# Relation of EDL Forces between Clay Particles Calculated by Different Methods
## 2 Analysis on the Previous Calculation Methods of EDL Force between Clay Particles
### Analysis  
In the literature, the boundary \(\Gamma\) in Equation (1) is often chosen to enclose a simply connected domain including one particle, as shown in Figure 1[23; 28; 29; 31; 33]. As there exists an equivalence between Equations (3) and (5), it seems that the total EDL force based  
Figure 1: Electrostatic stresses felt by charged clay particle in electrolyte solution.  
on Equation (1) will always be zero. This is obviously wrong while the reason has not yet been investigated [25; 29; 30; 33].  
An arbitrary complete enclosing boundary \(\Gamma\) has three parts \(\Gamma_{0}\), \(\Gamma_{1}\), and \(\Gamma_{2}\), as shown in Figure 2, which was adopted by [PERSON] and [PERSON] [28], while they used Equation (2) to calculate the osmotic force and thus did not mention the present issue. Accordingly, the integration in Equation (2) can be divided into three parts. The integrals on \(\Gamma_{1}\) and \(\Gamma_{2}\) are zero as the potentials on the opposite sides are the same, but the normal has the opposite sign. Therefore, the first integration in Equation (1) over the complete boundary \(\Gamma\) is that over \(\Gamma_{0}\) that encloses the usual simply connected domain. This has been studied in [28; 32].  
Similarly, the integration in Equation (4) can also be divided into three parts, that is:  
\[\oint_{\Gamma}(RM_{\text{j}})\,\text{d}\Gamma=\oint_{\Gamma_{0}}(RM_{\text{j} })\,\text{d}\Gamma_{0}+\oint_{\Gamma_{1}}(RM_{\text{j}})\,\text{d}\Gamma_{1}+ \oint_{\Gamma_{2}}(RM_{\text{j}})\,\text{d}\Gamma_{2} \tag{6}\]  
in which \(RM_{\text{j}}=\phi_{\text{j}}\phi_{\text{j}}n_{\text{i}}-\frac{1}{2}\phi_{ \text{k}}\phi_{\text{k}}n_{\text{j}}\). The third item on the RHS of Equation (6) is zero, as there are the same potential gradients but an opposite normal on the opposite sides of boundary \(\Gamma_{2}\), as shown in Figure 2. However, the first and the second item on the RHS of Equation (6) will not be zero in general. It is especially interesting to notice that the left side in Equation (6) is the minus osmotic repulsive force, while the first item on the RHS of Equation (6) is same as the second integration on the RHS of Equation (1) as boundary \(\Gamma_{0}\) just encloses the simply connected domain. Comparing Equation (6) with Equation (1), it can be concluded that the second item on the RHS of Equation (6), i.e., the integration of the Maxwell stress exactly on the surface of the clay particle, is in fact the total EDL force between clay particles.  
As a summary, both of the forces calculated by integrating the electrostatic force density or the gradient of Maxwell stress over a domain within which one clay particle resides, and the force calculated by the counterpart integration alone of the complete boundary enclosing a multiply connected domain in which the surrounding electrolyte and clay particle are separated, are the osmotic force. The integration of Maxwell stress along the partial boundary enclosing a simply connected domain including one particle represents the contribution of electrical force related to Maxwell stress, while the integration related to electrostatic force density along the same partial boundary denotes the osmotic force. In the following section, a numerical method is adopted to illustrate the above conclusions.  
Figure 2: Two inclined finite charged clay particles in an electrolyte solution.