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Kinetics of Crystallization and Segregation: Nucleation in Glass-Forming Systems

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Abstract

In the present chapter, the basic ideas underlying the theoretical description of nucleation and growth of a newly formed phase, in general, and in melts, in particular, are outlined. This is done for two purposes: The crystallization of glass-forming melts, its initiation and control or inhibition, respectively, determines to a large degree the possibility to transform a melt into a glass or to synthesize a glass-ceramic material with a predetermined structure and desired properties. On the other hand, crystallization processes and phase transformations in glass-forming melts can serve as model systems for the development and verification of different concepts concerning the kinetics of phase transformation processes also in other fields of science and technology. Consequently, the theoretical approaches outlined here are of significant importance for the understanding of the behavior not only of glass-forming systems but also of phase transitions, taking place in any other metastable system.

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Notes

  1. 1.

    A variety of new monographs devoted to the outline and extension of the classical concepts of nucleation and growth was published in the time since the publication of the first edition of the present book. Out of this spectrum, we would like to mention here, in particular, the following references: Kashchiev [439], Mutaftschiev [586], Milchev [558], Markov [534], Jackson [396], Kelton and Greer [450]; and a series of books edited by one of the authors (J.W.P. Schmelzer, Ed.): Schmelzer et al. [721], Schmelzer [695], Skripov and Faizullin [770], Baidakov [34], Ivanov [387], Slezov [773], Gusak et al. [286] as well as the series of workshop proceedings Schmelzer et al. [720]. For a current review on the state of affairs with respect to experimental and theoretical investigations of crystallization of glass-forming melts, see also the following articles: Fokin et al. [224], Schmelzer [696], Schmelzer [698], Zhuravlev et al. [958], Schmelzer [699], and Schmelzer and Schick [711]. Some particularly important to our point of view own contributions into this field are described briefly in the Chap. 14 and in some of the footnotes to this and subsequent chapters.

  2. 2.

    This assumption is based on the analysis of Gibbs’ equilibrium conditions determining the bulk properties of critical clusters and leading to this consequence.

  3. 3.

    An alternative approach to the the description of spatially inhomogeneous thermodynamic systems (denoted by us as generalized Gibbs approach) and its application to the determination of the properties of critical clusters was developed by the present authors and coworkers in the last decade starting with a paper published in 2000 (Schmelzer et al. [722]). For an illustration, the method is applied there to phase formation processes in solid and liquid solutions. However, it is applicable quite generally and not restricted to this particularly important but anyway special case. The presented there approach is a generalization of the classical Gibbs’ method. It is – like the classical Gibbs method – conceptually simple and directly applicable to real systems, but avoids its shortcomings. Central to this method was originally (later the consequences have been shown to follow directly from the modification of Gibbs classical approach developed by us) the formulation and application of a well-founded principle we denoted as generalized Ostwald’s rule in nucleation. The method allows one the determination of the dependence of the bulk properties of the critical clusters and the work of critical cluster formation in dependence on cluster size provided the bulk properties and the macroscopic values of the surface tension (at planar interfaces) for the possible different states of the system under consideration are known. As it turns out, in the framework of the generalized Gibbs approach the bulk properties – and as a consequence also the interfacial properties – of the critical clusters depend significantly on supersaturation (or the size of the critical clusters). Similarly to the van der Waals, Cahn, and Hilliard and density functional calculations in the determination of the work of critical cluster formation, the newly developed method reproduces the results of the classical Gibbs’ nucleation theory (involving the capillarity approximation) for small values of the supersaturation. However, in contrast to the classical and in agreement with van der Waals-type methods of descriptions of inhomogeneous systems, for initial states approaching the spinodal curve, the work of critical cluster formation, determined via the newly developed approach, is shown to tend to zero. As an immediate additional consequence, the method gives a more accurate description of the experimental results on nucleation rates also in the intermediate ranges of the initial supersaturation. This method was further developed then in a series of papers and applied also to the determination of the properties of sub- and supercritical clusters (Schmelzer et al. [725]; Schmelzer et al. [3]). The theoretical foundation of the generalized Gibbs approach is given in: Schmelzer et al. [729]. An overview on further developments and the application of this method to the analysis of experimental data is given in: Schmelzer and Abyzov [702]; Abyzov and Schmelzer [3]; Schmelzer [697]; Schmelzer et al. [728]; Abyzov et al. [4]; Schmelzer and Abyzov [703]; Schmelzer and Abyzov [704] (a more detailed overview is given in the Chap. 14). In the present book, however, we employ the classical approach treating the clusters widely as small aggregates with the bulk properties of the newly evolving macroscopic phases and introduce corrections via a curvature dependence of the surface tension.

  4. 4.

    Note as well that independent of the specific expression for the curvature dependence of the surface tension, always the equation

    $$\Delta G_{(\mathit{cluster})}^{(c)}(\sigma (R)) = \Delta G_{ (\mathit{cluster})}^{(c)}(\sigma (R \rightarrow \infty )){\left ( \frac{\sigma (R)} {\sigma (\infty )}\right )}^{3}$$

    holds, if the surface of tension is chosen as the dividing surface (cf. Parlange (1970) [625], Ulbricht, Schmelzer et al. (1988) [874]). Latter result was already well-known to J.W. Gibbs and the effects of a curvature dependence of the surface tension on nucleation have been also already discussed in detail by him [249].

  5. 5.

    A complete analytical theoretical description of nucleation-growth processes in solutions accounting for depletion effects – i.e. changes of the state of the ambient phase due to the formation of clusters of the newly evolving phase – is given in Slezov and Schmelzer [776, 778]; and in Slezov [773] as well as in cited there papers.

  6. 6.

    A derivation of this relation and some additional discussion can be found in Chap. 14.

  7. 7.

    For a more recent as compared with [775] formulation of these results including further developments, in particular, accounting for depletion effects on the course of first-order phase transitions, see the already cited references Slezov and Schmelzer [776, 778]; Slezov et al. [779]; and Slezov [773] and further cited there papers.

  8. 8.

    Indeed, in terms of the classical Gibbs approach, the bulk properties of the cluster phase are determined by Gibbs equilibrium conditions, equality of temperature in the cluster (specified by α) and ambient (β) phases, T α  = T β , and equality of chemical potentials, μ iα  = μ iβ , of all i = 1, 2, …, k components. Provided the cluster is considered to be of spherical shape and the surface of tension is chosen as the dividing surface, then the properties of the critical clusters (W c work of critical cluster formation; R c radius of the cluster referred to the surface of tension) are defined via

    $$W_{c} = \frac{1} {3}\sigma A_{c}\;,\qquad A_{c} = 4\pi R_{c}^{2}\;,\qquad p_{\alpha } - p_{\beta } = \frac{2\sigma } {R_{c}}\;.$$

    Provided W c is known (for example, from experiment or statistical-mechanical computations) then always R c and σ can be determined from above set of equations, i.e., one can always construct a spherical cluster – in terms of Gibbs thermodynamic theory – leading to the same value of the work of critical cluster formation as observed in experiment. Of course, in experiment the shape of the critical cluster can be much different as compared to a sphere or the cluster may be too small to allow one a thermodynamic description, anyway, such Gibbs’ model cluster can be uniquely defined. Similar considerations hold also in the case that the generalized Gibbs approach is employed in the analysis.

  9. 9.

    For a comprehensive discussion of this circle of problems, cf. Schmelzer [693, 694]. In these publications, a detailed analysis of the initial formulation of the nucleation theorem as given by Kashchiev in 1982 [436] is presented. In a next step, a new formulation of the nucleation theorem, mathematically widely equivalent to the form, as derived by Oxtoby and Kashchiev in 1994 [619] employing the classical Gibbs’s approach, is developed in above cited publications. This formulation is, however, more easily applicable to the interpretation of experimental results as compared with the original expressions given by Oxtoby and Kashchiev. It can be utilized straightforwardly also in cases where the original version cannot be employed and allows one, in addition, a variety of further theoretical developments. It is shown, moreover, first in the framework of Gibbs’s theory of heterogeneous systems that the nucleation theorem holds not only for critical clusters but for clusters of arbitrary sizes as well. The new formulation of the nucleation theorem is applied then to the analysis of different cases of phase formation. It is shown further on that an alternative thermodynamic derivation of the nucleation theorem both for clusters of critical as well as for arbitrary sizes can be given at certain specified conditions based on the van der Waals approach to the description of heterogeneous systems. The limits of validity of the nucleation theorem are analyzed with respect to its applicability to real systems; i.e., the problem is discussed to what extent the nucleation theorem may give an adequate description of properties of the real critical clusters determining the nucleation process in nature.

  10. 10.

    A detailed analysis of the peculiarities of nucleation in confined space and the derivation of conclusions concerning the general scenario of first-order phase transitions based on the generalized Gibbs approach was performed recently by one of the authors in cooperation with A. S. Abyzov (Schmelzer and Abyzov [703]). The general conclusions remain widely the same as discussed in the present section and derived here based on the classical Gibbs’ approach.

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  1. Institute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan S. Gutzow

  2. Institut für Physik, Universität Rostock, Rostock, Germany

    Jürn W. P. Schmelzer

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Gutzow, I.S., Schmelzer, J.W.P. (2013). Kinetics of Crystallization and Segregation: Nucleation in Glass-Forming Systems. In: The Vitreous State. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34633-0_6

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