Abstract

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Effect of different approaches to decouple the dependence of nuclei-liquid surface energy on size and temperature

Mara Cristina Freitas^I; Dalmo Inácio Galdez Costa^I; Aluísio Alves CabralII, ^* * e-mail: aluisio_cab@yahoo.com.br ; Adalto Rodrigues Gomes^I; José Manuel Rivas Mercury^{I, III}

^IDepartment of Mechanics and Materials - DMM - CEFET - MA, Brazil

^IIDepartment of Exact Sciences - DCE - CEFET - MA, Brazil

^IIIAcademic 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 Li₂O.2SiO₂ (LS₂) and Na₂O.2CaO.3SiO₂ (N₁C₂S₃). 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, nuclei-liquid 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^1-5. These tests consist of plotting Ln(Iη /T) vs. 1/TΔG_v², where the intercepts and slopes are proportional, respectively, to the pre-experimental 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^7-8; ii) the influence of H₂O content on crystal nucleation rates⁹; iii) the possible breakdown of the Stokes-Einstein equation at (ΔT/T_m ~ 0.5-0.6), which must not be applied to express the crystal nucleation kinetic as governed by viscous flow¹⁰; and iv) the absence of an experimental technique to accurately determine the surface energy of nanometric aggregates independently of nucleation experiments¹¹. 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 pre-exponential term. In this case, an increase in the temperature dependence of surface energy has been observed, as demonstrated by Turnbull¹², Spaepen¹³ and James¹.

On the other hand, Gutzow et al.¹⁴ demonstrated that the crystal-melt 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 crystal-melt surface energy should decrease with increasing temperature¹¹.

According to several researchers^16-17, 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 & Zanotto¹¹ applied the Tolman equation to decouple the temperature and size-dependent parts of surface energy from the homogeneous crystal nucleation kinetics of Li₂O.2SiO₂ (LS₂) and Na₂O.2CaO.3SiO₂ (N₁C₂S₃) 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 approach¹⁸, the expressions derived by Vogelsberger¹⁵ and Rasmussen¹⁹ were also applied to the crystal nucleation kinetics of LS₂ and N₁C₂S₃ silicate glasses to compare them with the σ_cl(T) results obtained by Fokin & Zanotto¹¹ using only Tolman's equation.

2. Theory

According to the Classical Nucleation Theory (CNT), the steady-state homogeneous nucleation rate (I_st) can be described as a function of temperature through the following expression¹²:

where I_st represents the number of nuclei formed per unit volume (m^-3/s), A is a weakly temperature-dependent 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 pre-exponential 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 Stokes-Einstein equation at deep undercooling (ΔT/T_m ~ 0.5-0.6) 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 crystal-melt 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 LS₂ and N₁C₂S₃ silicate glasses were selected from the literature, as indicated in Table 1.

The same values of δ used by Fokin & Zanotto¹¹ 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 Zanotto¹¹ 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 Zanotto¹¹ for a LS₂ 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 LS₂ and N₁C₂S₃ 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 LS₂ and N₁C₂S₃ 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 CNT¹⁴.

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 Stefan²³, 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 Appen²⁶ for a Li₂O-SiO₂ 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: LS₂ and N₁C₂S₃. 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 & Zanotto¹¹, 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 LS₂ glass.

Despite the lowest values of σ_lc(T), a more detailed analysis of the surface energy values was carried out for the LS₂ 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 (# 620249/2006-4), FINEP, CAPES and FAPEMA for their financial support of this investigation.

Received: October 27, 2008

Revised: February 14, 2009

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