**2. Model and Methods**

*2.1. Molecular Models*

Wyoming-type MMT, which has been widely simulated, was selected as the model clay in this study, as shown in Figure 1. The initial unit cell parameters are *a* = 5.23 Å and *b* = 9.06 Å, and the c-axis increases with increasing water content, the unit cell angles are *α* = *γ* = 90◦, *β* = 99◦. The x and y axes were chosen parallel to the clay layer, while the *z*-axis was orthogonal to the clay layer. The unit cell of MMT was replicated (4a × 2b × 1c) in the x, y, and z directions so that the supercell contained 8-unit cells with approximate dimensions in the XY plane of 21 Å × 18 Å and 9.6 Å in the *z*-axis. One of every 8 octahedral Al3+ atoms was substituted with an Mg2+ atom and one of every 4 tetrahedral Si4+ atoms was substituted with an Al3+ atom. The isomorphic substitutions were distributed randomly obeying Lowenstein's rule. This extent of substitution led to a layer charge of −0.75 e per unit cell and the unbalanced charge will be compensated by Na+, Ca2+, and Cs+ which are commonly adopted in engineering applications [12,35]. In addition, the interlayer space was randomly filled with a given number *n* of water molecules, the number *n* ranged from 0 to 160, and a total of 23 clay–water-ion systems will be evaluated. The structural formula of the simulated layer of Na- and Cs-MMT is M0.75{Si7.75Al0.25}[Al3.5Mg0.5]O20(OH)4·*n*H2O (M = Na or Cs), whereas for the Ca-MMT, it can be expressed as Ca0.375{Si7.75Al0.25}[Al3.5Mg0.5]O20(OH)4·*n*H2O.

**Figure 1.** Super-cell model of Na-montmorillonite (Mg octahedral are green, Al octahedral and tetrahedral purple, Si tetrahedral brown, O atoms red, H atoms white, and Na ions cyan).
