Abstract
Tests involving the Classical Nucleation Theory (CNT) often disregard the size dependence of surface energy. Thus, the surface energy of critical nuclei is assumed to be a macroscopic quantity that depends only on the temperature of a flat surface. However, because the size of critical nuclei changes with temperature, σcl(T) should be described as a function of both temperature and size of critical nuclei. The present work examines the temperature dependence of macroscopic surface energy, decoupling it from the size dependent part. Tolman, Rasmussen and Vogelsberger's equations are used to decouple the dependence of surface energy on size, using experimental data for the following silicate glasses Li2O.2SiO2 (LS2) and Na2O.2CaO.3SiO2 (N1C2S3). These equations are successful in obtaining a decrease in σcl(T), in agreement with theoretical predictions. For all the values of δ , Tolman's equation produces the lowest values of σcl(T). Nevertheless, they are very close to the liquid/vapor surface energy (σlv), which contradicts the Stefan's rule (i.e. σcl/σ lv << 1). Therefore, it is demonstrated that the assumption of the curvature dependence of surface energy does not suffice, per se, to explain the discrepancy between the experimental and theoretical values of nucleation rates.
crystal nucleation kinetics; nucleiliquid surface energy; decoupling equations
REGULAR ARTICLES
Effect of different approaches to decouple the dependence of nucleiliquid surface energy on size and temperature
Mara Cristina Freitas^{I}; Dalmo Inácio Galdez Costa^{I}; Aluísio Alves CabralII, ^{*} * email: aluisio_cab@yahoo.com.br ; Adalto Rodrigues Gomes^{I}; José Manuel Rivas Mercury^{I, III}
^{I}Department of Mechanics and Materials  DMM  CEFET  MA, Brazil
^{II}Department of Exact Sciences  DCE  CEFET  MA, Brazil
^{III}Academic Department of Chemistry  DAQ  CEFET  MA, Brazil
ABSTRACT
Tests involving the Classical Nucleation Theory (CNT) often disregard the size dependence of surface energy. Thus, the surface energy of critical nuclei is assumed to be a macroscopic quantity that depends only on the temperature of a flat surface. However, because the size of critical nuclei changes with temperature, σ_{cl}(T) should be described as a function of both temperature and size of critical nuclei. The present work examines the temperature dependence of macroscopic surface energy, decoupling it from the size dependent part. Tolman, Rasmussen and Vogelsberger's equations are used to decouple the dependence of surface energy on size, using experimental data for the following silicate glasses Li_{2}O.2SiO_{2} (LS_{2}) and Na_{2}O.2CaO.3SiO_{2} (N_{1}C_{2}S_{3}). These equations are successful in obtaining a decrease in σ_{cl}(T), in agreement with theoretical predictions. For all the values of δ , Tolman's equation produces the lowest values of σ_{cl}(T). Nevertheless, they are very close to the liquid/vapor surface energy (σ_{lv}), which contradicts the Stefan's rule (i.e. σ_{cl}/σ_{ lv} << 1). Therefore, it is demonstrated that the assumption of the curvature dependence of surface energy does not suffice, per se, to explain the discrepancy between the experimental and theoretical values of nucleation rates.
Keywords: crystal nucleation kinetics, nucleiliquid surface energy, decoupling equations
1. Introduction
The Classical Nucleation Theory (CNT) is well known for its good description of the temperature dependence of the nucleation rate, I(T). Nevertheless, CNT tests involving several silicate glasses that nucleate homogeneously have demonstrated that experimental crystal nucleation rates are much higher than theoretical ones^{15}. These tests consist of plotting Ln(Iη /T) vs. 1/TΔG_{v}^{2}, where the intercepts and slopes are proportional, respectively, to the preexperimental factor and the surface energy of a flat interface. According to Cabral Jr.^{6}, this discrepancy persists even if different free energy expressions are used. These tests are often based on the assumption that surface energy does not change as a function of temperature and nucleus size.
Several assumptions have been investigated to explain the reasons for these discrepancies: i) metastable phase formation^{78}; ii) the influence of H_{2}O content on crystal nucleation rates^{9}; iii) the possible breakdown of the StokesEinstein equation at (ΔT/T_{m} ~ 0.50.6), which must not be applied to express the crystal nucleation kinetic as governed by viscous flow^{10}; and iv) the absence of an experimental technique to accurately determine the surface energy of nanometric aggregates independently of nucleation experiments^{11}. Based on the latter assumption, the CNT assumes that surface energy is a macroscopic thermodynamic property with a value equal to that of a planar interface, σ_{∞} . In other words, surface energy is considered to be independent of nucleus size. This assumption is known as the capillarity approximation.
This discrepancy can be avoided by calculating a specific surface energy from experimental nucleation rate data at each temperature, using the theoretical value of the preexponential term. In this case, an increase in the temperature dependence of surface energy has been observed, as demonstrated by Turnbull^{12}, Spaepen^{13} and James^{1}.
On the other hand, Gutzow et al.^{14} demonstrated that the crystalmelt surface energy, σ_{cl}, should decrease with increasing temperature, mainly in cases where the molar volume of the liquid phase is higher than the corresponding one of the crystal phase. In addition, there is some experimental evidence that crystalmelt surface energy should decrease with increasing temperature^{11}.
According to several researchers^{1617}, a plot of σ_{cl}(T) may indicate a size effect, because the surface energy σ_{cl}(T, R) calculated from nucleation data refers to nuclei of critical size, R*. The latter parameter changes as a function of temperature.
Recently, Fokin & Zanotto^{11} applied the Tolman equation to decouple the temperature and sizedependent parts of surface energy from the homogeneous crystal nucleation kinetics of Li_{2}O.2SiO_{2} (LS_{2}) and Na_{2}O.2CaO.3SiO_{2} (N_{1}C_{2}S_{3}) stoichiometric silicate glasses. Based on experimental data of crystal nucleation rates, viscosity, and induction time, and on the difference between the volume free energies of glass and crystal, the authors demonstrated that the surface tension can decrease with temperature, since reasonable values were chosen for the Tolman parameter.
Taking into account the curvature dependence of surface tension, this paper focuses on the effects of the application of different expressions described in the literature on the decoupling of the temperature and size parts of surface energy. In addition to Tolman's approach^{18}, the expressions derived by Vogelsberger^{15} and Rasmussen^{19} were also applied to the crystal nucleation kinetics of LS_{2} and N_{1}C_{2}S_{3} silicate glasses to compare them with the σ_{cl}(T) results obtained by Fokin & Zanotto^{11} using only Tolman's equation.
2. Theory
According to the Classical Nucleation Theory (CNT), the steadystate homogeneous nucleation rate (I_{st}) can be described as a function of temperature through the following expression^{12}:
where I_{st} represents the number of nuclei formed per unit volume (m^{3}/s), A is a weakly temperaturedependent term, k is Boltzmann's constant, ΔG_{D} (J/mol) is the kinetic barrier for nucleation (which corresponds to the activation energy required to transport a structural unit through the nuclei/glass interface), and W* (J/mol) is the thermodynamic barrier for the formation of critical size nuclei.
The preexponential term A is given by:
where N is the number of molecules with size λ per unit of volume, ν_{o} is the vibration frequency of a structural unit for typical nucleation temperatures, σ_{∞} corresponds to the free energy per unit area of crystal/melt flat interface, and h is Planck's constant.
For spherical nuclei, W* is given by:
where ΔG_{V} (ΔG_{V} = ΔG/V_{m}) is the free energy change per unit volume of crystal, ΔG is the free energy change per mole, and V_{m} is the molar volume of the crystalline phase.
If one neglects a possible breakdown of the StokesEinstein equation at deep undercooling (ΔT/T_{m} ~ 0.50.6) and expresses the kinetic barrier in terms of viscosity, Equation (1) can be rewritten as:
If the kinetic barrier of nucleation, ΔG_{D}, is expressed in terms of the induction period of nucleation, t_{ind}, Equation (1) can be transformed into Equation (5):
From Equation (1), one can see that the crystal nucleation rates can be strongly influenced by the values of W*. Therefore, if one neglects the strain energy associated with the formation of critical nuclei, the overall thermodynamic work, W, can be written as:
Several approximated equations have been derived to describe the curvature dependence of the crystalmelt surface energy, σ = σ_{cl}(R):
where Tolman's parameter, δ , characterizes the width of the interfacial region between the coexisting phases (whose order is of atomic dimensions).
It should be noted that these decoupling equations comprise a large range of σ_{cl}(R) values. Nevertheless, Schmelzer et al.^{15} have demonstrated that the Tolman's (Equation (7)), Vogelsberger's (Equation (8)) and Rasmussen's (Equation (9)) expressions can be applied only to R >> δ , R >> 4δ and R >> 3δ, respectively. It should be noted that the negative and infinitive values of σ obtained for small values of R/δ can be neglected.
3. Calculations
Experimental data of crystal nucleation rates, viscosity, induction time and thermodynamic quantities for the LS_{2} and N_{1}C_{2}S_{3} silicate glasses were selected from the literature, as indicated in Table 1.
The same values of δ used by Fokin & Zanotto^{11} were also used here to evaluate σ_{cl}(R, T) through the decoupling equations for the silicate glasses investigated.
By combining Equation (6) and (7), one obtains:
From the condition (∂ W/∂ R)_{R*} = 0, and taking the positive root, one can find the critical radius as:
The corresponding thermodynamic barrier for nucleation (R > R*) can then be described as:
If one replaces Equation (12) with (4) or (5), only two parameters, σ_{¥} and d, will remain unknown in the resulting equation. In possession of all the experimental data  crystal nucleation rates, viscosity, induction time and difference in free energy per unit of volume  for each silicate glass investigated and at fixed values of δ , one can then determine the σ (T) dependence at different values of δ .
Similar procedures were employed to obtain the σ_{cl}(T) curves using the other decoupling equations (Equation (8) and(9)).
4. Results and Discussion
If one takes into account the dependence of surface tension on curvature, a significant quantitative change is expected in the work of critical cluster formation, as indicated in Figure 1. The typical values of the parameters used to calculate W as a function of the nucleus size for different approximations of σ_{cl}(R) are given in the caption of Figure 1.
It should be emphasized that to determine accurately the work of critical cluster formation, one should take into account the dependence of nuclei density on size, ρ (R)^{23}. However, simulations carried out by Fokin and Zanotto^{11} for a model glass demonstrated that W* is weakly affected by ρ (R).
Additionally, as the temperature changes, one also expects a decrease in crystal density with temperature, ρ (T). However, the results obtained by Fokin and Zanotto^{11} for a LS_{2} glass demonstrated that σ_{cl}(T) is weakly affected by ρ (T).
Hence, the calculations of σ_{cl}(R,T) presented in this paper disregarded the effects of ρ (R) and ρ (T). Figures 2 and 3 show the σ_{cl} vs. T curves obtained for the LS_{2} and N_{1}C_{2}S_{3} glasses using viscosity data. The error bars indicated in each figure are between 0.3 and 1%. As expected, the σ_{cl}(T) plots obtained from experimental induction time data showed a behavior similar to that depicted in Figures 2 and 3 of this paper. Nevertheless, only the σ_{cl}(T) curves obtained from viscosity data are presented on Figures 2 and 3.
As can be seen in Figures 2 and 3, the σ_{cl}(R, T) curves calculated from viscosity follow the same tendency, i.e., σ_{cl(Tolman)} < σ_{cl(Rasmussen)} < σ_{cl(Vogelsberger)}. This behavior is similar to that obtained by using the induction time as the kinetic barrier.
At temperatures below T_{g}, it is well known that the elastic strain resulting from the difference between the densities of glass and crystal can underestimate the driving force for crystallization and, hence, overestimate the thermodynamic barrier for nucleation, W*^{22}. Therefore, the slopes of the σ_{cl}(T) curves were estimated from temperatures higher than T_{g}. Figures 4 and 5 illustrate the behavior of dσ_{cl}/dT as a function of λ for LS_{2} and N_{1}C_{2}S_{3} glasses, respectively. The dotted lines serve to guide the eyes, while the solid lines indicate the values of Tolman's parameter at the point where dσ_{cl}/dT becomes negative.
From Figures 4 and 5, one can observe that dσ_{cl}/dT decreases gradually, becoming negative as δ increases. Therefore, physically reasonable values of the Tolman parameter can be chosen in such way that a decrease in surface tension is obtained, as predicted by the CNT^{14}.
Nevertheless, one must analyze the physical meaning of the σ_{cl} values, which can done by calculating the ratio of σ_{cl}/σ_{ lv} (σ_{lv} is the surface energy in the liquid). In line with Stefan^{23}, σ_{cl}/σ_{ lv}≅ ΔH_{cl}/ΔH_{lv} <<1; where ΔH_{cl} and ΔH_{lv} are, respectively, the melting enthalpy of the crystalline phase and the enthalpy of evaporation. Considering the σ_{lv} measured by Appen^{26} for a Li_{2}OSiO_{2} glass composition similar to the one investigated here, we obtained a plot of the σ_{cl}/σ_{ lv} ratio as a function of δ , as indicated in Figure 6.
Taking into account the Rasmussen and Vogelsberger equations, one can observe from Figure 6 that for values of the Tolman's parameter (δ > 6 x 10^{10} m) the nuclei/liquid interfacial tension (σ_{cl}) values are larger than the liquid/vapor ones (σ_{lv}) i.e. σ_{cl}/σ_{ lv}>>1. The reason for this result is not clearly understood at this time.
Applying Tolman's equation, one can observe that σ_{cl}/σ_{lv}→1. This result also contradicts the one given by Stefan's rule.
5. Conclusions
The analysis of the curvature dependence involved the following silicate glasses that exhibit internal nucleation kinetics: LS_{2} and N_{1}C_{2}S_{3}. In addition to the Tolman equation, the expressions proposed by Rasmussen and Vogelsberger were also applied to decouple the temperature and size parts of surface energy.
The σ_{lc}(T) curves obtained were plotted from experimental data of viscosity, crystal nucleation rate, thermodynamic free energy and induction time. The curves demonstrated that all the decoupling equations used in this work were successfully applied to produce decreasing temperature dependence, in agreement with the results obtained by Fokin & Zanotto^{11}, that applied only the Tolman equation. However, the lowest values of σ_{cl}(T) were obtained with Tolman's expression. This was reinforced by the σ_{cl}/σ_{ lv} curves plotted for a LS_{2} glass.
Despite the lowest values of σ_{lc}(T), a more detailed analysis of the surface energy values was carried out for the LS_{2} glass using Stefan's rule, which suggests that σ_{cl}/σ_{ lv} << 1. As can be seen in Figure 6, this ratio does not agree with that prediction.
Therefore, it is evident that the assumption of the curvature dependence of surface energy does not suffice, per se, to explain the discrepancy between the experimental and theoretical values of nucleation rates.
Acknowledgements
The authors gratefully acknowledge the Brazilian research funding agencies CNPQ (# 620249/20064), FINEP, CAPES and FAPEMA for their financial support of this investigation.
Received: October 27, 2008
Revised: February 14, 2009
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Publication Dates

Publication in this collection
18 May 2009 
Date of issue
Mar 2009
History

Reviewed
14 Feb 2009 
Received
27 Oct 2008