Baixe zanotto 1989 e outras Notas de estudo em PDF para Engenharia de Produção, somente na Docsity! Trends in homogeneous crystal nucleation in oxide glasses E. D. Zanotto* & M. C. Weinberg Department of Materials Science and Engineering, University of Arizona, Tucson, AZ 85712, USA Manuscript received 26 September 1988 Calculations are performed to estimate the temperature of maximum nucleation rate, Tmax for several oxide glasses. H is found that for one class of glasses for which Tmas<T, (where T, is the glass transition temper- ature ), homogeneous crystal nucleation has never been experimentally observed. The postulate thar the failure to detect homogeneous nucleation in these glasses is due to transient nucleation effects is tested for several cases. Ft is concluded that for some glasses, such as B,0; and Na,0.41,0,.6SiO, (albite), long transient times ap- pear to be a contributing factor to the prevention of homogeneous crystallisation. However, it is demon- strated that transient nucleation effects are not re- sponsible for the faci that homogeneous nucleation would not be observed in Na,0.2Si0; and PbO.SiO;,. Hence, the failure of glasses with low Tax to crystallise homogeneous!y cannot in general be attributed to trans- ient effects and one must conclude that their steady state nucleation rates are particularly small. Although a large amount of information exists regard- ing crystal nucleation in glasses, during the past 10-15 years the realisation has developed that several basic questions pertaining to nucleation in glasses remain unresolved.“-* This has come about, in part, due to a number of experimental studies in which homo- geneous crystal nucleation rates in simple glass com- positions have been measured.é 9 Hence there has been an effort to obtain homogeneous crystal nucle- ation data in as many systems as possible. During the course of this endeavour two difficulties have been encountered. First, homogeneous and het- erogeneous nucleation are not always unambiguously distinguishabie, and this problem has been addressed in several works;º 1 second, a very limited number of simple glass compositions appear to exhibit homo- geneous crystal nucleation. Although an explanation can be given for this with some systems, in general this finding is not understood. Homogeneous nucleation seems to appear at rather large undercoolings while certain compositions show a tendency towards surface nucleation at relatively small undercoolings and if the *On sabbatical leave from Universidade Federal de Sao Carlos, Brazil. 186 Physics and Chemistry of Glasses Vol. 30 No. 5 October 1989 surface nucleated crystallites also tend to exhibit rapid growth, then complete crystallisation of the sample may ensue at modest undercoolings. Under such circumstances deep undercooling of the glass is pre- vented, and hence it would be impossible to seek homogeneous crystal nucleation in such systems. For a number of glass compositions, however, deep undercoolings are possible with a minimal (or at least tolerable) amount of surface crystallisation in evidence. Under these circumstances it is feasible to seek for signs of homogeneous nucleation; a number of such experiments have been performed, and homo- geneous nucleation has been detected in only a small percentage ol the systems studied. Three conditions must be satisfied for homogeneous erystal nucleation to be observable (by common microscopic techniques): (i) the crystal growth rate must be sufficiently large (in some temperature region) so that the nuclei can be grown to a size where they are detectable; (ii) the steady state homogeneous nucle- ation rate must be sufficiently large (say greater than 10? m”? s-! in order to form a sufficient number of particles within a reasonable time; (iii) the transient times cannot be too long in the temperature region of significant nucieation. In most cases condition (i) is easily satisfied but there are a few notable exceptions, two of which will be mentioned herein. Hence, the detection of homogeneous nucleation usually depends upon the magnitude of the steady state rate or the length of transient times in the nucleation region. In the present work we observe a trend in the experimental results gathered to date. Homogeneous nucleation has been reported for glass compositions which have relatively low reduced glass transition tem- peratures, T,/T, and whose (predicted and observed) temperature at which maximum nuckation occurs, Tomas above the glass transition temperature. On the other hand, homogeneous nucleation has not been reported (at least for inorganic glasses) for those compositions for which the temperature of maximum (predicted) homogencous nucleation occurs below the glass transition temperature. It might be suspected that the latter compositions have long transient times and hence that the lack of observable homogeneous E. D. ZANOTTO & M. C. WEINBERG: HOMOGENEOUS NUCLEATION IN OXIDE GLASSES nucleation in such systems is due to transient effects; this hypothesis is analysed in detail for several compositions. Trends in the location of the nucleation curve For seven glass compositions James!" demonstrated that the experimental values of Tao, are always at or somewhat above T, and that T,/T is in the range 0:54 to 0:59. In a recent publication! ? one of the present authors shows that, for homogeneous nucleation, the calculated values Of Trax are also close to T, in agreement with James's observations; it was also demonstrated that the reverse also applies, i.e. if the predicted Tmas falls below T;, only surface (hetero- geneous) nucleation is observed. Here, we calculate Ta, for 12 oxide glasses. It should be emphasised, however, that the values for Fº predicted by classical nucleation theory are many orders of magnitude lower than the actual valuestt” but the predicted temperature dependence is well described by theory when the viscosity is used to estimate the kinetic barrier. Therefore, only the lo- cation Of Tax Will be computed. Location of the steady state maximum nucleation rate The steady state nucleation rate, according to classical theory, may be written? cr f=—"—exp(— W*/R 1 KT p( RT) [89] where C is a constant, T is temperature, q is the viscosity, R is the gas constant, and W* is the bulk free energy required to form a critical nucleus which, for spherical nuclei, is given by 16n9º 4 = “rar a where o denotes the liquid-crystal surface energy and AG, is the bulk free energy difference between liquid and crystal per unit volume. The temperature at which the nucleation rate is maximum can be obtained by solving the equation dr/dT=0. a reduced temperature, T,, is defined as T/Tw where E, is the melting temperature, and f is the entropy of fusion in units of R, then the temper- ature of maximum steady state nucleation rate may be found from s too Sicnma=o 63 where 12% +ylnT) AT) vin T, h=[ATU =) Tin TI (3 In deriving Equation (3), it has been assumed that the difference in specific heat betwcen crystal and liquid, AC,, is constant and y = AC,/AS,, (where AS, is the (4) Physics and Chemistry cf Glasses Vol. 30 No. 5 October 1989 entropy of fusion); furthermore, it was assumed that o is proportional to the enthalpy of fusion with «, the Turnbull ratio, ranging from 1/3 to 1/2; finally, AT=1—T. The viscosity was taken to be of the Fulcher form, Iny= b 6 nn=a+ qo (6 T em For Table 1 shows the thermodynamic and viscosity data employed in the present calculations and the results are shown in Figure 1, where the reduced maximum nucleation temperatures are plotted against To Tay the predictions are given as lines or geometrical figures rather than points. The extension along the Table 1. Thermodynamic and viscosity data for several glass forming systems To Aa Glass (kJ tdimole) A B n 1. Na,0.2C80.3S10, NC;S; 1564 BIO 486 4893 547 2 Lij0.250, 15, 1307 s7%0 E8L 1347 595 3 BaO 250, BS; 1693 37000 183 1702 795 4 GO; Go JI38T ISO 994 17960 O —680 16393 0 5 C4O.A1,0, 2510, CAS; 1826 135500 —585 6750 738 6. Na,0 2810, NS, 147 35500 064 2315 Sé 45190 7 Li0.P,05 LP 926 6100 —410 2000 462 & PO, Pp 853 21760 687 0071 O 27200 9. PPO.SIO, PS 1037 34000 60420 10. SiO, s 1996 15000 11. NajO.ALO, ÓSIO, NAS, 1380 55000 12 80, B 73 2600 login=4+BAT. T) Pas (1) togty) 2) logtm) Homogeneous nueleation Tola Heterogensous nucleation os 06 97 E Figure 1. Plot of the calculated values of reduced temperature of maximum nucleation frequency against reduced glass transition tem- perature interval (lines and geometrical figures) O experimental points for NC;S,, LS,, and BS, glasses 187 E. D. ZANOTTO & M. €. WEINBERG: HOMOGENEOUS NUCLEATION IN OXIDE GLASSES E LO poses. 100 years Inri 18 ft year LS 16 17 15 197 20 TaT Figure 3, Predicted values of transient times in the temperature range where the nucleation rates should be highest, ie. between the two tímits of Tuox rates are expected to be maximum, ie. between the two limits for Tnax- The induction periods increase with increasing T,, (reduced glass transition temper- ature), being quite short for glasses with low T, and extremely long for compositions with T, >075. Hence, onc might conclude that the fact that homo- gencous crystal nucleation could not be observed in these systems is due to the suppression of the actual nucleation rate in comparison to the steady state rate. This point is illustrated in Table 2 where the transient times at the temperature of maximum nucle- ation and the ratio of actual to steady state nucleation rates are listed for those systems possessing very long transient times: it may be observed that the nucleation rates are very small fractions of the steady states rates even after prolonged heating. These data seem to support the hypothesis that the absence of homo- geneous nucleation results from transient efiects and in the remainder of this work the plausibility of this argument is analysed in more detail and comparisons are made with experimental findings. Table 2. Predicted values for 3 = 1/2 Himas) System Tum es) Tio After 1 hours) B sd 39x 1075 16x 10 1º na 0x 1018 NAS, 04% 97 x 101 96x [0-0 n=3x10º LP 0595 12x 1010 TEXIOTL n=3x 108 NS; 0600 72x 10º 76x 1071 n=175 s 0690 DO x 108 1Oxt08 n=65 Ps 0575 8x 108 30x 1071 n=50 Physics and Chemistry of Classes Vol, 30 No. 5 October 1989 Analysis of transient effects Governing equations Here, the influence of transient effects upon the po- sition and magnitude of the maximum nucleation rate will be discussed. For times where transient nucleation effects are not negligible, the position of maximum nucleation is shifted to higher temperatures (relative to the temperature of the steady state maximum) and the magnitude of the nucleation rate is reduced. This behaviour is analysed for particular choices of the steady state nucleation rate and the transient time behaviour. If Kt, T) denotes the time dependent nucieation rate and IT) the steady state nucleation rate, then I=TANHEO (o) where f(t, 1) is a function which describes the transient nucleation and depends upon the transient time, r. The position of the maximum nucleation rate as a function of time can be found by taking the temper- ature derivative of Equation (10) and setting it to zero. One obtains dinfº/dT = x(dlnf/dx) (dInt/dT) (11) where x is the time scaled by the transient time. For the transient time behaviour we choose NO) =1+25 (exporta, (12) which is the expression derived by Kashchiev.(16) If Equation (12) is employed in conjunction with the following form for the logarithm of the relaxation time, Int=2in(T,-ND+2+bMT— T), (13) then the right side of Equation (11) becomes x(dlnf/dx) dinc/dT = [AT DAT = Teto) (14) and gog=2x 5 (It nexp(—n?o)/fo. (15) as If the reduced temperature, T;, is introduced and Equation (3) is used for the left side of Equation (11), then use of Equations (14) and (15) leads to b 16n2ºBjh acmmtos 272 b =etd [a = ir) a Solutions of Equation (16) for T, for varying x (or t) give the location of the maximum nucleation temper- ature as a function of time. The steady state maximum nucleation temperature is obtained from the solution of Equation (16) with the right side of this equation equal to zero. Tt will be of interest, also, to compare the maximum nucleation ratc at some reduced time, x*, with the 189 E. D. ZANOTTO & M. €. WEINBERG: HOMOGENEOUS NUCLEATION IN OXIDE GLASSES steady state maximum nucleation rate. If TH is a solution of Equation (16) at x*, then this ratio of maximum intensities is given as HKTEox) JT) * Ditt le ms, TO. 17 Td mol O TO (9 1f Nº denotes the number of particles nucleated at some temperature when heated for some time period under steady state conditions then, since N(1)= frade, it is possible to show that N ah a (pt “mn 2 (3) LÊ Cl eetro cla Nº x [1 nê 12 which is an expression for the reduction in the number of particles formed at a given temperature due to transient effects. The number density ratio, which is given by the ratio between the number formed at T, and the steady state number formed at Tm, (i.e. the number density analogue of Equation (17)), is also of interest and is given by Ns T) NE) NÁT) NÃ Tm) NT) NÚ Tuas) “O Pas) 2& ar explontx) a? «[uiien 5 - n (19) Shift in maximum nucleation temperature Now we employ Equations (13) and (16)(19) to predict the shift in peak nucleation rate position and the reduction in the magnitude of the nucleation rate (and particles formed) due to transient nucleation effects. Tt is of particular interest to investigate the behaviour of several systems whose steady state max- imum nucleation rates were predicted to fall below T, in order to test the hypothesis that transient nucle- ation is responsible for the absence of homogeneous nucleation in such systems. Attention will therefore be focused primarily upon three of the compositions (NS,, PS, and B) listed in Table 2; these were chosen because there is information in the literature regarding their crystal nucleation and growth behaviour. First, however, we point out that the results shown in Table 2 were generated on the assumption that a=1/2 but if one chooses «= 1/3, the predicted Table 3. Predicted values for u = 1/3 Tuma) System Tou (83 B 0705 17x 108 NAS, 0600 26x 108 LP 0640 10x 103 NS, 0660 92x 10! 5 0780 30x10º Ps 0615 50x10º 190 Physics and Chemistry of Glasses Vol. 30 No. 5 October 1989 values Of Tnax OCcur at much higher values and con- sequently the transient times at the temperatures of maximum nucleation rate will be much smaller (com- pare Tables 2 and 3). In this section we consider the worst case (a = 1/2) to analyse if transient effects can obstruct the detection of homogeneous nucleation. Results Nas0.280, (T, = 723-746 K) Figure 4 shows the variation in the temperature of maximum nueleation rate with heat treatment time for NS; glass; there is a marked decrease in Toa With increasingly longer times, reflecting the influence of transient nucleation in this temperature range (700-750 K). The calculated values of 7 and the ratio between the steady state nucleation rate at temperature T and the maximum rate 14 Ts) are shown in Figure 5. The transient period at Trax (688 K) is very long but at 30 K above Trax, it is reduced to about 1 h, while the steady state rate is lowered by only one order of magnitude. Hence, if the magnitude of 1 ( Tas) is large enough, homogeneous nucleation might be measured in NS, if one chooses an appropriate thermal treat- ment, say between 710 and 730 K for a few hours. Crystal nucleation in a NS, glass (with 0:3 mol% Sb,0,) was sought, exhaustively, by Hishinumal9) at 700 K for 220 h, at 770 K for 250 h, and at 820 K for 7% Tosa [KO no 70 gol, : 4 2 1 q 1 2 3 log (time in hours) Figure 4. Temperature of maximum nucleation rate as a function of heat treatmen: time for NS» glass é o 4 2. 2 «log (r in seconds) mol oe. gi 60 70 70 7 750 HO 750 Absolute temperature Figure 5. Calculated values of transient times and the ratio of the steady state nucleation rate ar temperature, T, to the maximum rate, T/Tuax), for NS» E. D.ZANOTTO & M. €. WEINBERG: HOMOGENEOUS NUCLEATION IN OXIDE GLASSES IT -2 “a -s º o lo (NR s + aR i fo. ! -8d | i tl a 190 Eu Time (h) Figure 6. Ratio of predicied nucleation rate to iss maximum value at 700 K for NS,. Similar ratio for number of crystals formed in time 140 h. For both one step and two step heat treatments crystal growth rates would have been sufficiently large to reveal any internal crystals if they existed but only surface crystallisation was observed. At 700K, the predicted nucieation rate would approach its steady state value after about 100--200 h, even for the worst possible condition (a = 1/2), as is shown by Figure 6, and thus homogeneous nucleation should have been detectable by Hishinuma's experiments, At the other two temperatures explored, 770 and 820 K, the transi- ent times are much shorter, but so are the steady state nucleation rates, rendering internal nucleation unobservable. PbO.SiO, (T,=674-695 K) Similar calculations were carried out for PS glasses, again for «= 1/2, and Figure 7 shows the transient times and the ratio between the steady state nucle- ation rate at temperature T and its maximum valve at Thasx: in this case the transient times at Taz (597 K) are very long. HE one chooses a higher temperature, say 650 K, where 7 would be greatly reduced, the steady state nuclcation rates would be too much decreased, as is shown by Figure 7. At 612 K, for instance, a heat mo A logtrin seconds) + fatia SO SO 610 GM 60 60 60 cão Absolute temperaturo Figure 7. Calculated values of transient times and the ratio of the steady state nucleation rate at temperature, T, to the maximum rate, P(Toaxh. for PS Physics and Chemistry of Glasses Vol, 30 No. 5 October 1989 treatment of approximately 2000 h would be required to bring the nucleation rates (and crystal number density) to within 2 orders of magnitude of their steady state values. However, for a = 1/3, Tras = 700K and the transient times would be only 5s, as is shown by Table 3, Hishinumal? subjected a PbO.SiO, glass to several heat treatments-—at 670 K for 140 h, from 710 to 730 K for 2200 b, at 760 K for 140h, and also at higher temperatures (780 to 900 K)—-and onty surface crystallisation was observed. If 1Y Trax) had been suficientiy high, internal nucleation should have been detected by Hishinuma's experiments (for « = 1/3); for a close to 1/2, homogeneous nucleation would not be measurable. B,0s (T,= 5530-564 K) B,O, is an interesting system because it has the longest induction period among the twelve compo- sitions studied, with a Toa, Of SIOK for «= 1/3 and 463K for x= 1/2. Both temperatures are well below T, and the induction periods are extremely long, 46 x 10º and 10!2h, respectively. For «= 1/2 no thermal path can be found which will lead to detectable homogeneous nucleation within a reasonable time period; for the other limiting case, = 1/3, heat treatments in the vicinity oí 530K for 1000h would be required il nucleation were to be detected. To the authors" knowledge, however, no one has been able to observe crystallisation (internal or surface) in B,O; glass, event when the melt or the external surface has been seeded with B,O, crystals. This indicates that the crystal growth rates are much too low and thus even if internal nucleation occurred, it would not be observed since the nuclei would not grow sufficiently. This is the most difficult case to analyse since 1 is very long, the growth rate is extremely small, and the nucleation rate is unknown. Other systems Albite (NAS,) glass also does not crystaltise, even when seeded, indicating a very low growth rate, For other glasses, such as SiO, and P,Os, the predicted Trax are below T, but the transient times are such that thermal paths can be found, so that they will show homogeneous nucieation. However, SiO, glass, which has been extensively studied due to its commercial importance, has only shown surface nucleation and this is also true for P,Os, although this has been studied much less. Summary The fact that homogencous nucleation cannot be observed in glasses may be related to one or more of the following causes: low nucleation rate, low growth rate, and long induction times. It has been shown that for many oxide glasses which do not exhibit homo- geneous crystal nucieation the predicted Tra will occur well below 7, leading to the reasonahle deduc- 191