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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 T (^) g , using experimental nucleation and growth data obtained for stoichiometric Li 2 O · 2SiO 2 and Na 2 O · 2CaO · 3SiO 2 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
I. INTRODUCTION
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 pre- ferred path of evolution—is inhibited to a sufficient degree. Therefore, a sound knowledge of nucleation and growth pro- cesses in glass-forming melts is a key issue to the under- standing 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 G (^) V 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 ef- fective diffusion coefficient D , which can be estimated from independent experimental determinations of crystal nucle- ation 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 ,
k (^) B T G (^) V^2 a^2 D
and via Eq. 2 from the rate of kinetically limited ballistic growth U ,1,
U = f
D
4 a 1 − exp−
G (^) V a^3 k (^) B T
(^) .^ ^2
In both equations, G (^) V 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 k (^) B 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 has^3
f
T (^) m − T T (^) m
where T (^) m 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 writ- ten in the form
U = f
D
G (^) V a^2 k (^) B T
THE JOURNAL OF CHEMICAL PHYSICS 126 , 234507 2007
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In Eqs. 1 and 4 , it is assumed so far that the thermo- dynamic 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 pro- cesses. Taking into account such considerations, we modify Eqs. 1 and 4 and write them in the following more gen- eral form:
=
k (^) B T G (^) V^2 I a^2 D I
U = f
D U
G (^) V U a^2 k (^) B T
The subscripts I and U assigned to the kinetic and thermo- dynamic parameters specify the fact that D and G (^) V 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 inter- vals corresponding to very deep undercoolings, which, in the case of lithium disilicate glass, includes the glass transition range.
II. ESTIMATE OF THE EFFECTIVE DIFFUSION COEFFICIENTS FOR CRYSTALLIZATION OF LITHIUM DISILICATE GLASS IN THE FRAMEWORK OF CNT
According to the classical nucleation theory CNT, which is based on Gibbs’ description of heterogeneous sys- tems, 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 G (^) V 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”
method^4 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 esti- mate the U T and T shown in Table I. In the case of Li 2 O · 2SiO 2 glass we employed direct experimental values of U T . Proper thermodynamic data for Li 2 O · 2SiO 2 and Na 2 O · 2CaO · 3SiO 2 glasses were taken from Ref. 15. For calculations employing Eq. 1 , a value of the specific sur- face energy equal to = 0.15 J / m 2 average value obtained from a previous analysis of experimental nucleation rates with a temperature dependent surface energy^15 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 calculated by Eq. 1 , D I , and Eq. 4 , D U , versus tempera- ture in normal and Arrhenian coordinates, respectively. There is a strong difference between D I and D U D I D U which reaches about three to four orders of magnitude de- pending on the assumed growth mechanism. Moreover, D I and D U have different temperature dependencies corre- sponding 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 Li 2 O · 2SiO (^2) exp−79.185+ 63 602.287/ T 693–773 5– exp31.740− 42 406.817/ T 713–783 11 and 12 Na 2 O · 2CaO · 3SiO (^2) exp−131.903+ 120 188.79/ T 843–877 13 and 14 exp31.400− 48 459.948/ T 853–953 13 and 14
FIG. 1. Diffusion coefficients of L 1 S 2 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 .
234507-2 Fokin et al. J. Chem. Phys. 126 , 234507 2007
IV. ESTIMATION OF THE THERMODYNAMIC DRIVING FORCE OF NUCLEATION
The coefficient K and correspondingly the thermody- namic 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 effec- tive diffusion coefficient so strongly see Eq. 3.98 in Ref. 16 or ii the critical cluster is a phase of the same compo- sition, e.g., a polymorphic form of LS 2. One should recall that situation ii occurs in the case of BaO · 2SiO 2 BS 2 glass crystallization, where the high temperature form of BS 2 crystals precipitates at low temperatures as the first phase.^20 This particular case, when
D I = D U 8
will be considered below. Using Eq. 8 together with Eqs. 5 and 6 and replac- ing G (^) V I by K G (^) V U one can then obtain the following equation for the coefficient K :
K =
f
G V U U
1/
. 9
Note that the knowledge of the size parameter a is not required for the determination of K. Figure 3 shows the val- ues 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 driv- ing force for critical cluster formation is smaller than that for LS 2 macrocrystals. Hence, as we concluded earlier, the criti- cal 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 T (^) m 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 . How- ever, 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 increas- ing 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 driv- ing 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 inter- esting to analyze, in addition, a system for which the com- position 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 between Na 2 O · 2CaO · 3SiO 2 N 1 C 2 S 3 and Na 2 O · 1CaO · 2SiO 2 N 1 C 1 S 2 , which belong to the field of solid solutions.^22 In the next section such analysis for a glass with a composition close to N 1 C 2 S 3 will be presented.
V. EFFECTIVE DIFFUSION COEFFICIENTS FOR CRYSTAL NUCLEATION AND GROWTH IN THE Na 2 O · 2CaO · 3SiO 2 GLASS
As we have shown earlier, the formation of Na 2 O · 2CaO · 3SiO 2 crystals in a glass of the same composi- tion occurs via the nucleation of a solid solution with a com- position strongly enriched in sodium, which differs consid- erably 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 tempera- tures 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 equation^1 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 N 1 C 2 S 3 phase, we must re- duce the thermodynamic driving force for nucleation and growth by the same coefficient K. Here the situation is dif- ferent 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 rewrit- ten as
k (^) B T G (^) V^2 U K^2
D I a^2
FIG. 3. Coefficient K estimated for L 1 S 2 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.
234507-4 Fokin et al. J. Chem. Phys. 126 , 234507 2007
U =
fD U
G (^) V U Ka^2 k (^) B T
Figure 4 shows the effective diffusion coefficients esti- mated from the time lag for nucleation Eq. 10 and crystal growth rates Eq. 11 —assuming different growth mechanisms—and using the thermodynamic driving force for Na 2 O · 2CaO · 3SiO 2 crystals the most stable phase, i.e., additionally assuming, K = 1. As for the lithium disilicate glass, a was taken as 1 Å and = 0.1 J / m 2 was taken from Ref. 15. The resulting values of D I are smaller than that for D U , i.e., the assumption K = 1 again leads to contradictions. However, as we already noted, the composition of critical nuclei and crystals to which crystal growth measurements refer is the same but different from that of the stable phase Na 2 O · 2CaO · 3SiO 2. Hence, the difference in the effective diffusion coefficients obtained in our computations results from the incorrect use of the thermodynamic driving force for N 1 C 2 S 3 crystals K = 1. Thus we must reduce G (^) V by a factor K 1. Assuming again that the condition given by Eq. 8 is fulfilled it holds with good accuracy for the N 1 C 2 S 3 glass, Eqs. 10 and 11 yield the following formula for the deter- mination of the coefficient K :
K =
f
U G V
Figure 5 presents K estimated via Eq. 12 for the normal growth mechanism. As for the lithium disilicate glass, K is less than 1 and increases with increasing temperature. It should be recalled here that the average value of this coeffi- cient in the temperature interval 580– 720 ° C estimated from the dissolution effect of subcritical crystals with increasing temperature is 0.13.^21 Taking into account the increase of K with temperature see Fig. 5 the values of K obtained in the present study are quite reasonable.
VI. FINAL COMMENTS
The starting point of the present paper consists of the analysis of the strong difference between the effective diffu- sion coefficients for nucleation and growth computed based
on experimental data and employing the classical theory of nucleation-growth processes. As previously mentioned, the classical theory employs the assumption that the bulk prop- erties of the clusters are independent of their sizes. The sub- sequent analysis showed, however, that the real situation is the opposite one. Differences in the diffusion coefficients are found for the considered systems only if the mentioned basic assumption of the classical theory of nucleation and growth is employed in the computations. If this assumption of con- stancy of bulk cluster properties is released, the experimental data demonstrate a completely different feature. For the con- sidered systems the effective diffusion coefficients are only slightly size dependent, but the state of the clusters is, in general, different from the state of the newly evolving cluster phase and size dependent. Note, however, that in the general case both thermodynamic and kinetic parameters of the clus- ters of the new phase may be size dependent.^24 The idea to employ kinetic coefficients estimated from nucleation data to evaluate the nucleation-growth kinetics and to perform computer simulations was proposed some decades ago and remains actual see, e.g., Refs. 25–29. This approach is based on the assumption that the properties of the newly evolving phase are size independent. However, the crystal growth rates estimated in this way often differ by several orders of magnitude from those measured experimen- tally. The results of the present paper assuming distinct ther- modynamic properties of nanosized critical clusters and mac- rocrystals could help one to understand such differences.
VII. CONCLUSIONS We found significant differences between the effective diffusion coefficients for nucleation and growth computed by the classical theory of nucleation-growth employing experi- mental data for nucleation time lag and crystal growth rates. The present analysis corroborates our previous findings based on other types of experiments and analysis for the same two glasses. They strongly support the point of view that, even for the so-called polymorphic crystallization, the
FIG. 4. Effective diffusion coefficients for N 1 C 2 S 3 glass estimated from time lag for nucleation 1 and crystal growth rates assuming different mechanisms: normal growth 2 and screw dislocation growth 3 .
FIG. 5. Coefficient K estimated for N 1 C 2 S 3 glass assuming normal growth. Dotted line shows the average value of K estimated in Ref. 21 for the temperature interval of 580– 720 ° C. Dashed lines are placed to guide the eyes.
234507-5 Diffusion coefficients for crystal nucleation J. Chem. Phys. 126 , 234507 2007
