(Parte 1 de 2)

Diffusion coefficients for crystal nucleation and growth in deeply undercooled glass-forming liquids

Vladimir M. Fokin S. I. Vavilov State Optical Institute, Ulitsa Babushkina 36-1, 193171 St. Petersburg, Russia

Jürn W. P. Schmelzer Institut für Physik, Universität Rostock, Universitätsplatz, 18051 Rostock, Germany

Marcio L. F. Nascimento and Edgar D. Zanotto Vitreous Materials Laboratory, Department of Materials Engineering, Federal University of São Carlos, São Carlos, 13595-905 São Carlos, Brazil

Received 14 March 2007; accepted 10 May 2007; published online 19 June 2007

We calculate, employing the classical theory of nucleation and growth, the effective diffusion coefficients controlling crystal nucleation of nanosize clusters and the subsequent growth of micron-size crystals at very deep undercoolings, below and above Tg, using experimental nucleation and growth data obtained for stoichiometric Li2O·2 SiO2 and Na2O·2 CaO·3 SiO2 glasses. The results show significant differences in the magnitude and temperature dependence of these kinetic coefficients. We explain this difference showing that the composition and/or structure of the nucleating critical clusters deviate from those of the stable crystalline phase. These results for diffusion coefficients corroborate our previous conclusion for the same glasses, based on different experiments, and support the view that, even for the so-called case of stoichiometric polymorphic crystallization, the nucleating phase may have a different composition and/or structure as compared to the parent glass and the evolving macroscopic crystalline phase. This finding gives a key to explain the discrepancies between calculated by classical nucleation theory and experimentally observed nucleation rates in these systems, in particular, and in deeply undercooled glass-forming liquids, in general. © 2007 American Institute of Physics. DOI: 10.1063/1.2746502

Crystal nucleation and growth determine the overall rate of crystallization. Glass formation on cooling a liquid or on varying any other appropriate thermodynamic parameter is only possible if crystallization—the thermodynamically preferred path of evolution—is inhibited to a sufficient degree. Therefore, a sound knowledge of nucleation and growth processes in glass-forming melts is a key issue to the understanding of vitrification. In addition, advanced glass-ceramic materials—polycrystalline materials produced by controlled nucleation and growth of glasses—can only be manufactured with a desired quality if these two kinetic processes are well understood. Besides the thermodynamic driving force for crystalliza- tion GV and the specific interfacial energy , the rate of diffusion of the different components of the melt ions that

build the crystalline phase with an average effective size a through the crystal/liquid interface determines the rates of nucleation and growth of the clusters of the newly evolving crystalline phase. The diffusion rate is determined by an effective diffusion coefficient D, which can be estimated from independent experimental determinations of crystal nucleation and growth kinetics.

The value of D can be evaluated via Eq. 1 from the time lag for formation of critical size clusters in nucleation , and via Eq. 2 from the rate of kinetically limited ballistic

U= f D


In both equations, GV is the thermodynamic driving force per unit volume of crystal, T is the absolute temperature, a is an average effective size parameter of the ions building the crystalline phase, and kB is Boltzmann’s constant. In Eq. 2 , f is a dimensionless parameter describing the different growth modes. For the normal growth mechanism f=1, whereas in the case of screw dislocation growth, one has3 where Tm is the melting point. For screw dislocation growth at deep undercoolings corresponding to the maximum of the nucleation rate f has values of the order of f 0.1. To a good approximation, the exponential term in Eq. 2 can be expanded into a Taylor series. Equation 2 can then be written in the form

In Eqs. 1 and 4 , it is assumed so far that the thermodynamic and kinetic parameters determining the rates of cluster formation and growth are independent of cluster size. Our own previous analyses and results of other authors show,4 however, that these parameters may vary with cluster size and, consequently, may be different for nucleation nanosize clusters and growth micron-size crystals processes. Taking into account such considerations, we modify Eqs. 1 and 4 and write them in the following more general form:

The subscripts I and U assigned to the kinetic and thermo-

dynamic parameters specify the fact that D and GV may be different for nucleation and growth, respectively.

In this paper we estimate D for crystal nucleation and growth by Eqs. 5 and 6 using independent experimental data on the time lag for homogeneous nucleation and crystal growth rates for two glass-forming liquids: lithium disilicate and sodium calcium metasilicate. Time-lag and crystal growth rates were measured in the same temperature intervals corresponding to very deep undercoolings, which, in the case of lithium disilicate glass, includes the glass transition range.

According to the classical nucleation theory CNT , which is based on Gibbs’ description of heterogeneous systems, the thermodynamic driving force for cluster nucleation and growth is considered to be size independent.1,2 In other words, the same phase is assumed to determine the state of the critical clusters and of macrocrystals. Therefore, in order to determine the values of the diffusion coefficient following this classical prescription, we employ the thermodynamic driving force GV U for the growth of macrocrystals the stable lithium disilicate crystal phase in our computations.

Its value can be determined directly by calorimetric methods. Data on nucleation time lag obtained by the “development” method4 and crystal growth rates taken from several sources were plotted in Arrhenian coordinates and approximated by a straight line. The results of this procedure were used to estimate the U T and T shown in Table I. In the case of

Li2O·2 SiO2 glass we employed direct experimental values of U T . Proper thermodynamic data for Li2O·2 SiO2 and

Na2O·2 CaO·3 SiO2 glasses were taken from Ref. 15. For calculations employing Eq. 1 , a value of the specific sur- face energy equal to =0.15 J/m2 average value obtained from a previous analysis of experimental nucleation rates with a temperature dependent surface energy15 was used. One can see from Eq. 1 that the precise value of does not strongly affect the estimate of D I . The value of the size parameter a was estimated as 1 Å.

Figures 1 and 2 show the effective diffusion coefficients pending on the assumed growth mechanism. Moreover, D I and D U have different temperature dependencies corresponding to activation enthalpies of 530 kJ/mol for the nucleation time lag and 370 kJ/mol for crystal growth. Thus

TABLE I. Time lag for nucleation and crystal growth rate vs temperature taken from a linear fit of an Arrhenian plot collected from literature data.

Time lag for nucleation s

Crystal growth rate m/s

Temperature interval T K References

FIG. 1. Diffusion coefficients of L1S2 glass estimated from the time lags for nucleation via Eq. 1 , and from growth rates for the normal growth and screw dislocation mechanisms via Eq. 4 .

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the use of the main assumptions of CNT resulted in a sig- nificant difference between D I and D U . Below we analyze possible reasons for this discrepancy.

In the framework of CNT one expects, in addition to above mentioned thermodynamic requirements, that the nucleation and growth processes are determined by one and the same diffusion coefficient irrespective of cluster size. Such assumption can be considered as a direct consequence of the basic assumption of classical theory that the state of a cluster does not depend on its size. The present result D I

D U thus challenges this main thermodynamic assumption of CNT employed in the derivation of Eqs. 1 and 2 .

Assuming that the critical clusters and the finally evolving macrophase have similar bulk properties, one should also expect that the effective diffusion coefficients are the same. In contrast to such expectation, however, the difference be- tween D I and D U reaches several orders of magnitude. According to Eq. 1 , it is obvious that the use of a different value of cannot eliminate this discrepancy. Hence one must look for other factors, which could be responsible for this result. Below we discuss several possible reasons.

It is reasonable to assume that in the case of small nearcritical clusters transport of matter from long distances is not required to sustain the aggregation process since a high enough quantity of each of the components is available in the immediate vicinity of the clusters.16 The formation of diffusion zones close to such clusters is thus unlikely. This conclusion implies that a kinetically limited or ballistic growth can be expected for small clusters independent of their composition. For the system considered, kinetically limited growth takes place also at the macroscopic stage of the phase transformation since the macrocrystals are lithium disilicate and, hence, their composition is the same as that of the parent liquid. Hence the same kinetics could be expected to govern the aggregation process for both small and large clusters. In this way, we come to the conclusion that the nucleating phase clusters of critical and near-critical sizes is not lithium disilicate since, otherwise, one should get D I

=D U . It should be recalled that in the early stages of crystalli- zation of this glass at deep undercoolings, metastable phases of 0.1–0.3 m have been detected by transmission electron

LS2 crystals. Metastable phase formation is, however, only one of the possible mechanisms that can explain deviations of the cluster compositions and/or structure for critical and near-critical sized clusters from the parameters reflecting the evolving macroscopic phase.4 Thus, at the present stage of analysis, we can conclude with certainty only that the composition and/or structure of the nucleating critical clusters deviate from that of lithium disilicate and, therefore, the thermodynamic driving force for critical cluster nucleation should be smaller than the driving force responsible for macroscopic crystal growth. Indeed the experimental values of the thermodynamic driving force GV U employed so far in our calculations refer to LS2 macrocrystals. Since LS2 is the thermodynamically most stable phase, the thermodynamic driving force for other thermodynamically possible phases and particularly for where 0 K 1. The value of K tends to unity when T

→Tm since at the melting temperature the critical size of the clusters tends to infinity, r*→ , i.e., the critical nucleus becomes a macrophase.

Once the bulk state of the clusters formed in the nucleation stage deviates from the values for the macroscopic phase, the estimate of the diffusion coefficients as performed above must be reconsidered. Since the effective diffusion coefficient depends on the compositions of the clusters and ambient phase,16 the difference between D I and D U could reflect a difference between the compositions of critical clus- ters and macrocrystals. It is important to emphasize that the real difference between these two types of effective diffusion coefficients would be lower if one assumes that the nucleat- ing phase is not LS2. In order to estimate D I via Eq. 1 one must then reduce the thermodynamic driving force for crys- tallization of the macrophase by the coefficient K 1. The introduction of K into Eq. 1 results in an increase of the effective diffusion coefficient estimated from the time lags for nucleation. However, in the general case one cannot cal- culate the precise value of D I since the value of K is unknown. Therefore, one can only claim that D I differs from

D U . In particular, a small value of K would considerably reduce the thermodynamic driving force for nucleation, and the inequality D I D U could be attained. Such situation could be expected to occur, e.g., if the near-critical clusters are liquidlike, as was proposed earlier in Refs. 1 and 19.

FIG. 2. Arrhenian plot of diffusion coefficients of L1S2 glass estimated from the time lags for nucleation 3 via Eq. 1 , and from growth rates for the

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The coefficient K and correspondingly the thermodynamic driving force for nucleation can be estimated only for the special case when the equality D I =D U is valid. This equality can be assumed to hold approximately if i the change of the cluster composition does not affect the effective diffusion coefficient so strongly see Eq. 3.98 in Ref. 16 or i the critical cluster is a phase of the same compo- sition, e.g., a polymorphic form of LS2. One should recall that situation i occurs in the case of BaO·2SiO2 BS2 glass crystallization, where the high temperature form of BS2 crystals precipitates at low temperatures as the first phase.20

This particular case, when

Note that the knowledge of the size parameter a is not required for the determination of K. Figure 3 shows the values of K versus temperature estimated via Eq. 9 for normal growth and screw dislocation growth. As we expected, the values of K are smaller than 1, i.e., the thermodynamic driving force for critical cluster formation is smaller than that for

LS2 macrocrystals. Hence, as we concluded earlier, the critical nucleus is not the stable lithium disilicate phase.

Another important feature of these results is the increase of K with temperature. Such behavior is quite reasonable since, as we already noted, at T Tm the value of K must approach 1. It is clear that the absolute values of K are only estimates being valid at the condition given by Eq. 8 . However, it should be recalled that the average value of K for the temperature interval of 440–640 °C estimated in Ref. 21 using the effect of dissolution of subcritical nuclei at increasing temperature was equal to 0.2. This value is close to that expected from Fig. 3 for normal growth. This result gives indirect evidence for the correctness of the assumption see Eq. 8 , used for the estimation of the coefficient K, even if the composition of the nucleating phase differs from that of the growing macroscopic phase. Thus, we can estimate the thermodynamic driving force for nucleation via Eq. 7 knowing the G U value governing the growth of the stable phase and the coefficient K. Consequently, it turns out that they are not the effective diffusion coefficients, but the driving force for crystallization that depends most significantly on the size of the crystallites of the newly evolving phase in the considered system.

However, since the correctness of Eq. 8 for lithium disilicate glass is only an reasonable assumption, it is interesting to analyze, in addition, a system for which the composition of critical and near-critical clusters is equal to that of the growing crystals and, respectively, the condition fixed by Eq. 8 is fulfilled with certainty. Such case is found in the early stage of crystallization of sodium calcium metasilicate glasses with compositions

Na2O·1 CaO·2 SiO2 N1C1S2 , which belong to the field of solid solutions.2 In the next section such analysis for a glass with a composition close to N1C2S3 will be presented.


As we have shown earlier, the formation of

Na2O·2 CaO·3 SiO2 crystals in a glass of the same composition occurs via the nucleation of a solid solution with a com- position strongly enriched in sodium, which differs considerably from the stoichiometric parent glass.23 With respect to the problem considered, it is important that the composition of critical and near-critical nuclei remain the same or only slightly changes in the initial stage of growth when the first nucleated crystals attain macroscopic sizes.14 We measured the crystal growth rate in this time interval at low temperatures close to the glass transition temperature together with the steady-state nucleation rate and time lag for nucleation. The kinetic parameters were estimated by fitting the crystal number density versus time curve by the Collins-Kashchiev equation1 that describes non-steady-state nucleation. Since the time lags for nucleation and the growth rates measured in this experiment refer to the same phase with a composition differing from that of the stable N1C2S3 phase, we must reduce the thermodynamic driving force for nucleation and growth by the same coefficient K. Here the situation is different from that of the lithium disilicate glass, for which only the thermodynamic driving force for nucleation was reduced, because the growth rate referred to the stable macrophase. By proceeding in such a way, Eqs. 5 and 6 can be rewritten as

FIG. 3. Coefficient K estimated for L1S2 glass for normal growth 1 and screw dislocation growth 2 . Dotted line shows the average value of K estimated in Ref. 21 for the temperature interval of 440–680 °C. Dashed lines are placed to guide the eyes.

(Parte 1 de 2)