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Review
Homogeneous crystal nucleation in silicate glasses:
A 40 years perspective
Vladimir M. Fokin
a,*
, Edgar D. Zanotto
b
, Nikolay S. Yuritsyn
c
Ju¨ rn 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 Sa˜o Carlos, 13565-905 Sa˜o Carlos, SP, Brazil c (^) Grebenshchikov Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24-2, 199155 St. Petersburg, Russia d (^) Institut fu¨ r Physik, Universita¨ t Rostock, 18051 Rostock, Germany
Received 18 June 2005; received in revised form 27 January 2006 Available online 24 May 2006
Dedicated to the memory of Peter F. James
Abstract
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. Ó 2006 Elsevier B.V. All rights reserved.
Keywords: Crystallization; Glass ceramics; Nucleation; Crystals; Glass transition; Oxide glasses; Silicates; Thermodynamics
- 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)
0022-3093/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.02.
- (^) Corresponding author. Address: ul. Nalichnay 21, ap.7, 199406 St. Petersburg, Russia. Tel.: +7 812 355 30 38. E-mail address: [email protected] (V.M. Fokin).
www.elsevier.com/locate/jnoncrysol
Journal of Non-Crystalline Solids 352 (2006) 2681–
method of vitrification consists in supercooling a liquid escaping crystallization. Thus, when a liquid is cooled down at sufficiently high rates, crystallization can occur to a lim- ited degree or can be completely arrested down to tempera- tures corresponding to very high viscosities, in the range g P 10 13 –10 12 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.
- 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
study. According to CNT (see, e.g., Refs. [12,13]), the steady-state homogeneous volume nucleation rate can be written as
Ist ¼ Io exp W (^) þ DG (^) D kBT
Io ¼ 2 N (^1)
kBT h
a^2 rcm kBT
ð 1 Þ
This equation determines the so-called steady-state nucle- ation rate, Ist , i.e., the number of supercritical clusters formed per unit time in a unit volume of the system. The pre-exponential term, Io , depends only weakly on tempera- ture (if compared to the exponential function) and varies between 10 41 and 10 43 m^3 s^1 for different condensed sys- tems [14]. In Eq. (1) kB and h are the Boltzmann and Planck constants, respectively; N 1 1/a^3 is the number of structural (formula) units, with a mean size a, per unit vol- ume of melt; rcm is specific surface free energy of the crit- ical nucleus-melt interface; DGD is the activation free energy for transfer of a ‘structural unit’ from the melt to a nucleus (kinetic barrier). To a first approximation, the ki- netic barrier for glass-forming liquids is often replaced by the activation free energy for viscous flow, DGg. W* is the thermodynamic barrier for nucleation, i.e., the increase in the free energy of a system due to the formation of a nu- cleus with critical size, r*. The critical nucleus size can be determined from the condition
oW or
¼ 0 ; W ¼ c 1 r^2 rcm c 2 r^3 DG (^) V ; ð 2 Þ
where DGV = Gl Gc is the difference between the free energies of liquid and crystal per unit volume of the crystal (i.e., the thermodynamic driving force for crystallization) and c 1 and c 2 are shape factors. In the case of a spherical nucleus, we obtain the expressions
r ¼
2 rcm DG (^) V
ð 3 Þ
and
W ¼
16 p 3
r^3 cm DG^2 V
: ð 4 Þ
The thermodynamic driving force for crystallization is given by
DG (^) V V (^) m ¼
DHm T (^) m ðT (^) m T Þ
Z (^) T (^) m
T
DC (^) p dT 0 þ T
Z (^) T (^) m
T
DC (^) p T 0 dT 0 ;
ð 5 Þ
where Vm is the molar volume, DHm and Tm are the molar heat of melting and the melting temperature of the crystal, respectively, and DC (^) p ¼ Cl p Cc p is the difference between the molar heat capacities of liquid and crystal at constant pressure. The experimental values of DGV are normally bounded by the approximations usually assigned to Turn- bull (Eq. (6)) and Hoffman (Eq. (7)) that assume DC (^) p = 0 and DC (^) p = constant, respectively [13],
DG (^) V ðT Þ ¼ DH (^) V 1
T
T (^) m
; ð 6 Þ
DG (^) V ðT Þ ¼ DH (^) V 1
T
T (^) m
T
T (^) m
: ð 7 Þ
Here DH (^) V is the melting enthalpy per unit volume of the crystal. One should note, however, that Eq. (6) was first employed by Thomson and Volmer (cf. Ref. [8]). Eq. (1) describes the time-independent steady-state nucleation. Such nucleation regime occurs if a stationary size distribution of the newly evolving subcritical (r < r* ) and critical (r = r* ) nuclei is established in the system. The cooling rates typically employed for glass formation from the melt, and the heating rates of small glass speci- mens to any given temperature T under investigation are commonly too high to maintain a steady-state distribution of nuclei in the system. Hence, some time period is needed for a reconstruction of the initial nuclei distribution towards the time-independent distribution corresponding to the temperature of study. During this period the nucle- ation rate varies and approaches a steady-state value given by Eq. (1). The time required to establish steady-state nucleation in a system is commonly denoted as the time-lag for nucle- ation, s. It characterizes the duration for the onset of the steady-state distribution, and hence the evolution of the nucleation rate, I(t), towards a steady-state value, Ist. In the cases when the initial concentration of critical and sub-critical nuclei may be neglected, s and I(t) can be expressed by Eqs. (8) and (9), respectively [15,16],
s ¼ 16 h p
rcm DG^2 V a^4
exp
DGD
kBT
; ð 8 Þ
IðtÞ ¼ Ist 1 þ 2
X^1
m¼ 1
ð 1 Þm^ exp m^2 t s
: ð 9 Þ
Integration of Eq. (9) results in the following expression for the time-dependence of the number of super-critical nuclei per unit volume of the system, NV,
N (^) VðtÞ Ists
t s
p^2 6
X^1
m¼ 1
ð 1 Þm m^2
exp m^2
t s
: ð 10 Þ
For sufficiently long times, t, as compared with s this expression can be approximated by
N (^) VðtÞ ¼ Ist t
p^2 6
s
: ð 11 Þ
For the experimental estimation of s, it is convenient to use the induction period, tind, defined via Eq. (12) as
s ¼
p^2
tind: ð 12 Þ
The induction period, tind , is easily determined as the inter- section of the asymptote (Eq. (11)) with the time-axis. An- other more correct way to estimate Ist and s is by fitting the experimental values of N(t) to Eq. (10).
2.3. 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 h þ 14 cos 3 h: ð 13 Þ
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, N 1 , 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 N S^
kBT h exp
W (^) U þ DGD kBT
: ð 14 Þ
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.
- 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 estimation
of 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 Ko¨ ster’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
process 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
Iðt^0 Þ
Z (^) t
t^0
Uðt^00 Þ dt^00
dt^0
; ð 19 Þ
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 3 t^4 4
: ð 20 Þ
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 exp½gN (^) oU 3 t^3 : ð 21 Þ
Avrami proposed that, in general, the following relation should be used:
aðtÞ ¼ 1 expðKt nÞ: ð 22 Þ
In typical applications, Eq. (22) is employed in the form
lnð lnð 1 aÞÞ ¼ ln K þ n ln t: ð 23 Þ
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 3 m; ð 24 Þ
where k and m are taken from the formulas N tk^ and r t m^ 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 2Na 2 O Æ CaO Æ 3SiO 2 [32] and Na 2 O Æ 2CaO Æ 3SiO 2 [33]. Good agreement was obtained between the values of gIU^3 (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].
- 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 Td
0 20 40 60 80 100
0
1x10^13
2x10^13
3x10^13
4x10^13
0 20 40 60 80 100
0
2x10^16
4x10^16
6x10^16
8x10^16
a
Li 2 O.2SiO 2 T (^) n=703 K Td =899 K
t (^) o t (^) ind
N, m
t, h t, h
2Na 2 O.CaO.3SiO 2 Tn =723 K Td =856 K
t (^) ind b
Fig. 3. Typical curves of the number density of Li 2 O Æ 2SiO 2 (a) and 2Na 2 O Æ CaO Æ 3SiO 2 (b) crystals in glasses of respective stoichiometric compositions versus time of nucleation obtained by the ‘development’ method [35,36].
(see Figs. 4 and 5). The total number of supercritical crys- tals, N, nucleated at a temperature, Tn , in a time, t, is given by
N ðT (^) n; r ðT (^) nÞ; tÞ ¼
Z (^) t
0
IðT (^) n; t^0 Þ dt^0 : ð 25 Þ
The number of crystals nucleated in the same conditions, but having sizes larger than the critical size, r* (Td ), and which are, consequently, capable to grow at Td , is given by
N ðT (^) n; r ðT (^) dÞ; tÞ ¼
Z (^) tt 0
0
IðT (^) n; t^0 Þ dt^0 ; ð 26 Þ
where to is the period of time that critical nuclei of size r* (Tn ) need in their growth to reach the size r* (Td ). This time interval is determined by
toðT (^) n; T (^) dÞ ¼
Z (^) r ðT (^) dÞ
r ðT (^) nÞ
dr UðT (^) n; rÞ
: ð 27 Þ
Eqs. (25) and (26) yield N ðT (^) n; r ðT (^) nÞ; tÞ ¼ N ðT (^) n; r ðT (^) dÞ; t þ toÞ: ð 28 Þ Hence, N(Tn , r* (Tn ), t) plots are similar to N(Tn , r* (Td ), t)- plots with the difference that the latter is shifted along the time-axis by a time to. Thus, the development method can provide the correct value of the steady-state nucleation rate, but overestimates the induction time for nucleation by to. The period during which heat treatment at the nucle- ation temperature Tn does not influence crystallization at Td can be identified with to (given by Eq. (27); here we neglect the time of the first critical nucleus formation). This time is indicated by an arrow in Fig. 3(a). According to Eq. (27), the higher the growth rate U at the nucleation temper- ature, Tn , and the closer is Td to Tn (r* (Tn ) is correspond- ingly closer to r* (Td )), the lower is to. Hence, for a strong overlap of the nucleation and growth rate curves, the value of to is not very high and can often be neglected. Fig. 3(b) confirms this assumption for a 2Na 2 O Æ CaO Æ 3SiO 2 glass. On the other hand, when the overlap of the nucleation and growth rate curves is weak, as observed for lithium disilicate glass, one has to reduce the measured value of tind (Tn , Td ) by a time to (Tn , Td ) (see Fig. 3(a)) to estimate tind (Tn ). The value of tind(Tn ) can be roughly estimated via extrapolation of the tind(Tn , Td )-values for the N(Tn , Td , t)-curves, obtained at different Td , to tind corre- sponding to Td = Tn. Fig. 5(a) presents examples of such N(Tn , Td , t)-curves for lithium disilicate glass. Fig. 5(b) shows the values of tind, taken from these curves, versus development temperature. When Td approaches Tn = 453 °C, tind is about 1.9 h (the average value of the linear and quadratic polynomial extrapolations). Hence, one can approximately estimate to as to (Td , Tn ) = tind(Tn , Td ) tind(Tn ), e.g., for Td = 530 °C and Tn = 453 °C to is about 0.9 h. A similar value is obtained by extrapolating the initial section of the N(t)-curve 1 (see also curve 5) to N = 0. Thus, according to Eq. (12), one can assume that Eq. (29) holds, i.e.,
sðT (^) nÞ ¼
p^2
ðt (^) indðT (^) n; T (^) dÞ toðT (^) n; T (^) dÞÞ: ð 29 Þ
Kinetic N(t)-curves, such as those presented by Fig. 3, can be plotted in dimensionless coordinates (N(T, t to )/ Ist (T)s(T) versus (t to)/s(T)). Fig. 6(a) shows that these coordinates allow one the combination of data for different glasses and different temperatures in the same plot. The experimental points are quite close to the theoretical master curve calculated with Eq. (10). This curve corresponds to increasingly higher nucleation rates towards the steady- state value, Ist. The evolution of the nucleation rate calcu- lated by Eq. (9) is shown in Fig. 6(b). As we already mentioned in Section 3.3, if one employs the single-stage method, the induction periods obtained from experimental N(Tn , t)-curves must be reduced by a period of time to rres/U(Tn ). An example of such curve, obtained for Au-catalyzed nucleation in NaPO 3 glass, is
900 950 1000 1050 1100
0
200
400
600
N, mm
Td , oC
Fig. 4. Number density of crystals versus development temperature in a lithium aluminum silicate glass subjected to nucleation treatment for 5 min at Tn = 785 °C [37].
(^0480 520 560 600 )
1
2
3
0 1 2 3 4 5 6 7 8
5.0x10^3
1.0x10^4
1.5x10^4
2.0x10^4
b
1.9 h
Tn=453oC
tind
, h
Td, oC
2.0x10 3
tind
a
1.0x10 3
N, mm
- N, mm -
t, h
1 2 3 4 5
Fig. 5. (a) Number density of Li 2 O Æ 2SiO 2 crystals developed at Td = 530 °C (curves 1 and 5), 560 °C (curve 2), 594 °C (curve 3), and 626 °C (curve 4) as a function of nucleation time at Tn = 453 °C [38]. (b) Induction time versus development temperature.
obtained 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 T 2. 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 T 1 for sizes less than r* (Td ) into the steady state cluster size distribution corresponding to T 2. 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 T 1 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 T 1 = 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 · 10 13 to 3 · 10 2 m^3 s^1 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 50 100 150 200 250
5.0x10 4
1.0x10 5
1.5x10 5
2.0x10 5
2.5x10 5 a^ 3
2
1
N, mm
t, min
0.0 0 50 100 150 200
5.0x10^2
1.0x10^3
1.5x10^3
2.0x10^3 b
I (^) st
I, mm
-3min
t, min
Fig. 9. (a) Number density of Li 2 O Æ 2SiO 2 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
- 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
2.0x10^4
4.0x10^4
6.0x10^4
5
4
3
2 1
N, mm
t, min Fig. 10. Number density of Li 2 O Æ 2SiO 2 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.
101
10 5
10 9
1013
(^4 )
2
1
Ist , m
-3^ s
T/Tm
Fig. 11. Steady-state nucleation rate versus reduced temperature for some stoichiometric glasses: (curve 1) 3MgO Æ Al 2 O 3 Æ 3SiO 2 [47]; (curve 2) Li 2 O Æ 2SiO 2 [35]; (curve 3) Na 2 O Æ 2CaO Æ 3SiO 2 [48]; (curve 4) 2Na 2 O Æ CaO Æ 3SiO 2 [36].
effects: 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 C 1
T (^) rð 1 T (^) rÞ^2
; C 1 ¼
16 p 3
a^3 STDHm RT (^) m ; T (^) r
T
T (^) m
ð 30 Þ
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) m= 3 N (^1) A=^3
ð 31 Þ
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
C 2
T (^) r T (^) or
; C 2
2 : 30 B
T (^) m
ffi 30 ðT (^) gr T (^) orÞ;
T (^) or T (^) o T (^) m
; T (^) gr T (^) g T (^) m
; ð 32 Þ
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 C 2 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 C 1 and C 2 , as indicated in the figure caption. One can see that the decrease in the kinetic barrier, caused by a decrease in C 2 at a fixed value of C 1 , 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 C 2 , by decreasing the parameter C 1 (which is proportional to aST and the reduced melting enthalpy DH r m ¼ DH (^) m=RT (^) mÞ, the value
of 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 H 2 O 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
0.2 0.4 0.
5
10
15
20
25
8
7
6
5
4 (^3 )
1
1/
log(I
, mst
-3s
-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 m^3 s^1 and following values of the parameters character- izing the temperature independent parts of the thermodynamic (C 1 ) and kinetic (C 2 ) barriers: C 1 = 6.5 (curves 1–5), 5.8 (curve 6), 5.1 (curve 7), 4. (curve 8); C 2 = 6 (curves 1 and 6–8), 4.8 (curve 2), 3.9 (curve 3), 2.8 (curve 4), 0 (curve 5).
performed by Turnbull in the 1960s (see, e.g., Ref. [60]). However, at that time, with the exception of the measure- ments of Tammann [61] and Mikhnevich [62] for organic liquids, nucleation rate data were not available in wide temperature ranges including Tmax. In order to verify the existence of a correlation between Imax and Tgr , as pro- posed here, an abundance of experimental points must be available. This is now the case (cf. Fig. 14).
- Nucleation rate data and CNT: some serious problems
5.1. Different approaches for the interpretation of experimental data by CNT
As shown in the previous sections, in its original form CNT provides a good qualitative description of nucleation rate data for silicate glasses, however, serious problems arise when one tries to employ this theory for a quantita- tive interpretation of experimental data. If one uses the Stokes–Einstein equation to connect the kinetic barrier of nucleation with the glass viscosity one can rewrite Eq. (1) for the steady-state nucleation rate as
Ist ¼ Kg
g exp
W
kBT
; Kg ¼ Io
h 4 l^3
; ð 33 Þ
where the size parameter l has the order of the Si–O bond length. Hereby, the diffusivity across the crystal/liquid interface is replaced by the volume diffusivity. The use of the Stokes–Einstein equation in Eq. (33) can be avoided if one estimates the kinetic barrier from the nucleation time-lag. In this case, Eq. (1) takes the following form:
Ist ¼ Ks
DG^2 V tind
exp
W
kBT
; Ks ¼ Io
8 hrcm 3 a^4 : ð 34 Þ
In the analysis of crystallization kinetics in glass-form- ing systems, it is commonly accepted – in accordance with CNT and Gibbs’ classical description of heterogeneous sys- tems – to use the properties of the newly evolving macro- phase as reference states for the description of the bulk properties of the critical nucleus. Additionally one has to properly specify the value of the specific interfacial energy, rcm. Since measurements of the interfacial energy of the crystals in their own melt are confronted with serious difficulties, one usually employs the easily measurable ther- modynamic driving force for crystallization of the macro- phase for the determination of the work of critical cluster formation. Hereby, rcm is commonly taken as a fit param- eter and is treated, to a first approximation, as a size-inde- pendent (capillarity approximation) and temperature independent quantity. The respective values of rcm are de- noted in Tables 1 and 2 as r cm. These approximations allow one to estimate both the magnitude of the pre-exponential term, Io , in Eq. (1) and the value of crystal-melt surface en- ergy, rcm , from a fit of experimental data (Ist , g, or tind ). According to Eqs. (33), (34), and (4), ln(Istg) and
lnðIsttindDG^2 V Þ versus 1=ðT DG^2 V Þ plots should yield straight lines. Their intercepts and slopes can be employed to eval- uate Io and r cm , respectively. However, these approxima- tions lead to the following problems:
(i) The use of Eq. (33) [55,63] and Eq. (34) [64] leads to drastic discrepancies between the experimental, Iexpo , and theoretical, Itheoo , values of the pre-exponential factor. This discrepancy was first observed for crystal nucleation in undercooled Ga [65] and Hg [66]. In order to illustrate this issue, Table 1 shows the ðIexpo =Itheoo Þ-ratio, and surface energy values for some stoichiometric silicate glasses calculated from lnðIsttindDG^2 V Þ versus 1=ðT DG^2 V Þ plots for temperatures above the glass transition range. To trace these plots, both the linear (Turnbull) approximation (Eq. (6)) and the experimental values (Eq. (5)) of the thermo- dynamic driving force for crystallization of the stable macro-phases were used. The discrepancy between theory and experiment is strongly affected by the choice of DGV (see also Appendix A, where an anal- ysis similar to that given in Ref. [13] is performed). The experimental values of DGV are close to Turn- bull’s approximation in the case of Li 2 O Æ 2SiO 2 glass, and to Hoffman’s approximation in the case of 2Na 2 O Æ 1CaO Æ 3SiO 2 glass. These equations normally bound the experimental values of DGV [13], and the ðIexpo =Itheoo Þ-ratio increases as one passes from Turn- bull’s to Hoffman’s approximation. However, inde- pendently of the particular choice of the expression of the thermodynamic driving force, i.e., with any reasonable approximation or with experimental val- ues of DGV, the mentioned discrepancy remains quite large.
Table 1 Ratio of experimental and theoretical pre-exponential, and surface energy values calculated by CNT for different glasses [40] Glass DCp = 0 DC (^) p = f(T) logðIexpo =Itheoo Þ r cm logðIexpo =Itheoo Þ r cm Li 2 O Æ 2SiO 2 15 0.19 19 0. Na 2 O Æ 2CaO Æ 3SiO 2 18 0.17 72 0. 2Na 2 O Æ CaO Æ 3SiO 2 27 0.15 139 0. The specific interfacial energy is given in J m^2.
Table 2 Liquid–crystal surface energies (in J m^2 ) calculated from nucleation and growth data [69] Glass rcm r cm r cm K rr cm Li 2 O Æ 2SiO 2 1.4 0.20 0.152–0. (450 °C < T < 485 °C)
0.19–0.23 0.050–0.
Na 2 O Æ 2CaO Æ 3SiO 2
1.5 0.18 0.099–0. (580 °C < T < 685 °C)
0.13 0.
(ii) The values of the surface energy, rcm , calculated as described above (in the deeply undercooled regime close to Tg ), are lower than the melt–vapor surface energy, rmv, which can be measured directly [67,68] (above the equilibrium melting point) by a factor of about 0.5–0.6. These values must then be corrected since rcm refers to nuclei of critical size, r* , while rmv refers to planar melt/vapor interfaces. In the case of lithium disilicate glass, for instance, corrections made with the Tolman equation, Eq. (35), for the size effect, increase this factor to 0.8 [69]. Such high values of rcm , as compared with rmv, strongly overestimate its real magnitude. Indeed, according to Stefan’s rule [70], one would expect the ratio rcm /rmv to be approximately equal to rcm /rmv ffi DHcm /DHmv 1, where DHcm DHm and DHmv are the melting enthalpy of the crystalline phase and enthalpy of evaporation, respectively.
It follows that the widespread believe – the driving force of critical cluster formation can be determined correctly via the classical Gibbs’ approach and all necessary corrections have to be incorporated into the theoretical description via the introduction of appropriate values of the specific inter- facial energy – is challenged by above given analysis and has to be reconsidered. In the following sections, possible reasons for the failure of CNT in application to a quantita- tive description of nucleation experiments will be analyzed in detail.
5.2. Temperature and size-dependence of the nucleus/liquid specific surface energy
The discrepancy between experimental and theoretical values of Io can be avoided if one calculates rcm from nucleation data (Ist and tind or g) employing the theoretical expression for Io. This procedure slightly decreases the val- ues of rcm and leads to a weak increase of rcm with increas- ing temperature [71] (dr/dT (0.06–0.16) · 10 ^3 J/m 2 K) regardless of the way of estimating the kinetic barrier. As far as we know, Turnbull [66] was the first to draw atten- tion to this fact. At a first sight such kind of temperature dependence of rcm (i.e., an increase of the surface tension with increasing temperature obtained via the mentioned treatment of nucleation experiments) is in conflict with the theoretical expectations of most, but not all, authors (see the discussion below). Commonly the opinion is favored that, from a thermodynamic point of view, a decrease of rcm (for planar interfaces (r 1 )) with tempera- ture should be expected [72–74], at least, in the temperature range where crystallization processes may occur [74]. It fol- lows that we are confronted here with a contradiction between the discussed interpretation of experimental results and general theoretical expectations. As will be shown now this contradiction can be partly removed by taking into account a possible curvature (or nucleus size) dependence of the surface energy. Recall that
the specific surface energy estimated from nucleation rate data refers to nuclei of critical size. Curvature corrections are expected to reduce the effective value of the surface energy. When the critical nucleus size increases with increasing temperature, the effect of curvature corrections decreases (see Eq. (35)), leading to higher effective values of the surface energy. To a first approximation, Tolman’s equation (that was originally derived for a liquid drop in equilibrium with its vapor) can be used to decouple these size and temperature effects. The Tolman equation reads
rðr Þ ¼
r 1 1 þ (^2) rd
(^) ; ð 35 Þ
where the Tolman parameter d is a measure of the (un- known) width of the interfacial region between the coexis- ting phases. Employing this relation, the work of formation of a spherical critical nucleus may be written as
W (^) ¼
16 p 3
r^31
1 þ (^2) rd
DG^2 V
; ð 36 Þ
where
r ¼ 2 r 1 DGV
2 d ð 37 Þ
holds. Fig. 15 shows the average values of (dr 1 /dT) at T P Tg versus the Tolman parameter. Using experimental nucle- ation data for Li 2 O Æ 2SiO 2 glass, fits of r 1 have been per- formed for different values of d employing Eq. (34). For this glass, as d increases (dr 1 /dT) progressively decreases and becomes negative for d > 2.4 · 10 ^10 m. Thus, reason- able values of the Tolman parameter may be chosen such that r 1 decreases with increasing temperature, in line with
0 1x10 -10^ 2x10-10 3x10-10^ 4x10-
-1x10-
0
1x10-
2x10-
∞ dσ
/dT, J/m
2 K
δ, m Fig. 15. (dr 1 /dT) versus Tolman’s parameter for Li 2 O Æ 2SiO 2 crystals in a glass of the same composition. The kinetic barrier for nucleation was estimated from the nucleation time-lag.
values 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, DG (^) V, with the respective value for the macro-phase, DG 1 , as
DG (^) V ¼ KðrÞDG 1 : ð 43 Þ
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 rr cm the true value of the surface energy estimated with account of Eq. (43) and takes into consideration that U DGV, the following equations connecting rr cm with rcm and r cm are obtained from Eqs. (41) and (42)
rr cm ¼ KðrÞ^2 rcm; rr cm ¼ KðrÞ^2 =^3 r cm: ð 44 Þ
Eq. (44) yield
K ¼
r cm rcm
: ð 45 Þ
Thus, both methods provide the same value of crystal/ melt surface energy if the reduced thermodynamic driving force, DG (^) V = K(r)DG 1 , 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 rr cm (see Table 2) is smaller than that of r cm 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 not
and 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 dn (^) ir þ r dA þ C dc; ð 46 Þ
where U is the internal energy, S the entropy, T the temper- ature, li the chemical potential, n (^) i 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
c = (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
U (^) r ¼ T (^) rSr þ
X
lirn (^) ir þ rA: ð 47 Þ
A combination of both equations yield the Gibbs adsorp- tion equation in the general form
Sr dT (^) r þ
X
n (^) ir dlir þ A dr ¼ C dc: ð 48 Þ
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
lian (^) ia þ T (^) bSb pbV (^) b
þ
X
libn (^) ib þ T (^) rSr þ
X
lirn (^) ir þ rA: ð 49 Þ
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; ð 50 Þ
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 state
parameters 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
2 r r ð 51 Þ
is commonly of minor importance here although the pres- sure differences may be large. With the numerical estimates pb = pat 10 5 N/m 2 , r 0.1 J/m 2 , r* 10 ^9 m (at high under-cooling), we get Dp 2 · 10 8 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 /(DG (^) V) 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
i.e., for metastable initial states of the ambient phase. The results of the classical Gibbs approach to the determination of the properties of the critical clusters are given by a dot- ted curve (curve 3). It is evident that the composition of the critical clusters – determined in such a way – practically does not depend on supersaturation and is widely equal to the value in the newly evolving macroscopic phase. The full curve (curve 2) shows the results for the cluster composition in the center of the critical cluster as obtained via the van der Waals and Cahn–Hilliard square gradient approximation as described by Eq. (52). It is evident that both approaches lead, in general, to very different results. Qualitatively similar results are obtained when the van der Waals and Cahn–Hilliard approach or more advanced density functional computations are applied to the descrip- tion of crystallization [77,89,90], i.e., the state of the critical cluster differs, in general, from the state of the newly evolv- ing macroscopic phase. Both Gibbs’ and the van der Waals or more advanced density functional methods of description of thermody- namically heterogeneous systems are considered commonly as well-established theories. Nevertheless, only one of them (if any) can be correct in the prediction of the properties of the critical clusters. Moreover, the Gibbs and van der Waals approaches lead to contradicting each other results in the description of the behavior of phase separating sys- tems in the vicinity of the classical spinodal curve (cf. Ref. [91]). In this way, one is confronted here with internal con- tradictions in two well-established theories, which must be, hopefully, resolved. The question which of both mentioned theories describes more correctly the properties of the critical clus-
ters can be answered from a theoretical point of view based on the analysis of the results of computer simulation meth- ods of phase formation processes in model systems [92–96]. The respective analyses show that critical clusters do have properties, in general, significantly different from the prop- erties of the newly evolving macroscopic phases (although in some particular cases also results are obtained which are in agreement with the classical Gibbs approach). In this way, computer simulation methods support, in general, the van der Waals or alternative density functional approaches for the description of heterogeneous systems. Consequently, we can conclude that the majority of exist- ing theoretical approaches for the determination of the work of critical cluster formation gives strong support to the point of view that the state of the critical clusters may significantly differ from the state of the newly evolving macroscopic phases. Consequently, in order to obtain correct expressions for the work of critical cluster formation in the interpreta- tion of experimental results one has to account for a cluster size dependence not only of the surface properties of the crit- ical clusters but also of their bulk properties.
5.4.3. A generalization of Gibbs’ classical theory Having reached such conclusion, immediately the ques- tion arises whether it is necessary to abandon the classical Gibbs approach at all or whether it is possible to modify it in such a way that it retains its advantages (use of macro- scopic properties of the phases of interest for a determination of the work of critical cluster formation) but overcomes its shortcomings (incorrect determination of the bulk proper- ties of the critical clusters) discussed above. As it turns out such generalization of Gibbs’ thermodynamic theory can be really performed. It was initiated several years ago based initially on a generalization of Ostwald’s rule of stages in application to nucleation. This generalization of Ostwald’s rule was formulated as follows [97]: ‘Those classes of critical clusters determine the process of the transformation, which correspond to a minimum work of critical cluster formation (as compared with all other possible alternative structures and compositions, which may be formed at the given thermo- dynamic constraints)’. This concept was then employed in order to develop a new approach for the determination of the work of critical cluster formation and the determination of critical cluster properties based on a generalization of Gibbs’ classical approach [79,98]. In such a generalization of Gibbs’ theory, we followed again Gibbs’ method of dividing surfaces but started with the analysis of the question how to formulate a thermody- namic description of heterogeneous systems (clusters or ensembles of clusters in the otherwise homogeneous ambi- ent phase) for well-defined non-equilibrium states, when both the clusters and the ambient phase are in an internal thermodynamic equilibrium but the system as a whole is not. Having a look at Eq. (49), immediately the question arises then, how to determine the values of the superficial temperature and chemical potentials for any well-defined non-equilibrium states of the heterogeneous systems under
Fig. 16. Composition of the critical cluster, xa, as a function of the supersaturation for segregation processes in solutions [80]. The molar fraction, x, of the segregating component in the ambient solution can be considered as a measure of supersaturation, which varies in the range between the binodal (xb) and spinodal (xsp ) curves. The dotted curve (curve 3) refers to results of computations of the critical cluster parameters obtained via Gibbs’ classical method; the dashed curve (curve 1) to the newly developed generalized Gibbs approach and the full curve (curve 2) to results of density functional calculations of the density in the center of the critical cluster obtained via the van der Waals square gradient method.
consideration. 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; ð 53 Þ
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)
dU (^) r ¼ T (^) r dSr þ
X
lir dn (^) ir þ r dA þ C dc þ
X
/ia duia; ð 54 Þ
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
n (^) ir dlib þ A dr ¼ C dc þ
X
/ia duia: ð 55 Þ
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 2 r pa pb
X
qiaðlia libÞ saðT (^) a T (^) bÞ
h i ;
ð 56 Þ
lia lib ¼ ð 3 =r Þðor=oqiaÞ; ð 57 Þ
T (^) a T (^) b ¼ ð 3 =r Þðor=osaÞ: ð 58 Þ
Here 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
X X
Nijðuia uibÞðuja ujbÞ: ð 59 Þ
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 13 rA; ð 60 Þ 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
