Neural Network Modelling on Contact Angles of Liquid Metals and Oxide Ceramics

Abstract

A neural network model was developed in this paper to predict the contact angles of 21 metals and 14 solid oxides. 15 factors were used in the neural network model to distinguish different metal and oxide categories and experimental conditions. With 1120 contact angle values as the learning data, the neural network model was successfully developed. It can properly reproduce the experimental data on contact angles of molten metals and solid oxides under various conditions. Specifically, only three predictions among the total 1155 predictions were over 20% deviation from the experimental data. All the predictions on the 35 test data are within 20% deviation from the experimental values. Factors such as oxygen partial pressure and surface tension of molten metal were found to be important for a good model prediction. With the developed model, contact angle values of Fe and CeO₂ were predicted.

1. Introduction

Wettability between a molten metal and a solid oxide ceramic substrate, which is usually characterized by a contact angle value, is of great importance in material manufacturing. In metal production processes, it can directly influence the corrosion of ceramic refractories which acts as the container materials.^1,2) Also, it can affect the agglomeration of oxide inclusions in molten steel which is formed during the deoxidation process in steel production.^3,4) A large contact angle between oxide inclusions and molten steel can enhance the inclusion agglomeration to form large size inclusions, which could be easily separated from a molten steel into the top slag due to a large buoyancy effect. Therefore, inclusion agglomeration is an important way to improve the steel cleanness. However, sometimes, a good wettability between a molten steel and oxide inclusions is preferred, for example in the steel welding field, in order to form fine metal structures, where the suspended micro size oxides in steel could work as nucleation cites.⁵⁾

Besides metal production processes, wettability as well as contact angle between molten metal and solid ceramics is also an important technological parameter in evaluating the adhesion work of a metal-oxide interface, which also reflects the strength of metal/ceramic bonds.⁶⁾ This is due to that, experimentally, the work of adhesion so far can only be evaluated by measuring the contact angle, between molten metals and solid oxides, and the surface energy of a molten metal. The adhesion work is the required energy to reversibly separate the interface to form two new surfaces. Under the condition of no chemical reactions and interfacial adsorption equilibrium, the work of adhesion can be calculated by using the following equation:   

$${\text{W}}_{ad}={\sigma }_{SV}+{\sigma }_{LV}-{\sigma }_{SL}$$(1)

where σ_SV is the surface tension of a solid ceramic, σ_LV is the surface tension of a molten metal and σ_SL is the interfacial tension between the solid and the liquid. At the triple point of the molten metal, gas and solid ceramic, as shown in Fig. 1, the balance among the surface tension of molten metal, surface tension of solid ceramic and the interfacial tension of the molten metal and the solid ceramic is given by Young’s equation:   

$${\sigma }_{LV}cos{\theta }_{SL}={\sigma }_{SV}-{\sigma }_{SL}$$(2)

where θ_SL is the contact angle of a molten metal and a solid ceramic. By combining Eqs. (1) and (2), the work of adhesion can be obtained by measuring the molten metal surface tension and the contact angle, namely Young-Dupré equation as follows:   

$${\text{W}}_{ad}={\sigma }_{LV}\left(1+cos{\theta }_{SL}\right)$$(3)

Therefore, the contact angle of a solid ceramic and a molten metal is often required, when the work of adhesion is experimentally evaluated. The knowledge of wettability as well as adhesion work is very important for the joining of dissimilar materials in various aspects such as solid oxide fuel cells, ceramic supported catalysts, thermal barrier coatings, metal-ceramic structural joining, and so on.^{6,7,8,9,10,11)}

Fig. 1.

Schematic of the contact angle between a liquid and a solid substrate.

In the past, wettability of molten metals and solid ceramics have been vastly investigated not only due to its importance in material manufacturing, but also due to its theoretical interest in academic field. Sessile drop method is commonly used in measurements of contact angles. Such measurements are normally carried out under a high temperature and a strictly controlled atmosphere due to a high melting point of some metals and their high affinity with oxygen. This imposes experimental difficulties since an extremely low oxygen partial pressure is difficult to be accurately guaranteed. This also leads to deviations of some experimental data among different measurements. It is widely recognized that temperature and oxygen partial pressure are the mutual factors that influences the contact angle of various metal/ceramic system. Generally, the contact angle value decreases with an increased experimental temperature.¹²⁾ Oxygen partial pressure in an experimental atmosphere is one of the most important factors which can significantly influence the contact angels of molten metals and oxide ceramics.^2,6) This is due to that oxygen is an interfacial active element. The adsorption of oxygen at the metal/ceramic interface or metal surface can reduce the interfacial energy and the metal surface tension. By using the Langmuir adsorption isotherm and Gibbs adsorption isotherm, the dependences of surface tensions of some metals as well as the interfacial tensions between some metals and ceramics or slags on oxygen activity in the metal bulk have been established, and it can be expressed by using the following general equation:^13,14)   

$$\sigma ={\sigma }^P-\text{RT}{\mathrm{\Gamma }}^0\text{ln}\left(1+K{a}_{\text{O}}\right)$$(4)

where σ is the interfacial or surface tension, σ^P is the surface tension of a pure metal, Γ⁰ is the adsorption of oxygen at a full coverage, K is the adsorption equilibrium under a certain temperature, and a_O is the oxygen activity in a metal bulk.

Since the contact angle or wettability reflects the interfacial energy, the adhesion work as well as the atomic bonding at the interface, thus its value should directly relate to the categories of metals and ceramics as well as their physical properties. Also, impurities both in metals, for example sulfur and oxygen in molten Fe, and in ceramics might also influence contact angle measurements. In addition, the contact angle between a molten metal and a single crystal oxide normally has a smaller value compared to the polycrystalline oxide solid substrate. Furthermore, the contact angel for different crystallographic orientations of single crystal ceramic also show some difference. These are due to the difference in the surface energy of solid ceramics.^15,16) In addition, the contact angle was found to decrease with a decreased bandgap energy of ceramics materials. This has been observed in several metals such as Cu, Sn, Si, and Au on various ceramics.⁷⁾ Furthermore, due to a linear correlation between the bandgap energy and the formation enthalpy of solid oxides, the dependence of the contact angle on the formation enthalpy was also found for some metals, such as Cu, CO, Ni, Fe, Ag and Sn on different ceramic materials.⁷⁾ In addition, the surface roughness of solid ceramics was also found to influence the contact angle.¹⁷⁾ Finally, sometimes, the contact angle shows a dynamic change over time due to chemical reactions occurring at the metal/ceramic interface.¹⁸⁾ Thus, the contact angle in such a case is apparent contact angle, which is also affected by the reaction energy and the non-equilibrium adsorption at the interface.

In a summary, contact angle or wettability between molten metals and oxide ceramics is very important both for material manufacturing and for understanding the interface phenomena in various fields. The contact angle value not only directly depends on material characteristics, but is also very sensitive to experimental conditions. Currently, the knowledge on the interface of metals and ceramics is still limited. There is no a universal equation or model which can predict the contact angles between different metals and different ceramics. Since many factors as previously mentioned can affect the contact angle and experimental measurements are normally carried out under various conditions, it is difficult to find the correlations among various factors and the measured contact angle for different metal/ceramic systems. In recent years, neural network modelling has become an efficient way to find the universal correlations among various factors. The current authors have successfully predict the temperature dependent coefficients of metal surface tension, surface tensions and electrical conductivities of molten slag by using neural network models.^19,20,21,22) In this paper, 1155 contact angle data for 21 categories of metals and 14 categories of oxides were collected from over 120 research articles.^{2,8,15,16,18,23-142)} During these contact angle measurements, interfacial reactions may happen in some metal/oxide systems and the reaction products may affect the contact angle values. In such cases, the measured contact angle value is the apparent contact angle, which also represents the characteristics of a metal in contact with an oxide. As a first step, all the collected data both for the reactive system and non reactive system were used, with the aim to build up a neural network model to predict the contact angles or apparent contact angles among 21 metals and 14 oxide ceramics. Therefore, it will be a universal model for the contact angle predictions which are not seen in public literature.

2. Neural Network Model of Contact Angle Prediction

2.1. Neural Network Model Structure

The detailed explanation of neural network computations can be found elsewhere.^18,19,20,21) Figure 2 shows the schematic of a neural network model, which includes an input layer, a middle layer and an output layer. Firstly, a forward propagation scheme was used. The value of each hidden unit in the middle layer was calculated by using a sigmoid function where the function input was the summation of the product of each unit and its weight in the input layer, as shown in Eq. (5). Similarly, the unit value in the output layer was obtained by using the unit value in the middle layer and its weight as shown in Eq. (6).   

$$a_k=f\left({\sum }_{i=1}^n{x}_i{W}_{ki}-W_{k0}\right)$$(5)

  

$$\hat{y}=f\left({\sum }_{j=1}^n{a}_j{V}_j-V_0\right)$$(6)

where a_k or a_j is the middle layer neuron, x_i is the input value of the selected features, W_ki is the weight of the input x_i in order to get the middle unit value a_k, $\hat{y}$ is the output value, and V_j is the weight of the middle layer unit a_j to obtain the output value. Finally, W_k0 and V₀ are the threshold for the middle layer and the output layer, respectively. In the learning process of the neural network model, the initial values of the weights are randomly chosen. With those weight values and the input data x_i, the neural network model can predict the output value $\hat{y}$ by Eqs. (5) and (6). Then, a back propagation approach was used to minimize the Root mean squared error (RMSE) between the output value $\hat{y}$ and the teaching data y which is the experimental data of contact angle. The RMSE value was calculated by using the following equation:   

$$\text{RMSE}=\sqrt{{\sum }_{i=1}^N\frac{{({\theta }_{Cal}-{\theta }_{Exp})}^2}{N}}$$(7)

where θ_Cal is the model predicted contact angle, θ_Exp is the experimental measured contact angle, and N is the number of the data used in the model leaning.

Fig. 2.

Schematic of a neural network model.

Each weight was updated by using the following equations:   

$$V_j\left(\text{New}\right)=V_j\left(\text{Old}\right)-\eta \frac{\partial E}{\partial V_j}$$(8)

  

$$W_{ki}\left(\text{New}\right)=W_{ki}\left(\text{Old}\right)-\eta \frac{\partial E}{\partial W_{ki}}$$(9)

where η is the learning rate. Through the repeated forward and backward propagation process, the weight values are adjusted to reduce the RMSE error value. The iteration is stopped until a certain error criterion is satisfied. This is the whole learning process of a neural network model. In this study, the software for neural network computations was developed by the Research Center of Computational Mechanics at Osaka University and Sumitomo Metal Industries Ltd.

2.2. Factors on Wettability of Metal and Solid Oxide

Wettability of liquid metals on solid oxides is a very complex phenomenon and is very sensitive to experimental conditions, which can be affected by various factors such as oxygen partial pressure, temperature, oxide category, metal category, roughness of solid surface, and so on. In this study, 15 factors were used in the neural network model: 1) physical properties of metals (atomic mass, density, melting point T_m, surface energy at experimental temperature and oxygen partial pressure, equilibrium partial pressure P_M); 2) oxide physical properties (molecular weight, surface energy, formation free energy, bandgap energy, dissolution energy, equilibrium oxygen partial pressure P_O); 3) experimental conditions (temperature T and oxygen partial pressure P); 4) mixed features (ratio of T/T_m and P/P_M). Metal viscosity was not used as an input parameter of metal physical properties, since this can be reflected by the input parameters in our model such as metal category parameter, experimental temperature, and so on. Surface energies of oxides are calculated by using the empirical equations shown in Table 1. Surface energies of metals are obtained from experimental data and the values from Iida’s empirical equation¹⁴⁷⁾ were used when the experimental data are not available. All the data can be found in the Appendix data file of this paper.

Table 1. Surface free energies of solid oxides.

Oxide	Surface energy of solid oxide [mJ/m2]	Oxide	Surface energy of solid oxide [mJ/m2]
Al2O3	2559−0.784 T143)	Cr2O3	925−0.2 T143)
MgO	2600−0.476 T14)	Fe3O4	2450−0.49 T143)
CaO	2200−0.381 T143)	Ti2O3	Supposing same of Cr2O3
SiO2 (solid)	952−0.193 T143)	TiO	2631−0.476 T *)
SiO2 (glass)	304+0.031(T−2073)144)	NiO	2473−0.497 T *)
BeO	2140−0.321 T143)	CoO	2331−0.468 T *)
ZrO2	1432−0.431 T84)	ZnO	610−0.1 T112)
HfO2	1940−0.218 T *)	CdO	530−0.1 T112)
UO2	1507−0.346 T84)	Y2O3	2278−0.391 T84)
ThO2	1562−0.240 T145)	Sc2O3	2289−0.347 T *)
TiO2	800−0.167 T143)	CeO2	2465−0.563 T146)
La2O3	Supposing same as Y2O3

Note: [*]  indicates the calculated value by Bruce’s method¹⁴³⁾ in this work.

2.3. Neural Network Model Learning

Contact angles of liquid metals and solid oxides have extensively been studied. 1155 data for different metals and oxides are collected from literature to carry out neural network modeling study. 35 data points were randomly selected as the test data and the remaining 1120 data points were used as the learning data. These data together with the input factor values are shown in the Appendix data file.

With 1120 teaching data, an extensive neural network learning study has been carried out to find a good reproduction of the measured contact angle values. The averaged value of the contact angle is 119.46° for the teaching data. The performance of the neural network model can be evaluated by its ability to reproduce the experimental data, and also by the RMSE between experimental data and model predictions.

The number of unit from 5 to 25 in the middle layer of the neural network was tested. It was found that the unit number is a very important factor for the prediction accuracy and 21 units in the middle layer give the best predictions among the tested unit number. In addition, the iteration number also shows an important influence on the model prediction accuracy. 3000 000 iteration steps give a good reproduction on the measured contact angle values. A further increase of the iteration steps did not significantly reduce the RMSE value. Furthermore, with 21 middle units after 3000 000 iteration steps, the RMSE values decreased to 6.09°, which is also the lowest one among the tries. After the learning process, the neural network model was built up and the weight and threshold values, namely W_ki and V_j in Eqs. (5) and (6) were obtained. They are shown in Tables 2 and 3.

Table 2. Values of the weight W_ki in Eq. (5) for different features from the input layer to the middle layer.

i k	0 (Threshold)	1 (Metal Atom weight)	2 (Oxide Molecular weight)	3 (Metal density)	4 (Experimental temperature)	5 (Metal melting point)	6 (Ratio of T/Tm)	7 (LOG(P))
1	−13.157618	3.677158	3.48651	−14.107415	2.574135	−17.913869	2.902398	0.574981
2	47.110285	−21.589817	6.799588	−70.296503	28.146439	34.359573	27.082195	−25.389288
3	−9.106048	−15.952521	40.377653	34.405175	−88.484703	−4.5841	−0.076351	−10.091526
4	−8.114525	6.907063	0.307229	−6.613052	9.295134	10.38418	−2.501382	3.145005
5	98.392421	−23.803545	−6.368767	48.27999	35.06948	95.581382	−12.596492	15.0841
6	−10.488686	−3.890282	−1.27701	−44.883533	32.538177	44.324579	30.790406	−21.887752
7	33.2439	−10.675697	−0.57965	8.552528	8.026159	−5.240994	−44.475128	11.096651
8	18.63742	39.798917	21.9272	−36.714755	−14.472718	23.289496	−28.447732	−6.15118
9	−21.75575	9.467659	6.239868	−13.2249	12.958909	13.18242	9.82509	−8.794957
10	30.264045	−39.531026	27.917596	39.247042	−64.443924	9.387938	38.874968	57.534985
11	10.919098	−12.517449	−4.467588	21.41222	−6.579781	22.590998	17.905201	7.080767
12	42.290284	−15.388507	−20.571126	9.323671	−16.426145	10.9247	−7.426906	−23.161685
13	9.799758	9.114833	7.936738	−4.527459	−9.036674	19.377565	3.370816	6.992657
14	59.047097	2.923191	−37.727032	50.887432	3.572813	−58.816803	6.331843	68.052101
15	34.082905	−77.578552	5.285045	12.119268	−3.216109	−19.221038	−19.250123	19.506847
16	9.788786	−9.248943	−6.325944	−16.095361	−30.59308	26.909809	18.958583	−7.015278
17	14.200113	26.785361	−9.611796	−8.48018	−32.542539	35.579447	9.395738	6.441902
18	21.576107	4.672164	−2.713963	−34.140328	14.60989	−3.850233	−2.150464	13.441372
19	−11.637671	−7.001087	18.043826	72.001438	88.037251	−62.420027	−7.680913	−19.413479
20	−41.714638	−8.478709	−12.774374	−31.672134	−93.09468	−13.882372	−53.127338	1.73913
21	3.804798	−0.477909	3.735287	−23.010611	11.253506	−16.52848	−45.485753	−5.765915

i k	8 (Metal Interfacial energy)	9 (Oxide interfacial energy)	10 (Oxide formation energy)	11 (LOG(PM))	12 (LOG(P/PM))	13 (LOG(PO))	14 (Bandgap energy)	15 (Oxide dissolution energy)
1	2.050167	10.84492	−0.609292	15.869274	−3.600181	−27.074504	0.033413	−26.208471
2	−11.20246	65.973282	16.718941	−86.756519	33.483557	36.79991	62.889528	−0.562409
3	−36.983219	−30.518024	13.642751	−7.277257	13.479553	20.976547	−0.718593	−21.54
4	9.920983	−5.788876	11.224826	−0.820302	8.477692	−34.357162	−25.186081	−7.974449
5	−3.946517	4.868449	32.525316	6.303241	−90.918353	−25.308515	−7.95869	3.023236
6	−57.252722	−55.984978	−34.479118	27.977644	−21.34108	−15.160078	7.830161	3.231294
7	7.939572	−3.493621	−3.672774	−16.967271	−10.267838	36.686299	−0.001635	30.290145
8	8.354464	−35.948102	8.017438	−22.422144	2.308555	21.24372	−13.9619	−27.888862
9	−13.625163	−0.641735	−1.676073	−0.630806	17.912072	−39.227855	6.086247	−13.300356
10	4.691315	14.153655	35.135617	−4.524546	−13.143878	−31.745499	−18.279539	−77.681988
11	−2.153919	2.712234	5.163552	−4.472245	−3.44453	−7.50477	−6.571267	37.474244
12	31.648869	−9.710927	3.846651	−8.147836	−32.721458	59.047098	−17.657495	7.215088
13	−10.889103	−6.004571	−11.706284	13.806547	−19.147614	−5.526528	1.032376	−13.915532
14	−13.009811	15.771479	14.220046	−14.215645	−25.996128	25.470305	−17.229281	80.903148
15	12.392228	−16.518216	−14.802167	6.832304	−31.639359	49.910673	−2.307799	−12.02405
16	13.948549	−6.950136	3.768944	27.434044	−36.406257	−9.384346	−4.688249	−7.548123
17	−7.719352	−1.439564	−17.393727	18.057316	−28.177995	−22.894115	9.118224	47.220679
18	4.154217	−4.037693	−8.378663	−21.758904	3.766087	40.710607	−4.615127	67.112829
19	14.277134	−40.110433	−38.98086	−13.414004	19.276226	−19.849681	13.213904	110.763986-
20	0.742577	12.53868	−6.042357	37.564996	3.204716	−12.809977	13.172762	31.724272
21	7.887097	−8.319309	−0.413945	−10.069599	4.063288	26.77074	−32.243309	49.83766

Table 3. Values of the weights V_j in Eq. (6) from the middle layer unit to the output layer.

j	Vj
0	−0.125496
1	2.122034
2	0.600867
3	1.414296
4	−3.1663
5	−1.438243
6	1.437859
7	3.960889
8	3.829335
9	−2.271777
10	−2.177227
11	1.383199
12	2.218971
13	−1.377287
14	−0.858219
15	−2.188219
16	−2.792127
17	1.341027
18	−0.628638
19	0.958893
20	1.28891
21	2.692402

3. Results and Discussion

3.1. Neural Network Prediction and its Comparison with Experimental Data

Figure 3 shows the comparison of the neural network predictions and the experimental data for both the 1120 learning data and the 35 test data. It can be seen that only 3 predictions are slightly over 20% deviation from experimental values among the total 1155 data, and 10 predictions are over 15% deviation from the experimental data. Furthermore, the scattered data points have been explained in Fig. 3. The causes for the low prediction accuracy data are generally as follows: 1) the lack of enough experimental data for that metal and oxide category, which limits a good neural network learning for that specific metal and oxide; 2) the accuracy of the experimental data is in doubt. In addition, Fig. 4 shows the predicted value distribution of the contact angle with different deviation percentages from the experimental data in the range of 120° to 130°. It can be seen that over 95% of predictions are within 10% deviation from the experimental data. Due to that the contact angle or wettability between molten metals and solid oxides is sensitive to the experimental conditions, experimental measured data are often in a large deviation. Oxygen partial pressure in atmosphere and the impurities in both liquid metals and solid oxides are normally the reasons for such data scattering. Therefore, the above developed neural network model can properly reproduce the experimental data with a reasonably good accuracy. Furthermore, it is a general model for 21 kinds of metals and 14 kinds of solid oxides. Currently, there is no such a general model or an empirical equation, which can predict the contact angles of so many different kinds of metals and oxides. Therefore, the current model is very valuable and provides an efficient way to evaluate the contact angle of different metals and oxides.

Fig. 3.

Comparison of neural network predictions and experimental data for both the learning data and the test data. (Online version in color.)

Fig. 4.

Distribution of prediction deviation from experimental data for the contact angle in the range of 120° to 130°. (Online version in color.)

3.2. Factors Influence on Wettability

In the above, 15 features were used to build up the neural network model. Some factors were mainly used to recognize different metal and oxide category. They are the physical property features for metals and oxides such as atom weight or molecular weight, density, melting point, and so on. Furthermore, factors such as the oxide formation energy and the bandgap energy can also distinguish different oxides. In addition, atom interaction could be reflected by the oxide formation energy and melting point. As is known, the surface tension of solid oxide, surface tension of liquid metal, interfacial tension between metal and oxide, and the contact angle have a relationship based on the force balance as shown in Eq. (2). Therefore, the surface tensions of both metals and oxides were a direct way to be used as the input factors to predict the contact angle values.

Some factors such as surface tension, oxide formation energy are dependent on temperature or surfactant elements such as oxygen. Therefore, experimental conditions must also be the input factors in the model. Also, different metals or oxides may have similar values of these properties, which are dependent on temperature and oxygen pressure, among different measurements. This is due to that the experimental temperature and oxygen partial pressure are often different for different measurements. Therefore, it is important to let the neural network model distinguish different materials and different experimental conditions, as much as possible, by using enough factors. A direct plot of all the 1155 experimental data of contact angle values as a change of each factor cannot find a general change trend. This is due to that it cannot guarantee all the other factors the same while one factor was changed among all the 1155 measurements, and experiment measurements are carried out in various conditions. Therefore, we ignored one feature each time and carried out the neural network modeling study, with 21 middle layer units and 300000 iteration steps. This aims to generally show the influence of the ignored factor in the neural network model on the contact angle predictions.

Figure 5 shows the influence of the ignored factor on the neural network model performance. It can be seen that the metal melting point in Fig. 5(f) has only a little influence on the model predictions. The influences of the oxide surface energy in Fig. 5(b), bandgap energy in Fig. 5(e) and the oxide formation energy in Fig. 5(h) on the model prediction are also small. However, the oxygen partial pressure in Fig. 5(c) shows a large influence on the predictions. This means that it is an important factor for the contact angle predictions. Metal surface energy in Fig. 5(a), experimental temperature in Fig. 5(d), and metal equilibrium corresponding oxygen partial pressure in Fig. 5(g) show some influence on the model predictions in a similar level. It is required to mention that the metal surface energy and metal equilibrium oxygen pressure are also a function of temperature. Furthermore, the metal surface energy is also influeced by the oxygen partial pressure. In order to clearly show the infulence of experimental oxygen partial pressure on the contact angle, we plotted the contact angle data of Fe-oxides varying with oxygen partial pressure, as shown in Fig. 6, while all the other factors are kept the same. It can be seen that the contact angle is significantly affacted by the oxygen partial pressure. The values decrease with an increased oxygen partial pressure when the oxygen pressure is larger than 10⁻¹⁷ atm. However, the change trends for different oxides are different. Specifically, the contact angle of Fe–Al₂O₃ with 0.3 wt% of SiO₂ shows a fast change over oxygen partial pressure, while the contact angle of Fe–Al₂O₃ with 0.01 wt% of SiO₂ shows a slow change. Contact angle of Fe–MgO oxide also shows a decrease as the increase of the oxygen partial pressure. However, when the oxygen pressure is lower than 10⁻¹⁷ atm, the contact angle value increases with an increased oxygen partial pressure. Therefore, oxygen pressure is an important parameter for the contact angle. As for the neural network model, the expriemntal oxygen partial pressure also influence the parameter of the metal surface energy. The expeirmental temperature parameter affects the metal surface energy, oxide surface energy, oxide formation energy, equilibrium oxygen partial pressure, and so on. Therefore, these parameters also reflect the influence of the experimental temperature in the neural network model. This is also the reason that simply ignoring one factor of the total 15 factors did not significantly change the model performance. The other factors can still work to distuigish the metal and oxide catagergies and the experimental conditions. Therefore, the current developed neural network model did not show a significant deterioried performace even one of the factors is ignored.

Fig. 5.

Influence of one ignored factor on the neural network model performance. (Online version in color.)

Fig. 6.

Contact angle change with oxygen partial pressure for Fe-oxide system. (Online version in color.)

3.3. Model Predictions on Contact Angle of Fe and CeO₂

The aim of developing such a neural network model is to predict contact angles of different metals and oxides under various conditions. So far, the contact angle data of Fe and CeO₂ are very rare. In this study, no contact angle data of Fe and CeO₂ were available to train the neural network model. However, such information is important for the production of Rare Earth Metal alloyed steel. Therefore, the developed neural network model was used to evaluate the contact angles of Fe and CeO₂ in the following to show the model performance. Different oxygen partial pressures and temperatures were considered. Table 4 shows the input data and the predicted contact angles by using the developed neural network model. It can be seen that the contact angle value is estimated to be 150° when the oxygen partial pressure was 10⁻¹⁵ atm and the temperature was 1873 K. The value decreases with an increased oxygen partial pressure, with the value of around 118° under conditions of 1873 K and 10⁻¹³ atm, while the predicted value increases to 147° when the temperature is 1823 K under the same oxygen partial pressure condition. For the oxygen pressure increasing from 10⁻¹³ to 10⁻¹¹ atm, the predicted value firstly decreases to 107° and then increases to 117° under the temperature of 1873 K. This behavior may be due to the deviation of the model predictions. However, considering that there is no training data of Fe and CeO₂ for the developed neural network model, this kind of deviation may be unavoidable. In future, reliable data are still required to further improve the neural network model.

Table 4. Input data for Fe and CeO₂ and the predicted contact angles by the developed model.

A	B	C	D	E	F	G	H	I	J	K	L	M	N	O	P
55.84	172	6.98	1873	1811	1.0342	−15.0000	1775	1410.5	−615099	−8.0188	−6.9812	−17.15	0.266	19.10	150.46
55.84	172	6.98	1873	1811	1.0342	−12.9536	1737.5	1410.5	−615099	−8.0188	−4.9348	−17.15	0.266	19.10	118.12
55.84	172	6.98	1873	1811	1.0342	−11.9994	1656.7	1410.5	−615099	−8.0188	−3.9806	−17.15	0.266	19.10	107.53
55.84	172	6.98	1873	1811	1.0342	−10.9536	1368.3	1410.5	−615099	−8.0188	−2.9348	−17.15	0.266	19.10	117.32
55.84	172	7.02	1823	1811	1.0066	−12.9536	1860	1438.7	−625374	−8.4383	−4.5617	−16.65	0.291	19.41	147.78

Note: A-Fe atom number; B-CeO₂ molecular weight; C-Fe density, g/cm³; D-Temperature, K; E-Metal melting point, K; F-Ratio of D/E; G-LOG(Oxygen pressure); H-Metal surface tension, mJ/m²; I-Oxide surface tension, mJ/m²; J-Oxide formation energy, J; K-LOG(Fe equilibrium oxygen pressure); L-(G-K); M-LOG(Oxide equilibrium oxygen pressure); N-Oxide bandgap energy, eV; O-Oxide dissolution energy divided by R and T; P-Predicted contact angle value.

4. Conclusions

A neural network model was built up in this study to predict the contact angles of different metals and oxides. This is a universal model which can be used for the contact angle predictions of 21 metals and 14 oxides. 15 factors were used in the model to distinguish different metal and oxide categories. The results show that the current model can properly reproduce the experimental data, where only 3 predictions among the total 1155 predictions are over 20% deviation from the experimental data. All the predictions on the test data were within 20% deviation from the experimental value. Furthermore, 95.5% of data in the range of 120° to 130° were within 10% deviation from the experimental values. These illustrate a good performance of the current model. Furthermore, factors such as oxygen partial pressure and surface tension of molten metal were found to be important for a good prediction. In addition, contact angles of Fe and CeO₂ were predicted, with the value of around 150° under 1873 K and oxygen partial pressure of 10⁻¹⁵ atm.

Acknowledgments

Peiyuan NI wants to thank JSPS (Japan Society for the Promotion of Science) who ever supported his stay at Osaka University as a JSPS International Research Fellow. Also, thanks for the support from the Fundamental Research Funds for the Central Universities (N2025019) and the National Natural Science Foundation of China (Grant No. 51704062).

References

• 1)   E.  Ricci and  R.  Novakovic: Gold Bull., 34 (2001), 41.
• 2)   K.  Mukai and  Z.  Yuan: Mater. Trans. JIM, 41 (2000), 338.
• 3)   K.  Sasai: ISIJ Int., 56 (2016), 1013.
• 4)   K.  Nogi,  N.  Iwamoto and  K.  Ogino: Trans. JWRI, 20 (1991), 165.
• 5)   Y.  Okazaki,  H.  Ishida,  K.  Suenaga and  T.  Hidaka: Weld. Int., 26 (2012), 593.
• 6)   E.  Saiz,  R. M.  Cannon and  A. P.  Tomsia: Annu. Rev. Mater. Res., 38 (2008), 197.
• 7)   J.  Li: Mater. Chem. Phys., 47 (1997), 126.
• 8)   F.  Valenza,  M. L.  Muolo,  A.  Passerone and  A. M.  Glaeser: J. Eur. Ceram. Soc., 33 (2013), 539.
• 9)   I.  Villarreal,  C.  Jacobson,  A.  Leming,  Y.  Matus,  S.  Visco and  L.  De Jonghe: Electrochem. Solid-State Lett., 6 (2003), A178.
• 10)   I. A.  Fisher and  A. T.  Bell: J. Catal., 178 (1998), 153.
• 11)   N. P.  Padture,  M.  Gell and  E. H.  Jordan: Science, 296 (2002), 280.
• 12)   J. E.  McDonald and  J. G.  Eberhart: Trans. Metall. Soc. AIME, 233 (1965), 512.
• 13)   G. R.  Belton: Metall. Trans. B, 7 (1976), 35.
• 14)   H.  Sun,  K.  Nakashima and  K.  Mori: ISIJ Int., 37 (1997), 323.
• 15)   S.-Y.  Cho and  J.-H.  Lee: Korean J. Mater. Res., 18 (2008), 18.
• 16)   K.  Nogi,  M.  Tsujimoto,  K.  Ogino and  N.  Iwamoto: Acta Metall. Mater., 40 (1992), 1045.
• 17)   R. N.  Wenzel: Ind. Eng. Chem., 28 (1936), 988.
• 18)   A. I.  Timofeev and  V. P.  Chentsov: Russ. J. Phys. Chem. A, 89 (2015), 114.
• 19)   M.  Hanao,  M.  Kawamoto,  T.  Tanaka and  M.  Nakamoto: ISIJ Int., 46 (2006), 346.
• 20)   M.  Nakamoto,  M.  Hanao,  T.  Tanaka,  M.  Kawamoto,  L.  Holappa and  M.  Hämäläinen: ISIJ Int., 47 (2007), 1075.
• 21)   Y.  Haraguchi,  M.  Nakamoto,  M.  Suzuki,  K.  Fuji-ta and  T.  Tanaka: ISIJ Int., 58 (2018), 1007.
• 22)   J.  Yeon,  P.  Ni,  M.  Nakamoto and  T.  Tanaka: Calphad, 64 (2019), 267.
• 23)   K.  Ogino,  K.  Nogi and  Y.  Koshida: Tetsu-to-Hagané, 59 (1973), 1380 (in Japanese).
• 24)   K.  Ogino,  A.  Adachi and  K.  Nogi: Tetsu-to-Hagané, 59 (1973), 1237 (in Japanese).
• 25)   K.  Ogino,  K.  Nogi and  O.  Yamase: Tetsu-to-Hagané, 66 (1980), 179 (in Japanese).
• 26)   K.  Nogi and  K.  Ogino: Can. Metall. Q., 22 (1983), 19.
• 27)   K.  Ogino,  K.  Nogi and  S.  Kinashi: Tetsu-to-Hagané, 60 (1974), S124 (in Japanese).
• 28)   K.  Ogino,  H.  Taimatsu and  F.  Nakatani: J. Jpn. Inst. Met., 46 (1982), 957 (in Japanese).
• 29)   K.  Ogino,  H.  Taimatsu and  F.  Nakatani: J. Jpn. Inst. Met., 46 (1982), 1049 (in Japanese).
• 30)   H.  Taimatsu,  K.  Ogino and  F.  Nakatani: J. Jpn. Inst. Met., 50 (1986), 176 (in Japanese).
• 31)   K.  Ogino and  K.  Nogi: Tetsu-to-Hagané, 73 (1986), S112 (in Japanese).
• 32)   K.  Nogi,  K.  Ogino and  T.  Kurachi: Tetsu-to-Hagané, 74 (1988), 648 (in Japanese).
• 33)   K.  Ogino and  H.  Taimatsu: J. Jpn. Inst. Met., 43 (1979), 871 (in Japanese).
• 34)   K.  Nogi,  K.  Oishi and  K.  Ogino: J. Jpn. Inst. Met., 52 (1988), 72 (in Japanese).
• 35)   H.  Taimatsu,  M.  Abe,  F.  Nakatani and  K.  Ogino: J. Jpn. Inst. Met., 49 (1985), 523 (in Japanese).
• 36)   K.  Nogi,  H.  Takeda and  K.  Ogino: J. Jpn. Inst. Met., 53 (1989), 927 (in Japanese).
• 37)   K.  Nogi and  K.  Ogino: J. High Temp. Soc., 15 (1989), 204 (in Japanese).
• 38)   K.  Nogi and  K.  Ogino: J. High Temp. Soc., 16 (1990), 20 (in Japanese).
• 39)   K.  Nogi and  K.  Ogino: Refractories (Taikabutsu), 41 (1989), 27 (in Japanese).
• 40)   P.  Shen,  H.  Fujii and  K.  Nogi: J. Mater. Res., 20 (2005), 940.
• 41)   J.-H.  Lee,  H.  Ishimura and  T.  Tanaka: Scr. Mater., 54 (2006), 1369.
• 42)   T.  Yoshikawa,  K.  Motosugi,  T.  Tanaka and  M.  Ueda: Tetsu-to-Hagané, 97 (2011), 361 (in Japanese).
• 43)   K.  Mukai,  H.  Sakao and  K.  Sano: J. Jpn. Inst. Met., 31 (1967), 923 (in Japanese).
• 44)   N.  Takiuchi,  T.  Taniguchi,  N.  Shinozaki and  K.  Mukai: J. Jpn. Inst. Met., 55 (1991), 44 (in Japanese).
• 45)   N.  Takiuchi,  T.  Taniguchi,  Y.  Tanaka,  N.  Shinozaki and  K.  Mukai: J. Jpn. Inst. Met., 55 (1991), 180 (in Japanese).
• 46)   K.  Nakashima,  K.  Takihira,  K.  Mori and  N.  Shinozaki: J. Jpn. Inst. Met., 55 (1991), 1199 (in Japanese).
• 47)   N.  Shinozaki,  N.  Echida,  K.  Mukai,  Y.  Takahashi and  Y.  Tanaka: Tetsu-to-Hagané, 80 (1994), 748 (in Japanese).
• 48)   N.  Shinozaki,  H.  Mizuguchi and  K.  Mukai: J. Jpn. Inst. Met., 62 (1998), 1177 (in Japanese).
• 49)   N.  Shinozaki,  M.  Sonoda and  K.  Mukai: J. Jpn. Inst. Met., 62 (1998), 71 (in Japanese).
• 50)   N.  Shinozaki,  N.  Fukami,  H.  Kaku and  K.  Mukai: J. Jpn. Inst. Met., 63 (1999), 1009 (in Japanese).
• 51)   H.  Matsumoto,  M. R.  Locatelli,  K.  Nakashima,  A. M.  Glaeser and  K.  Mori: Mater. Trans. JIM, 36 (1995), 555.
• 52)   K.  Nakashima,  H.  Matsumoto and  K.  Mori: Acta Mater., 48 (2000), 4677.
• 53)   J.  Zhu,  A.  Kamiya,  T.  Yamada,  W.  Shi,  K.  Naganuma and  K.  Mukai: Mater. Sci. Eng. A, 327 (2002), 117.
• 54)   Z.-F.  Yuan,  K.  Mukai,  K.  Takagi and  M.  Ohtaka: J. Jpn. Inst. Met., 65 (2001), 21 (in Japanese).
• 55)   Z.-F.  Yuan,  K.  Mukai,  K.  Takagi,  M.  Ohtaka,  W.-L.  Huang and  Q.-S.  Liu: J. Colloid Interface Sci., 254 (2002), 338.
• 56)   M.  Kishimoto,  K.  Mori and  Y.  Kawai: J. Jpn. Inst. Met., 48 (1984), 413 (in Japanese).
• 57)   H.  Hashikura,  Y.  Morisada,  H.  Fujii,  K.  Kanda,  K.  Kubota and  M.  Isaka: J. Jpn. Soc. Precis. Eng., 79 (2013), 547 (in Japanese).
• 58)   K.  Aratani and  Y.  Tamai: J. Ceram. Assoc. Jpn., 89 (1981), 480 (in Japanese).
• 59)   V. N.  Eremenko,  Y. V.  Naidich and  A. A.  Nosonovich: Russ. J. Phys. Chem., 34 (1960), 566.
• 60)   V. N.  Eremenko: The Role of Surface Phenomena in Metallurgy, ed. by V. N. Eremenko, Consultants Bureau, New York, (1963), 1.
• 61)   V. N.  Eremenko,  V. I.  Nizhenko and  L. I.  Sklyarenko: Sov. Powder Metall. Met. Ceram., 4 (1965), 372.
• 62)   Y. V.  Naidich: Kontaktnie Javlenia ν Metallicheskikh Rasplabakh (Contact Phenomena in Metallic Melts), Naukova Dumka, Kiev, (1972), 62 (in Russian).
• 63)   J. V.  Naidich: Prog. Surf. Membr. Sci., 14 (1981), 353.
• 64)   Y. V.  Naidich and  V. S.  Zhuravlev: Refractories, 15 (1974), 55.
• 65)   J. V.  Naidich and  J. N.  Chuvashov: J. Mater. Sci., 18 (1983), 2071.
• 66)   J. V.  Naidich,  V. S.  Zhuravljov and  N. I.  Frumina: J. Mater. Sci., 25 (1990), 1895.
• 67)   V. S.  Zhuravlev,  N. A.  Krasovskaya,  V. S.  Sudovtsova and  Y. V.  Naidich: Proc. 2nd Int. Conf. High Temperature Capillarity, Foundary Research Institute, Cracow, Poland, (1997), 158.
• 68)   A. V.  Durov,  Y. V.  Naidich and  B. D.  Kostyuk: J. Mater. Sci., 40 (2005), 2173.
• 69)   O. V.  Durov and  T. V.  Sydorenko: Proc. Symp. A, Advanced Composite Materials: Technologies, Properties, Applications E-MRS Fall Meeting 2012, Ukrainian Materials Research Society, Ukraine, (2012), 121
• 70)   A. V.  Durov: Powder Metall. Met. Ceram., 50 (2011), 552.
• 71)   M.  Humenik, Jr. and  W. D.  Kingery: J. Am. Ceram. Soc., 37 (1954), 18.
• 72)   C. R.  Manning, Jr. and  R. F.  Stoops: J. Am. Ceram. Soc., 51 (1968), 415.
• 73)   B. G.  Chernov: Izv. Vuz. Chern. Metall., (1983), 4 (in Russian).
• 74)   S.  Nakano and  M.  Ohtani: J. Jpn. Inst. Met., 34 (1970), 562 (in Japanese).
• 75)   P.  Nikolopoulos,  S.  Nazaré and  F.  Thümmler: J. Nucl. Mater., 71 (1977), 89.
• 76)   P.  Nikolopoulos and  G.  Ondracek: J. Nucl. Mater., 98 (1981), 306.
• 77)   P.  Nikolopoulos and  S.  Agathopoulos: J. Eur. Ceram. Soc., 10 (1992), 415.
• 78)   P.  Nikolopoulos: J. Mater. Sci., 20 (1985), 3993.
• 79)   G.  Angelopoulos,  P.  Nikolopoulos and  U.  Jauch: Materialwiss. Werkstofftech. (Mater. Sci. Eng. Technol.), 19 (1988), 168.
• 80)   P.  Nikolopoulos and  D.  Sotiropolou: J. Mater. Sci. Lett., 6 (1987), 1429.
• 81)   P.  Nikolopoulos,  G.  Ondracek and  D.  Sotiropoulou: Ceram. Int., 15 (1989), 201.
• 82)   D.  Sotiropoulou and  P.  Nikolopoulos: J. Mater. Sci., 28 (1993), 356.
• 83)   A.  Tsoga and  P.  Nikolopoulos: J. Mater. Sci., 31 (1996), 5409.
• 84)   G.  Triantafyllou,  G. N.  Angelopoulos and  P.  Nikolopoulos: J. Mater. Sci., 45 (2010), 2015.
• 85)   N. F.  Gregorenko,  A. I.  Stegny,  I. E.  Kasich-Pilipenko,  Y. V.  Naidich and  V. V.  Pasichny: Proc. 1st Int. Conf. High Temperature Capillarity, Reproprint, Bratislava, (1995), 123.
• 86)   A. C. D.  Chaklader,  A. M.  Armstrong and  S. K.  Misra: J. Am. Ceram. Soc., 51 (1968), 630.
• 87)   T. E.  O’Brien and  A. C. D.  Chaklader: J. Am. Ceram. Soc., 57 (1974), 329.
• 88)   H.  Taimatsu,  T.  Tani and  H.  Kaneko: J. Mater. Sci., 31 (1996), 6383.
• 89)   H.  Nakae and  Y.  Matsuda: J. Jpn. Foundry Eng. Soc., 71 (1999), 28.
• 90)   B. C.  Allen and  W. D.  Kingery: Trans. Metall. Soc. AIME, 215 (1959), 30.
• 91)   J. G.  Duh,  W. S.  Chien and  B. S.  Chiou: J. Mater. Sci. Lett., 8 (1989), 405.
• 92)   D. H.  Bradhurst and  A. S.  Buchanan: J. Phys. Chem., 63 (1959), 1486.
• 93)   S. I.  Popel,  L. A.  Smirnov,  B. V.  Tsarevskii,  N. K.  Dzhemilev and  A. I.  Pastukhov: Izv. Akad. Nauk SSSR Met., (1965), 62 (in Russian).
• 94)   I.  Jimbo and  A. W.  Cramb: ISIJ Int., 32 (1992), 26.
• 95)   H.  Shibata,  X.  Jiang,  M.  Valdez and  A. W.  Cramb: Metall. Mater. Trans. B, 35 (2004), 179.
• 96)   M.  Demeri,  M.  Farag and  J.  Heasley: J. Mater. Sci., 9 (1974), 683.
• 97)   R.  Sangiorgi,  A.  Passerone and  R.  Minisini: Metal Science Research Vol. 14, Plenum Press, New York, (1986), 445.
• 98)   R.  Sangiorgi,  M. L.  Muolo and  N.  Eustathopoulos: Proc. 1st Int. Conf. on High Temparature Capillarity, Reproprint, Bratislava, (1995), 148.
• 99)   V.  Merlin and  N.  Eustathopoulos: J. Mater. Sci., 30 (1995), 3619.
• 100)   P.  Kritsalis,  J. G.  Li,  L.  Coudurier and  N.  Eustathopoulos: J. Mater. Sci. Lett., 9 (1990), 1332.
• 101)   M. L.  Muolo,  F.  Valenza,  A.  Passerone and  D.  Passerone: Mater. Sci. Eng. A, 495 (2008), 153.
• 102)   I.  Rivollet,  D.  Chatain and  N.  Eustathopoulos: Acta Metall., 35 (1987), 835.
• 103)   V.  Ghetta,  J.  Fouletier and  D.  Chatain: Acta Mater., 44 (1996), 1927.
• 104)   V.  De Jonghe,  D.  Chatain,  I.  Rivollet and  N.  Eustathopoulos: J. Chim. Phys., 87 (1990), 1623.
• 105)   I.  Harter,  P.  Dusserre,  T.  Duffar,  J.-Ph.  Nabot and  N.  Eustathopoulos: J. Cryst. Growth, 131 (1993), 157.
• 106)   J.-G.  Li: Scr. Metall. Mater., 30 (1994), 337.
• 107)   J.-G.  Li and  J.  Gao: Mater. Lett., 60 (2006), 1323.
• 108)   E. N.  Hodkin: J. Nucl. Mater., 97 (1981), 27.
• 109)   S.  Barzilai,  M.  Aizenshtein,  E.  Shapiro-Tsoref,  N.  Froumin and  N.  Frage: Int. J. Adhes. Adhes., 27 (2007), 358.
• 110)   E. L.  Dubinin,  S. G.  Gushchin,  A. I.  Timofeev and  L. V.  Uzberg: Известия Высщих Заведений Цветная Металлургия (Izv. Vyssh. Uchebn. Zaved., Tsvetn. Metall., Proc. Higher Schools Nonferr. Metall.), 5 (1976), 151.
• 111)   S. G.  Gushchin,  N. A.  Vatolin,  É. L.  Dubinin and  A. I.  Timofeev: Refractories, 18 (1977), 739.
• 112)   D. T.  Livey and  P.  Murray: Proc. Warmfeste und Korrosionbeständige Sinterwerkstoffe, 2. Plansee Seminar, Metallwerk Plansee GmbH., Reutte, (1956), 375.
• 113)   R.  Sangiorgi,  M. L.  Muolo,  D.  Chatain and  N.  Eustathopoulos: J. Am. Ceram. Soc., 71 (1988), 742.
• 114)   I.  Rivollet,  D.  Chatain and  N.  Eustathopoulos: J. Mater. Sci., 25 (1990), 3179.
• 115)   J.-G.  Li,  D.  Chatain,  L.  Coudurier and  N.  Eustathopoulos: J. Mater. Sci. Lett., 7 (1988), 961.
• 116)   S. P.  Mehrotra and  A. C. D.  Chaklader: Metall. Trans. B, 16 (1985), 567.
• 117)   G.  Levi,  C.  Scheu and  W.  Kaplan: Interface Sci., 9 (2001), 213.
• 118)   U. M.  Ahmad and  L. E.  Murr: J. Mater. Sci., 11 (1976), 224.
• 119)   H.  Fukui,  J.  Hasegawa,  I.  Ohta and  S.  Matumura: J. Jpn. Prosthodont. Soc., 21 (1977), 106.
• 120)   F. L.  Harding and  D. R.  Rossington: J. Am. Ceram. Soc., 53 (1970), 87.
• 121)   A. J.  Mcevoy,  R. H.  Williams and  I. G.  Higginbotham: J. Mater. Sci., 11 (1976), 297.
• 122)   C. S.  Kanetkar,  A. S.  Kacar and  D. M.  Stefanescu: Metall. Trans. A, 19 (1988), 1833.
• 123)   D.  Chatain,  F.  Chabert,  V.  Ghetta and  J.  Fouletier: J. Am. Ceram. Soc., 76 (1993), 1568.
• 124)   D.  Chatain,  F.  Chabert,  V.  Ghetta and  J.  Fouletier: J. Am. Ceram. Soc., 77 (1994), 197.
• 125)   M.  Diemer,  A.  Neubrand,  K. P.  Trumble and  J.  Rödel: J. Am. Ceram. Soc., 82 (1999), 2825.
• 126)   A.  Amirfazli,  D.  Chatain and  A. W.  Neumann: Colloids Surf. A, 142 (1998), 183.
• 127)   B.  van Muu,  H.-W.  Fenzke and  G.  Neuhof: Neue Hütte, 29 (1984), 128.
• 128)   E.  Kapilashrami,  A. K.  Lahiri,  A. W.  Cramb and  S.  Seetharaman: Metall. Mater. Trans. B, 34 (2003), 647.
• 129)   A.-H.  Liu,  B.-S.  Li,  Y.-W.  Sui and  J.-J.  Guo: Mater. Lett., 89 (2012), 299.
• 130)   M.  Nakamura,  R.  Harada,  T.  Togaya,  Y.  Kuwamura,  K.  Ichiki,  M.  Ohmori and  T.  Degawa: Proc. 11th Spring Academic Seminar, The Japanese Society for Dental Materials and Devices, Tokyo, (1988), 7.
• 131)   T.  Kusakawa,  S.  Yoshida and  J.  Takeuchi: Tetsu-to-Hagané, 57 (1971), S464 (in Japanese).
• 132)   C.  Xuan,  H.  Shibata,  S.  Sukenaga,  P. G.  Jönsson and  K.  Nakajima: ISIJ Int., 55 (2015), 1882.
• 133)   N.  Kaiser,  A.  Cröll,  F. R.  Szofran,  S. D.  Cobb and  K. W.  Benz: J. Cryst. Growth, 231 (2001), 448.
• 134)   M.  Hida,  A.  Sakakibara and  H.  Kamiyabu: J. Jpn. Inst. Met., 53 (1989), 1263 (in Japanese).
• 135)   P.  Nicolopoulos and  G.  Ondracek: Verbundwerkstoffe, Deutsche Gesellschaft für Metallkunde (DGM), Oberursel, (1981), 391.
• 136)   M.  Ueki,  M.  Naka and  I.  Okamoto: J. Mater. Sci. Lett., 5 (1986), 1261.
• 137)   J. J.  Brennan and  J. A.  Pask: J. Am. Ceram. Soc., 51 (1968), 569.
• 138)   P.  Kritsalis,  V.  Merlin,  L.  Coudurier and  N.  Eustathopoulos: Acta Metall. Mater., 40 (1992), 1167.
• 139)   D.  Chatain,  I.  Rivollet and  N.  Eustathopoulos: J. Chim. Phys., 83 (1986), 561 (in French).
• 140)   Z.  Amondarain,  L.  Kolbeinsen and  J. L.  Arana: ISIJ Int., 51 (2011), 733.
• 141)   S. E.  Volkov,  V. A.  Mchedlishvili and  A. M.  Samarin: Доклады Академий Наук СССР (Dokl. Akad. Nauk SSSR, Proc. USSR. Sci. Acad.), 149 (1963), 1131 (in Russian).
• 142)   N.  Sobczak,  R.  Nowak,  R.  Asthana and  R.  Rurgert: Scr. Mater., 62 (2010), 949.
• 143)   R. H.  Bruce: Science of Ceramics, Vol. 2, ed. by G. H. Stewart, Academic Press, London, (1965), 359.
• 144)   W. D.  Kingery: J. Am. Ceram. Soc., 42 (1959), 6.
• 145)   T.  Inoue and  H.-J.  Matzke: J. Am. Ceram. Soc., 64 (1981), 355.
• 146)   N.  Zouvelou,  X.  Mantzouris and  P.  Nikolopoulos: Mater. Sci. Eng. A, 495 (2008), 54.
• 147)   T.  Iida and  R. I. L.  Guthrie: The physical Properties of Liquid Metals, Clarendon Press, Oxford, (1993), 113.