# Relation of EDL Forces between Clay Particles Calculated by Different Methods
## 3 Illustration
### Discussion  
Comparing the calculated forces between the second, third, and fourth column in Table 3, which correspond to Equation (2), Equation (3), and Equation (5), respectively, it can be seen that the forces calculated by Equations (2) and (3) are almost the same except for the last case shown of \(\alpha=60^{\circ}\), \(d=1.0\), and \(\phi_{0}=10\) in Table 3(C2) where a large surface potential and inclination of a clay particle likely lead to a highly nonuniform potential field. The equivalence between the forces calculated by Equations (3) and (5) can only be observed for the case of \(\alpha=0^{\circ}\), \(d=1.0\), and \(\phi_{0}=5\) with a relatively large integral domain shown in Table 3, and the deviation between them increases with increasing angle and potential and decreasing integral domain. This results from the different accuracy of Equations (2), (3) and (5), as explained in Section 3.2. \begin{table}
\begin{tabular}{c c} \hline \hline
**Cases of Integral Domain** & **Description of the Chosen Domain** \\ \hline (1) & The symmetrical half of the numerical model \\ (2) & Middle plane is the bottom boundary, and \(x_{R}=y_{U}=x_{L}=2\) \\ (3) & Middle plane is the bottom boundary, and \(x_{R}=y_{U}=x_{L}=1\) \\ (4) & \(x_{R}=x_{L}=y_{U}=y_{D}=0.5\) while middle plane is the bottom boundary if \(y_{D}<0.5\) \\ (5) & \(x_{R}=x_{L}=y_{U}=y_{D}=0.1\) \\ (6) & \(x_{R}=x_{L}=y_{U}=y_{D}=0.01\) \\ (7) & \(x_{R}=x_{L}=y_{U}=y_{D}=0.001\) \\ (8) & \(x_{R}=x_{L}=y_{U}=y_{D}=0.0001\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cases of the chosen integral domain. +++ ==ERROR: No output for this page==
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Cases of Integral Domain**[FOOTNOTE:]Footnote 1: \(-\iint_{\Omega}\rho_{\text{comb}}(\phi_{1})\phi_{1}\Delta\Omega\) & \(-\iint_{\Omega}\rho_{\text{comb}}(\phi_{1})\Delta\Omega\) & \(\iint_{\Omega}\rho_{\phi}\rho_{\phi}\Delta\  
Comparing the forces between the sixth and last column shown in Table 3, it can be observed that the last column, i.e., the integration of Maxwell stress along a boundary approximate to \(\Gamma_{1}\), is almost the same as the total EDL force in the sixth column, which is calculated using Equation (1). Similarly, the deviation between them increases nonlinearly with surface potential, angle, and distance. Therefore, it can be seen that the integration of Maxwell stress just along the surface of the clay particle is indeed the total EDL force between clay particles, while that along the complete boundary enclosing a multiply connected domain is the osmotic repulsive force. For the ideal case of two infinite parallel charged plates, [PERSON] and [PERSON] [1] used Equation (2) choosing a partial enclosing boundary similar to \(\Gamma_{0}\), as shown in Figure 2, which corresponds to a simply connected domain bounded by the mid-plane and infinity. This method was widely adopted in the later studies [5; 10; 13; 23; 28; 29; 31; 33; 34]. On the contrary, [PERSON] and [PERSON] [41] used Equation (4) choosing an enclosing boundary identical to \(\Gamma_{1}\), as shown in Figure 2, which is just the surface of the enclosed particle. For the ideal case of infinite parallel particles, analytical solutions based on the above two methods are the same [33]. For the actual case of particles with finite size, the contribution of [PERSON]'s stress was usually taken into account numerically by Equation (4) choosing the enclosing boundary similar to \(\Gamma_{0}\)[8; 17; 23] without considering the possible adverse effect of numerical accuracy. However, as illustrated in Table 3, the calculated contribution of [PERSON]'s stress is prone to be affected by the selection of an integral boundary. As an example, the calculated EDL forces for the case of \(\alpha=60^{\circ}\), \(d=1.0\), and \(\phi_{0}=10\) with different integral boundaries listed in Table 2 are shown in Figure 6 in which the characteristic distances between the integral boundary and particle surface presented in Table 3 are also shown.