New insights on the thermodynamic barrier for nucleation in glasses: The case of lithium disilicate
Introduction
The investigation of glass crystallization kinetics is of great interest from both practical and theoretical points of view. In many aspects, the first stage of the process – crystal nucleation – determines the pathways of crystallization. Existing theories provide a qualitative description of both the temperature and time dependencies of the nucleation rate. However, they fail if a quantitative prediction of these dependencies is attempted. Such quantitative discrepancies arise possibly because the classical approaches do not adequately take into account several effects. Such effects may be a primary precipitation of metastable phases [1], the formation of solid solutions with compositions unlike those of the parent glass [2], the (unknown) size or temperature dependence of the nucleus/melt surface energy, σ [3], or the effect of elastic strain energy [4]. A neglect of such possibilities may result in erroneous estimations of the thermodynamic barrier to nucleation (the work of critical nucleus formation, W*) and, consequently, in an inaccurate description of the nucleation kinetics.
Above mentioned difficulties are partly due to the absence of direct experimental data about the properties of the critical nuclei in inorganic glass-forming melts. The absence of such data is understandable because, at high undercoolings required to enter the range of measurable nucleation rates, the size of the critical nuclei is only a few nanometers. The small sizes of these nuclei hinder the application of direct experimental methods to specify their thermodynamic properties, which are required to estimate W*.
For these reasons, in analyzing nucleation experiments, it is common practice to follow the classical nucleation theory (CNT). According to CNT, the properties of the critical nucleus are widely identified with those of the corresponding macro-phase. Assuming this condition, the driving force for nucleation, ΔG, can be estimated based on knowledge of the properties of the respective macroscopic phases. Therefore, in accordance with CNT, ΔG values measured for the macro-phase are usually applied to critical nuclei.
Another problem, which cannot be solved easily, is the determination of the appropriate value of the specific interfacial energy. It is well-known that existing measurements of the surface energy of macro-crystals in their own melt are confronted with serious difficulties. The problem is even more complex if one tries to determine the value of σ for a cluster of critical size. Since W* is expressed as a combination of both ΔG and σ, any error in determining these quantities will lead to incorrect values of W*.
However, when the theory is applied to experiment, ΔG is often determined in the manner as described above. This choice may or may not correspond to the real situation. Any inconsistency between the employed value of the thermodynamic driving force and the one corresponding to the real physical situation can be corrected – with respect to the proper determination of W* – by choosing an appropriate value of the surface energy. In such typical approach, σ plays then the role of a fit parameter. This procedure allows one to reproduce theoretically (at least, to some extent) the correct values of the nucleation rates. However, the use of macro-properties for the specification of the properties of critical nuclei gives rise to a number of additional problems when nucleation experiments are analyzed within the framework of CNT. One of these internal contradictions is the significant difference between the values of the surface energy calculated from nucleation data on one side and from the growth rate of nuclei with sizes slightly exceeding the critical one. This problem was resolved by the reduction of the value of thermodynamic driving force as compared with the respective macroscopic samples [5].
From a more general theoretical point of view, both surface energy and thermodynamic driving force for critical and near-critical nuclei have to be considered as unknown quantities. Moreover, in analyzing nucleation rates, the possible effect of elastic strain (caused by the difference between the densities of melt and crystal) on the thermodynamic driving force is usually not taken into consideration. However, at high undercoolings, especially close to the glass transition region, the driving force may be diminished due to elastic strain, which, according to [6], [7], [8], can strongly diminish the nucleation rate.
In the present paper, we analyze the thermodynamic barrier for nucleation calculated from a fit (to CNT) of both steady-state nucleation rates and induction periods in lithium disilicate glass. However, in contrast to the conventional approach, in this procedure both the thermodynamic driving force and surface energy are initially considered as unknown quantities. In addition, we estimate the elastic strain energy from the deviation between the experimentally and theoretically determined values of the thermodynamic barrier at low temperatures (close to and below the glass transition range).
Section snippets
Basic equations
According to the classical theory, the steady-state nucleation rate, Ist, can be written in the following general form [9]:where ΔGD is the free activation energy for the transport of a ‘structural’ unit across the nucleus/melt interface (kinetic barrier), W* is the work of critical nucleus formation (thermodynamic barrier), ΔG is the thermodynamic driving force for the phase transition, g is a shape factor equal to 16π/3 in the case of spherical
Experimental data on nucleation and calculations
Fig. 1 presents our own data and data from the literature [3] on the steady-state nucleation rates (a) and induction periods of nucleation (b) in lithium disilicate glass. The values of tind are taken from a linear approximation of the dependence of ln(tind) on 1/T. Both sets of data (Ist and tind as functions of temperature) are then employed in our calculations.
According to X-ray diffraction experiments, in advanced stages of the transformation, the crystalline phase is lithium disilicate, Li2
Driving force of transformation and interfacial energy of critical clusters
The fit to Eq. (2) of the work of critical cluster formation, W*(T), from experimental steady-state nucleation rates and induction periods, leads to the following conclusions:
- (i)
The specific surface energy of the critical nuclei shows a slight positive temperature dependence, i.e., σ = 0.020 + 0.0438 × 10−3 T; σ in J/m2, T in K.
- (ii)
The thermodynamic driving force for the formation of a critical nucleus is considerably lower than the respective value for the corresponding macro-phase (P3 = 0.197).
Conclusion (i)
Estimate of the elastic stresses effect on nucleation
A remarkable deviation of the W*(T) dependence (shown by open circles in Fig. 2) from the theoretical curve (shown by a dashed line in Fig. 2) is observed at temperatures below Tg. As mentioned earlier, this deviation can be interpreted as the result of the reduction of the thermodynamic driving force by elastic strains. These strains evolve in the course of the formation of the crystals owing to the difference between the melt and crystalline phase densities. Such a reduction of the driving
Discussion
In the present analysis, we consider two groups of results as particularly interesting for discussion. These results are: (i) the estimation of the thermodynamic properties of critical nuclei at relatively high temperatures, when the undercooled melt can be treated, to a first approximation, as a Newtonian liquid and (ii) the estimation of the effect of elastic strain on nucleation at low temperatures, when the melt reveals viscoelastic properties.
(i) According to the fit of the thermodynamic
Conclusions
The thermodynamic driving force for the formation of critical nuclei and their specific surface energy are considerably lower than the respective quantities for the corresponding macro-phase. This fact has not been taken into account in previous analyses of crystal nucleation kinetics in glasses.
If, as suggested by different authors, at temperatures close to the glass transition, the effective diffusion coefficient (which is responsible for the rate of molecular aggregation) and the viscosity
Acknowledgement
E. D. Zanotto, V. M. Fokin and J. W. P. Schmelzer are highly appreciative the financial support of CNPq-Cyted, Pronex and Fapesp (Brazil).
References (23)
- et al.
J. Non-Cryst. Solids
(2003) - et al.
J. Non-Cryst. Solids
(2003) - et al.
J. Non-Cryst. Solids
(2000) - et al.
J. Non-Cryst. Solids
(2000) - et al.
J. Non-Cryst. Solids
(2003) - et al.
J. Non-Cryst. Solids
(2004) J. Non-Cryst. Solids
(2000)- et al.
J. Cryst. Growth
(1974) - et al.
J. Non-Cryst. Solids
(1985) Surf. Sci.
(1969)
J. Colloid Interf. Sci.
Cited by (36)
Measurement of nucleation rate of ibuprofen in ionic liquid using induction time method
2019, Journal of Crystal GrowthStructure and ionic conductivity of nitrated lithium disilicate (LiSiON) glasses
2018, Materials Chemistry and PhysicsThe effect of elastic stresses on the thermodynamic barrier for crystal nucleation
2016, Journal of Non-Crystalline SolidsCitation Excerpt :Such anomalous behavior of Wc(T) and its possible origin and consequences have already been addressed e.g. in [16,17]. The first studies on this effect were performed by some of the authors of this work in [18–20]. At that time, only a few examples of glass-forming liquids exhibiting such behavior were known; hence, they were considered to be anomalous exceptions.
Crystallization of glass-forming liquids: Maxima of nucleation, growth, and overall crystallization rates
2015, Journal of Non-Crystalline SolidsLag time to crystal nucleation of supercooled lithium disilicate melts: A test of the classical nucleation theory
2015, Journal of Non-Crystalline Solids