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Nucleation Rates in Silicate Glasses: Homogeneous vs. Heterogeneous, Notas de estudo de Engenharia de Produção

The challenges in quantitatively describing nucleation rates in silicate glasses using the cnt theory. It highlights issues such as discrepancies between calculated and measured rates, high crystal nuclei/melt surface energy, temperature dependence, and surface energy and time-lag discrepancies. The document also introduces homogeneous and heterogeneous nucleation, providing equations for thermodynamic barriers and steady-state nucleation rates. It concludes by mentioning the importance of crystal size distribution analysis and the influence of pre-existing nuclei on nucleation rates.

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Baixe Nucleation Rates in Silicate Glasses: Homogeneous vs. Heterogeneous e outras Notas de estudo em PDF para Engenharia de Produção, somente na Docsity! www.elsevier.com/locate/jnoncrysol Journal of Non-Crystalline Solids 352 (2006) 2681–2714Review Homogeneous crystal nucleation in silicate glasses: A 40 years perspective Vladimir M. Fokin a,*, Edgar D. Zanotto b, Nikolay S. Yuritsyn c, Jürn 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 São Carlos, 13565-905 São Carlos, SP, Brazil c Grebenshchikov Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24-2, 199155 St. Petersburg, Russia d Institut für Physik, Universität 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. JamesAbstract 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 temper- ature 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 under- lying CNT, the bulk thermodynamic properties of the critical nuclei generally differ from those of the corresponding macro-phase result- ing 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; Thermodynamics0022-3093/$ - 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).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) 2682 V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714method 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 lim- ited degree or can be completely arrested down to tempera- tures 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 rearrange- ments required to keep the liquid in the appropriate meta- stable equilibrium state cannot follow any more the change of temperature. This process of freezing-in the struc- ture of an undercooled liquid transforming it into a glass is commonly denoted as glass transition. Typical glass-form- ing liquids, such as silicate melts, are usually characterized by: (i) relatively high viscosities (g > 100 Pa s) at the melting point or liquidus and (ii) a steep increase of the viscosity with decreasing temperature. These properties favor vitrifi- cation. The mechanism above sketched leads to the conclu- sion 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 lat- ter evolution process, as will be shown below, involves overcoming of a thermodynamic potential barrier. At room temperature glasses can exist for extremely long peri- ods 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 sur- face and sometimes in the bulk via heterogeneous or homo- geneous nucleation (see below). Nucleation, or the process of formation of the precur- sors of the crystalline phases, may occur by different mech- anisms. Commonly one divides these processes into homogeneous and heterogeneous nucleation. Homoge- neous nucleation is a stochastic process occurring with the same probability in any given volume (or surface) ele- ment. Alternatively, nucleation occurring on preferred nucleation sites, e.g., such as pre-existing interfaces, previ- ously 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 nucle- ation can sometimes be observed at deep undercoolings (T/Tm < 0.6) because glass-forming melts are excellent sol- vents 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 pro- duced by mankind, having its origin about 6000 years ago in ancient Mesopotamia [2], but are still gaining tech- nological importance. It is evident from the above discussion that crystalliza- tion 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, con- trolled 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 homo- geneous 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 exper- imentally determine nucleation rates. Section 4 is devoted to experimental findings concerning transient and steady- state 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 exper- imentally 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 sys- tems developed by Gibbs [4]. Following Gibbs, a real inho- mogeneous system is replaced by a model system consisting V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714 26852.3. Heterogeneous nucleation The existence of foreign solid particles and phase boundaries may favor nucleation. This effect is due mainly to the diminished thermodynamic barrier as compared to that for homogeneous nucleation, owing to a decrease of the contribution of the effective surface energy to the work of critical cluster formation. For example, the thermody- namic barrier for nucleation in the case of condensation on planar interfaces is given by [12] W het ¼ W U; U ¼ 12 34 cos hþ 14 cos 3 h: ð13Þ Depending on the value of the wetting angle, h, the param- eter U varies from zero to unity. The value of U depends on the mechanism of nucleation catalysis. In order to adapt the expression for the steady-state nucleation rate, Eq. (1), to the description of heterogeneous nucleation, the number of ‘structural’ units per unit vol- ume, N1, which appears in the pre-exponential term of Eq. (1), must be replaced by the number, NS, of ‘structural units’ in contact with the catalyzing surface per unit vol- ume. Hence, in the case of heterogeneous nucleation, the following equation can be written for the steady-state nucleation rate: Ihetst ffi NS kBT h exp W Uþ DGD kBT   : ð14Þ Catalyzing surfaces may be represented, for instance, by dispersed solid particles that act as nucleation sites. In this case, their curvature and number may strongly affect the nucleation kinetics [14,17]. The exhaustion of available nucleation sites due to crystal nucleation leads to satura- tion of the kinetic curve N versus t. If, however, for some reason such saturation is not achieved, the knowledge of the N(t)-dependence is not sufficient to conclude what type of nucleation took place. 3. Experimental methods to estimate nucleation rates 3.1. General problem At high undercoolings corresponding to the range of measurable homogeneous (volume) nucleation rates in typ- ical glass-forming liquids, the critical nuclei are undetect- able by common experimental techniques, hence they must first be developed to a visible size to allow one to determine (e.g., using a microscope) their number density, N, as a function of time, allowing then to estimate the nucleation rate as I = dN/dt. In order to perform such task, different methods have been developed. 3.2. Double-stage (‘development’) method If the overlapping of the nucleation and growth rate curves is weak (i.e., the crystal growth rates are very low at temperatures corresponding to high nucleation rates), the observation of the nucleated crystals and the estimationof the crystal number density is a quite difficult task. For these cases, about a hundred years ago, Gustav Tammann (who was studying crystallization of organic liquids) pro- posed the following procedure, which is now known as the Tammann or ‘development’ method [18]. Crystals nucleated at a low temperature, Tn, are grown up to micro- scopic sizes at a higher temperature, Td > Tn. The develop- ment temperature Td has to meet the following conditions for nucleation (I) and growth (U) rates: I(Td) I(Tn) and U(Td) U(Tn). After a lapse of seventy years, Ito et al. [19] and Filipovich and Kalinina [20] independently applied Tammann’s method to the study of crystal nucleation kinetics in lithium disilicate glasses. Since then, this method has been widely employed for glass crystallization studies. Some problem inherent in this method and connected with the possible dissolution of some part of the originally formed (at the nucleation temperature) nuclei at the devel- opment temperature will be discussed later. 3.3. Single-stage methods 3.3.1. The direct method When there is considerable overlap of the I(T) and U(T)- curves, the number density of crystals can be measured directly after single-stage heat treatments at Tn. Then, the obtained N(Tn, t)-curve will be shifted (relatively to the true one) to higher times by a time to = (rres  r*)/U(Tn) ffi rres/ U(Tn) that is needed to grow the crystals up to the micro- scope resolution limit, e = 2rres [21]. Finally, one must cor- rect the number densities to account for stereological errors. This procedure will be described in Section 3.4. 3.3.2. Crystal size distribution analysis Continuous nucleation and growth normally result in a broad distribution of crystal sizes, i.e., the first nucleated crystal has the largest size and so forth. If the crystal growth rate is known, one can calculate the ‘birth dates’ of crystals belonging to different size groups and then plot a N(t)-curve. Toschev and Gutzow derived the basic for- mulas relating the size distribution of spherical isolated particles embedded in a continuous matrix with that of their circular intersections on a sample cross-section for both steady-state and transient volume nucleation [22]. For surface crystallization the size distribution is easily constructed from direct measurements. This method, known as Köster’s method, also works in the case of heter- ogeneous nucleation from a finite number of active centers when the latter are depleted in a relative short time, and further advancement of crystallization only occurs via crys- tal growth. It has been systematically employed to study the surface nucleation rates in metallic [23] and silicate glasses [24]. 3.4. Stereological corrections The use of reflected light microscopy can lead to large errors in the determination of the number of crystals per 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 f ε /D M Fig. 2. Fractional underestimation of the number of spherical particles versus the ratio between the microscope resolution limit and the largest particle diameter. Solid and dashed curves refer to cases (i) and (ii), respectively. 2686 V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714unit volume due to the stereological methods employed to calculate volume properties (size distributions, numbers, etc.) based on statistical evaluations performed on cross- sections through the specimens. Thus a significant fraction of the cut crystals (in the cross-sections) can be smaller than the resolution limit of the microscope used, which may lead to an underestimation of the crystal numbers and, consequently, of the determined values of nucle- ation rates. In Refs. [25,26], equations were derived for the fractional underestimation, f, of the number of spheri- cal particles per unit volume and of the nucleation rates, as obtained from stereological techniques for reflected light microscopy or SEM, for typical cases of crystal nucle- ation in glasses. The following two cases bound the most common experimental situations: (i) a monodisperse system of spherical particles that can result from instan- taneous heterogeneous nucleation; (ii) a uniform size dis- tribution of spherical particles from the critical size to DM, where DM is the largest diameter of the clusters in the distribution. Such distribution is typical for simulta- neous nucleation and growth with constant rates in a single-stage heat treatment. The equations for these cases are: Case (i). Monodisperse systems: f ¼ 2 p arcsinðr1Þ; ð15Þ Case (ii). Uniform size distribution from the critical size to DM: f ¼ 1 2 p cos h1½1 lnð1þ sin h1Þ þ h1 þ r1 ln r1  r1½  : ð16Þ In above equations, h1 = arccosr1, r1 e/DM, and e is the resolution limit of the microscope used. Comparison with experimental nucleation data for two silicate glasses dem- onstrated that these equations predict well the observed underestimations of the number of spherical particles. Fig. 2 shows the function f for cases (i) and (ii). To minimize these errors employing reflected light opti- cal microscopy methods, one should use high magnification objective lenses or SEM. Alternatively, transmission meth- ods could be used because they lead to much smaller errors than reflection techniques. Similar underestimates occur when one tries to deter- mine volume fractions crystallized, and these may be sub- jected to significant errors when the largest grain size of the distribution is close to the microscope resolution limit [26]. For transformations occurring from a fixed number of nuclei, the systematic errors are smaller than those observed in the continuous nucleation case, but can still be significant when reflected light microscopy is used. Transmission methods are more time-consuming, but lead to much smaller errors than reflection techniques.3.5. Probabilistic approach for the analysis of the nucleation process For the sake of completeness we should briefly mention a method based on the stochastic nature of nucleation [27]. The appearance of critical nuclei is a stochastic event that can be characterized by an average waiting period, s, s ¼ 1 IV ; ð17Þ where I is the nucleation rate and V is the volume of the system under study. Since the probability of critical nucleus formation due to a successful series of attachment and separation reactions is very low, nucleation can be treated as a Poissonian process. Hence the probability of appearance of one critical nucleus in a time period s1 is P 1ðs1Þ ¼ ks1 expðks1Þ; ð18Þ where k ¼ 1=s. In cases of high nucleation rates, their measurement is normally limited to relatively low undercoolings that corre- spond to high values of the crystal growth rate. Thus, the first few super-critical nuclei trigger crystallization of the whole sample. Fitting the experimental distribution of waiting times of the first nucleus, s1, to Eq. (18) one can estimate an aver- age waiting period, s, and then the nucleation rate from Eq. (17). Such analysis has been employed, e.g., for metals dis- persed in the form of small drops when the use of other meth- ods is connected with difficulties (see, e.g., [13,28]). 3.6. Overall crystallization kinetics Crystal nucleation followed by subsequent growth results in the overall crystallization of the sample. This V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714 2687process can be described by determining the volume frac- tion of the transformed phase, a(t). The formal theory of overall-crystallization kinetics under isothermal conditions was developed in the late 1930s by Kolmogorov [29], John- son and Mehl [30], and Avrami [31], and is well-known as the JMAK theory. According to this theory, the volume fraction of the new phase is given by aðtÞ ¼ 1 exp g Z t 0 Iðt0Þ Z t t0 Uðt00Þdt00  3 dt0 " # ; ð19Þ where g is a shape factor, which is equal to 4p/3 for spher- ical crystals. If the nucleation (I) and growth (U) rates are constant throughout the transformation (e.g., steady-state homogeneous stoichiometric nucleation), Eq. (19) can be rewritten as aðtÞ ¼ 1 exp  gIU 3t4 4   : ð20Þ When the number of growing crystals, No, does not change with time (as it is typical for fast heterogeneous nucleation on a finite number of active sites), Eq. (19) transforms to aðtÞ ¼ 1 exp½gNoU 3t3 : ð21Þ Avrami proposed that, in general, the following relation should be used: aðtÞ ¼ 1 expðKtnÞ: ð22Þ In typical applications, Eq. (22) is employed in the form lnð lnð1 aÞÞ ¼ ln K þ n ln t: ð23Þ The values of K and n can be estimated then by fitting the experimental data of a(t) to Eq. (23). Thus the coefficient K includes I and U, or No and U. The Avrami coefficient, n, depends on both nucleation and growth mechanisms, and can be written for the case of three-dimensional growth as n ¼ k þ 3m; ð24Þ0 20 40 60 80 100 0 1x1013 2x1013 3x1013 4x1013 a Li2O.2SiO2 T n =703 K T d =899 K t o t ind N , m -3 t, h Fig. 3. Typical curves of the number density of Li2O Æ2SiO2 (a) and 2Na2O ÆC versus time of nucleation obtained by the ‘development’ method [35,36].where k and m are taken from the formulas N  tk and r  tm describing the variation of crystal number (N) and crystal size (r) with time. The knowledge of the Avrami coefficient, n, is helpful to understand the mechanism of phase transformation at a given temperature. When it is possible to independently measure the crystal growth rate, one can then calculate the nucleation rate from the coefficient K. This method is not as precise as direct measurements, but can give useful information about nucleation in advanced stages of crystal- lization, when the application of other methods is hindered (see Section 5). For the simplest cases of constant nucleation rate (or constant number of crystals) and linear growth, Eqs. (20) and (21) have been tested by using Ist, U, and No data inde- pendently measured by optical microscopy in glasses of stoichiometric compositions 2Na2O ÆCaO Æ3SiO2 [32] and Na2O Æ2CaO Æ3SiO2 [33]. Good agreement was obtained between the values of gIU3 (or gNoU 3), calculated from fit- ting the a(t)-data to the JMAK equation, and directly mea- sured values. Recently, the JMAK-equation was also successfully employed, together with measured crystal growth rates, to estimate extremely high nucleation rates in a stoichiometric glass of fresnoite composition [34]. 4. Interpretation of nucleation experiments by the classical nucleation theory 4.1. Non-steady state (transient) nucleation 4.1.1. Estimation of the time-lag in nucleation Typical N(Tn,Td, t)-curves obtained by the ‘develop- ment’ method are shown in Fig. 3. As we already men- tioned, only the nuclei that achieve the critical size, r*(Td), during heat treatment at Tn can grow at the devel- opment temperature Td. The other nuclei have a high prob- ability to dissolve at Td. As the result, the number of crystals nucleated at given conditions and developed at Td has, strictly speaking, to decrease with increasing Td0 20 40 60 80 100 0 2x1016 4x1016 6x1016 8x1016 t, h 2Na2O.CaO.3SiO2 Tn=723 K Td=856 K t ind b aO Æ3SiO2 (b) crystals in glasses of respective stoichiometric compositions 0 50 100 150 200 250 0.0 5.0x104 1.0x105 1.5x105 2.0x105 2.5x105 a 3 2 1 N , m m -3 t, min 0 50 100 150 200 0.0 5.0x102 1.0x103 1.5x103 2.0x103 b Ist I, m m -3 m in -1 t, min Fig. 9. (a) Number density of Li2O Æ2SiO2 crystals obtained via the ‘development’ method (Td = 626 C) versus time of nucleation at Tn = 465 C. Curve 1 refers to the quenched glass. Curves 2 and 3 refer to glasses subjected to preliminary treatment at T = 430 C for 65 h (curve 2) and 89 h (curve 3) [43]. (b) Nucleation rate versus time. Solid and dashed lines correspond to curves 1 and 3 from (a), respectively. 0 20 40 60 80 100 0.0 2.0x104 4.0x104 6.0x104 5 4 3 2 1 N , m m -3 t, min Fig. 10. Number density of Li2O Æ2SiO2 crystals obtained via the ‘development’ method (Td = 626 C) versus time of nucleation at Tn = 485 C. Curve 1 quenched glass, curves 2–5 glasses subjected to preliminary treatment at T = 473 C (curve 2), 451 C (curve 3), 440 C (curve 4), and 430 C (curve 5) for the following times: t = 0.75 h (curve 2), 4.5 h (curve 3), 18 h (curve 4), and 65 h (curve 5) which exceed the time- lags at T [43]. 0.5 0.6 101 105 10 9 1013 4 3 2 1 I s t, m -3 s- 1 T/Tm Fig. 11. Steady-state nucleation rate versus reduced temperature for some stoichiometric glasses: (curve 1) 3MgO ÆAl2O3 Æ3SiO2 [47]; (curve 2) Li2O Æ2SiO2 [35]; (curve 3) Na2O Æ2CaO Æ3SiO2 [48]; (curve 4) 2Na2O Æ CaO Æ3SiO2 [36]. 2690 V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714obtained by the ‘development’ method at Td = 626 C. Curves 2 and 3 demonstrate, as compared with curve 1, a strong increase in the number of crystals, and only for times higher than about 120 min the nucleation rate reaches steady-state conditions corresponding to the tem- perature T2. The evolution of the nucleation rate corre- sponding to curve 3 is shown in Fig. 9(b). Such unusual behavior of the nucleation kinetics is caused by the transition of an initial distribution formed at T1 for sizes less than r*(Td) into the steady state cluster size distribution corresponding to T2. Since the number of nuclei having sizes r P r*(T) increases with decreasing tem- perature, down to T = Tm/3, a strengthening of the effect of the preliminary heat treatment with decrease of T1 should be expected. This is indeed the case as shown in Fig. 10. The presented effects of the multistage heat treatments were well-described by the numerical modeling of the cluster evolution performed in the framework of the classical nucleation theory [44–46] with the exception of the heat treatments involving the temperature T1 = 430 C [45]. Since the values of the parameters needed for the simula- tions were estimated via a fitting procedure this disagree- ment could be caused by the error in the Ist(430 C) estimation or viscosity data taken from other authors. Nev- ertheless, the simulations clearly show that the nucleation kinetics is governed by the evolution of the nuclei distribution. 4.2. Steady-state nucleation 4.2.1. Temperature dependence of steady-state nucleation rates Some examples of steady-state nucleation rates, Ist, mea- sured from the slope of the linear part of the N(t)-plots, such as those shown in Fig. 3, are presented in Fig. 11 as a function of reduced temperature. The values of Ist(T)pass through a maximum at a temperature Tmax. The magnitudes of Ist(Tmax) Imax vary from 5 · 1013 to 3 · 102 m3 s1 and cover practically the whole range of avail- able measurements of nucleation rates in silicate glasses with stoichiometric compositions. The reason for the existence of the nucleation rate max- imum follows from a simple analysis of Eq. (1). Since the pre-exponential term, Io, depends only weakly on tempera- ture, the temperature dependence of the nucleation rate is determined mainly by the thermodynamic and kinetic bar- riers for nucleation. A temperature decrease produces two 0.2 0.4 0.6 5 10 15 20 25 8 7 6 5 4 3 2 1 1/3 lo g( I st , m -3 s-1 ) T/Tm Fig. 12. Temperature dependence of homogeneous nucleation rates. The curves were calculated with Eqs. (1), (30), and (32) with a pre-exponential term Io = 10 42 m3 s1 and following values of the parameters character- izing the temperature independent parts of the thermodynamic (C1) and kinetic (C2) barriers: C1 = 6.5 (curves 1–5), 5.8 (curve 6), 5.1 (curve 7), 4.5 (curve 8); C2 = 6 (curves 1 and 6–8), 4.8 (curve 2), 3.9 (curve 3), 2.8 (curve 4), 0 (curve 5). V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714 2691effects: a decrease of the thermodynamic barrier due to an increase in the thermodynamic driving force for crystalliza- tion, leading to a higher nucleation rate, and an increase of the kinetic barrier, leading to a lower nucleation rate (the kinetic barrier is, as mentioned earlier, often replaced by the activation free energy for viscous flow). As a result of these two opposite tendencies, one finds a maximum of the steady-state nucleation rate at a temperature Tmax, which is located well below Tm. Eq. (4) for the thermodynamic barrier can be rewritten as W  kBT ¼ C1 1 T rð1 T rÞ2 ; C1 ¼ 16p 3 a3STDHm RT m ; T r T T m : ð30Þ Here we used the linear approximation for the thermody- namic driving force, Eq. (6), and the following semi-empir- ical equation: rcm ¼ aST DH m V 2=3m N 1=3 A ð31Þ for the specific surface energy of the nucleus/melt interface proposed by Skapski and Turnbull [49,50]. In Eq. (31), DHm is the melting enthalpy per mole, Vm is the molar vol- ume, NA is Avogadro’s number, and aST is an empirical dimensionless coefficient, smaller than unity, reflecting the fact that surface atoms have less neighbors than bulk atoms. Assuming that DGD is of the same order of magni- tude as the activation free energy for viscous flow, DGg, one can write the kinetic barrier as DGDðT Þ kBT ¼ C2 T r  T or ; C2 2:30B T m ffi 30ðT gr  T orÞ; T or T o T m ; T gr T g T m ; ð32Þ where To and B are the empirical coefficients of the Vogel– Fulcher–Tammann (VFT) equation and Tg is the glass transition temperature. The application of the VFT-rela- tion implies the assumption of a temperature-dependent activation free energy, DGg. In the definition of C2 we took into account the fact that DGg/(kBT) ffi 30 at T = Tg. Fig. 12 shows Ist(Tr)-curves calculated with Eqs. (1), (30), and (32), reasonable estimates of the pre-exponential term and values of the parameters C1 and C2, as indicated in the figure caption. One can see that the decrease in the kinetic barrier, caused by a decrease in C2 at a fixed value of C1, results in a shift of the nucleation rate maximum to lower temperatures (cf. curves 1–4). The reduced tempera- ture Tr T/Tm = 1/3 is a lower limit to T maxr T max=T m obtained when the kinetic barrier tends to zero (cf. curve 5). This shift is accompanied by a strong increase in the magnitude of I(Tmax) Imax. When the thermodynamic barrier is diminished, at fixed values of C2, by decreasing the parameter C1 (which is proportional to aST and the reduced melting enthalpy DH rm ¼ DH m=RT mÞ, the valueof Imax also increases (curves 1 and 6–8), but the value of Tmax shifts to higher temperatures. The effect of variation of the kinetic barrier on the nucle- ation rate can be qualitatively illustrated for lithium disili- cate [51] and sodium metasilicate [52] glasses with different H2O content (a few percent of water often result in a signif- icant decrease of viscosity) as shown in Fig. 13. A decrease in the thermodynamic barrier can be also caused by a decrease in the effective crystal/melt interfacial energy as in the case of heterogeneous nucleation. As a result, as was shown in Ref. [53], the temperature Tmax for heteroge- neous surface nucleation is displaced to higher values as compared with homogeneous nucleation. 4.3. Correlation between nucleation rate and glass transition temperature The methods discussed in Section 3 to measure nucle- ation kinetics are both difficult to perform and time con- suming. Also, owing to several restrictions, they cannot always be employed. Hence, the knowledge of any correla- tion between nucleation rate and easily measurable proper- ties of glasses is highly desirable. As one example, well before the development of nucleation theory for condensed systems, Tammann called attention to the following ten- dency: the higher the melt viscosity at the melting temper- ature, the lower is its crystallizability [54]. Almost eighty years after Tammann’s pioneering research work, James [55] and Zanotto [56], based on numerous experimental nucleation rate data for several silicate glasses, concluded that glasses having a reduced glass transition temperature, Tgr Tg/Tm, higher than 400 420 440 460 480 500 10 100 1000 a 3 2 1 I st , m m -3 m in -1 T,oC 560 600 640 680 1000 10000 100000 2 1 b I st , m m -3 m in -1 T, oC Fig. 13. Temperature dependencies of the steady-state nucleation rates in Li2O Æ2SiO2 [51] (a) and Na2O Æ2CaO Æ3SiO2 [52] (b) glasses containing different amounts of H2O: (a) 0.05 mol% (curve 1), 0.12 mol% (curve 2), and 0.20 mol% (curve 3); (b) 0.01 mol% (curve 1), 0.2 mol% (curve 2). 2692 V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–27140.58–0.60, display only surface (mostly heterogeneous) crystallization; while glasses showing volume (homogeneous) nucleation have values Tgr < 0.58–0.60. Since at tempera- tures T < Tm the nucleation rate is always positive, the absence of volume nucleation for glasses having Tgr > 0.60 merely indicates undetectable nucleation on labora- tory time/size scales. Hence, an increase in the nucleation rate with decreasing Tgr could be expected. Indeed, a dras- tic increase of the magnitude of Imax with decreasing Tgr has been demonstrated by Deubener [57]. Fig. 14 presents a plot of the Imax(Tgr)-dependence, which has been extended in Ref. [58] and in the present work. In a rela- tively narrow range of Tgr (from 0.47 to 0.58) shown by 55 glasses of stoichiometric and non-stoichiometric compo- sitions, belonging to eight different silicate systems, the nucleation rates drop by about 17 orders of magnitude! When0.48 0.52 0.56 0.60 -5 0 5 10 15 20 2 1 1Li2O.1.27SiO2 2BaO. TiO . 2 2SiO 2 3MgO. Al2O3.3SiO2 2Na2O.CaO.3SiO2 Na2O.2CaO.3SiO2 CaO.SiO 2 Li2O.2SiO2 BaO.2SiO 2 lo g( I m ax , m -3 s-1 ) Tg / Tm/L Fig. 14. Maximum nucleation rate as a function of reduced glass transition temperature for 55 silicate glasses. The lines are calculated from CNT with C1 = 4.5 (curves 1) and 6.5 (curves 3). Solid lines refer to C2 = 4.5 and Tor = Tgr  C2/30; dashed lines to Tor = 0.4 [58].Tgr increases, the kinetic inhibition of nucleation proceeds at higher temperatures and at higher values of the thermo- dynamic barrier due to lower values of the thermodynamic driving force. As a consequence, nucleation becomes prac- tically undetectable at Tgr > 0.58. This result confirms the findings of James [55] and Zanotto [56]. The lines in Fig. 14 are calculated from CNT (Eqs. (1), (30), and (32)) with reasonable values of the parameters C1 and C2 indi- cated in the figure caption. Remember that C1 and C2 char- acterize the temperature independent parts of the thermodynamic and kinetic barriers for nucleation, respec- tively. Since Eq. (32) contains two independent parameters C2 and Tor, the viscosity and, correspondingly, Tgr, was varied in two different ways, by keeping either C2 (solid line) or Tor (dashed line) fixed. In the most interesting tem- perature range (0.5 < Tr < 0.6) these different ways of vary- ing Tgr lead to similar results. The lines reflect correctly the experimentally observed general trend. However, in apply- ing the mentioned rule to particular systems one has to act with some precaution since a substantial variation of the thermodynamic barrier can result in a considerable varia- tion of Imax for glasses having similar values of Tgr. For instance, fresnoite (2BaO ÆTiO2 Æ2SiO2) and wollastonite (CaO ÆSiO2) glasses have Tgr about 0.57, while the values of the parameter aST are 0.4 and 0.6, respectively. The lat- ter fact leads to a strong difference in the values of the ther- modynamic barriers and correspondingly to a strong difference in Imax. Also nucleation of metastable phases, such as BaO Æ2SiO2, is possible as shown in Ref. [59]. An important parameter is the location of Tmax. It is commonly accepted that Tmax is close to Tg. However, it was shown in Ref. [58] that the ratio Tmax/Tg depends on Tgr. Tmax/Tg is higher than one (i.e., Tmax exceeds Tg) at low Tgr, approaches one at about Tgr  0.55, and then becomes smaller than one. This trend results in an addi- tional increase of the kinetic barrier at Tmax with increasing Tgr caused by the increase of g(Tmax). Computations of Ist(T) temperature dependencies simi- lar to those published in Ref. [58] and presented here were V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714 2695the theoretical predictions of Refs. [72,73]. Similar results were obtained for a Na2O Æ2CaO Æ3SiO2 glass [71]. For completeness of the discussion, we would like to mention also another interpretation of the increase of rcm with increasing temperature widely discussed in Ref. [75]. The argumentation is based on model considerations sup- posing an increased ordering of the liquid near the crystal. These ideas were expressed first by Turnbull [66] and result in an entropy decrease. Employing some plausible assump- tions, the positive temperature coefficient of rcm can be accounted for then by the mentioned entropy loss. Run- ning ahead we could also suppose that the temperature dependence of rcm is the result of a possible change of the critical nucleus composition and/or structure with its size. However, regardless of the above possible interpretations the values of the specific surface energy estimated from nucleation rate data in the framework of the classical Gibbs’ approach remain too high when compared with the respective melt–vapor surface energies. Consequently, the problem posed at the end of the preceding section remains unsolved by these considerations. 5.3. Estimation of crystal/liquid specific surface energies via dissolution of subcritical nuclei Essentially all known methods to determine the nucleus- undercooled liquid surface energy are based on nucleation experiments involving certain additional assumptions. However, in order to test the classical nucleation theory or to make theoretical predictions, independent estimates of the specific surface energy are required. Such an inde- pendent method of estimating rcm for clusters of near-crit- ical sizes has been developed recently [69]. The results are summarized below. The new method is based on the dissolution phenome- non (discussed in Sections 3 and 4) of subcritical nuclei with an increase in temperature. As we already have shown, an N(Tn, r*(Tn), t)-plot coincides with the N(Tn, r*(Td), t)-plot, with the only difference that the latter is shifted along the time-axis by a time to (Eq. (27)). Then, kinetic N(Tn, t)-curves obtained with different development temperatures Td1 and Td2 > Td1 should be shifted with respect to each other by a time Dto = to2  to1. Fig. 5 shows an example of such kinetic curves. The following equation: Dto ¼ Z rðT d2Þ rðT d1Þ dr UðT n; rÞ ¼ 1 UðT n;1Þ rðT d2Þ  rðT d1Þ  þ rðT nÞ ln rðT d2Þ  rðT nÞ rðT d1Þ  rðT nÞ   ð38Þ was derived in Ref. [38] to estimate this shift. In the deriva- tion of Eq. (38) a size-dependent crystal growth velocity [76] was used of the form UðT ; rÞ ¼ UðT ;1Þ 1 rðT Þ r   : ð39ÞEmploying Eq. (3) for the critical nucleus size and assum- ing that rcm depends only slightly on temperature, Eq. (38) can be rewritten as rcm ¼ 1 2 DtoUðT n;1Þ 1 DGV ðT d2Þ  1DGV ðT d1Þ þ 1 DGV ðT nÞ ln 1 DGV ðT d2Þ  1DGV ðT nÞ 1 DGV ðT d1Þ  1DGV ðT nÞ    ð40Þ Hence, it is possible to calculate the average value of rcm in the temperature range Tn–Td2 from experimental values of Dto, U(Tn,1) and DGV. Note that in doing so neither nucle- ation rate nor time-lag data are required. The values of rcm calculated by this method for Li2O Æ2SiO2 and Na2O Æ2- CaO Æ3SiO2 glasses are collected in Table 2, which also shows values estimated with the assumption of a size and temperature independent specific surface energy, rcm (see also Table 1) and rcm employing the theoretical values of Io. The values of rcm calculated via Eq. (40) significantly exceed the corresponding values calculated from a fit of nucleation rate data to CNT (rcm, r  cm). According to CNT such high values of rcm lead to vanishing nucleation rates. However, nucleation processes do occur and are in- deed observed in deeply undercooled glasses! In order to find out the origin of this discrepancy, one should realize that the methods discussed above do not provide us with the surface energy directly, but instead only give its combination with the thermodynamic driving force. In particular, rcm is calculated from the measured values of Dto and U(Tn,1) via (see Eq. (40)) Dto ¼ 2 U rcmf 1 DGV   ð41Þ and rcm (as well as r  cm) from the thermodynamic barrier for nucleation W   rcm 3 DG2V : ð42Þ One should recall again that, in line with Gibbs’ thermo- dynamic description of heterogeneous systems, the thermo- dynamic driving force for crystallization of macro-crystals has been used to estimate the surface energy of critical and near-critical nuclei. Provided, this assumption is correct then we obtain correct values of the specific interfacial energy. However, if this assumption occurs to be incorrect then also the estimates of the surface energy are not cor- rect. In such case, in order to arrive at correct values of the work of critical cluster formation for nucleation, the value of the surface energy has to be chosen appropriately becoming merely a fit parameter. Hence, the above dis- cussed discrepancy may result from the difference between the macroscopic values of the thermodynamic driving force, DG1, employed and the correct driving force of critical cluster formation and growth, DGV, which is deter- mined by the real physical state of the critical and near- critical clusters. Since the identity of the driving force of critical cluster formation with the respective macroscopic 2696 V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714values is the only assumption employed in the analysis it has to be removed in order to solve the discussed in this and earlier sections discrepancies. Then we have to admit that the bulk properties of critical and near-critical clusters do not coincide with the properties of the respective macro- scopic phases and are not determined correctly employing Gibbs’ classical thermodynamic approach. As a direct con- sequence from this assumption, it follows that both surface energy and thermodynamic driving force must be consid- ered as unknown quantities. Let us analyze now the above mentioned results intro- ducing a coefficient K(r) that connects the (supposed) real thermodynamic driving force, DGV, with the respective value for the macro-phase, DG1, as DGV ¼ KðrÞDG1: ð43Þ The coefficient K(r) reflects the fact that the thermody- namic driving force for critical nuclei may differ from that of the corresponding macro-phase. If one denotes by rrcm the true value of the surface energy estimated with account of Eq. (43) and takes into consideration that U  DGV, the following equations connecting rrcm with rcm and r  cm are obtained from Eqs. (41) and (42) rrcm ¼ KðrÞ 2rcm; r r cm ¼ KðrÞ 2=3rcm: ð44Þ Eq. (44) yield K ¼ r  cm rcm  2=3 : ð45Þ Thus, both methods provide the same value of crystal/ melt surface energy if the reduced thermodynamic driving force, DGV = K(r)DG1, is employed. The values of K pre- sented in Table 2 show a considerable reduction of the ther- modynamic driving force for nucleation and growth of critical and near-critical nuclei as compared with that for the macro-crystal growth (K < 1). Employing this self-con- sistently determined value of the driving force, different estimates for the specific surface energy are obtained as compared with the case when the classical Gibbs’ approach for the determination of the driving force is used. It should be emphasized that the value of rrcm (see Table 2) is smaller than that of rcm and r  cm. Hence, in this way, the decrease of the thermodynamic driving force results in values of the interfacial energy that are significantly more reasonable (taking Stefan’s rule into account). We can conclude, con- sequently, that the discussed so far grave problems in the theoretical interpretation of crystallization can be removed if one assumes that the state of critical and near-critical clusters is different from the state of the newly evolving macro-phase. That is the classical Gibbs’ approach does not give, consequently, in general a correct description of the bulk properties of critical and near-critical clusters. Arriving at such conclusion, two classes of problems arise: First, one has to discuss whether there exist alterna- tive theoretical concepts favoring this point of view or notand whether it is possible to generalize eventually Gibbs’ approach in order to remove mentioned defect in Gibbs’ classical treatment. Second, one has to search for the phys- ical origin of such differences in the state of the critical clus- ters as compared with the respective bulk phases and for additional arguments and experimental results confirming such point of view. Such analysis will be performed in the subsequent sections. 5.4. Bulk properties of critical clusters and properties of the newly evolving macroscopic phase: some results of theoretical analyses 5.4.1. Gibbs’ theory of heterogeneous systems: basic postulates, advantages and shortcomings In the theoretical interpretation of experimental results on the dynamics of first-order phase transitions starting from metastable initial states, up to now the classical nucle- ation theory has been predominantly employed treating the respective process in terms of cluster formation and growth and employing Gibbs’ theory of capillarity. This preference is due to the advantage of Gibbs’ approach to the descrip- tion of thermodynamically heterogeneous systems allowing one to determine the parameters of the critical clusters and the work of critical cluster formation in the nucleation rate expression in a relatively simple way which is based on the knowledge of macroscopic bulk and surface properties of the ambient and newly evolving phases. In his classical analysis [4], Gibbs describes heteroge- neous systems (in application to the problems under consid- eration, we discuss a cluster of a newly evolving phase in the ambient phase) via an idealized model system. In this model, the real system is described as consisting of two homoge- neous phases divided by a mathematically sharp interface. The thermodynamic characteristics of the system are repre- sented as the sum of the contributions of both homogeneous phases and correction terms, the so-called superficial quan- tities, which are assigned to the interface. They reflect the diffuseness of the interface in the framework of Gibbs’ model approach. In contrast to alternative statements [77,78] we believe that such approach is theoretically well-founded and correct provided one is able to determine the superficial quantities in an appropriate way for any real system. In order to further develop the theoretical concept attempting to solve this task, Gibbs formulated a funda- mental equation for the superficial (or interfacial) thermo- dynamic parameters (specified by the subscript r) which is widely similar to the fundamental equation for homoge- neous bulk phases. For spherical interfaces we restrict our considerations to, it reads [4] dUr ¼ T r dSr þ X lir dnir þ rdAþ C dc; ð46Þ where U is the internal energy, S the entropy, T the temper- ature, li the chemical potential, ni the number of particles or moles of the different components (i = 1,2, . . . ,k), r the surface or interfacial tension, A the surface area, and V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714 2697c = (1/R) the curvature of the considered surface element, while C is a thermodynamic parameter determining the magnitude of changes of the internal energy with variations of the curvature of the considered surface element. R is the radius of curvature of the considered surface element. An integration of this equation results in Ur ¼ T rSr þ X lirnir þ rA: ð47Þ A combination of both equations yield the Gibbs adsorp- tion equation in the general form Sr dT r þ X nir dlir þ Adr ¼ C dc: ð48Þ In order to assign well-defined values to the superficial quantities and cluster size, as an essential requirement of Gibbs’ theory the location of the dividing surface has to be specified. In application to nucleation processes, usually the surface of tension is employed. It is defined, utilizing Gibbs’ fundamental equation for the superficial quantities, via the equation C = 0. For this particular dividing surface, the surface tension does not depend explicitly on the curva- ture. Moreover, it follows that in the classical Gibbs’ approach the surface tension depends on (k + 1) indepen- dent state variables. With Eq. (47) and the well-known expressions for the internal energy of homogeneous bulk phases, we get the following expression for the internal energy of the whole system (e.g., [79–81]) U ¼ T aSa  paV a þ X liania þ T bSb  pbV b þ X libnib þ T rSr þ X lirnir þ rA: ð49Þ Here p is the pressure, V the volume, the subscript a spec- ifies the parameters of the cluster phase, the subscript b re- fers to the parameters of the ambient phase. In application to nucleation, the state of the ambient phase is known. In this way, in order to employ Gibbs’ the- ory, the bulk state of the cluster phase has to be specified. This procedure is performed in Gibbs’ classical treatment for equilibrium states of heterogeneous substances, exclu- sively (the title of his paper, Ref. [4], is ‘On the equilibrium of heterogeneous substances’), a cluster of critical size in the ambient phase being a particular realization of a ther- modynamic equilibrium state. By employing the general conditions for thermodynamic equilibrium [4], two of the three basic sets of the equilibrium conditions are obtained T a ¼ T b ¼ T r; lia ¼ lib ¼ lir; i ¼ 1; 2; . . . ; k; ð50Þ allowing one to uniquely determine the state parameters of the cluster phase from the knowledge of the state of the ambient phase. The bulk properties of the critical clusters of the newly evolving phase are determined, consequently, in Gibbs’ approach uniquely via the equilibrium conditions Eq. (50) for temperature and chemical potentials of the differ- ent components in the two coexisting bulk phases. Hereby the question is not posed whether or not these state param- eters represent a correct description of the bulk stateparameters of the cluster. It is commonly believed that this is the case. However, Gibbs himself made a comment that, in general, the properties of the critical clusters may differ from the predictions obtained in his approach. It follows further from the Gibbs method that, for the critical clus- ters, the interfacial tension referred to the surface of ten- sion is uniquely determined by the state parameters of either the ambient or the cluster phase (cf. Eqs. (48) and (50)). Consequently, once the parameters of the ambient phase are given, the surface tension does not depend – according to Gibbs’ classical method – on the state para- meters of the cluster phase. Moreover, the superficial temperature and chemical potentials are determined by the respective parameters of the bulk phases as well. As it turns out [80–82], Gibbs’ method leads to state parameters of the critical cluster’s bulk phase which are widely identical, at least, in application to phase formation in condensed phases, to the properties of the newly evolv- ing macroscopic phases. Modifications of these properties, due to differences in the pressure of small clusters as com- pared with the equilibrium coexistence of both phases at planar interfaces, as given by the Young–Laplace equation (the third equilibrium condition), pa  pb ¼ 2r r ð51Þ is commonly of minor importance here although the pres- sure differences may be large. With the numerical estimates pb = pat  105 N/m2, r  0.1 J/m2, r*  10 9 m (at high under-cooling), we get Dp  2 · 108 Pa or 2000pat. How- ever, the effect of pressure on the density is small due to the low compressibility of the cluster bulk phase. This re- sult – the wide similarity of the properties of the critical cluster with the properties of the evolving macroscopic phases – is an essential general feature of Gibbs’ classical theory not only in application to crystallization. It leads – as discussed in detail here above – to contradictions in the interpretation of experimental results and as we will see below to contradictions with the results of computer simulations and density functional computations of the properties of critical clusters showing a quite different behavior, in particular, for higher supersaturations. So, why Gibbs’ theory can be applied at all to nucleation? The following answer can be given. In application to nucleation, not the knowledge of the properties of the critical clusters is commonly of major interest but instead the value of the work of critical cluster formation, W*. This quantity is determined in Gibbs’ description generally via W* / r 3/(pa  pb)2 [4] or in a fre- quently good approximation via W* / r 3/(DGV) 2 (cf. Eq. (4)). For any state of the ambient phase, the driving force of critical cluster formation, which can be considered to be proportional to either (pa  pb) or DGV, is determined uniquely via the equilibrium conditions Eq. (50). In this way, as far as the process proceeds via nucleation with a well-defined value of the work of critical cluster forma- tion, one can always find a value of the interfacial tension 2700 V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714consideration. Since these parameters cannot be deter- mined independently of the parameters of the coexisting bulk phases, we postulated long ago [99] that generally the conditions T b ¼ T r; lib ¼ lir; i ¼ 1; 2; . . . ; k; ð53Þ must hold. In other words, it is assumed that the superficial temperature and chemical potentials are determined widely by the properties of the ambient phase (with known prop- erties). Note that the bulk state parameters of the cluster phase may vary independently and may have so far arbitrary values. Employing such condition and the fundamental equation for the superficial quantities Eq. (46) as formu- lated by Gibbs, the interfacial tension (referred to the sur- face of tension) becomes then a function of the state parameters of the ambient phase exclusively. However, for non-equilibrium states the interfacial tension has to de- pend, in general, not only on the properties of the ambient but also on all intensive state parameters of the cluster phase. This set of intensive state parameters of the cluster phase we denote here as {uia}. In order to be able to de- scribe such additional dependence, Gibbs’ fundamental equation Eq. (46) has to be generalized resulting in (see also [79,82] for further details) dUr ¼ T r dSr þ X lir dnir þ rdAþ C dcþ X /ia duia; ð54Þ where /ia are parameters determining the magnitude of variations of the superficial internal energy with respect to variations of the bulk state of the cluster phase. Since all parameters uia of the cluster phase, entering Eq. (54), are intensive quantities, the expression for the superficial internal energy Eq. (47) and also for the thermo- dynamic potentials are formally not changed as compared with Gibbs’ original approach. In contrast, the generalized Gibbs’ adsorption equation reads now Sr dT b þ X nir dlib þ Adr ¼ C dcþ X /ia duia: ð55Þ In the generalization of Gibbs’ approach, the interfacial tension can and must be considered consequently as a func- tion both of the intensive state variables of the ambient and the cluster phases and curvature. For the surface of tension (defined also in the generalized Gibbs approach via C = 0) an explicit curvature dependence of the surface tension does not occur, again. Having at ones disposal the thermodynamic potentials for the respective non-equilibrium states, the equilibrium conditions are obtained by known procedures employed already by Gibbs in his classical model approach [4]. They differ from the equilibrium conditions derived by Gibbs and read, in general, r ¼ 2r pa  pb  X qiaðlia  libÞ  saðT a  T bÞ h i ; . ð56Þ lia  lib ¼ ð3=rÞðor=oqiaÞ; ð57Þ T a  T b ¼ ð3=rÞðor=osaÞ: ð58ÞHere p is the pressure, q the volume density of the (i = 1,2, . . . ,k) different components in the system, s is the vol- ume density of the entropy. The subscript a specifies, again, the parameters of the cluster, while b refers to the param- eters of the ambient phase. In order to determine the parameters of the critical clus- ters, one has to know the values of the surface tension (or the specific interfacial energy). In the simplest case [79,82,98,100], it can be expressed as a quadratic form in the differences of the state parameters of the ambient ({uib}) and cluster ({uia}) phases as r ¼ XX Nijðuia  uibÞðuja  ujbÞ: ð59Þ The values of the parameters Nij can be determined then from the knowledge of the specific interfacial energy for phase coexistence at planar interfaces. As it turns out, the work of critical cluster formation can be written generally again in the well-known classical form W  ¼ 13rA; ð60Þ where A* is the surface area of the critical cluster. Note however that the results for the numerical values for the work of critical cluster formation are different in both dis- cussed classical and generalized Gibbs’ approaches since the state parameters of the clusters differ in these two methods. In general, the parameters of the critical clusters as obtained via the generalized Gibbs approach differ signifi- cantly from the parameters obtained following the classical Gibbs method. However, for phase equilibrium of macro- scopic systems, the equilibrium conditions derived in the generalized Gibbs approach coincide with Gibbs’ classical expressions (here the radius of the critical clusters tends to infinity and the classical Gibbs equilibrium conditions are obtained as a special case). Note that Gibbs’ classical equilibrium conditions are retained in the above given gen- eralized equations also as a limiting case when the deriva- tives of the interfacial specific energy with respect to the intensive state parameters of the cluster phase are set equal to zero. Employing the generalized Gibbs’ approach to the determination of critical cluster properties for a variety of phase-separating systems (segregation in solutions [80], condensation and boiling in one-component fluids [81], boiling in multi-component fluids [82]) it has been shown that the predictions concerning the properties of critical clusters and the work of critical cluster formation, derived in the generalized Gibbs’ approach, are in agreement with van der Waals’ and more advanced density functional methods of determination retaining, on the other hand, the simplicity in applications similarly to the classical Gibbs method as an additional advantage. For example, in Fig. 16 the composition of the critical clusters as obtained via the generalized Gibbs approach is shown by a dashed curve (curve 1). For small supersaturations, the results of all mentioned approaches agree, however, when V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714 2701the whole range of initial supersaturations is considered and especially for large supersaturations the results of the generalized Gibbs’ approach are similar to the results obtained via square gradient density functional computa- tions and deviate significantly from the results of Gibbs’ classical approach. Such kind of behavior is essential in order to guarantee the vanishing of the surface free energy and of the work of critical cluster formation near the clas- sical spinodal curve, two features commonly considered as essential for a correct description of nucleation and which are not described by the classical approach when the capil- larity approximation is utilized [88]. It can be shown fur- ther in a general way [99] that the classical Gibbs approach employing in addition the capillarity approxima- tion as a rule overestimates the work of critical cluster for- mation and, in general, significantly. Recently the generalized Gibbs’ approach was further extended [91,101–104] to allow the description not only of nucleation but also of growth and dissolution processes taking into account changes of the bulk and surface state parameters of the clusters as a function of supersaturation and size. Hereby a criterion was advanced to allow one the quantitative determination of the changes in the bulk and surface properties of the clusters in the course of their growth. As a first application, this new theory of growth and dissolution processes was applied to the analysis of segregation in solutions. However, the method is generally applicable. In the framework of this approach, the change of a variety of thermodynamic and kinetic properties with cluster size has been determined for the first time such as the change of the surface tension, the driving force of clus- ter growth, the dependence of the effective diffusion coeffi- cients on cluster size, etc. As it turns out the respective thermodynamic and kinetic parameters may change signif- icantly in dependence on cluster size. In this way, the esti- mates of these parameters obtained from nucleation data may not be appropriate for the description of growth pro- cesses of clusters of macroscopic sizes and vice versa. This result gives a new key to the solution of the problems posed by Granasy and James [105] that growth rates computed with values of kinetic coefficients obtained from nucleation data may lead to deviations between theory and experiment reaching several orders of magnitude. Even peculiarities in the evolution of the cluster size distributions – like the development of bimodal distributions in intermediate states of the nucleation-growth process and unexpected properties – may be explained straightforwardly based on these concepts [102,104,106,107]. Thus, in a correct theo- retical treatment not only deviations of the composition of the critical nuclei from those of the respective macro- scopic phases, but also variations in the composition of the sub- and supercritical crystals have to be and can be accounted for. The extension of these concepts in application to crystal- lization is in progress. Here, in addition to changes in com- position and density also possible differences in the structure of the critical clusters (and their mutual interde-pendence with concentration fluctuations [12,88,108,109]), as compared with the state of the crystalline macro-phase, and its possible change in the course of the growth of the supercritical crystallites have to be taken into consideration (cf., e.g., [110–112]). 5.4.4. Discussion Let us first briefly summarize the results of the preceding subsection: In order to develop a consistent theoretical method of determination of the properties of the critical clusters, we have generalized Gibbs’ theory starting with the thermodynamic description of non-equilibrium states and including in this way into the theoretical schema the possibility of description of clusters of sub- and supercriti- cal sizes in the ambient phase. In order to realize such task, Gibbs’ fundamental equation for the superficial thermody- namic state parameters was generalized to allow one, in particular, an incorporation into the theory of the depen- dence of the interfacial or surface tension both on the state parameters of the ambient and the newly evolving cluster phases, respectively. Such essential additional step in the generalization of Gibbs’ classical approach was not done in earlier own work [99] and also not in the two (to the knowledge of the authors) existing alternative generaliza- tions of Gibbs’ theory to non-equilibrium states (see [113]). By this reason, in latter mentioned approaches [99,113] the equilibrium conditions retain the same form as in the classical Gibbs’ approach. Following the generalized Gibbs’ approach, it is possible to determine the properties of the critical clusters in a new way. We arrive at relations, which are, in general, different as compared with the predictions of the classical Gibbs approach. The respective results are – for model systems – in agreement with density functional computations and results of computer simulations. Moreover, since we have formulated a consistent description of clusters in thermo- dynamically non-equilibrium states, regular methods can be and are developed to determine also the properties of clusters of sub- and supercritical sizes in dependence on supersaturation and their sizes. In this way, a new tool for the description of nucleation-growth processes, in gen- eral, and crystallization processes in glass-forming liquids, in particular, has been developed allowing one to interpret a variety of experimental findings from a new point of view [91,102–104,111]. As an alternative non-classical method of theoretical treatment of crystallization going back already to van der Waals [83,84], the van der Waals and Cahn–Hilliard square gradient density functional approach is employed presently intensively for the interpretation of nucleation in crystalli- zation processes [77,87–89,114]. These studies are supple- mented by the analysis of nucleation-growth processes based on so-called phase field models, a dynamic extension of the van der Waals and Cahn–Hilliard approach [78,87,115–118], allowing one the determination of the evo- lution of the order-parameter fields with time. These types of analyses are confronted, however, with one principal 2702 V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714problem, which has to be taken into consideration – as it seems to us – more carefully in future. This problem is the prediction – in the framework of mentioned van der Waals and Cahn–Hilliard type approaches – of spinodal curves in melt-crystallization. More then three decades ago, Skripov and Baidakov [119], based on the analysis of experimental and computer simulation data – advanced the conjecture about the non- existence of a spinodal curve in one-component melt crystallization processes (or widely equivalent to them polymorphic transformations where liquid and crystal phases have the same composition). It was emphasized that this statement is in agreement with the point of view of the non-existence of a critical point in liquid–solid phase equi- libriums and of a necessarily discontinuous transition between liquid and crystal [120]. A further detailed proof of this statement in a period of about 30 years resulted in a confirmation of its validity [74,121]. An additional sup- port of such point of view can be obtained from the anal- ysis of experimental data on crystallization processes of liquids, in general, and glass-forming melts, in particular. Such analysis does not give any indication on the existence of spinodal curves in crystallization processes of the con- sidered type [12]. The latter conclusion is supported, for example, by Oxtoby [87,88] and Granasy and James [77]. However, density functional theories of crystallization predict in a variety of cases the existence of spinodal curves. Since such kind of behavior is not found by exper- iments, parameters are chosen that transfer the spinodal into parameter regions, where – due to the high viscosity – phase formation processes cannot occur [77,87,122,123]. A spinodal type behavior is also predicted in some cases by Granasy’s so-called diffuse interface theory and even close to the glass transition temperature [124]. Provided – as we believe – the conjecture of Skripov and Baidakov is correct, the prediction of a spinodal in the mentioned the- ories leads to some serious doubts into their applicability to melt crystallization, at least, in the present form. A theory cannot be correct if it predicts – not as an exception but as a rule – phenomena, which are absolutely not observed in nature. By the above discussed reasons, a further detailed analysis of the basic ideas and limitations of density func- tional approaches in application to melt crystallization seems to be absolutely essential. Completing the discussion on the limitations of the classical Gibbs approach to the description of the proper- ties of critical clusters, we would like to add a few comments on the so-called ‘nucleation theorem’ [125–128] employed frequently in order to determine the proper- ties of critical clusters based on nucleation rate data [88,94,96,129,130]. In an approximate form and for one- component systems, the content of this theorem can be formulated as [125] dW =dDl  n; ð61Þ i.e., derivatives of the work of critical cluster formation (or the steady-state nucleation rate) with respect to the stateparameters of the ambient phase allow one to determine the parameters of the critical clusters. Relations of this type – derived in the framework of Gibbs’ classical theory and employing the capillarity approximation – have been known for a long time. The increased interest in dependen- cies of such type resulted from the statements by Kashchiev [125] that the nucleation theorem is valid independent of the method employed for the thermodynamic description and valid for any kind of phase transformation and size of the critical clusters considered. However, the indepen- dence of the mentioned relation on the way of description of the clusters is questionable already on general argumen- tations. For example, Einstein noted in a conversation with Heisenberg on the foundations of quantum mechanics that it is the theory which determines what can be measured. In a detailed analysis of the results of Ref. [125] it has been shown recently in detail [127,128] that all above mentioned statements concerning Eq. (61) are not correct. In an extension of the analysis of Ref. [125], Oxtoby and Kashchiev developed similar relations in application to multi-component systems [126]. In this analysis, Gibbs’ classical theory of thermodynamically heterogeneous sys- tems was employed without introducing any additional assumptions like the capillarity approximation, i.e., the assumption that the surface tension of critical clusters is equal to the respective value for an equilibrium coexistence of both phases at planar interfaces. Consequently, the mentioned generalizations of the nucleation theorem are of the same level of validity in application to experiment as the classical Gibbs approach. They can describe the parameters of the real critical clusters correctly only as far as Gibbs’ classical method is adequate to the consid- ered particular situation. Having in mind the above dis- cussed limitations of Gibbs’ classical approach in the description of the parameters of critical clusters, men- tioned generalizations of the nucleation theorem do not supply us, in general, with a description of the real critical clusters but merely with a description of Gibbs’ model clusters resulting in the same value of the work of critical cluster formation as for the real critical clusters. Conse- quently, also the correctly derived – in the framework of the classical Gibbs’ approach – versions of the nucleation theorem do not describe, in general, the parameters of the real critical clusters. Since the generalized Gibbs approach allows one a determination of the parameters of the critical clusters, that is, for model systems, in agreement with density functional computations and computer simulation studies, it is of interest to prove whether dependencies similar to the ‘nucleation theorem’ can be formulated also in this gener- alization of the classical Gibbs approach. The respective work is in progress. Finally, we would like to note that there exist also approaches connecting the deviations of the experimental data on crystallization and growth with the effect of static disorder in the melts [131] or the existence of so-called floppy and rigid modes in glasses [132–134]. V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714 2705Deviations of the composition of the smallest crystals (50 nm) from that of the ambient glass have also been observed for surface crystallization of l-cordierite in a glass of cordierite composition. But the composition of the largest l-cordierite crystals (>1 lm) was equal to that of the parent glass [139]. Variations of the crystal composi- tions during phase transformation were also found in CaO–Al2O3–SiO2 glasses [140]. A direct experimental proof of changes of crystal composition with size in crystal- lization of Ni(P)-particles in hypoeutectic Ni–P amorphous alloys was recently reported in Refs. [106,107]. All mentioned results give a further experimental confir- mation of the thesis of a considerable variation of the prop- erties of the clusters in the course of their evolution corroborating the predictions of the generalized Gibbs’ approach. 5.6. Independent estimate of the time-lag for nucleation from nucleation and growth kinetics It was correctly claimed in Ref. [141] that another prob- lem may occur in the treatment of nucleation-growth pro- cess in glasses. For a glass with a composition close to lithium disilicate, it was shown in Ref. [141] that the induc- tion time for crystal growth, tgr, estimated (as illustrated by Fig. 22) from a R  t plot, where R is the size of the largest crystal experimentally observed, and t the time elapsed from the beginning of the nucleation-growth process, strongly exceeds the induction period for nucleation (tind ¼ 6p2 s, see Eq. (11)). Latter value was estimated from an N  t plot obtained by the ‘development’ method. How- ever, if crystal nucleation and growth rates refer to the for- mation of the same phase, tgr and tind are expected to be similar [21]. In other words, it is reasonable to assume that after an elapsed time tgr the first supercritical nuclei have0 100 200 300 400 500 600 700 800 0 10000 20000 30000 5 10 15 20 25 30 t ind t gr N , m m - 3 t , h 2R . μ m 0 2 4 6 8 10 0 10000 20000 30000 t indN , m m - 3 t , h Fig. 22. Number density of crystals, N, and size of the largest crystals, 2R, versus time of heat treatment at T = 460 C for a lithium silicate glass with 35.1 mol% Li2O. The inset shows the N  t data on a larger scale. The solid line was plotted with Eq. (10) and the dashed line is a linear fit of the 2R(t)-data [142].formed, which then deterministically grow up to sizes visi- ble under an optical microscope. The discrepancy in induction times reported in Ref. [141] has also been observed for lithium silicate glasses con- taining 32.6–38.4 mol% Li2O [142] belonging to the compo- sition range where solid solution crystals precipitate via homogeneous nucleation [143,144]. An example of N  t and R  t plots for lithium silicate glass with 35.1 mol% Li2O at T = 460 C is shown in Fig. 22, while Fig. 23 shows the time parameters tind and tgr estimated at different tem- peratures for lithium silicate glasses with 33.5 and 32.6 mol% Li2O. Since the N  t curve was obtained by the ‘development’ method (see Section 3.2), tind is overesti- mated as compared with the correct value corresponding to the nucleation temperature. (In Ref. [145] measurements of nucleation and growth rates and corresponding time-lags in lithium disilicate glass were undertaken using single- stage heat treatments at a relatively high temperature, 500 C > Tmax = 455 C. The estimated (extrapolated) nucleation time-lag was considerably higher than that obtained by the ‘development’ method. We now think that this result was probably due to insufficient stereological corrections of the crystal number density of the samples subjected to single-stage treatments; see Section 3.4.) Thus, the tgr/tind ratios experimentally obtained in the cited refer- ences are only a lower bound for the difference between the real induction periods. To correct the value of tgr to partly resolve the above discussed problem, an attempt was undertaken in Ref. [141] to account for the effect of a size dependent growth rate. However, the discrepancy between induction times independently estimated from nucleation and growth experiments remained too high. By this reason, it was suggested that initially nucleation of metastable450 500 550 600 650 1 10 100 1000 1 2 tgr t ind t in d , t g r , m in T, oC Fig. 23. Induction periods for crystal nucleation, tind, and for crystal growth, tgr, versus temperature for lithium silicate glasses with 32.6 mol% (points 2) Li2O [142] and 33.5 mol% (points 1) Li2O [141]. tind were taken from N versus t plots obtained by the ‘development method’ (they are thus overestimated, see text) while tgr were estimated from single-stage experiments at each temperature. 2706 V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714phase crystals take place, which grow more slowly than the macroscopic crystals of the stable phase. Weinberg [146] questioned the conclusions of Ref. [141] with the argument that the induction time for growth cannot be uniquely determined because it depends on the cluster size for which the measurements are performed. He also stated that the induction time for growth becomes unbounded even for measurements performed at large clus- ter sizes. Strictly speaking those arguments are correct, but since the growth rate tends to time-independent values fairly rapidly with increasing R (see, e.g., Eq. (39) or Eq. (62) and Fig. 24), the induction time also tends to a practi- cally finite value when the measurements are extended to large (optical microscopy scale) crystal sizes. Consequently, we believe that the comparison of induction times indepen- dently obtained by nucleation and growth experiments can be a useful tool, and, in principle, allows one to draw con- clusions similar to those of Ref. [141]. Nevertheless, the results and analysis of induction times for growth deserve some comments. The analysis carried out in Ref. [141] was based on the solution of macroscopic growth equations starting with an initial cluster radius equal to the critical cluster size. With such initial condition, the induction time for growth tends to infinity indepen- dently of any particular growth mechanism, since U(r*) = 0, and the numerical integration employed in Ref. [141] could not resolve this problem. In other words, the macroscopic growth equation is not valid for R = r* and cannot be employed to describe the change of the nuclei size close to the critical one. Recall that according to the Zeldovich–Frenkel equation, in the vicinity of the critical cluster size the ‘motion’ of the clusters in cluster size space is mainly governed by diffusion-like processes in clus- ter size space under the action of the concentration gradi- ent with respect to the cluster size distribution function, and thus it is not governed by the thermodynamic driving force, as it is the case in deterministic growth. In addition,20 40 60 80 100 0.0 0.4 0.8 1.2 1.6 1τ /r * d R /d t R/r* Fig. 24. Crystal growth rate (dR/dt) in (r * /s1)-units versus reduced crystal size according to Eq. (62).the discrepancy observed in Ref. [141] could also be explained in different ways, and not only via the assump- tion of formation of metastable phases. To reconsider the above mentioned problem from a dif- ferent perspective, we employed an analytical solution of the Fokker–Planck or Frenkel–Zeldovich equation describ- ing nucleation-growth process (cf. Ref. [147]). According to this analysis, for nuclei with sizes larger than two critical sizes, R > 2r*, the following relation holds: ŝ ¼ 3 5 bR þ lnðbR  1Þ þ 2 3 h i : ð62Þ In Eq. (62) the following dimensionless variables are used: bR R r ; ŝ t s1 : ð63Þ Here s1 is the period of time needed to establish a steady- state cluster size distribution in a range of cluster sizes slightly exceeding the critical size, i.e., it is practically equal to the time required to establish a steady-state nucleation rate for clusters of critical sizes. Recall that, according to Eq. (9) or (10), to practically establish a steady-state nucle- ation rate a time period about 5s is required (see Fig. 6(b)). Hence the following relation between s1 and s exists s1 ffi 5s: ð64Þ It should be emphasized that Eq. (62) was derived with the following (strong) assumptions commonly employed in CNT: (i) The bulk state of the clusters is independent on their sizes and is identical to that of the newly evolving macroscopic phase; (ii) The mechanism of cluster growth does not depend on cluster size, and growth is kinetically limited. The term ‘kinetically limited’ refers to the ballistic growth mechanism, where the growth process is only lim- ited by diffusion across the interface, and does not depend on bulk diffusion, as it is the case, for instance, for signifi- cant compositional differences between the liquid phase and growing crystal. The experimental R(t) data were fitted to Eq. (62) using s1 and r* as fit parameters [142]. Fig. 25 shows the result of such calculations. In this way, in order to arrive at the R(t)- dependence we did not use any macroscopic growth equa- tion, but relied instead on an analytical solution of the Frenkel–Zeldovich equation, which gives a correct descrip- tion of the evolution of the cluster ensemble. In addition, in our approach, we do not determine an induction time for growth, but instead determine the time-lag for nucleation by fitting experimental growth data to the nonlinear Eq. (62). Hence, even if Weinberg’s comments [146] about the impossibility of defining tgr from R(t) curves are strictly correct, they do not affect our analysis. The value of s1 exceeds the corresponding nucleation induction time, 5s, estimated from the N  t curve, by 0 2x10-6 4x10-6 6x10-6 103 10 4 10 5 10 6 τ 1 r * (4-5)τ τ t, s R, m Fig. 25. Time necessary for a crystal to achieve a size R in a lithium silicate glass with 32.6 mol% Li2O at 460 C. The full curve was plotted by Eq. (62) using s1 and r* as fit parameters. The coordinates of the open star show the fit parameters s1 and r*. The circles refer to the experimental data. The dashed horizontal line shows the value of the time-lag for nucleation s estimated by a fit of the N  t data to Eq. (10). The crosshatched band corresponds to the time when the nucleation rate achieves 95–99% of its steady-state value [142]. V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714 2707about one order of magnitude (see Fig. 25). However, pro- vided the conditions (i) and (ii) are fulfilled, one expects that s1 must be equal to about 5s, see Eq. (64), since both s1 and s refer to nucleation kinetics. This discrepancy leads to the following conclusion: at least one or both of the assumptions underlying the derivation of Eq. (62) are not valid. In order to explain the present results, one should recall the assumptions made in the derivation of the above equa- tions. In particular, one can assume that the compositions of near-critical clusters deviate from those of the macro- scopic crystals to which the crystal size measurements refer. Since in the advanced stages of crystallization the compo- sition of the macro-crystals coincides with those of the ambient melt, this assumption leads to the conclusion that growth of near-critical nuclei is limited by diffusion and is thus not kinetically determined. Moreover, as shown in the analysis of a model system [102], the size dependence of the cluster composition results in a cluster size dependence of a variety of thermodynamic and kinetic parameters (driving force, surface tension, effective diffusion coefficients, and growth rates). These deviations are not taken into account in the derivation of Eq. (62). Consequently, the mentioned deviations can be inter- preted as an additional indication that the classical approach to the description of nucleation-growth processes is insufficient for an interpretation of experimental results on crystallization in lithium disilicate glasses. One of the possible solutions is the assumption of a size (and eventu- ally structure) dependent composition of the crystallites. For completeness we should also to mention an alternativeapproach [148,149] connecting the possible deviations with possible (cluster-size dependent) solute depletion and vol- ume diffusion in nucleation. Taking into account the results of the generalized Gibbs’ approach, density functional studies and computer simulation methods of the properties of critical clusters, the first interpretation of the deviations between the time-lag established in two independent ways (being the result of the change of both cluster properties and growth kinetics in dependence on their sizes) seems to us to be a more convincing explanation. The questions under which conditions and in what way nuclei change their properties and the growth mechanism are not trivial to answer, especially if these changes occur in the early stages of crystallization. However, the transfor- mation must finally lead to the formation of a stable mac- rophase with well-defined properties. One of the possible and often assumed ways to account for such effects – the formation of metastable phases – will be discussed in Sec- tion 5.7 However, metastable phase formation is not the only possible but a very particular explanation for such kind of behavior (see, e.g., Section 5.5). The analysis of already mentioned model system (segregation in regular solutions [102]) shows that clusters may continuously change their properties with their sizes and do not have the properties of some fictive metastable phase. Such expla- nation for the observed discrepancy is more general and could be ascribed to the formation of different transient phases more or less continuously changing their properties in dependence on cluster size. 5.7. On the possible role of metastable phases in nucleation As mentioned in Sections 5.3 and 5.5 the precipitation of metastable phases in the early stages of nucleation may be one of the reasons for the deviation of the critical nuclei properties (e.g., composition) from that of the evolving (stable) macro-phase. The formation of metastable phases is consistent with the original formulation of Ostwald’s Rule of Stages according to that, ‘if the supersaturated state has been spontaneously removed then, instead of a solid phase, which under the given conditions is thermody- namically stable, a less stable phase will be formed’ [150]. Note that Ostwald restricted his formulation to the possi- ble result of the transformation not specifying the bulk state of the critical clusters as done in the generalization of this rule as given above (see Section 5.4). Implicitly it is assumed in his formulation – and also in its theoretical foundation as developed first by Stranski and Totomanov [151] – that the critical clusters have properties equivalent to the properties of one of the finite number of phases which can exist in a macroscopic form, at least, in a meta- stable state at the given conditions. Ostwald’s rule is corroborated by the following thermo- dynamic considerations. Employing the Skapski–Turnbull equation, Eq. (31), to estimate the crystal/liquid interfacial energy, one can show that the thermodynamic barrier for 2710 V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714formulation of the basic concepts of CNT. However, fol- lowing Gibbs’ ideas in the description of thermodynami- cally heterogeneous systems, in the search for the solution of this problem the properties of the critical clus- ters have been commonly identified with the properties of the newly evolving macroscopic phases. Exclusively under such assumption, the supersaturation (or driving force) can be considered – at constant pressure – as a function only of temperature. As a consequence, in most attempts to reconcile theoretical and experimental results attention was predominantly directed to the determination of the size-dependence of the specific interfacial energy. In con- trast, it follows from the present review that the main prob- lem regarding the application of CNT for a quantitative description of nucleation kinetics in glass-forming liquids consists primarily in the adequate description of the bulk properties of the critical nuclei. Of course, a deviation of the bulk properties of the critical clusters as compared with the newly evolving macroscopic phases also leads to mod- ifications of the specific interfacial energy. However, the resulting variation of the specific interfacial energy – due to changes in the bulk properties of the critical clusters as compared with the newly evolving macroscopic phase – is only a secondary factor that must be, of course, also ade- quately incorporated into the theory. Therefore, the circle of problems one has to solve for the theoretical description of nucleation is enlarged. On the other hand, a new meth- odology – the generalized Gibbs approach – that allows one to overcome the mentioned problems, which cannot be resolved following the classical concepts of Gibbs, has been recently developed. Direct experimental methods usually employed to study micron-sized or larger crystals cannot be used for nuclei of critical sizes, which are only of a few nanometers in the temperature range of interest. This is one of the reasons why one typically follows Gibbs’ description of heteroge- neous systems and assigns the thermodynamic properties (particularly the thermodynamic driving force for crystalli- zation) of the macro-phases to the critical nuclei, thus assuming that the critical nuclei and the evolving stable macro-phase can be characterized by similar bulk state parameters. However, since the thermodynamic barrier for nucleation includes both the thermodynamic driving force and the nucleus-melt surface energy, a maximum thermodynamic driving force (corresponding to the stable phase) is not a necessary condition to attain the lowest value of the thermodynamic barrier and, correspondingly, the highest value of the nucleation rate. Moreover, the thermodynamic properties of the critical nuclei can be affected by elastic stresses arising from differences between the densities of the nucleus and the melt. Hence, one can suppose that, in some cases, the deviation of the composi- tion of the nuclei from those of the stable phase may be accompanied by an approach of the nuclei density to that of the melt. In such cases, the effect of elastic stresses is reduced and, correspondingly, a decrease in the thermody- namic barrier for formation of such nuclei (as comparedwith the respective value for the stable phase) could be expected. Thus, elastic stress effects can considerably com- plicate the thermodynamics of nucleation and extend the variety of possible structures and compositions of the crit- ical nuclei. Since, with rare exceptions, direct measurements of the characteristic properties of critical nuclei are inaccessible, it is rather difficult or impossible to attribute the measured nucleation rates to defined crystal phases. It seems that such situation will not change in the near future. Moreover, taking into account density functional studies, computer simulations and theoretical analyses connected with the generalization of Ostwald’s rule of stages, it is even ques- tionable whether the critical clusters have structures and compositions resembling those of the possible macroscopic phases that may evolve in the system under consideration. As shown here, there is some remarkable evidence – partly presented in this review – for the existence of considerable differences between the properties of near-critical nuclei and those of the respective stable macroscopic phases. Glasses of stoichiometric compositions have been used as model systems in a variety of studies of crystal nucle- ation. Such choice was made hoping that it should be pos- sible to treat such systems as one-component systems. However, it now became clear that a stoichiometric glass composition, equal to the composition of the evolving crys- talline phase, does not guarantee that the nuclei have the same composition. Therefore, systematic investigations of nucleation rates versus glass compositions are of great interest allowing us to understand the true nature of nucle- ation in glasses. The great value of such analysis is rein- forced if the crystal growth rates are also measured in the same temperature range. In this way, additional informa- tion can be accumulated allowing one to reveal both the crystal nucleation and growth mechanisms operating in the systems under study. On the other hand, further development of the classical theories of nucleation and growth – aimed to describe not only critical nuclei formation, but also its subsequent growth, including the possible evolution of their composi- tion – may allow us to develop a more adequate description of phase transformation kinetics. Here we drew attention to a new approach to the description both of nucleation and growth – the generalized Gibbs’ approach – which has been developed in recent years and already demonstrated its power in the analysis of phase formation in different sys- tems. Existing different alternative theories and modifica- tions of CNT and their further developments will show which of them will be most successful in treating nucle- ation-growth phenomena in crystallization. However, in order to be successful in the description of experimental data on nucleation and growth, any of the proposed theo- ries – and this is one of the main conclusions of the present review – must be able to appropriately describe the depen- dence of the properties of the critical clusters on the state of the ambient glass-forming melt and the change of the state of the crystallites with their sizes both in dissolution and 1.0x10-20 1.5x10-20 2.6x10-20 62 64 66 68 70 72 a ln (I st t in d ΔG 2 V ,J 2 m -9 ) 1/ΔGv 2T, m6J-2K-1 600 700 800 900 1000 1100 1200 1300 1400 0 1x108 2x108 3x108 4x108 5x108 Li2O2SiO2 b T m ax = 73 3 K T m = 13 07 K Eq.6 G Eq.1A Eq.2A Eq.3A Eq.7 ΔG v, J/ m 3 T, K 1.5x10-20 2.0x10-20 2.5x10-20 4.0x10-20 69 70 71 72 73 74 75 1/ΔGv 2T, m 6J-2K-1 c ln (I st t in d ΔG 2 , J2 m -9 ) 800 900 1000 1100 1200 1300 1400 1500 1600 0 1x108 2x108 3x108 d T m ax = 87 0 K Na2O2CaO3SiO2 T m = 15 64 K ΔG v, J/ m 3 T, K 2.00E-020 4.00E-020 6.00E-020 8.00E-020 66 68 70 72 74 76 78 1/ΔGv 2T , m6J-2K-1 e ln (I st t in d ΔG 2 V ,J 2 m -9 ) 700 800 900 1000 1100 1200 1300 1400 1500 0.0 5.0x107 1.0x108 1.5x108 2.0x108 2.5x108 2Na2OCaO3SiO2 f T m ax = 77 8 K T m = 14 48 KΔG v, J/ m 3 T, K 2.00E-020 3.00E-020 4.00E-020 7.00E-020 44 46 48 50 52 54 56 g ln (I st η, Jm -6 ) 1/ΔGv 2T , m6J-2K-1 900 1000 1100 1200 1300 1400 1500 1600 1700 0,0 5,0x107 1,0x108 1,5x108 2,0x108 h T m ax = 97 3 K T m = 16 93 K Ba2O2SiO2 ΔG v, J/ m 3 T, K Fig. A1. Analysis of nucleation data with different expressions for the thermodynamic driving force. (b,d, f,h): thermodynamic driving force versus temperature; (a,c, e): lnðI sttindDG2V Þ; (g): lnðI stgÞ versus 1=DG2V T . Opened circles are plotted employing the experimental values of the thermodynamic driving force. V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714 2711 Table A1 Ratio of experimental and theoretical pre-exponential terms, and surface energy for different glasses [40] calculated by fitting nucleation data to CNT employing experimental and approximate values of the thermodynamic driving force Li2O Æ2SiO2 Na2O Æ2CaO Æ3SiO2 2Na2O ÆCaO Æ3SiO2 BaO Æ2SiO2 a rcm log I exp o I theoo   rcm log I exp o I theoo   rcm log I exp o I theoo   rcm log I exp o I theoo   Eq. (6) 0.19 15 0.17 18 0.15 27 0.13 8 Eq. (A.1) 0.20 27 0.18 30 0.16 46 0.13 14 Eq. (A.2) 0.20 45 0.19 51 0.17 79 0.13 23 Eq. (A.3) 0.25 113 0.22 156 0.14 43 Experiment 0.20 19 0.19 72 0.17 139 The specific interfacial energy is given in J m2. a Viscosity was used to calculate Iexpo and r  cm. 2712 V.M. Fokin et al. / Journal of Non-Crystalline Solids 352 (2006) 2681–2714growth processes. We believe the analysis of the size-depen- dence of the cluster properties and their theoretical inter- pretation may lead to new exciting developments in the field of crystal nucleation of glasses, with a variety of new applications. Thus, despite the fact that numerous analyses of crystallization kinetics and mechanisms of sili- cate and other glasses have been performed for decades, they are expected to remain a highly interesting subject for both fundamental and applied research on nucleation and phase transformations in general. Acknowledgements The authors thank Fapesp, Capes and CNPq (Brazil) for their financial support of this work. Appendix A The experimental values of the thermodynamic driving force for crystallization given by Eq. (5) is bounded by a linear approximation (Eq. (6)), commonly denoted as Turnbull’s formula, and by the approximation of Hoffman (Eq. (7)), see Fig. A1(b), (d), and (f). Eq. (6) directly fol- lows from Eq. (5) in the case of DCp = 0. The Hoffman equation assumes DCp = constant and some additional simplifications. There are other approximations that pre- dict values of DGV located inside the range given by Eqs. (6) and (7). Some of them, taken from Ref. [13], are DGV ¼ DHV DT T m 7T T m þ 6T   ; ðA:1Þ DGV ¼ DHV DT T m  cDSm DT  T ln T m T    ; ðA:2Þ DGV ¼ DHV DT T m 2T T m þ T : ðA:3Þ Fig. A1(b), (d), (f), and (h) shows the values of DGV versus temperature calculated with Eqs. (6), (A.1), (A.2), (A.3), and (7). The value of c in Eq. (A.2) was chosen equal to 0.8. Experimental data on DGV are also shown for Li2O Æ2- SiO2, Na2O Æ2CaO Æ3SiO2 and 2Na2O Æ1CaO Æ3SiO2 glasses. Different approximations for the thermodynamic driving force were used to plot the nucleation rates as shown inFig. A1(a), (c), (e), and (g). The intercepts and slopes of the linear fits at T > Tg were employed to estimate I exp o and rcm. These parameters are listed in Table A1. According to Table A1 the discrepancy between experi- mental and theoretical values of Io is always drastic and becomes even stronger when the DGV(T)-function becomes weaker, while rcm depends only weakly on the choice of a particular expression for the thermodynamic driving force. References [1] J.W.P. Schmelzer, in: J.W.P. Schmelzer, G. Röpke, V.B. Priezzhev (Eds.), Nucleation Theory and Applications, Joint Institute for Nuclear Research Publishing Department, Dubna, Russia, 1999, p. 1. [2] J.M.F. Navaro, El Vidrio, CSIC, Madrid, Spain, 1991. [3] W. Höland, G. Beall, Glass-ceramic technology, American Ceramic Society, 2002. [4] J.W. Gibbs, The Collected Works, Thermodynamics, vol. 1, Longmans & Green, New York, 1928. [5] R. Kaischew, I.N. Stranski, Z. Phys. Chem. B 26 (1934) 317. [6] M. Volmer, A. Weber, Z. Phys. Chem. 119 (1926) 277. [7] R. Becker, W. Döring, Ann. Phys. 24 (1935) 719; R. Becker, Ann. Phys. 32 (1938) 128. [8] M. Volmer, Kinetik der Phasenbildung, Steinkopf, Dresden, 1939. [9] J. Frenkel, Kinetic Theory of Liquids, Oxford University Press, Oxford, 1946. [10] D. Turnbull, J.C. Fisher, J. Chem. Phys. 17 (1949) 71. [11] H. Reiss, J. Chem. Phys. 18 (1950) 840. [12] I. Gutzow, J. Schmelzer, The Vitreous State: Thermodynamics, Structure, Rheology and Crystallization, Springer, Berlin, 1995. [13] K.F. Kelton, Solid State Phys. 45 (1991) 75. [14] J.W. Christian, The Theory of Transformations in Metals and Alloys. Part I, Pergamon Press, Oxford, 1981. [15] F.C. Collins, Z. Electrochem. 59 (1955) 404. [16] D. Kashchiev, Surf. Sci. 14 (1969) 209. [17] V.N. Filipovich, V.M. Fokin, N.S. Yuritsyn, A.M. Kalinina, Thermochim. Acta 280&281 (1996) 205. [18] G. Tammann, Z. Phys. Chem. 25 (1898) 441. [19] M. Ito, T. Sakaino, T. Moriya, Bull. Tokyo Inst. Technol. 88 (1968) 127. [20] V.N. Filipovich, A.M. Kalinina, Izv. Akad. Nauk USSR, Neorgan. Mat. 4 (1968) 1532 (in Russian). [21] I. Gutzow, Contemp. Phys. 21 (1980) 121, 243. [22] S. Toschev, I. Gutzow, Phys. Status Solidi 24 (1967) 349. [23] U. Köster, Mater. Sci. Eng. 97 (1988) 183. [24] V.M. Fokin, N.S. Yuritsyn, V.N. Filipovich, A.M. Kalinina, J. Non- Cryst. Solids 219 (1997) 37. [25] E.D. Zanotto, P.F. James, J. Non-Cryst. Solids 124 (1990) 86.
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