**UFRJ**

# schmelzer 2006

(Parte **2** de 4)

For liquids of suﬃciently low viscosity, the Stokes–Einstein/Eyring equation is fulﬁlled and the characteristic times of molecular motion, s, are of the same order of mag- nitude as Maxwell’s relaxation time, sR. Hence, one can get the following estimate for the Young modulus of the liquid, ð1 þ cmÞ ﬃ kBT. ð21Þ

Substitution of Eqs. (18) and (21) into Eq. (17) yields resulting in ln e

In this way, the eﬀect of elastic stresses can be ignored as far as Eqs. (18) and (21) are fulﬁlled.

However, when the decoupling temperature, Td,i s approached in cooling the liquid, molecular motion in the melt (which determines the rate of crystal growth) changes from liquid-like to solid-like. At Td, viscous relaxation and molecular motion decouple, the ratio of the char- acteristic time-scales sR/s increases exponentially with decreasing temperature, the Stokes–Einstein/Eyring equa- tion does not hold any more and, according to Eq. (20), the ratio (kBT/Dgd0) tends to zero. In order to analyze the dependence of the ratio e/e0 in this alternative case, we employ Eq. (20) and rewrite Eq. (17) as

Since, below Td with a further decrease of temperature, the ratio (s/sR) decreases exponentially, the parameter e tends to e0. This result is independent on any assumptions concerning the temperature dependence of the stress parame- ter, e0, itself, it is exclusively a consequence of the breakdown of the Stokes–Einstein/Eyring equation below

Td. A similar analysis, as outlined above in detail for normal growth, can be made for any other modes of growth. The only diﬀerence in the resulting equations is that the number ‘4’ in Eqs. (23) and (24) has to be replaced by (4/f). Since f has generally ﬁnite positive values less than one, the conclusions remain the same. Note also that a similar analysis with qualitatively equivalent results can be easily performed for the case of stretched exponential relaxation, as given by Eq. (12). Obviously, the only diﬀerence is that a replacement lnðe0=eÞ)½ lnðe0=eÞ 1=b has to be made in Eq. (17). We thus conclude that the results of the present anal- ysis are independent on the speciﬁc mechanisms of crystal growth and stress relaxation.

Analogous conclusions have been derived in our previous analyses of stress eﬀects on nucleation. Similarly to crystal growth, elastic stresses do not have any eﬀect on nucleation as far as the Stokes–Einstein/Eyring equation is fulﬁlled. However, in the temperature range below the decoupling

J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443 437 temperature, Td, elastic stress eﬀects may be of considerable importance. It follows that elastic stresses can be of signiﬁ- cance both for critical crystallite nucleation – as shown earlier [5,6,15,16] – and for the description of crystal growth, in both cases, in the same range of temperatures, T 6 Td. Similarly, as for nucleation, the approach developed here allows one to account for a continuous transition from a behavior of the ambient phase typical for a Newtonian liquid (for temperatures at and above the temperature of decoupling) to a behavior typical for a Hookean solid (for temperatures near and below Tg). In order to demonstrate these results, we will analyze in the next section, as an exam- ple, stress eﬀects on crystal growth in lithium disilicate glass.

4. Application: crystal growth rates in lithium disilicate glass

According to Eq. (9), for lithium disilicate the eﬀective stress parameter e, determining the eﬀect of elastic stresses on crystal growth, does not exceed, in the vicinity of Tg,a value of about 4% of the thermodynamic driving force, Dl.

By this reason, we are aware that the magnitude of the eﬀect of elastic stress on crystal growth will not be strong for this particular system. Despite this disadvantage, the general features of the theory developed above can be clearly demonstrated. The choice of this particular system has the advantages that crystal growth rate data are available over a wide range of temperature and, moreover, we can compare stress eﬀects on crystal growth with the results of the analysis of stress eﬀects on nucleation performed earlier by us for this particular system [15,16]. The availability of such additional information will allow us to derive some additional general conclusions not only concerning the relative magnitudes of the eﬀect of stress on nucleation, on one side, and crystal growth, on the other, but also of more general nature.

Assuming, again, that stress relaxation proceeds in accordance with Maxwell’s law (cf. Eq. (1)), Eq. (17) allows us to determine the ratio (e/e0) and, as a next step, the value of e as a function of temperature. The parameters of lithium disilicate glass are taken from the previous analyses of the eﬀect of elastic stresses on crystal nucleation [15,16]. By this reason, we only present here the ﬁnal results without a detailed discussion on how the respective dependencies were obtained.

The thermodynamic driving force of crystal growth in the absence of stresses, Dl, was taken from the experimental investigations performed by Takahashi and Yoshi [29]. The molar mass equals M = 150 g mol 1, the density of the glass and the crystalline phase are qglass = 2.35 g cm 3 and qcrystal = 2.45 g cm 3, respectively, resulting in a value of the misﬁt parameter equal to d = 0.04255. The characteris- tic size of the building units of the crystalline phase equals

1307 K and the glass-transition temperature equals Tg = 728 K. For the Poisson ratios of glass-forming melt and crystal, we take cm = c = 0.23.

To the best of our knowledge, experimental data on the dependence of Em on temperature are not available for the system under consideration. By this reason, we have set the Young’s modulus of the ambient phase equal to the modulus of the crystal, Ec = 76 GPa. There exists a variety of experimental data and general theoretical argu- ments indicating the existence of a considerable increase of Young’s modulus of glass-forming melts with decreasing temperature near the respective temperature of vitriﬁca- tion, Tg [6,17,24–28]. The incorporation of such eﬀects (as done, for example, in Ref. [6]) does not aﬀect the results of the present analysis and is, therefore, omitted here.

The temperature dependencies of the viscosity (in Pa s) and the diﬀusion coeﬃcient (in m2s 1) are interpolated via the following equations and

In both equations, the temperature is given in Kelvin.

For the determination of the eﬀective diﬀusion coeﬃcient, which determines the rate of crystallization as a function of temperature, we employed measurements of the nucleation time-lag. Such data are available for tempera- tures in the range 693 K 6 T 6 763 K (Tg 725 K). The results were interpolated in such a way as to fulﬁl the

Stokes–Einstein/Eyring relation for temperatures above

Td = 1.2Tg. This approach is corroborated by the comparison of diﬀusion coeﬃcients calculated from viscosity via the Stokes–Einstein/Eyring equation and from growth rates of lithium disilicate crystals in the melt of the same composition [30]. They indicate the possible existence of decoupling of diﬀusion and relaxation near Td = 1.2Tg

(see also Fig. 4). The Eyring ratio (kBT/gDd0) versus temperature, shown in Fig. 1, demonstrates signiﬁcant devia- tions from the Stokes–Einstein/Eyring behavior for low temperatures.

The ratio (e/e0) and the eﬀective stress parameter e versus temperature, obtained with Eqs. (25) and (26), are shown in Figs. 2 and 3 by full curves. One ﬁrst sees that, for temperatures in the range 0.46 6 T/Tm 6 0.49, the system switches from a liquid-like behavior (where elastic stresses are negligible) to a behavior that is typical for a Hookean solid. However, this transition is found here in a temperature range, where growth rates cannot be experimentally measured in reasonable time-scales (note that the glass-transition temperature for lithium disilicate is Tg/

Tm ﬃ 0.557). Therefore, for the considered system, elastic stresses seem to have no sizeable eﬀect on crystal growth rates, U, in the range of temperatures where U can be measured.

However, in the present analysis we employed values of the diﬀusion coeﬃcient calculated from data on the timelag for nucleation. In this procedure, classical nucleation

438 J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443

theory has been employed in order to determine the size of the critical clusters as a function of undercooling and normal growth was assumed in describing aggregation processes to clusters of critical sizes. Latter assumption has no eﬀect on the results of the computations of the temperature dependence of eﬀective stress parameter. Indeed, employing time-lag data for the determination of the kinetic coeﬃcients and assuming other modes of growth, actually the product Df in Eq. (2) is computed and in Eq. (17), we have to replace D by Df as well leaving the results of the computations of the stress parameters unchanged. However, taking into account well-known limitations of the classical nucleation theory in application to crystallization [31–3], we have to check whether the estimates of the eﬀective diﬀusion coeﬃcients, obtained in this way, describe crystal growth rates correctly. For these purposes, in Fig. 4 the linear growth rate U, determined by diﬀerent methods, is shown as a function of temperature. The dashed curve (1) represents the results if viscosity-data (according to Eq. (25)) are employed for the determination of the diﬀusion coeﬃcient and growth rates. The full curve gives the respective data obtained with the diﬀusion coeﬃcient – derived from time-lag measurements (cf. [15])– without (full curve (2)) and with (circles (3)) an account of elastic stress eﬀects. As we expected and in agreement k T/

D d

Fig. 1. Eyring ratio, (kBT/gDd0), as a function of temperature. To a good approximation, for T > Td ﬃ 1:2Tg, the Stokes–Einstein/Eyring equation is fulﬁlled. For T 6 Td, signiﬁcant deviations are found and the Eyring ratio tends to zero with decreasing temperature.

Fig. 2. Ratio (e/e0) as a function of temperature. The full curve shows the results of the computations if the diﬀusion coeﬃcient is determined via Eq.

(26). The dashed curves correspond to possible corrections of the diﬀusion coeﬃcient by a factor 102 (see text). In both cases, the approach of the eﬀective stress parameter to zero is found for temperatures, where Young’s modulus retains values typical for the crystal.

5.00E-022

1.00E-021

1.50E-021 2.00E-021

Fig. 3. Eﬀective elastic stress parameter e versus temperature. The full curve shows the results of the computations if the diﬀusion coeﬃcient is determined via Eq. (26). The dashed curves correspond to possible corrections of the diﬀusion coeﬃcient by a factor 102 (see text).

1E-17 1E-15 1E-13 1E-1 1E-9 1E-7 1E-5 1E-3

T U, m /s

Fig. 4. Crystal growth rate, U, versus temperature. Dashed curve: viscosity (Eq. (25)) and the Stokes–Einstein/Eyring equation (Eq. (18)) are employed for the determination of the diﬀusion coeﬃcient; full curve: data obtained with the diﬀusion coeﬃcient – derived from time-lag measurements (cf. [15]) – without (full curve) and with (circles) an account of elastic stress eﬀects. In addition, available experimental data are shown [30]. In order to describe adequately the experimental data in the vicinity of Tg, the diﬀusion coeﬃcient should increase by a factor of the order 102. Available experimental data denoted by stars.

J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443 439 with the results of computations shown in Figs. 2 and 3, elastic stresses do not have any sizeable eﬀect on the growth rate in the considered range of temperatures. In addition, available experimental data on crystal growth rates are shown [30]. Deviations of experimental crystal growth data and pre- dictions in the range of temperatures 0.8 < T/Tm <1 , employing viscosity measurements described by Eq. (25), can be resolved easily: the authors, which performed the analyses in this range, report lower (by a factor 1.5–2) values of viscosity for their systems. Consequently, employing their viscosity data, their growth data are described appropriately by diﬀusion coeﬃcients determined via the Stokes– Einstein/Eyring equation. Fig. 4 shows also that, in order to describe adequately the experimental growth rate data

in the vicinity of Tg, the diﬀusion coeﬃcient – obtained from time-lag data via classical theory – has to be increased by a factor 102 to reproduce the experimental results. The origin of this deviation in the considered range of temperatures can be twofold. First, one can suppose that the classical nucleation theory does not give an appropriate description of the properties and the parameters of the critical clusters. This point of view is supported by the dramatic deviations between the predictions of the classical theory and experimental rates of critical crystallite formation (cf. Refs. [16,31–3]). Such point of view gets additional support also from a generalization of Gibbs’ approach as developed in recent years (cf. Refs. [34–36] for an overview). On the other hand, the mentioned theoretical analyses and investigations by other authors, discussed there, show that the kinetic parameters determining nucleation and growth processes may depend on cluster size. In this way, the existing deviation in the estimates of the eﬀective diﬀusion coeﬃcient can be considered a reﬂection of existing problems in the description of crystallization kinetics which do not have found a satisfactory general solution so far.

In analyzing the eﬀect of elastic stresses on nucleation in lithium disilicate, in Ref. [16] we determined the driving force of crystallization and the speciﬁc interfacial energy from experimental data on time-lag and nucleation rates. In our analysis, both driving force and surface energy were considered as unknown functions of temperature. These results can be employed to determine the diﬀusion coeﬃcient from time-lag data, again, but in an alternative way as compared with classical nucleation theory. The resulting dependence of diﬀusion coeﬃcient on temperature is shown in Fig. 5 by a dashed curve. It turns out that it is one order of magnitude larger than the estimates obtained via the classical nucleation theory.

Provided we assume that time-lag data and their interpretation by the classical theory underestimate the diﬀusion coeﬃcient by the mentioned factor, then we have to repeat the computations, but with higher values of the diffusion coeﬃcients. The results of such computations are shown in Figs. 2 and 3 by dashed curves. Qualitatively, the results of the previous computations, given in Figs. 2 and 3 by full curves, are not changed, however, the respective curves are moved into a range of considerably higher temperatures (see the dashed curves in Figs. 2 and 3). Such a modiﬁcation of the values of the diﬀusion coeﬃcients leads, consequently, to a similar transition of the eﬀect of elastic stresses both on growth and also on nucleation (cf. the analysis in Ref. [15]).

5. Discussion

In the present analysis, we developed a general method to treat the eﬀect of elastic stresses on crystal growth taking into account both stress evolution and stress relaxation. We assumed normal growth and Maxwellian relaxation. However, the procedure developed can be similarly performed for any other mechanism of crystal growth and/ or relaxation with qualitatively equivalent results.

We have shown that in the range of temperatures where the Stokes–Einstein/Eyring equation is fulﬁlled, stresses relax too fast to allow for a signiﬁcant inﬂuence on the growth rates. However, as soon as this condition is violated, elastic stresses may be important for the description of crystal growth and have, in general, to be accounted for. In the example analyzed here, elastic stress eﬀects are of importance only in temperature ranges where the growth rate is too low to be experimentally detectable. In this way, the question arises, for which classes of systems elastic eﬀects may be of particular importance?

As mentioned above, as soon as the Stokes–Einstein/

Eyring equation is fulﬁlled, elastic stresses cannot have a signiﬁcant eﬀect on crystal growth. Consequently, the higher the deviations from the Stokes–Einstein/Eyring log(D , m/s)

Fig. 5. Eﬀective diﬀusion coeﬃcient, determining the rate of crystallization, calculated from nucleation time-lag data employing the classical nucleation theory (full curve) and data from a ﬁt of both driving force of crystallization and speciﬁc interfacial energy as performed in Ref. [16] (dashed curve).

440 J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443 equation immediately below the decoupling temperature, the higher is the possible eﬀect of elastic stresses. With respect to the results shown in Fig. 1, we can reformulate this statements as follows: the eﬀect of elastic stresses increases with increasing rate of approach to zero of the

Eyring ratio kBT/(Dgd0). Second, the eﬀect of elastic stresses depends signiﬁcantly on the ratio of the thermodynamic driving force Dl(Tg) and the elastic stress parameter, e0. An overview of the spectrum of possible values of this ratio for diﬀerent classes of glass-forming liquids is given in Ref. [37]. As it turns out, the stress parameter e0 can be comparable in magnitude with Dl. This is a necessary condition for a signiﬁcant

dependence of the growth rate on stress.

(Parte **2** de 4)