate to be around 460°C from their measurement of the number density of nuclei. From DTA measurement, Marotta et al. [1] and Ray and Day [2] reported the temperature with maximum nucleation rate to be 455°C and 453°C, respectively. The present value agrees with these reports well. 3.2. Dependence of TC on heat treatment time Fig. 3 shows the dependence of TC on heat treatment time. The heat treatment temperature was 460°C. In order to show the variation of TC for short heat treatment time clearly, the scale of heat
Fig. 3. Variation of TC of Li2 O 2SiO2 glass heat-treated at 460°C with heat-treatment time.
treatment time is extended on the upper abscissa. The TC of the as-quenched glass was 681.9°C. Mishima et al. [13] and Matusita et al. [14] performed the DTA for the as-quenched Li2 O 2SiO2 glass at the same heating rate as the present study, and reported TC to be 683.7°C and 680°C, respectively. The present TC is close to these values. TC decreased rapidly with heat treatment time at the ®rst stage, and gradually the rate of TC decrease became low. However, it is impossible to evaluate the variation of the number density of nuclei with time from Fig. 3, directly. 3.3. Relationship between TC and the number density In order to obtain a relationship between TC and the number density, Li2 O 2SiO2 glasses were heat treated under the conditions (0.5±20 h, 455±497°C), for which the number density was reported [10], and their TC was measured. The result is shown in Table 1, and the relationship between TC and the number density is shown in Fig. 4. The coecient `c' shown in Table 1 and Fig. 4 is a parameter in the equation of crystal growth rate, Eq. (5), as discussed in Section 4.1. It is found that there is a linear relationship between TC and logarithmic number density, log N . The marks in Fig. 4 corre-
T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
67
Table 1 Relationship between the number density and TC for Li2 O 2SiO2 glass heat-treated under various conditions Heat-treatment conditions
N m 3 [10]
TC °C
c m Pa 1
Temperature
Time
455°C
4.0 h 8.0 h 15.0 h
1.531013 5.191013 11.601013
646.1 639.0 632.3
0.193 0.164 0.159
464°C
1.0 h 2.0 h 4.0 h 8.0 h 15.0 h
0.271013 0.951013 2.311013 4.981013 9.651013
658.1 647.1 643.5 636.1 632.4
0.233 0.219 0.184 0.184 0.169
481°C
0.5 h 1.0 h 3.0 h 20.0 h
0.151013 0.301013 0.861013 5.741013
660.3 655.4 648.1 635.9
0.265 0.245 0.219 0.177
497°C
5.0 h 8.0 h
0.301013 0.491013
651.9 651.7
0.274 0.234
It should be noted that Eq. (1) is valid only for the TC measured with the heating rate of 10°C min 1 because TC depends on the heating rate. The standard deviation of the dierence between log N calculated from Eq. (1) and experimental one is 0.10, which corresponds to 25% for N. The error in the literature is about 20% [10], indicating that the number density of nuclei can be evaluated by Eq. (1) with close accuracy to that measured by direct measurement.
4. Discussion 4.1. Calculation of crystallization process on DTA Fig. 4. Relationship between the number density of nuclei and TC for Li2 O 2SiO2 glass heat-treated under various conditions. Solid line is a guide for eye and dashed line was calculated according to Eq. (6).
spond to glasses heat treated at dierent temperatures. The relationship between TC and the number density is expressed by a straight line independent of the heat treatment temperature. The log N is expressed by Eq. (1) according to linear regression: log N m
3
s 1
0:06698TC °C 56:38 0:10: 1
The authors calculated TC of 47Na2 O 51SiO2 2ZrO2 glasses with various number densities of nuclei by estimating the crystal growth rate, and evaluated the number density of nuclei from TC obtained by DTA [6]. In that work, however, the experimental data of the number density of nuclei for this glass was not available, and only the estimation for relative number density of nuclei against that for the as-casted glass was possible. Here, the same calculation was performed for Li2 O 2SiO2 glass, for which the nucleation data are available.
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T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
Now, it is assumed that nuclei are spherical and grow isotropically on DTA heating. The variation of the volume fraction of crystals, a, is expressed by Eq. (2) [15]: da 4pNr2 1
a dr;
2
where N is the number density of nuclei and r the crystal radius. By integrating Eq. (2), Eq. (3) is obtained. r is a function of temperature, T, as shown in Eq. (4), where T0 and r0 are room temperature and the radius before DTA measurement, respectively. The crystal growth rate, U, is expressed as Eq. (5) [16] for convenience. 4 a 1 exp pNr3 ; 3 3 1 r r0 Q c U 1 g
Z
T
T0
U dT ;
exp
DHm T Tm RTTm
4 ;
5
where g denotes the viscosity, DHm the enthalpy change at melting, R the gas constant, Tm the melting temperature, and c a constant, Q the heating rate, dT =dt. The number of nuclei formed during DTA measurement is considered negligible here. TC is de®ned by the following equation; 2 da 4pN r 1 a dT 2 TC Q 4pN 2 3 dU 2U 2 U r r 0: 6 Q dT Q The decrease of TC with the increase of the number density of nuclei will be shown in the following. A stricter analysis for this decrease was presented by Weinberg [17,18]. The variation of d2 a=dT 2 as a function of temperature for the glass with the number density of N1 is shown in Fig. 5 by solid line. TC of this glass is T1 . When the number density of nuclei increases from N1 to N2 , the shift of d2 a=dT 2 at T1 determines whether TC increases or decreases. For example, the decrease of d2 a=dT 2 at T1 means the curve of d2 a=dT 2 changes to the dotted curve in Fig. 5, and TC decreases to be T2 .
Fig. 5. Variation of d2 a=dT 2 as a function of temperature for glasses with dierent number density of nuclei.
The variation of d2 a=dT 2 with the increase of the number density is expressed by Eq. (7): o d2 a=dT 2 1 d2 a 4pN 4p 2 3 r : 7 r 1 a U oN N dT 2 Q Q From the above equation, it is clear that o d2 a=dT 2 =oN is negative at T1 , then TC decreased by the increase of the number density of nuclei. Calculation of TC was performed using the following physical properties of Li2 O 2SiO2 glass to obtain the temperature which satis®es Eq. (6). Tm is 1034°C and DHm is 27:0 kJ mol 1 SiO2 . The viscosity data by Zanotto et al. [10] were used. The value of c is unknown and was determined so that the calculated TC agrees with measured one shown in Table 1, where the calculated c is also shown. The value of c ranges between 0.159 and 0:274 m Pa 1 , and tends to decrease with the increases of the number density of nuclei. A reason for this variation of c might be a dependence of crystal growth rate on the size of nuclei because size dependence was not considered in Eq. (5). Kelton and Greer [19] showed the size dependence of crystal growth rate is expressed as Eq. (8): 1=3 16D 3V V 2rA U 2 DGv sinh ; 8 4p 2kB T r k where D is the interfacial diusivity, k the molecular jump distance, V the molecular volume, DGv the free energy decrease per unit volume, and r is
T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
the interfacial energy. The growth rate of the crystal with size close to the critical radius increases as it grows, and ®nally reaches a constant value. If the number density of nuclei was high, the size of nuclei near TC would be so small that the growth rate of crystals at that size might be smaller than that of the glass with low number density of nuclei. As a result, c might be evaluated to be smaller as the number density of nuclei increases. Since the crystal growth rate calculated by Eq. (8) is 2.6 times larger than the experimental value, this equation was not employed for the calculation of TC . Deubener et al. [20] evaluated the crystal growth rate of Li2 O 2SiO2 glass at 490°C using Eq. (8) and stated that the size-dependent growth rates of crystals become dominant for a radius <0.1 lm. When r 0:1 lm and the number density of nuclei is 1015 m 3 which is the maximum value estimated under present experimental conditions, the volume fraction of crystals is less than 10 5 which is too small to give an eect on TC . Therefore, size-dependent growth rates does not account for the variation of c. It is clear that the temperature dependence of crystal growth rate should be evaluated more precisely to obtain the constant c. The crystal growth rate calculated from Eq. (5) with c 0:209 m Pa 1 , which is the average value in Table 1, was compared with experimental data in Fig. 6 [12,20±22]. The calculation agrees well with the reported values, which means that the average value of c based on crystal growth rate calculations is reasonable. In the following calculation, this average value of 0:209 m Pa 1 was used. The calculated relationship between the number density of nuclei, N, and TC is shown by a dashed line in Fig. 4, which agrees with the relationship obtained by experiments. 4.2. Eect of the size of nuclei on TC As shown in Fig. 2, TC drops again at 560°C after showing a minimum at 460°C. This decrease in TC would not mean an increase of the number density of nuclei because the temperature pro®le of nucleation rate generally shows one maximum. Since the sample after the heat treatment at 560°C for 20 h was partially crystallized, the eect of the size of
69
Fig. 6. Variation of crystal growth rate of Li2 O 2SiO2 glass with temperature. Solid line was obtained from Eq. (5) and marks are reported values [12,20±22].
nuclei on TC was considered. A similar decrease in TC was observed for 30Li2 O 70SiO2 glass [23]. As discussed in the previous section, the variation of d2 a=dT 2 with the increase of r was considered. The sign of d2 a=dT 2 is same as that in the brackets in Eq. (6). Then, the variation of F de®ned by Eq. (9) as a function of r was considered instead of d2 a=dT 2 . The partial derivative of F with respect to r is expressed by Eq. (10). F oF or
4pN 2 3 dU 2U 2 U r r ; Q dT Q 12pN 2 2 dU U r : Q dT
9
10
The ®rst term in Eq. (10) is negative, and the other positive since crystal growth rate increases with the temperature under 920°C [24]. Fig. 7 shows the variation of F and oF =or with r. The value rC is the radius of nuclei at T1 which is TC for this glass, and r1 is the solution for oF =or 0. Since oF =or is positive at r 0 and decreases monotonically with the increase of r, rC is always greater than r1 and oF =or is negative at rC . Therefore, o d2 a=dT 2 =or is negative at T1 , then it is found that TC decreased with the increase of the radius of nuclei before DTA measurement.
70
T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
the experiments would be smaller than this calculation because there is a distribution in the size of nuclei and the average size is smaller than that for this calculation. Therefore, the eect of the growth of nuclei during heat treatment on TC is neglected in the present study and it is reasonable to use TC as a measure of the number density of nuclei. 4.3. Eect of nuclei formed during DTA measurement on TC
Fig. 7. Variation of F and oF =or as a function of crystal size.
The variation of TC with the change of r0 for various number densities of nuclei was calculated as shown in Fig. 8. The decrease of TC is small when the number density of nuclei is low and it becomes signi®cant as the number density of nuclei increases. However, the decrease of TC for the heat treatment for 120 h at 460°C (r0 2:7 10 7 m) is only 1°C even when the number density of nuclei is 1015 m 3 which is maximum value in the present study. The decrease of TC obtained by
Fig. 8. The eect of the size of nuclei on TC .
Mishima et al. [25] measured the number density of nuclei in as-quenched Li2 O 2SiO2 glass melted at 1300°C and reported it as 2 109 m 3 which is much smaller than the calculated value (5:1 1010 m 3 ) from measured TC according to Eq. (1). The TC corresponding to 2 109 m 3 is 764.3°C. The dierence of TC would be explained by the formation of nuclei during DTA. Now the number of nuclei formed during DTA measurement is estimated. The induction time for nucleation at low temperature is so long that the nucleation at this temperature range during DTA measurement was neglected and the induction time was estimated to be zero at a temperature higher than 470°C ac-
Fig. 9. The nucleation rate of Li2 O 2SiO2 glass as a function of temperature [8,10,26,27]. Solid line is a guide for the eye.
T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
cording to Zanotto et al. [10]. Then, only the nucleation above 470°C was considered. Fig. 9 shows the steady state nucleation rate (experimental data) above 460°C [8,10,26,27]. Nucleation rate, I, is expressed as Eq. (11) by leastsquare method. Eq. (11) is shown in Fig. 9 by a solid line. I mm
3
s 1 exp
8:395 10 4 T 2 0:7349T 138:64 :
11
By integrating Eq. (11) from 470°C to 600°C with heating rate of 10°C min 1 , the number density of nuclei formed during DTA measurement was estimated to be 1:5 1011 m 3 . Calculated TC with this value was 696.3°C, which is in good agreement with measurements. When the number density of nuclei is lower than 1:5 1011 m 3 , it is dicult to evaluate it by DTA with good accuracy. 4.4. Evaluation of the number density of nuclei from TC The TC shown in Fig. 3 was converted to the number density of nuclei by using Eq. (1) and the data are replotted in Fig. 10. The number density of nuclei increases linearly with heat-treatment
71
time. From the slope and the intercept, the nucleation rate and the induction time at 460°C were calculated to be 2:5 109 m 3 s 1 and 0.2 h, respectively. The calculated nucleation rate agrees well with the reported value (2:8 109 m 3 s 1 at 460°C) shown in Fig. 9. It was shown that the steady state nucleation continues for 120 h. 5. Conclusion DTA measurements were performed on the Li2 O 2SiO2 glasses that had been heat-treated under the same conditions as in the literature which reported the number density of nuclei formed under these conditions. The relationship between the crystallization peak temperature and the number density of nuclei was obtained and some factors having eects on the relationship were considered. It was shown from the calculation of crystallization process in DTA measurement that the growth of nuclei during the heat-treatment for nucleation makes TC shift toward lower temperature. This tendency appears strongly when the number density of nuclei is high, and it was found that this eect was negligible in this study. The number density of nuclei formed during a DTA measurement was estimated to be 1:5 1011 m 3 and the evaluation of the number density of nuclei of this order or less would have low accuracy. The nucleation rate evaluated from a crystallization peak temperature measurement agrees well with the literature. This method is useful to evaluate the number density of nuclei for highly nucleated glasses and it is expected to be applied to other glass systems. References
Fig. 10. Variation of the number density of nuclei evaluated from TC for Li2 O 2SiO2 glass with heat treatment time.
[1] A. Marotta, A. Buri, F. Branda, S. Saiello, in: J.H. Simmons, D.R. Uhlmann, G.H. Beall (Eds.), Nucleation and Crystallization in Glasses, vol. 4 Advances in Ceramics, American Ceramic Society, Westerville, OH, 1982, p. 146. [2] C.S. Ray, D.E. Day, J. Am. Ceram. Soc. 73 (1990) 439. [3] C.S. Ray, X. Fang, D.E. Day, J. Am. Ceram. Soc. 83 (2000) 865. [4] T. Wakasugi, M.C. Weinberg, R. Ota, J. Soc. Mater. Res. Japan 48 (1999) 535.
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[5] T. Wakasugi, L.L. Burgner, M.C. Weinberg, J. Non-Cryst. Solids 244 (1999) 63. [6] T. Wakasugi, R. Ota, J. Non-Cryst. Solids 274 (2000) 175. [7] K. Matsushita, M. Tashiro, J. Non-Cryst. Solids 11 (1973) 471. [8] P.F. James, Phys. Chem. Glasses 15 (1974) 95. [9] V.M. Fokin, V.N. Filipovich, A.M. Kalinina, Fiz. Khim. Stekla 3 (1977) 129. [10] E.D. Zanotto, P.F. James, J. Non-Cryst. Solids 74 (1985) 373. [11] P.F. James, J. Non-Cryst. Solids 73 (1985) 517. [12] M. Tomozawa, Phys. Chem. Glasses 14 (1973) 112. [13] N. Mishima, R. Ota, T. Wakasugi, J. Fukunaga, J. NonCryst. Solids 19 (1996) 197. [14] K. Matusita, S. Sakka, T. Maki, M. Tashiro, J. Mater. Sci. 10 (1975) 94. [15] K. Matusita, S. Sakka, Y. Matsui, J. Mater. Sci. 10 (1975) 961. [16] H. Scholze, Glass, Springer, New York, 1991, p. 64. [17] M.C. Weinberg, J. Am. Ceram. Soc. 74 (1991) 1905.
[18] M.C. Weinberg, J. Non-Cryst. Solids 127 (1991) 151. [19] K.F. Kelton, A.L. Greer, J. Non-Cryst. Solids 79 (1986) 295. [20] J. Deubener, R. Bruckner, M. Sternitzke, J. Non-Cryst. Solids 163 (1993) 1. [21] T. Ogura, R. Hayashi, M. Kodama, J. Ceram. Soc. Jpn. 76 (1968) 277. [22] C.J.R. Gonzalez-Oliver, P.S. Johnson, P.F. James, J. Mater. Sci. 14 (1979) 1159. [23] T. Wakasugi, T. Kadoguchi, R. Ota, J. Soc. Mater. Res. Japan 50 (2001) 608. [24] K. Matusita, M. Tashiro, Yogyo-Kyokai-Shi 81 (1973) 500. [25] N. Mishima, R. Ota, T. Wakasugi, J. Fukunaga, Zairyo 44 (1995) 687. [26] J.J. Tuzzeo, PhD Thesis, Ohio State University, USA, 1976. [27] V.M. Fokin, V.N. Filipovich, A.M. Kalinina, Fiz. Khim. Stekla 3 (1977) 129.
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Evaluation of the number density of nuclei in Li2O 2SiO2 glass by DTA method Takashi Wakasugi *, Takuya Kadoguchi, Rikuo Ota Department of Chemistry and Materials Technology, Kyoto Institute of Technology Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan Received 4 December 2000
Abstract DTA measurements were performed to obtain the crystallization peak temperature, TC , for Li2 O 2SiO2 glass heattreated for nucleation. The relationship between TC and the number density of nuclei was obtained. Calculation of TC was performed by simulating the crystallization process, and the eects of the size and the number density of nuclei formed during DTA measurement on TC was evaluated. The calculated relationship between the number density of nuclei and TC agreed with the experimental one, and it was shown that the eects of the size and the number density of nuclei formed during DTA measurements is negligible in the evaluation of the number density of nuclei by TC . The steady state nucleation rate for the Li2 O 2SiO2 glass evaluated by DTA agreed well with the literature. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Since many glasses lose their desirable properties when they crystallize, the stability of glasses under usage conditions is very important to the application of glass materials. On the other hand, the properties of crystallized glass-ceramics depend on their crystallization process. The crystallization process is composed of nucleation and crystal growth, and the investigation of the nucleation behavior of glasses provides an important component to the production of glass materials including crystallized glass-ceramics. For the investigation of nucleation, it is necessary to count the number of nuclei in the glass.
* Corresponding author. Tel.: +81-75 724 7575; fax: +81-75 724 7580. E-mail address: [email protected] (T. Wakasugi).
Since the size of nuclei in glasses heat-treated for nucleation is too small for a direct microscopic observation, the glasses are heat treated again at higher temperature for a short time to make the nuclei grow to a detectable size. This time-consuming technique has been commonly used for the investigation of nucleation. Recently a new technique using dierential thermal analysis (DTA) has been proposed. Marrotta et al. [1] measured crystallization peak temperature by DTA and used the inverse function as a measure of the number density of nuclei, N. Ray and Day [2] measured the crystallization peak temperature, the height of the crystallization peak and the width of the crystallization peak at half maximum, and stated that it was preferable to use the height of crystallization peak as a measure of the number density of nuclei in a glass. This technique is a convenient way to know the variation of the relative number density of nuclei and to
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 7 1 8 - 9
T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
®nd the temperature at which nucleation becomes maximum. However, the absolute value of the number density cannot be obtained because the variation of crystal growth rate with temperature, which is necessary to analyze the crystallization process during DTA measurement, is not available. The latest report by Ray et al. [3] proposed a unique method to determine the rate of nucleation and crystal growth in glasses. Prior to the present study, the authors succeeded in evaluating the temperature dependence of nucleation rate semi-quantitatively by studying the relationship between crystallization peak temperature TC and heat treatment time for nucleation [4,5]. Also the relative value of the number density of nuclei was evaluated from TC by estimating the temperature dependence of crystal growth rate [6]. For further investigation of nucleation behavior, more quantitative analysis is required to determine the relationship between the number density of nuclei and TC . For this purpose, Li2 O 2SiO2 glass was selected in this study because various data necessary for the analysis of nucleation behavior are available for this glass [7±11]. This glass was heat treated under various conditions and TC was measured by DTA. The relationship between the number density of nuclei and TC was obtained and the variation of TC with heat treatment was discussed.
65
reduce the eect of surface nucleation, bulk samples were used on DTA runs and the surface of samples was ground before the DTA measurements. The release of the heat of crystallization makes the heating rate larger and causes TC shift to higher temperatures. TC tended to increase as the sample weight increased because of the increase of the heat of crystallization relative to the system. Generally, the sample with a large number of nuclei showed a large crystallization peak, which made the heating rate increase. Therefore, the weight of sample was adjusted in the range 10±20 mg depending on its nucleation conditions, so that the reproducibility of temperature in TC was 2°C.
3. Result 3.1. Dependence of TC on heat treatment temperature Fig. 1 shows DTA curves for Li2 O 2SiO2 glasses heat-treated for 20 h at the temperatures between 420°C and 540°C. After the heat treatment for 20 h at 560°C, the glass became opaque slightly and its crystallization peak was smaller than others. There are two possibilities for this
2. Experimental Li2 O 2SiO2 glass was prepared from reagent grade of Li2 CO3 and SiO2 . Well mixed batches were melted in a Pt crucible in an electric furnace at 1300°C for 1 h. The melts were stirred several times by moving the crucible to obtain homogeneous glasses and quenched by pressing the melt between steel plates to a thickness of about 1 mm. The glass was heat treated at temperatures between 400°C and 560°C and was cut to be about 2 mm square for DTA measurement to obtain TC . Though the shape of the samples was not the same for all samples, the dierence did not cause a measurable eect on TC . DTA measurements were conducted with a heating rate of 10°C min 1 using a Pt crucible (5 mm£ 5 mm). In order to
Fig. 1. DTA curves of Li2 O 2SiO2 glass heat-treated at various temperature for 20 h.
66
T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
Fig. 2. Variation of TC for Li2 O 2SiO2 glass heat-treated for 20 h with heat-treatment temperature.
observation: crystallization or phase separation. In this case, a partial crystallization during the heat treatment would proceed because the glass composition was out of the immiscibility region [12]. Fig. 2 shows the variation of TC with heat treatment temperature for Li2 O 2SiO2 glass heat treated for 20 h. The sample heat-treated at 560°C was partially crystallized before the DTA measurement. As discussed later, partially crystallized samples tend to show low TC . TC decreased with the increase of heat treatment temperature and showed a minimum at 460°C and increased after that. According to Marotta [1], this means the number density of nuclei formed for 20 h becomes maximum at 460°C. Many investigators reported the temperature with maximum nucleation rate to be around 460°C from their measurement of the number density of nuclei. From DTA measurement, Marotta et al. [1] and Ray and Day [2] reported the temperature with maximum nucleation rate to be 455°C and 453°C, respectively. The present value agrees with these reports well. 3.2. Dependence of TC on heat treatment time Fig. 3 shows the dependence of TC on heat treatment time. The heat treatment temperature was 460°C. In order to show the variation of TC for short heat treatment time clearly, the scale of heat
Fig. 3. Variation of TC of Li2 O 2SiO2 glass heat-treated at 460°C with heat-treatment time.
treatment time is extended on the upper abscissa. The TC of the as-quenched glass was 681.9°C. Mishima et al. [13] and Matusita et al. [14] performed the DTA for the as-quenched Li2 O 2SiO2 glass at the same heating rate as the present study, and reported TC to be 683.7°C and 680°C, respectively. The present TC is close to these values. TC decreased rapidly with heat treatment time at the ®rst stage, and gradually the rate of TC decrease became low. However, it is impossible to evaluate the variation of the number density of nuclei with time from Fig. 3, directly. 3.3. Relationship between TC and the number density In order to obtain a relationship between TC and the number density, Li2 O 2SiO2 glasses were heat treated under the conditions (0.5±20 h, 455±497°C), for which the number density was reported [10], and their TC was measured. The result is shown in Table 1, and the relationship between TC and the number density is shown in Fig. 4. The coecient `c' shown in Table 1 and Fig. 4 is a parameter in the equation of crystal growth rate, Eq. (5), as discussed in Section 4.1. It is found that there is a linear relationship between TC and logarithmic number density, log N . The marks in Fig. 4 corre-
T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
67
Table 1 Relationship between the number density and TC for Li2 O 2SiO2 glass heat-treated under various conditions Heat-treatment conditions
N m 3 [10]
TC °C
c m Pa 1
Temperature
Time
455°C
4.0 h 8.0 h 15.0 h
1.531013 5.191013 11.601013
646.1 639.0 632.3
0.193 0.164 0.159
464°C
1.0 h 2.0 h 4.0 h 8.0 h 15.0 h
0.271013 0.951013 2.311013 4.981013 9.651013
658.1 647.1 643.5 636.1 632.4
0.233 0.219 0.184 0.184 0.169
481°C
0.5 h 1.0 h 3.0 h 20.0 h
0.151013 0.301013 0.861013 5.741013
660.3 655.4 648.1 635.9
0.265 0.245 0.219 0.177
497°C
5.0 h 8.0 h
0.301013 0.491013
651.9 651.7
0.274 0.234
It should be noted that Eq. (1) is valid only for the TC measured with the heating rate of 10°C min 1 because TC depends on the heating rate. The standard deviation of the dierence between log N calculated from Eq. (1) and experimental one is 0.10, which corresponds to 25% for N. The error in the literature is about 20% [10], indicating that the number density of nuclei can be evaluated by Eq. (1) with close accuracy to that measured by direct measurement.
4. Discussion 4.1. Calculation of crystallization process on DTA Fig. 4. Relationship between the number density of nuclei and TC for Li2 O 2SiO2 glass heat-treated under various conditions. Solid line is a guide for eye and dashed line was calculated according to Eq. (6).
spond to glasses heat treated at dierent temperatures. The relationship between TC and the number density is expressed by a straight line independent of the heat treatment temperature. The log N is expressed by Eq. (1) according to linear regression: log N m
3
s 1
0:06698TC °C 56:38 0:10: 1
The authors calculated TC of 47Na2 O 51SiO2 2ZrO2 glasses with various number densities of nuclei by estimating the crystal growth rate, and evaluated the number density of nuclei from TC obtained by DTA [6]. In that work, however, the experimental data of the number density of nuclei for this glass was not available, and only the estimation for relative number density of nuclei against that for the as-casted glass was possible. Here, the same calculation was performed for Li2 O 2SiO2 glass, for which the nucleation data are available.
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T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
Now, it is assumed that nuclei are spherical and grow isotropically on DTA heating. The variation of the volume fraction of crystals, a, is expressed by Eq. (2) [15]: da 4pNr2 1
a dr;
2
where N is the number density of nuclei and r the crystal radius. By integrating Eq. (2), Eq. (3) is obtained. r is a function of temperature, T, as shown in Eq. (4), where T0 and r0 are room temperature and the radius before DTA measurement, respectively. The crystal growth rate, U, is expressed as Eq. (5) [16] for convenience. 4 a 1 exp pNr3 ; 3 3 1 r r0 Q c U 1 g
Z
T
T0
U dT ;
exp
DHm T Tm RTTm
4 ;
5
where g denotes the viscosity, DHm the enthalpy change at melting, R the gas constant, Tm the melting temperature, and c a constant, Q the heating rate, dT =dt. The number of nuclei formed during DTA measurement is considered negligible here. TC is de®ned by the following equation; 2 da 4pN r 1 a dT 2 TC Q 4pN 2 3 dU 2U 2 U r r 0: 6 Q dT Q The decrease of TC with the increase of the number density of nuclei will be shown in the following. A stricter analysis for this decrease was presented by Weinberg [17,18]. The variation of d2 a=dT 2 as a function of temperature for the glass with the number density of N1 is shown in Fig. 5 by solid line. TC of this glass is T1 . When the number density of nuclei increases from N1 to N2 , the shift of d2 a=dT 2 at T1 determines whether TC increases or decreases. For example, the decrease of d2 a=dT 2 at T1 means the curve of d2 a=dT 2 changes to the dotted curve in Fig. 5, and TC decreases to be T2 .
Fig. 5. Variation of d2 a=dT 2 as a function of temperature for glasses with dierent number density of nuclei.
The variation of d2 a=dT 2 with the increase of the number density is expressed by Eq. (7): o d2 a=dT 2 1 d2 a 4pN 4p 2 3 r : 7 r 1 a U oN N dT 2 Q Q From the above equation, it is clear that o d2 a=dT 2 =oN is negative at T1 , then TC decreased by the increase of the number density of nuclei. Calculation of TC was performed using the following physical properties of Li2 O 2SiO2 glass to obtain the temperature which satis®es Eq. (6). Tm is 1034°C and DHm is 27:0 kJ mol 1 SiO2 . The viscosity data by Zanotto et al. [10] were used. The value of c is unknown and was determined so that the calculated TC agrees with measured one shown in Table 1, where the calculated c is also shown. The value of c ranges between 0.159 and 0:274 m Pa 1 , and tends to decrease with the increases of the number density of nuclei. A reason for this variation of c might be a dependence of crystal growth rate on the size of nuclei because size dependence was not considered in Eq. (5). Kelton and Greer [19] showed the size dependence of crystal growth rate is expressed as Eq. (8): 1=3 16D 3V V 2rA U 2 DGv sinh ; 8 4p 2kB T r k where D is the interfacial diusivity, k the molecular jump distance, V the molecular volume, DGv the free energy decrease per unit volume, and r is
T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
the interfacial energy. The growth rate of the crystal with size close to the critical radius increases as it grows, and ®nally reaches a constant value. If the number density of nuclei was high, the size of nuclei near TC would be so small that the growth rate of crystals at that size might be smaller than that of the glass with low number density of nuclei. As a result, c might be evaluated to be smaller as the number density of nuclei increases. Since the crystal growth rate calculated by Eq. (8) is 2.6 times larger than the experimental value, this equation was not employed for the calculation of TC . Deubener et al. [20] evaluated the crystal growth rate of Li2 O 2SiO2 glass at 490°C using Eq. (8) and stated that the size-dependent growth rates of crystals become dominant for a radius <0.1 lm. When r 0:1 lm and the number density of nuclei is 1015 m 3 which is the maximum value estimated under present experimental conditions, the volume fraction of crystals is less than 10 5 which is too small to give an eect on TC . Therefore, size-dependent growth rates does not account for the variation of c. It is clear that the temperature dependence of crystal growth rate should be evaluated more precisely to obtain the constant c. The crystal growth rate calculated from Eq. (5) with c 0:209 m Pa 1 , which is the average value in Table 1, was compared with experimental data in Fig. 6 [12,20±22]. The calculation agrees well with the reported values, which means that the average value of c based on crystal growth rate calculations is reasonable. In the following calculation, this average value of 0:209 m Pa 1 was used. The calculated relationship between the number density of nuclei, N, and TC is shown by a dashed line in Fig. 4, which agrees with the relationship obtained by experiments. 4.2. Eect of the size of nuclei on TC As shown in Fig. 2, TC drops again at 560°C after showing a minimum at 460°C. This decrease in TC would not mean an increase of the number density of nuclei because the temperature pro®le of nucleation rate generally shows one maximum. Since the sample after the heat treatment at 560°C for 20 h was partially crystallized, the eect of the size of
69
Fig. 6. Variation of crystal growth rate of Li2 O 2SiO2 glass with temperature. Solid line was obtained from Eq. (5) and marks are reported values [12,20±22].
nuclei on TC was considered. A similar decrease in TC was observed for 30Li2 O 70SiO2 glass [23]. As discussed in the previous section, the variation of d2 a=dT 2 with the increase of r was considered. The sign of d2 a=dT 2 is same as that in the brackets in Eq. (6). Then, the variation of F de®ned by Eq. (9) as a function of r was considered instead of d2 a=dT 2 . The partial derivative of F with respect to r is expressed by Eq. (10). F oF or
4pN 2 3 dU 2U 2 U r r ; Q dT Q 12pN 2 2 dU U r : Q dT
9
10
The ®rst term in Eq. (10) is negative, and the other positive since crystal growth rate increases with the temperature under 920°C [24]. Fig. 7 shows the variation of F and oF =or with r. The value rC is the radius of nuclei at T1 which is TC for this glass, and r1 is the solution for oF =or 0. Since oF =or is positive at r 0 and decreases monotonically with the increase of r, rC is always greater than r1 and oF =or is negative at rC . Therefore, o d2 a=dT 2 =or is negative at T1 , then it is found that TC decreased with the increase of the radius of nuclei before DTA measurement.
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T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
the experiments would be smaller than this calculation because there is a distribution in the size of nuclei and the average size is smaller than that for this calculation. Therefore, the eect of the growth of nuclei during heat treatment on TC is neglected in the present study and it is reasonable to use TC as a measure of the number density of nuclei. 4.3. Eect of nuclei formed during DTA measurement on TC
Fig. 7. Variation of F and oF =or as a function of crystal size.
The variation of TC with the change of r0 for various number densities of nuclei was calculated as shown in Fig. 8. The decrease of TC is small when the number density of nuclei is low and it becomes signi®cant as the number density of nuclei increases. However, the decrease of TC for the heat treatment for 120 h at 460°C (r0 2:7 10 7 m) is only 1°C even when the number density of nuclei is 1015 m 3 which is maximum value in the present study. The decrease of TC obtained by
Fig. 8. The eect of the size of nuclei on TC .
Mishima et al. [25] measured the number density of nuclei in as-quenched Li2 O 2SiO2 glass melted at 1300°C and reported it as 2 109 m 3 which is much smaller than the calculated value (5:1 1010 m 3 ) from measured TC according to Eq. (1). The TC corresponding to 2 109 m 3 is 764.3°C. The dierence of TC would be explained by the formation of nuclei during DTA. Now the number of nuclei formed during DTA measurement is estimated. The induction time for nucleation at low temperature is so long that the nucleation at this temperature range during DTA measurement was neglected and the induction time was estimated to be zero at a temperature higher than 470°C ac-
Fig. 9. The nucleation rate of Li2 O 2SiO2 glass as a function of temperature [8,10,26,27]. Solid line is a guide for the eye.
T. Wakasugi et al. / Journal of Non-Crystalline Solids 290 (2001) 64±72
cording to Zanotto et al. [10]. Then, only the nucleation above 470°C was considered. Fig. 9 shows the steady state nucleation rate (experimental data) above 460°C [8,10,26,27]. Nucleation rate, I, is expressed as Eq. (11) by leastsquare method. Eq. (11) is shown in Fig. 9 by a solid line. I mm
3
s 1 exp
8:395 10 4 T 2 0:7349T 138:64 :
11
By integrating Eq. (11) from 470°C to 600°C with heating rate of 10°C min 1 , the number density of nuclei formed during DTA measurement was estimated to be 1:5 1011 m 3 . Calculated TC with this value was 696.3°C, which is in good agreement with measurements. When the number density of nuclei is lower than 1:5 1011 m 3 , it is dicult to evaluate it by DTA with good accuracy. 4.4. Evaluation of the number density of nuclei from TC The TC shown in Fig. 3 was converted to the number density of nuclei by using Eq. (1) and the data are replotted in Fig. 10. The number density of nuclei increases linearly with heat-treatment
71
time. From the slope and the intercept, the nucleation rate and the induction time at 460°C were calculated to be 2:5 109 m 3 s 1 and 0.2 h, respectively. The calculated nucleation rate agrees well with the reported value (2:8 109 m 3 s 1 at 460°C) shown in Fig. 9. It was shown that the steady state nucleation continues for 120 h. 5. Conclusion DTA measurements were performed on the Li2 O 2SiO2 glasses that had been heat-treated under the same conditions as in the literature which reported the number density of nuclei formed under these conditions. The relationship between the crystallization peak temperature and the number density of nuclei was obtained and some factors having eects on the relationship were considered. It was shown from the calculation of crystallization process in DTA measurement that the growth of nuclei during the heat-treatment for nucleation makes TC shift toward lower temperature. This tendency appears strongly when the number density of nuclei is high, and it was found that this eect was negligible in this study. The number density of nuclei formed during a DTA measurement was estimated to be 1:5 1011 m 3 and the evaluation of the number density of nuclei of this order or less would have low accuracy. The nucleation rate evaluated from a crystallization peak temperature measurement agrees well with the literature. This method is useful to evaluate the number density of nuclei for highly nucleated glasses and it is expected to be applied to other glass systems. References
Fig. 10. Variation of the number density of nuclei evaluated from TC for Li2 O 2SiO2 glass with heat treatment time.
[1] A. Marotta, A. Buri, F. Branda, S. Saiello, in: J.H. Simmons, D.R. Uhlmann, G.H. Beall (Eds.), Nucleation and Crystallization in Glasses, vol. 4 Advances in Ceramics, American Ceramic Society, Westerville, OH, 1982, p. 146. [2] C.S. Ray, D.E. Day, J. Am. Ceram. Soc. 73 (1990) 439. [3] C.S. Ray, X. Fang, D.E. Day, J. Am. Ceram. Soc. 83 (2000) 865. [4] T. Wakasugi, M.C. Weinberg, R. Ota, J. Soc. Mater. Res. Japan 48 (1999) 535.
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[5] T. Wakasugi, L.L. Burgner, M.C. Weinberg, J. Non-Cryst. Solids 244 (1999) 63. [6] T. Wakasugi, R. Ota, J. Non-Cryst. Solids 274 (2000) 175. [7] K. Matsushita, M. Tashiro, J. Non-Cryst. Solids 11 (1973) 471. [8] P.F. James, Phys. Chem. Glasses 15 (1974) 95. [9] V.M. Fokin, V.N. Filipovich, A.M. Kalinina, Fiz. Khim. Stekla 3 (1977) 129. [10] E.D. Zanotto, P.F. James, J. Non-Cryst. Solids 74 (1985) 373. [11] P.F. James, J. Non-Cryst. Solids 73 (1985) 517. [12] M. Tomozawa, Phys. Chem. Glasses 14 (1973) 112. [13] N. Mishima, R. Ota, T. Wakasugi, J. Fukunaga, J. NonCryst. Solids 19 (1996) 197. [14] K. Matusita, S. Sakka, T. Maki, M. Tashiro, J. Mater. Sci. 10 (1975) 94. [15] K. Matusita, S. Sakka, Y. Matsui, J. Mater. Sci. 10 (1975) 961. [16] H. Scholze, Glass, Springer, New York, 1991, p. 64. [17] M.C. Weinberg, J. Am. Ceram. Soc. 74 (1991) 1905.
[18] M.C. Weinberg, J. Non-Cryst. Solids 127 (1991) 151. [19] K.F. Kelton, A.L. Greer, J. Non-Cryst. Solids 79 (1986) 295. [20] J. Deubener, R. Bruckner, M. Sternitzke, J. Non-Cryst. Solids 163 (1993) 1. [21] T. Ogura, R. Hayashi, M. Kodama, J. Ceram. Soc. Jpn. 76 (1968) 277. [22] C.J.R. Gonzalez-Oliver, P.S. Johnson, P.F. James, J. Mater. Sci. 14 (1979) 1159. [23] T. Wakasugi, T. Kadoguchi, R. Ota, J. Soc. Mater. Res. Japan 50 (2001) 608. [24] K. Matusita, M. Tashiro, Yogyo-Kyokai-Shi 81 (1973) 500. [25] N. Mishima, R. Ota, T. Wakasugi, J. Fukunaga, Zairyo 44 (1995) 687. [26] J.J. Tuzzeo, PhD Thesis, Ohio State University, USA, 1976. [27] V.M. Fokin, V.N. Filipovich, A.M. Kalinina, Fiz. Khim. Stekla 3 (1977) 129.