Computational intelligence method of estimating solid-liquid interfacial energy of materials at their melting temperatures

Abstract

The reversible work required in forming an interface between a crystal and its coexisting liquid plays significant roles in many phase transformation and controls many processes such as nucleation, crystal growth, surface roughening and surface melting among others. Despite these significances, its experimental determination is difficult and only few experimental results are available in the literatures. This present work aims at circumventing these experimental challenges by developing a computational intelligence (CI) based model that relates solid-liquid interfacial energies of materials with their melting temperatures using support vector regression (SVR) with test-set cross validation optimization technique. The results of the developed CI-based model show persistent closeness to the few available experimental data than other compared existing theoretical models such as Miedema and den Broeder model, Granasy and Tegze model, Jiang combined model and Ewing model. The outstanding performance of the developed CI-based model as well as its implementation which only needs the value of melting temperature of the concerned material, is of immense importance in circumventing the experimental challenges in the practical attainment of equilibrium between a crystal and its melt for solid-liquid interfacial energy determination.

Keywords

Solid-liquid interfacial energy computational intelligence based model support vector regression and melting temperature

1 Introduction

The reversible work needed to create an interface between a crystal and its melt is referred to as solid-liquid interfacial energy (γ_SL). It controls many important processes involving nucleation, crystal growth, surface roughening and surface melting among others [1]. It dictates the temperature at which solid nucleates homogenously or heterogeneously and plays vital role in many phase transformation [2 –7]. Despite the significances of γ_SL, its determination is associated with difficulties and challenges. Most experimental methods of determining γ_SL involve the use of Gibbs-Thomson equation which indicates that the equilibrium between a solid bounded by an element of interface and its melt is attained at a temperature T_eq differs from its melting temperature T_m while keeping other intensive variables constant [5, 8]. Gibbs-Thomson equation is presented in Equation 1. $T_{m} - T_{eq} \equiv Δ T = γ_{SL} (\frac{1}{r_{1}} + \frac{1}{r_{2}}) (\frac{V_{sl}}{S_{l} - S_{s}})$ (1)

Where r₁ and r₂ represent the interface principal radii of curvature measured in the solid while V_sl, S_l and S_s represent partial molar values with respect to the ith component of the volume of solid, liquid’s entropy and solid’s entropy respectively. γ_SL is directly obtained from Gibbs-Thomson equation for known values of principal radii of curvature for a system at equilibrium by experimental measurement of ΔT. Although, the difference between solid and liquid entropy at melting temperature is referred as the entropy of fusion and can be easily connected with only melting temperature using a novel learning algorithm discuss in this present work.

Implementation of Gibbs-Thomson equation is very challenging due to the difficulty in the practical attainment of equilibrium between a solid and its melt. Small fluctuation can easily melt or solidify a crystal of solid bounded by its melt at equilibrium temperature which subjects the system to a condition of unstable equilibrium. Attempts to overcome these challenges are reported elsewhere [8]. In order to facilitate proper implementation of Gibbs-Thomson equation, many experimental techniques such as the depression of melting points of small crystals [9], the shape of grain-boundary-grooves [10, 11] and dihedral angles [12] among others were afterward developed.

The early theoretical attempts of estimating γ_SL includes the empirical formulation put forward by Turnbull based on the classical nucleation theory and nucleation experiment which proposed that γ_SL is proportional to its melting enthalpy [13]. Turnbull empirical relation was later recognized to underestimate γ_SL which is consequent upon omission of overall vibrational melting entropy [14]. The developed CI-based model puts vibrational melting entropy and melting enthalpy of materials into consideration through the use of melting temperature which connect the two quantities.

Another model that accounts for the observed reduced entropy in the interface region was proposed using radial distribution function [15]. The formulation is presented in Equation (2) $γ_{SL}^{'} = \frac{1}{4 V_{s}} {dH}_{m} + \frac{N_{0} bk}{V_{l}} \int_{0}^{b} g (r) Ing (r) dr$ (2)

Where d, V_s, g (r) , H_m, b, k, N₀ and V_l represent the crystal atomic diameter, g-atom volume of the crystal, radial distribution function of the liquid, melting enthalpy, cut-off distance above which the radial distribution function shows insignificant deviation from unity, Boltzmann’s constant, Avogadro’s number and the molar volume of liquid respectively.

The approach adopted by Miedema and den Broeder [16] gives $γ_{SL}^{″} = \frac{0.211 H_{m}}{\sqrt[3]{(V_{s}^{2} N_{0})}} + \frac{5.2 X 10^{- 8} T}{V_{s}^{2 / 3}}$ (3)

And that of Gránásy and Tegze [12] results into $γ_{SL}^{‴} = \frac{\emptyset (H_{m} + {TS}_{m})}{2 (\sqrt[3]{V_{s}^{2} N_{0}})}$ (4)

Where Ø and S_m represent the geometric factor and melting entropy respectively.

Combination of Gibbs-Thomson equation and the size-dependent melting temperature model yields [1] $γ_{SL}^{⁗} (T) = Ø \frac{Δ H_{m} {\frac{7 T}{T_{m} + 6 T}}^{2}}{\sqrt[3]{V_{s}^{2} N_{0}}}$ (5)

On the perspective of reduction in the entropy of the liquid in the interface region of a crystal and its melt which is due to the restriction in the positioning of atoms and accountability of the entropy of fusion (ΔS) for this reduction at melting point, non-dimensional constant (Ø) with a range of value between 0.83 and 0.89 [17] could be related with γ_SL as expressed in Equation (6) with inclusion of the entropy of fusion (ΔS) which is melting temperature dependent. $Ø = \frac{γ_{SL} A_{m}}{T_{m} Δ S}$ (6)

Where A_m represents molar surface area of solid which depends on the solid crystal structure.

The formulation presented in Equation (6) assumes a smooth surface interface (which is non-smooth interface in the real sense) and supposes the density of atoms at the interface as an average of the densities of atoms within the bulk of solid and liquid (which is not usually the case) [17]. The working principle of the developed CI-based model can be taught of as a way of recognizing inspired pattern that accommodates non-smooth nature of the surface interface as well as its actual density. It develops pattern for each crystal structure with the aid of support vectors during training using their melting temperatures and further generalize the pattern for different crystal structures.

Computational intelligence techniques are computational methodologies and approaches that address complex problems and circumvent experimental difficulties. It involves the use of intelligent algorithms to learn and recognize patterns that govern and control complex systems. These algorithms effectively handle uncertainty, noise and other challenges associated with experimental description of the systems. Computational intelligence tools include Support vector machines, fuzzy logic and evolutionary computation among others [18 –24]. Among these tools, SVR derived from support vector machines has unique features and properties that make it a successful tool in the field of material science [25, 26]. It has distinct and excellent generalization as well as estimation accuracy when developed using few experimental data samples and descriptive features [27, 28]. The incorporation of the use of kernel ensures that features can easily be transformed into high dimensional features space where linear regression can be performed which not only allows efficient computation but permits the construction of non-linear decision boundaries contributing to the robustness and the flexibility of the proposed model. Also, SVR was developed from well-grounded and sound theory before its implementation and application which allow optimum formulation of its solution and ensures easy adaptation of its parameters. This is in contrast to other computational algorithms such Artificial Neural Network (ANN) which follows a more heuristic path. Furthermore, SVR maintains excellent performance in the face of dearth of training data while ANN over-fits. Additionally, the use of structural minimization principle in SVR ensures good generalization accuracy and gives it an edge over ANN which uses empirical risk minimization. The uniqueness of SVR as compared to other algorithms is reported elsewhere [18, 29]. These properties make it a right choice in this present work. The results of developed CI-based model are consistent with the available experimental data and the results of the described broken-bound models. The ease with which the developed CI-based model estimates γ_SL of materials using only melting temperature is an edge over other compared models.

Calculation of the absolute percentage errors between the available experimental data and the estimated γ_SL using the developed CI-based models and other theoretical models, show that CI-based estimated γ_SL are characterized with low absolute percentage error than other compared theoretical models.

The remaining part of this work is organized as follows: Section 2 describes the proposed CI based model and the evaluation of its generalization performance. Section 3 deals with experimental set up and descriptions of the computational and optimization methodology employed. The results of the developed CI-based model are discussed in Section 4 and the results are compared with that of existing theoretical models. Section 5 concludes the work.

2 Description of the proposed computational intelligence tool

Support vector regression is a special class of algorithm first introduced by Vladimir and uses structural risk minimization inductive principles to attain its excellent generalization on limited number of learning patterns [30]. It is characterized with absence of local minima and utilization of kernels. In its motivation to optimize and seek the generalization bounds, its defines the loss function called epsilon intensive loss function which ignores possible errors located within certain distance of the true value. Epsilon intensive loss function ensures the existence of global minimum in addition to its optimization of generalization bound. Among the most important characteristic features of SVR is its ability to maintain excellent generalization performance in the presence of small subset of training points which gives significant computational advantages [31]. In developing and implementing SVR algorithm, the input melting temperatures T_m of materials are mapped with the aid of nonlinear mapping function (polynomial kernel function) onto k-dimensional feature space where linear model f (T_m, ω) presented in Equation (7) is constructed. $f (T_{m}) = \sum_{i = 1}^{k} α_{i} g_{i} (T_{m}) + z$ (7)

Where g_i (T_m) , i = 1, …, k represents a set of nonlinear transformations while z denotes the bias term which can be dropped by preprocessing the data to zero mean.

Loss function L (γ_SL, f (T_m, ω)) which measures the quality of estimation of solid-liquid interfacial energy is defined by Equation (8).

$\begin{matrix} L (γ_SL, f (T_m, ω)) \\ = {\begin{matrix} 0, if | γ_{SL} - f (T_{m}, ω) | \leq ɛ \\ | γ_{SL} - f (T_{m}, ω) | - ɛ, otherwise \end{matrix} \end{matrix}$ (8)

The empirical risk r_emp then becomes $r_{emp} (ω) = \frac{1}{m} \sum_{i = 1}^{m} L_{ɛ} (γ_{{SL}_{i}}, f (T_{m_{i}}, ω))$ (9)

SVR reduces the complexity of the model through minimization of Euclidian norm ∥ω ∥ ² and measures the deviation of training data outside the loss function zone by introducing non-negative variables $(ξ_{i} and ξ_{i}^{*}, i = 1, \dots, m)$ called slack variables. The optimization problem is presented in Equations (10) and (11) with regularization or penalty factor C. $Minimize \frac{1}{2} {∥ ω ∥}^{2} + C \sum_{i = 1}^{m} (ξ_{i} + ξ_{i}^{*})$ (10) $\begin{matrix} Subject to \\ {\begin{matrix} γ_{{SL}_{i}} - f (T_{m_{i}}, ω) \leq ɛ + ξ_{i}^{*} \\ f (T_{m_{i}}, ω) - γ_{{SL}_{i}} \leq ɛ + ξ_{i} \\ ξ_{i}, ξ_{i}^{*} \geq 0, i = 1, \dots, m \end{matrix} \end{matrix}$ (11)

The essence of the penalty factor as indicated in Equation (10) is to control the trade-off between the complexity of the model and the degree to which the deviation larger than epsilon is allowed. Transformation of the optimization problem into dual problem gives solution described by Equation (12) with inclusion of kernel function K (T_m, T_{m
_i}) expressed in Equation (13).

$\begin{matrix} f (T_{m_{i}}) \\ = \sum_{i = 1}^{n_{support vectors}} (α_{i} - α_{i}^{*}) K (T_{m_{i}}, T_{m}), \\ 0 \leq α_{i}^{*} < C, 0 \leq α_{i} \leq C \end{matrix}$ (12) $K (T_{m}, T_{m_{i}}) = \sum_{i = 1}^{k} g_{i} (T_{m}) g_{i} (T_{m_{i}})$ (13)

Where $α_{i} and α_{i}^{*}$ are the Lagrangian multiplier employed in solving the optimization problem.

The estimation accuracy of SVR depends significantly on the optimal selection of penalty factor C, loss function ɛ and kernel option of the chosen kernel function. Epsilon influences the number of support vectors utilized in constructing the regression function and controls the width of ɛ - insensitive zone used in fitting the training samples [23, 32].

2.1 Evaluation of the estimation accuracy of the developed CI-based model

The estimation accuracy of the developed CI-based model was evaluated using coefficient of correlation (CC), root mean square error (RMSE), mean absolute error (MAE) and absolute percentage error (APE). The estimation accuracy evaluation parameters are formulated using relation 14, 15, 16 and 17 respectively. $CC = 1 - \sum_{j = 1}^{m} \frac{Δ γ_{j}}{γ_{\exp}^{2}}$ (14) $RMSE = \sqrt{\frac{1}{m} \sum_{j = 1}^{m} {(Δ γ_{j})}^{2}}$ (15) $MAE = \frac{1}{m} \sum_{j = 1}^{m} | Δ γ_{j} |$ (16) $APE = | \frac{Δ γ_{j}}{γ_{\exp}} | \times 100$ (17)

Where γ_exp, Δγ and m represent experimental solid-liquid interfacial energy, error (difference between the experimental and estimated solid-liquid interfacial energy) and number of data points respectively.

High value of CC and low values of RMSE, MAE and APE indicate excellent estimation accuracy of the model. Table 1 presents these determinants of the estimation accuracy of the developed CI-based model.

3 Experimental set up and description

3.1 Dataset description

Thirteen experimental data of metals was used in the modeling stage of the developed CI-based model. The experimental data was drawn from the results of different experimental techniques such as dihedral angle technique [12], contact angle technique [1] and depression of melting points of small crystal technique [8, 9]. The computational intelligence technique utilized in this present work has demonstrated its excellent estimation and generalization accuracy in material properties estimation using few data –points and few descriptive features [25 , 31]. Its tendency to generalize well when developed with few data-points is also highlighted in this present work as the results of the developed CI-based model show consistent closeness with the experimental values and characterized with high CC as well as low RMSE and MAE. The dataset is presented in Table 2while its statistical analysis including the values of mean, median, maximum and minimum are presented in Table 3 and further shows the consistency in the dataset.

3.2 Development of CI-based model

CI-based model was developed using mathematical description of SVR detailed in section 2 within MATLAB computing environment. Efficient computation was ensured by normalization and randomization of the dataset presented in Table 2 before implementing it for model development. The dataset was further separated into two sets, training set and testing set, in the ratio of eight to two. The training set of data was used to learn the relation between the experimental values of solid-liquid interfacial energy and their melting temperatures with the aid of developed SVR source code using best parameters obtained through test-set-cross validation technique described in Section 3.3. The efficiency and the estimation capacity of the trained model was assessed and validated using the testing set of data. The stability of the proposed model was investigated and verified by running the experiment five times and each run takes about five seconds. The algorithm for the developmental stages of CI-based model is presented in algorithm 1.

3.3 Test-set-cross validation technique of optimization

The optimization of the developed CI-based model was achieved by tuning user defined parameters (penalty factor, epsilon loss function, and hyper-parameter and kernel option) described in Section 2 using test-set-cross validation technique. It involves monitoring the value of the determinants of estimation accuracy (CC, RMSE and MAE) for a group of penalty factor (bound on the Langrangian multiplier), kernel option, epsilon and hyper-parameter (conditioning parameter for QP methods) for every generated training and testing sample of data. The experiment is repeated for each of the available kernel function with an incremental step in the values of the determinant of estimation accuracy until optimum performance is attained by the model. The final optimum parameters were used to learn the relation between γ_SL and T_m. Further description of test-set-cross validation method is contained in our previous work [25, 31]. The model parameters and the corresponding optimum values are presented in Table 4.

4 Results and discussion

4.1 Estimation accuracy of the developed CI-based model

The learning and validation phase of the developed CI-based model are characterized with high correlation coefficients illustrated in Figs. 1 and 2 respectively and tabulated in Table 1. The developed model also has low RMSE and low MAE. In addition to this, measured values are very close to the estimated values as would be seen in the following subsections. This is a strong indication that the developed model has excellent ability to estimate γ_SL of materials that are difficult to measure experimentally.

During the validation stage of the developed CI-based model, 100% coefficient of correlation is obtained as depicted in Fig. 2. This high coefficient of correlation indicates that the developed model can estimate γ_SL of materials with the aid of their melting temperatures with high degree of precision.

4.2 Comparison of absolute percentage error of the estimated γ_SL using CI-based model and other theoretical models

We further justify the excellent estimation accuracy of the developed CI-based model by comparing the absolute percentage error of its results with that of other theoretical models. Figure 3 depicts the comparison. For Zn, Tl and Al metals, CI-based model has absolute percentage error of 0.13%, 0.15% and 0.92% respectively, Jiang combined model has 26.44%, 35.82% and 21.09% respectively while Granasy and Tegze model gives 44.82%, 16.41% and 28.13% respectively. Furthermore, CI-based model has 0.06%, 3.94%, 5.13% and 0.32% error for Pt, Pb, Au and Bi metals respectively while Ewing model give 3.4%, 34.21%, 11.58% and 3.28% respectively. Persistence closeness of the results of the developed CI-based model makes it a viable model of circumventing experimental challenges associated with γ_SL measurement.

4.3 Interfacial energy obtained from the proposed CI-based model

We present the estimated γ_SL using the developed CI-based model and the results of other existing theoretical models as well as the available experimental data [1] in Table 5. As illustrated in Table 5, the developed CI-based model estimates γ_SL for Zn, Tl and Al as 87 mJm^- 2, 67 mJm^- 2 and 129 mJm^- 2 respectively while their experimental values are reported [1] as 87 mJm^- 2, 67 mJm^- 2 and 128 mJm^- 2 respectively using dihedral angle technique [13]. Other existing theoretical models such as Jiang combined model [1] reports 110 mJm^- 2, 43 mJm^- 2 and 155 mJm^- 2 for the metals respectively while Granasy and Tegze model [17] gives 126 mJm^- 2, 56 mJm^- 2 and 164 mJm^- 2 for the respectivemetals.

In the same vein, the developed CI-based model estimates γ_SL for Pt, Pb, Au and Bi metals as 323 mJm^- 2, 73 mJm^- 2, 200 mJm^- 2 and 61 mJm^- 2 respectively while the experimentally measured values are 323 mJm^- 2, 76 mJm^- 2, 190 mJm^- 2 and 61 mJm^- 2 respectively. For these four metals, Ewing model [17] estimates 312 mJm^- 2, 50 mJm^- 2, 168 mJm^- 2 and 59 mJm^- 2 respectively while Miedema and den Broeder model gives 331 mJm^- 2, 58 mJm^- 2, 199 mJm^- 2 and 74 mJm^- 2 respectively [16]. Generally, CI-based model has excellent estimation accuracy as compared to other models.

5 Conclusion

Relationship between solid-liquid interfacial energies and melting temperatures of metals is established using support vector regression computational intelligence technique resulting in the development of the CI-based model. Estimation accuracy of the developed CI-based model was assessed through comparison of its estimated values with the available experimental data. The results of the developed CI-based model are more consistent with the experimental values than other existing theoretical models. Absolute percentage error of the CI-based estimated solid-liquid interfacial energies are lower than that of Miedema and den Broeder model, Granasy and Tegze model, Jiang combined model and Ewing model using experimental data obtained through dihedral angle method. A database of solid-liquid interfacial energies of materials can easily be established using the developed CI-based model. Excellent estimation accuracy of the developed CI-based model and its outstanding performance indicate its potential to circumvent experimental challenges associate with solid-liquid interfacial energies measurement.

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Computational intelligence method of estimating solid-liquid interfacial energy of materials at their melting temperatures

Abstract

Keywords

1 Introduction

3.1 Dataset description

3.2 Development of CI-based model

3.3 Test-set-cross validation technique of optimization

4 Results and discussion

4.1 Estimation accuracy of the developed CI-based model

4.2 Comparison of absolute percentage error of the estimated γ SL using CI-based model and other theoretical models

4.3 Interfacial energy obtained from the proposed CI-based model

5 Conclusion

References

4.2 Comparison of absolute percentage error of the estimated γ_SL using CI-based model and other theoretical models