**UFRJ**

# schmelzer 2006

(Parte **3** de 4)

Third, despite the uncertainties connected with the applicability of the classical nucleation theory to crystallization of glass-forming liquids, one can expect that the eﬀect of stresses on nucleation will be, in most cases, much more signiﬁcant than on growth. Indeed, the steady-state nucleation rate depends exponentially on the driving force squared via

kBTðDlÞ2 where r is the nucleus/melt surface energy and I0 a parameter determined mainly by kinetic factors. In contrast, according to Eq. (2), the dependence of the growth rate on the driving force and thus, on the eﬀect of stresses, is rather weak. Elastic stresses can have an eﬀect on growth only as soon as e0 is comparable in magnitude with Dl. For the case of lithium disilicate, for example, elastic stres- ses can reduce the growth rate by less than 2% at 420 C

(this estimate results if one determines e0 via Eq. (9) with

Em = Ec neglecting stress relaxation). However, as shown in Refs. [15,16], the eﬀect on the nucleation rate can reach several orders of magnitude. Hence, if elastic stresses have a signiﬁcant eﬀect on growth rates, homogeneous nucleation in the bulk will be totally suppressed, in general. Vice versa, in glasses which crystallize by homogeneous nucleation (i.e. in glasses having a reduced glass-transition tem- perature Tg/Tm 6 0.6 (cf. [38])) elastic stresses will not aﬀect signiﬁcantly the growth rates.

In more detail, the kind of dependence of the growth rates on the thermodynamic driving force is illustrated in Fig. 6. In this ﬁgure, the value of the thermodynamic factor,

in the expression for the growth rate, Eq. (2), is shown as a function of (a) the reduced thermodynamic driving force,

Dl/(kBT), and (b) the reduced melting entropy, DSmr = DHm/(kBTm). In Eq. (28), DHm is the melting enthalpy per particle, Tr = T/Tm and the Volmer–Turnbull equation

(5) was employed. In Fig. 6(b), the full curve shows the respective values of this quantity at Td, while the dashed curve refers to the value of Uth at Tg. Here we assume

Tg = (2/3)Tm and Td = 1.2Tg. The graph presented in Fig. 6(a) is valid for any value of the reduced melting entro- py, DSmr = DHm/(kBTm). In order to ﬁnd out the temperature, corresponding to a given point along the curve, one has to know the value of DSmr (which is a characteristic of the substance under consideration).

As we have shown above, elastic stresses can aﬀect crystal growth rates only at temperatures below the decoupling temperature T 6 Td ﬃ 1.2Tg. It is evident from Fig. 6(a) that, when Td corresponds to the right part of the plot where the growth rate weakly depends on Dl, elastic stres- ses can diminish the growth rate only if the values of the stress parameter e0 are close to or exceed the thermody- namic driving force. However, if Td corresponds to the left part of the plot, then the growth rate is highly sensitive to slight changes of Dl and an eﬀect of elastic stresses can be expected to occur even then if the magnitude of e0 is considerably smaller as compared with Dl.

Since the temperature scale in Fig. 6(a) depends on DSmr we plotted Uth at Td and Tg as a function of DSmr

(Fig. 6(b)). Hereby we estimated Tg/Tm as 2/3 taking into account the above made conclusion. The interval of the

DSmr variation, employed in Fig. 6(b), is typical for silicate glasses.

According to the solid and dotted lines in Fig. 6(b), the

Uth values at Td and Tg and, consequently, the position of Td and Tg on the plot of Fig. 6(a) strongly depend on DSmr.

Since the values of Uth in the range from 0.0 to 0.8 correspond to a strong dependence of the growth rate on the thermodynamic driving force (see Fig. 6(a)), the eﬀect of elastic stresses on growth rates can be expected to be signif- icant in glasses with DSmr lower than about 5. Therefore,

T 1-exp(-

Δμ / k T)

ΔH /k T

1-exp(-

Δμ /k

Δμ/k T 246

Fig. 6. Value of the thermodynamic factor, 1 exp( Dl/(kBT)), in Eq. (2) versus (a) reduced thermodynamic driving force, Dl/(kBT), and (b) reduced melting enthalpy, DHm/(kBT). In (b) the full curve shows the respective values of this quantity at Td, while the dashed curve refers to its value at Tg. Here Tg is taken to be equal to Tg = (2/3)Tm and Td = 1.2Tg.

J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443 441 the probability of having important elastic stress eﬀects increases with decreasing DSmr. Taking into account Jackson’s [39] criterion for crystal growth mechanisms, we can suppose that normal growth is more aﬀected by elastic stresses than screw dislocation and 2D-surface nucleation mediated growth. By this reason, we performed the respective analyses here explicitly for the case of normal growth.

In the present analysis, we considered the eﬀect of elastic stresses on the growth of crystals of macroscopic sizes. The results can also be employed for the analysis of growth of crystallites of near-critical sizes taking into account, in addition, interfacial contributions to the thermodynamic potential. In this case, we have to add a term rA into Eq. (4) where r is the speciﬁc interfacial energy and A the surface area of the growing aggregate. As the result, the eﬀective driving force of crystal growth would contain then an additional term due to interfacial eﬀects proportional to n1/3.

6. Possible modiﬁcations and generalizations

In the present and preceding papers [5,6,15,16],w ea nalyzed the eﬀect of elastic stress evolution and relaxation on nucleation and crystal growth. This analysis was based on two basic assumptions: (i) The state of the newly evolving phase does not depend on the size of the aggregates considered and is widely equivalent to the state of the stable macroscopic phase. (i) Elastic stresses are due to misﬁt eﬀects and grow linearly with the size of the newly evolving phase (Nabarro model). The basic result of the analysis is that – as soon as the Stokes–Einstein/Eyring equation is not fulﬁlled – elastic stresses may have a signiﬁcant eﬀect both on nucleation and cluster growth. The magnitude of this eﬀect depends, of course, on a variety of additional factors, however, the principal possibility of such eﬀect has always to be taken into account.

Posing the question about possible modiﬁcations and generalization of the theory developed, one has to answer the question whether the assumptions of the theory are generally fulﬁlled or not. A detailed analysis of experimental data on nucleation of glass-forming silicate melts proves that the classical nucleation theory leads to serious problems in treating nucleation data [31–3]. These problems can be resolved in the framework of a newly developed approach to the description of nucleation and growth processes [34–36,40,41]. The basic advantages of this so-called generalized Gibbs’ approach consists in its ability to determine theoretically the dependence of the state parameters both of critical, sub- and supercritical clusters on supersaturation and cluster size. Hereby it turns out that the state of the clusters is essentially cluster size and supersaturation dependent. As a consequence, a variety of thermodynamic and kinetic parameters (surface energy, diﬀusion coeﬃcients, driving force for cluster formation, growth rates) become cluster size dependent as well. Such a size-dependence occurs (due to the variation of the bulk properties with cluster size) even then if one neglects surface energy terms in the thermodynamic description. In this way, the problem arises to treat theoretically stress evolution and stress relaxation going beyond the classical theory of nucleation and growth and relying on mentioned generalized Gibbs’ approach. A ﬁrst attempt to proceed in this direction can be found in Ref. [16]. The analysis shows that the eﬀect of elastic stresses on nucleation is increased as compared with the case when basic assumptions of classical nucleation theory are employed in the analysis.

Another limitation of the present model is connected with the assumption that stresses are due to misﬁt eﬀects and can be described by Nabarro-type dependencies. As already mentioned, in the case of segregation in solutions, when the segregating component has a much higher diﬀusivity than the ambient phase particles, stresses may evolve growing proportional to (V V0)2. Here V is the volume of the cluster of the new phase and V0 some initial volume, where such kind of stresses become dominant. A detailed analysis shows that such model of stress evolution may be eﬀective for suﬃciently large clusters when the above mentioned condition is fulﬁlled. In such situations, several time or length scale parameters (connected with the large diﬀerences in the partial diﬀusion coeﬃcients of the diﬀerent components of the system) exist and decoupling of relaxation and growth is not connected with the Stokes– Einstein/Eyring equation, but with the diﬀerences in the diﬀusivities. Consequently, in a further generalization of the theory one has to specify more clearly the conditions at which Nabarro-type stresses will be eventually replaced by mechanisms of stress evolution resulting in a more rapid increase of stress with the size of the clusters or regions of the newly evolving phase.

7. Conclusions

As far as the Stokes–Einstein/Eyring equation is fulﬁlled, elastic stresses do not have any eﬀect on crystal growth. However, below the temperature of decoupling of diﬀusion and viscous ﬂow, when the Stokes–Einstein/ Eyring equation breaks down, stresses may have a signiﬁcant inﬂuence on crystal growth. The present results therefore challenge the widespread argument that internal stresses relax too fast to aﬀect crystal growth in glass-forming liquids and could help to explain the often observed lack of agreement between model predictions (which do not take stresses into account) and experimental crystal growth data.

Acknowledgments

The authors would like to express their gratitude to the

Deutsche Forschungsgemeinschaft (DFG), the State of Sao Paulo Research Foundation FAPESP (Grants 03/12617-0; 9/00871-2; 2003/03575-2), CNPq and Pronex for ﬁnancial support. The critical comments of M. L. F. Nascimento of LaMaV-UFSCar are fully appreciated.

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

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(Parte **3** de 4)