*2.4. Interfacial Force Model*

Considering the interfacial force **F***s*, the interparticle potential force is defined using the space derivative of potential *E* **r***ij* . **F***s* is localized at the liquid interface by applying it to the liquid elements in the transition region of the interface. The force per unit area **F***s<sup>i</sup>* is then converted into force per unit volume using the expression [36]:

$$\langle \mathbf{F}\_{s} \rangle\_{i} = -2\sigma\_{i} \Big| \mathbf{r}\_{i\bar{j}} \Big|\_{0}^{2} \left( \sum\_{j=1}^{N} E\{ |\mathbf{r}\_{i\bar{j}}| \} \right)^{-1} \cdot \sum\_{j=1}^{N} \frac{\partial E\{ |\mathbf{r}\_{i\bar{j}}| \}}{\partial \mathbf{r}} \frac{\mathbf{r}\_{i\bar{j}}}{|\mathbf{r}\_{i\bar{j}}|} \tag{11}$$

where σ*<sup>i</sup>* is the surface tension or interfacial tension of particle *i*. The Fowkes hypothesis is considered in calculating the interfacial force on the multiphase boundary [37]. The Fowkes hypothesis explains that, in a system in which two immiscible liquid phases (liquid iron and molten slag) are in contact, the elements present at the two-phase interface are subject to forces. At the interface between liquid iron and molten slag, liquid iron interface elements receive the attractive force σ*<sup>m</sup>* equivalent to the "surface tension" of liquid iron and the dispersion force σ*<sup>D</sup>* from molten slag. The force acting on the interface elements of the molten slag can be described similarly. Hence, the interfacial tension σ*ms* is expressed as follows:

$$
\sigma\_{\text{mfs}} = \sigma\_{\text{m}} + \sigma\_{\text{s}} - 2\sigma\_{\text{D}} \tag{12}
$$

This simple hypothesis indicates that the unknown dispersion force and interfacial tension can be calculated explicitly by applying the surface tension as the input and the interfacial tension of the two liquid phases in contact as the conditions. An immiscible blend of liquid iron and molten slag contacting the coke plate is considered, as illustrated in Figure 1. In terms of the tension balance on the solid-gas-liquid triple line in which liquid iron, coke, and gas are in contact with one another, the surface tension σ*<sup>m</sup>* of the liquid iron, surface tension σ*<sup>c</sup>* of the solid phase, and the solid-liquid interfacial tension σ*mc* are assumed to be balanced by the contact angle θ*m*. In other words, Young's equation reflects a horizontal balance among the interfacial tensions: σ*<sup>m</sup>* cos θ*<sup>m</sup>* + σ*mc* = σc. Here, the unknown solid surface tension and the solid-liquid interfacial tension are eliminated from Young's and Fowkes' equations to obtain the following equation:

$$2\cos\theta\_m = 2\frac{\sigma\_D}{\sigma\_m} - 1\tag{13}$$

**Figure 1.** Two liquid droplets in contact on a flat, solid surface. These forces balance one another.

Equation (13) indicates that θ*<sup>m</sup>* is determined by the surface tension of the liquid phase and the dispersion force acting between the different phases. The dispersion force is explicitly defined by this equation, and the static contact angle can be calculated using the potential interparticle model. Furthermore, considering other triple lines, such as that existing between the molten slag, the solid, and the gas, and that between the two liquid phases and the solid, the following equation is obtained:

$$\cos\theta\_{\rm ms} = \frac{\sigma\_{\rm m}}{\sigma\_{\rm m\rm s}} \cos\theta\_{\rm m} - \frac{\sigma\_{\rm s}}{\sigma\_{\rm m\rm s}} \cos\theta\_{\rm s} \tag{14}$$

In Equation (14), θ*<sup>s</sup>* is the contact angle between the molten slag and solid plate, and θ*ms* is the contact angle between the two liquid phases and the solid plate. This equation indicates that the liquid iron-molten slag-solid contact angle θ*ms* is represented by the contact angles θ*<sup>m</sup>* and θ*s*.
