Solving the Classical Nucleation Theory with respect to the surface energy

TL;DR

This paper introduces a numerical method using the Lambert W function to determine surface energy in Classical Nucleation Theory without assuming a temperature-dependence model, improving accuracy over traditional regression approaches.

Contribution

A novel numerical solution to solve Classical Nucleation Theory for surface energy without assuming its temperature dependence, using Lambert W function.

Findings

The Lambert W-based method accurately determines surface energy.

The method outperforms traditional regression techniques.

No assumption of temperature dependence is needed.

Abstract

An essential parameter of the Classical Nucleation Theory (CNT) is the surface energy between a critical-size nucleus and the ambient phase, $\sigma$. In condensed matter, this parameter cannot be experimentally determined independently of CNT. A common practice to obtain $\sigma$ is to assume a model for its temperature-dependence and perform a regression of the CNT equation against experimental nucleation data. The drawback of this practice is that assuming the temperature-dependence of $\sigma$ adds a bias to the analysis. Nonetheless, this practice is common because an analytical solution of the Classical Nucleation Theory with respect to $\sigma$ is not possible considering common expressions of this theory. In this article, a general numerical solution to this problem using the Lambert $W$ function is proposed, tested, and compared with typical regression methods. The major advantage of the proposed method is that there is no need to assume a model for the temperature-dependence of $\sigma$.

Figures

Tables

Table 1. Table 1: Parameters a 𝑎 a , p 𝑝 p , b 𝑏 b , and q 𝑞 q for Eqs. ( 2 ) and ( 3 ). The parameters b 𝑏 b and q 𝑞 q are the same for both equations.

	$a$	$p$	$b$	$q$
Eq. (2)	$\sqrt{kT}{\left[J\eta {\lambda }^5\right]}^{-1}$	$1/2$	$-\frac{16\pi }{3\Delta G_V^{2}kT}$	$3$
Eq. (3)	$16\sqrt{kT}{\left[3J\Delta G_V^2\tau {\lambda }^6\right]}^{-1}$	$3/2$

Table 2. Table 2: Crystal nucleation data and properties of interest of \ce Li2B4O7  [ 18 ] . Uncertainty in J 𝐽 J , τ 𝜏 \tau , and η 𝜂 \eta is one standard deviation. Uncertainty in J 𝐽 J and τ 𝜏 \tau is the standard deviation obtained in the parameters of the non-linear regression of the Collins–Kashchiev equation  [ 23 , 24 ] (see  [ 18 ] for more details). Uncertainty in η 𝜂 \eta is the confidence band of the shear viscosity regression.

$T$	$J\times {10}^{-7}$	$\tau$	$\Delta G_V\times {10}^{-8}$	$\eta \times {10}^{-9}$
[$K$]	[$m^{-3}{s}^{-1}$]	[$s$]	[$J{m}^{-3}$]	[$\mathrm{Pa}s$]
773	90(1)	1323(8)	$-4.20$	54(2)
783	64(1)	877(9)	$-4.15$	7.0(2)
793	6.9(1)	140(10)	$-4.10$	1.10(4)

Table 3. Table 3: Values of σ 𝜎 \sigma computed using data from Table  2 , considering λ = 𝜆 absent \lambda=   4.86 × 10 − 10   m times 4.86E-10 m 4.86\text{\times}{10}^{-10}\text{\,}\mathrm{m} and uncertainty of 2   K times 2 K 2\text{\,}\mathrm{K} in T 𝑇 T . T 𝑇 T in K K \mathrm{K} and σ 𝜎 \sigma in J ​ m − 2 J superscript m 2 \mathrm{J}\,\mathrm{m}^{-2} . Values obtained using branch − 1 -1 -1 of the Lambert W 𝑊 W function.

	Eq. (2)	Eq. (3)
$T$	$\sigma$	$\sigma$
773	0.1621(2)	0.1608(1)
783	0.1649(2)	0.1614(2)
793	0.1697(2)	0.1664(2)

Table 4. Table 4: Reported values of J 𝐽 J and calculated values of J 𝐽 J using different methods. See the text for a description of each method. T 𝑇 T in K K \mathrm{K} and J 𝐽 J in m − 3 ​ s − 1 superscript m 3 superscript s 1 \mathrm{m}^{-3}\,\mathrm{s}^{-1} .

	$J\times {10}^{-7}$
$T$	Reported	Method 1	Method 2	Method 3
773	90(1)	110	90	90
783	64(1)	45	64	64
793	6.9(1)	8.1	31	6.9

Equations

$$J=J_{0}DZ\exp \left(-\frac{W^{\ast }}{kT}\right)$$

$$W^{\ast }=\frac{16\pi {\sigma }^3}{3\Delta G_V^2},$$

$$J=\frac{\sqrt{kT\sigma }}{\eta {\lambda }^5}\exp \left(-\frac{16\pi {\sigma }^3}{3kT\Delta G_V^2}\right)$$

$$J=\frac{16\sqrt{kT{\sigma }^3}}{3\tau \Delta G_V^2{\lambda }^6}\exp \left(-\frac{16\pi {\sigma }^3}{3kT\Delta G_V^2}\right)$$

$$z=W_n(z\exp (z)).$$

$$a{\sigma }^p\exp (b{\sigma }^q)=1,$$

$$\sigma ={\left[\frac{p}{bq}{W}_n\left(\frac{bq}{p}{a}^{-\frac{q}{p}}\right)\right]}^{\frac{1}{q}}.$$

Full text

††journal: Journal of Non-Crystalline Solids. Publisher version at dx.doi.org/10.1016/j.jnoncrysol.2019.02.006t1t1footnotetext: © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

1 Classical Nucleation Theory

The Classical Nucleation Theory (CNT) formulation shown in Eq. (1) is based on the work of Gibbs [1] with further modifications from other authors [2, 3, 4, 5, 6]. In this equation, $J$ is the nucleation rate, $J_{0}$ is a pre-exponential factor, $D$ is the effective diffusion coefficient related to the mobility of the units that participate in the nucleation process, $Z$ is the Zeldovich factor [5], $W^{*}$ is the work of formation of a critical-size nucleus, $k$ is the Boltzmann constant, and $T$ is the absolute temperature.

[TABLE]

Some expressions exist for the parameters $J_{0}$, $D$, $Z$, and $W^{*}$ [7]. These expressions depend on specific considerations about the nucleation process. For instance, if the critical-size nucleus is considered to be spherical and isotropic, then

[TABLE]

where $\sigma$ is the surface energy between the nucleus and the ambient phase, and $\Delta G_{V}$ is the change in the Gibbs free energy (per unit of volume) when the ambient phase transforms into the phase of the nucleus [8, 9].

Eqs. (2) and (3) show two expressions for Eq. (1) found in the literature [8, 7, 10, 11, 12], considering a plethora of assumptions discussed therein.

[TABLE]

[TABLE]

In the previous expressions, $\eta$ is the shear viscosity of the ambient phase, $\lambda$ is the jump distance for diffusion estimated by the size of the units that diffuse during the nucleation process, and $\tau$ is the time-lag of the nucleation process.

Eqs. (2) and (3) cannot be solved analytically for $\sigma$. The objective of this short communication is to show how to solve this kind of problem numerically using the Lambert $W$ function [13].

The domain of Eqs. (2) and (3) is dependent on the assumptions used to obtain them; here they will be used only for experimental nucleation rate data above the reported temperature of maximum nucleation. Fortunately, the method proposed here is general and can also be used for other expressions of $J$.

2 Lambert $W$ function

The Lambert $W$ function (also known as the omega function) is defined as the inverse function of $f(z)=z\exp(z)$, with $z$ being any complex number, thus

[TABLE]

The $W$ is a multivalued function with $n$ branches, and one must select an integer $n$ to solve it. Only $n=$0$$ and $-1$ produce real results, the only ones of interest to the problem here, considering there is no physical meaning for complex values of $\sigma$. Figure 1 shows the real values of $W$ for these two branches.

Eqs. (2) and (3) can be written in the form

[TABLE]

where $a$, $p$, $b$, and $q$ are non-zero parameters that depend on the nucleation expression in consideration (see Table 1). A solution of Eq. (4) for $\sigma$ is

[TABLE]

With the solution shown in (5), it is possible to solve expressions (2) and (3) for $\sigma$. In fact, any equation that can be written in the form of (4) can be solved for $\sigma$. This solution may also be used, for example, when dealing with crystal growth mediated by secondary surface nucleation (see Ref. [14]). The fact that a solution is possible does not imply that it is non-complex. In some cases, a real solution may be impossible.

Considering that $\sigma$ is expected to increase monotonically with respect to the temperature in the range where nucleation data are available [15], the only physically valid branch of $W$ in this scenario is branch $-1$.

Several free and commercial numerical programs have an implementation of the $W$ function. Scipy [16], for instance, is a free and open-source module for the Python programming language that has this function built in with the name lambertW, available in the scipy.special submodule. For an iterative approach that can be performed using a spreadsheet program, see Ref. [17].

3 Test on experimental data

Table 2 shows crystal nucleation data measured in a supercooled \ceLi2B4O7 liquid (lithium tetraborate, also known as lithium diborate), as well as other properties of interest [18]. With these data and the solution shown in Eq. (5), the value of $\sigma$ was computed considering the two expressions for $J$ that are common in the literature (Eqs. (2) and (3)). The results are shown in Table 3. The computed $\sigma$ values are within the expected range for this parameter ($0.1$ to $0.25\text{\,}\mathrm{J}\,\mathrm{m}^{-2}$)  [19, 20, 21, 10, 22].

4 Comparison with regression methods

Another approach to obtain $\sigma$ from experimental $J(T)$ data is via regression. A method popularized by James [25] consists of linearizing Eq. (2) by plotting $\ln\left(J\eta T^{-1/2}\right)$ versus $\left[T\Delta G_{V}^{2}\right]^{-1}$, with the underlying assumption that $\sigma$ and $\lambda$ are temperature-independent. With a linear regression of the data in the linearized plot, $\sigma$ can be obtained from the slope of the regression, which is equal to -16\pi\sigma^{3}$$\left[3k\right]^{-1}, and $\lambda$ can be obtained from the intercept of the regression, which is equal to $\ln\left(\sqrt{kT\sigma}\lambda^{-5}\right)$.

Values of $\sigma=$ $0.32(1)\text{\,}\mathrm{J}\,\mathrm{m}^{-2}$ and $\lambda=$ $1(5)\text{\times}{10}^{-32}\text{\,}\mathrm{m}$ were obtained using this method with the data shown in Table 2. The value of $\sigma$ is two times higher than the value obtained using the Lambert $W$ function. The value of $\lambda$ is 22 orders of magnitude smaller than what is expected, which is a discrepancy that is well documented in the literature [8, 10]. Rarely discussed, however, is the uncertainty in $\lambda$ when using this method, which is massive for the dataset investigated here.

As a consistency test, Table 4 shows the computed value of $J$ when putting the values of $\sigma$ and $\lambda$ obtained using this method back into Eq. (2). The computed value is shown in the column “Method 1”. The difference between the reported and the computed value is up to $42\text{\,}\mathrm{\char 37\relax}$ for the temperature of $783\text{\,}\mathrm{K}$.

A similar approach using Eq. (3) instead of Eq. (2) was proposed by Weinberg and Zanotto [26], which yields similar results (not shown here). One advantage of these two regression approaches (advanced by James, Weinberg, and Zanotto) is that no assumption must be made on the value of $\lambda$. However, the resulting uncertainty in $\lambda$ is enough to discard this apparent advantage in this case. Moreover, assuming $\sigma$ to be temperature-independent may be a reasonable approximation only for sufficiently small temperature ranges.

Another regression approach is to assume some model for the temperature-dependence of $\sigma$ and use it to perform a non-linear regression of CNT. The expression $\sigma=$2.8\text{\times}{10}^{-4}$T-$0.054$$ is obtained if we consider Eq. (2), $\lambda=$ $4.86\text{\times}{10}^{-10}\text{\,}\mathrm{m}$, and assume that $\sigma$ has a linear temperature-dependence. The computed values of $J$ using this method are shown in the column “Method 2” in Table 4. The difference between the reported and the computed value is $78\text{\,}\mathrm{\char 37\relax}$ for the temperature of $793\text{\,}\mathrm{K}$.

Finally, Table 4 also shows the calculated values of $J$ using the method proposed in this article, column “Method 3”. It is no surprise that these values are equal to the experimental values because no assumption concerning the temperature-dependence of $\sigma$ is necessary to use this new method; thus it gives the exact value of $\sigma$ to bring theory and experiment to an agreement, point by point. The hope is that with fewer assumptions we can reach new insights into this problem.

5 Conclusion

An analytical solution for the surface energy may not be possible depending on the form of Classical Nucleation Theory that is used. Here I demonstrate a numerical solution (using the Lambert $W$ function) to solve the Classical Nucleation Theory with respect to the surface energy. The advantage of this numerical approach, in contrast with previous methods, is that no assumption on the temperature-dependence of the surface energy is necessary.

Acknowledgements

Daniel R. Cassar gratefully acknowledges the São Paulo Research Foundation for the financial support (FAPESP grant number 2017/12491-0) and Carolina B. Zanelli for the English revision.