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Homogeneous crystal nucleation in silicate glasses: A 40 years perspective

Vladimir M. Fokin a,*, Edgar D. Zanotto b, Nikolay S. Yuritsyn c, Jurn W.P. Schmelzer d a Vavilov State Optical Institute, ul. Babushkina 36-1, 193171 St. Petersburg, Russia b LaMaV – Vitreous Materials Laboratory, Federal University of Sao Carlos, 13565-905 Sao Carlos, SP, Brazil c Grebenshchikov Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24-2, 199155 St. Petersburg, Russia d Institut fur Physik, Universitat Rostock, 18051 Rostock, Germany

Received 18 June 2005; received in revised form 27 January 2006 Available online 24 May 2006

Dedicated to the memory of Peter F. James


We review a plethora of relevant experimental results on internal homogeneous crystal nucleation in silicate glasses obtained in the last four decades, and their analyses in the framework of the classical nucleation theory (CNT). The basic assumptions and equations of CNT are outlined. Particular attention is devoted to the analysis of the properties of the critical nuclei, which, to a large extent, govern nucleation kinetics. The main methods employed to measure nucleation rates are described and the possible errors in the determination of the crystal number density (and, correspondingly, in nucleation rates) are discussed. The basic regularities of both time and temperature dependencies of nucleation rates are illustrated by numerous experimental data. Experimental evidence for a correlation between maximum nucleation rates and reduced glass transition temperatures is presented and theoretically justified. Special attention is given to serious problems that arise in the quantitative description of nucleation rates when using the CNT, for instance: the dramatic discrepancy between calculated and measured nucleation rates; the high value of the crystal nuclei/melt surface energy, rcm, if compared to the expected value estimated via Stefan’s rule; the increase of rcm with increasing temperature; and the discrepancies between the values of the surface energy and the time-lag for nucleation when independently estimated from nucleation and growth kinetics. The analysis of the above mentioned problems leads to the following conclusion: in contrast to Gibbs’ description of heterogeneous systems underlying CNT, the bulk thermodynamic properties of the critical nuclei generally differ from those of the corresponding macro-phase resulting simultaneously in significant differences of the surface properties as compared with the respective parameters of the planar interfaces. In particular, direct experimental evidence is presented for compositional changes of the crystal nuclei during formation of the critical nuclei and their growth from critical to macro-sizes. In addition, detailed examinations of crystal nucleation and growth kinetics show a decrease of both the thermodynamic driving force for nucleation and of the critical nuclei/liquid interfacial energy, as compared with the respective properties of the macro-phase. However, despite significant progress in understanding crystal nucleation in glasses in the past four decades, many problems still exist and this is likely to remain a highly interesting subject for both fundamental and applied research for a long time. 2006 Elsevier B.V. All rights reserved.

Keywords: Crystallization; Glass ceramics; Nucleation; Crystals; Glass transition; Oxide glasses; Silicates; Thermodynamics

1. Introduction

Glasses can be defined as non-crystalline solids that undergo a glass transition in the course of their preparation. One of the most important and traditional (but not the only)

02-3093/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.02.074

* Corresponding author. Address: ul. Nalichnay 21, ap.7, 199406 St. Petersburg, Russia. Tel.: +7 812 355 30 38. E-mail address: vfokin@pisem.net (V.M. Fokin).

w.elsevier.com/locate/jnoncrysol Journal of Non-Crystalline Solids 352 (2006) 2681–2714 method of vitrification consists in supercooling a liquid escaping crystallization. Thus, when a liquid is cooled down at sufficiently high rates, crystallization can occur to a limited degree or can be completely arrested down to temperatures corresponding to very high viscosities, in the range g P 1013–1012 Pa s g(Tg), where Tg is the glass transition temperature. Below this temperature, the viscosity is so high that large-scale atomic rearrangements of the system are no longer possible within the time-scale of typical experiments, and the structure freezes-in, i.e., the structural rearrangements required to keep the liquid in the appropriate metastable equilibrium state cannot follow any more the change of temperature. This process of freezing-in the structure of an undercooled liquid transforming it into a glass is commonly denoted as glass transition. Typical glass-forming liquids, such as silicate melts, are usually characterized by: (i) relatively high viscosities (g > 100 Pa s) at the melting point or liquidus and (i) a steep increase of the viscosity with decreasing temperature. These properties favor vitrification. The mechanism above sketched leads to the conclusion that the glass structure must be similar to that of the parent undercooled liquid at temperatures near Tg and, indeed, this similarity has been experimentally observed.

Glass is thermodynamically unstable with respect to the undercooled liquid, i.e., there is no energy barrier between the glass and its corresponding undercooled (metastable) liquid. At a first glance, the high stability of the glassy state reflects only a relaxation problem; the system cannot evolve to a metastable state due to the kinetic inhibition of this process at low temperatures. On heating, relaxation of the glass structure may occur to reach first a metastable liquid state corresponding to the given temperature and then, eventually, go over into the crystalline state. The latter evolution process, as will be shown below, involves overcoming of a thermodynamic potential barrier. At room temperature glasses can exist for extremely long periods of time because their high viscosity inhibits structural rearrangements required for crystal nucleation and growth. However, when a glass is heat-treated for a sufficiently long time at temperatures within or above the glass transition range, devitrification readily starts, as a rule, from the surface and sometimes in the bulk via heterogeneous or homogeneous nucleation (see below).

Nucleation, or the process of formation of the precursors of the crystalline phases, may occur by different mechanisms. Commonly one divides these processes into homogeneous and heterogeneous nucleation. Homogeneous nucleation is a stochastic process occurring with the same probability in any given volume (or surface) element. Alternatively, nucleation occurring on preferred nucleation sites, e.g., such as pre-existing interfaces, previously nucleated phases, and surface defects, is denoted as heterogeneous nucleation. Depending on the location where nucleation takes places, volume (bulk) and surface crystallization can be distinguished.

Glass-forming melts are interesting models for studies of nucleation, growth and overall crystallization phenomena.

Their high viscosities result in relatively low (measurable) rates of crystallization, which may permit detailed studies of nucleation and growth kinetics. Homogeneous nucleation can sometimes be observed at deep undercoolings

(T/Tm < 0.6) because glass-forming melts are excellent solvents for solid impurities that thus only exist as ionic spe- cies when the liquid is vitrified. In addition, the rapid increase of viscosity with decreasing temperature makes it possible to ‘freeze-in’ different states of the crystallization process by quenching previously heat-treated specimens to room temperature. Hence, as it was figuratively said in Ref. [1], ‘glasses did and may serve as the Drosophila of nucleation theory in order to test different approaches’. Moreover, silicate glass is one of the oldest materials produced by mankind, having its origin about 6000 years ago in ancient Mesopotamia [2], but are still gaining technological importance.

It is evident from the above discussion that crystallization and glass formation are competitive processes. In this way, in order to avoid uncontrolled crystallization of glassy articles one needs to know the main factors that govern crystal nucleation and growth. On the other hand, controlled nucleation and crystallization of glasses underlay the production of glass-ceramics invented in the mid- 1950s [3], which are widely used in both domestic and high-technology applications. By the foregoing reasons, the investigation of glass crystallization kinetics is of great interest from both practical and theoretical points of view. Since, in many respects, the nucleation stage determines the pathways of overall crystallization, in this review we will focus our attention on nucleation, with particular emphasis on the analysis of relevant experimental results in the framework of the classical nucleation theory (CNT). Hereby we will restrict ourselves to selected data for homogeneous nucleation obtained mainly with silicate glasses.

The present paper is organized as follows: In Section 2, the basic equations of CNT are briefly summarized, which are then employed for nucleation data analysis. Section 3 presents the main methods that may by employed to experimentally determine nucleation rates. Section 4 is devoted to experimental findings concerning transient and steadystate crystal nucleation in glasses. In particular, evidence for a strong correlation between nucleation rates and reduced glass transition temperature is given. An analysis of the problems arising in the application of CNT to experimentally observed nucleation rate data is performed in Section 5. The paper is completed by concluding remarks.

2. Basic assumptions and equations of classical nucleation theory (CNT)

2.1. Historical notes

In its original form, classical nucleation theory is based on the thermodynamic description of heterogeneous systems developed by Gibbs [4]. Following Gibbs, a real inhomogeneous system is replaced by a model system consisting

2682 V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714 of two homogeneous phases divided by a mathematical surface of zero thickness. While the properties of the ambient phase are known, the bulk properties of the critical clusters are determined via Gibbs’ equilibrium conditions. A detailed analysis shows that the cluster bulk properties determined in such way are widely identical to the properties of the newly evolving macroscopic phase coexisting in stable equilibrium with the ambient phase at a planar interface. The free energy of the heterogeneous system – consisting of a cluster of the newly evolving phase in the ambient phase – is expressed as the sum of the bulk contributions of the nucleus and the ambient phase. These bulk terms are supplemented by interfacial contributions, the main one is given by the product of the interfacial area and specific surface energy.

When applying the theory to cluster formation, these surface terms initially result in an increase of the characteristic thermodynamic potential, which leads to the existence of a critical cluster size. Only clusters with sizes larger than the critical size are capable to grow up to macroscopic dimensions in a deterministic way. The change of the characteristic thermodynamic potential resulting from the formation of clusters of critical size is commonly denoted as the work of critical cluster formation. This quantity reflects the thermodynamic aspects in the description of nucleation.

In addition to thermodynamic aspects of nucleation, the dynamics of cluster formation and growth must be appro- priately incorporated into the theory. Different approaches have been employed depending on the particular problem being analyzed. The application of CNT to the formation of crystals originates from the work of Kaischew and Stranski [5]. These authors investigated this problem for the case of crystal formation from supersaturated vapor employing the approach developed by Volmer and Weber [6] for vapor condensation. Further advances in CNT including nucleation in the condensed systems, which are the focus of the present review, were connected with the work of Becker and Doring [7], Volmer [8], Frenkel [9], Turnbull and Fisher [10], Reiss [1] and others. Photographs of some of these pioneers of nucleation theory are shown in Fig. 1.

According to CNT, the description of homogeneous and heterogeneous nucleation can be basically performed by the same methods. We will present first the results for homogeneous nucleation and afterwards will introduce the modifications required to account for the effect of insoluble solid impurities and interfaces that may lead to heterogeneous nucleation.

2.2. Homogeneous nucleation

As we already discussed, homogeneous nucleation supposes the same probability of critical nucleus formation in any given volume or surface element of the system under

Fig. 1. From top left to right bottom: J.W. Gibbs, G. Tammann, M. Volmer, R. Kaischew, J. Frenkel, and D. Turnbull.

V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714 2683 study. According to CNT (see, e.g., Refs. [12,13]), the steady-state homogeneous volume nucleation rate can be written as

Io ¼ 2N1 kBTh a2rcm

This equation determines the so-called steady-state nucle- ation rate, Ist, i.e., the number of supercritical clusters formed per unit time in a unit volume of the system. The pre-exponential term, Io, depends only weakly on temperature (if compared to the exponential function) and varies tems [14]. In Eq. (1) kB and h are the Boltzmann and

Planck constants, respectively; N1 1/a3 is the number of structural (formula) units, with a mean size a, per unit vol- ume of melt; rcm is specific surface free energy of the crit- ical nucleus-melt interface; DGD is the activation free energy for transfer of a ‘structural unit’ from the melt to a nucleus (kinetic barrier). To a first approximation, the kinetic barrier for glass-forming liquids is often replaced by the activation free energy for viscous flow, DGg. W* is the thermodynamic barrier for nucleation, i.e., the increase in the free energy of a system due to the formation of a nu- cleus with critical size, r*. The critical nucleus size can be determined from the condition

where DGV = Gl Gc is the difference between the free energies of liquid and crystal per unit volume of the crystal

(i.e., the thermodynamic driving force for crystallization)

and c1 and c2 are shape factors. In the case of a spherical nucleus, we obtain the expressions



The thermodynamic driving force for crystallization is given by


DCpdT0 þ T Z T where Vm is the molar volume, DHm and Tm are the molar heat of melting and the melting temperature of the crystal,

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