## 3 Results and Discussion  
### Swelling and Thermodynamics Analysis  
To obtain the water content corresponding to the thermodynamic steady-state in the hydration process of MMT, the strategy of simulating 23 different water content systems was adopted. The interlayer space of the MMT lamellae increases with the increase of water content, and this swelling process can be described by using the parameter of basal spacing. The basal spacing can be obtained from the view of water content by averaging the system volume during the production stage of simulation and defined as in Equation (13):  
\[d_{001}=\frac{\langle V\rangle}{\langle a\rangle\langle b\rangle\sin\langle\alpha\rangle} \tag{13}\]  
where \(<\)\(V\)\(>\) denotes the statistically averaged volume, and \(<\)\(a\)\(>\)\(<\)\(b\)\(>\)\(<\)\(a\)\(>\) are the statistically averaged parameters of the simulation supercell. Figure 2 illustrates the variation of basal spacing of different hydrated montmorillonite systems as a function of water content from this study and available results from the literature. In the early stage of crystalline swelling, the basal spacing increases from step to step, while the step characteristic (plateau) disappears, and shows approximately linear variation in the following stage of osmotic swelling. The swelling curves of MMT with different compensation cations are different. The basal spacing of Cs-MMT is greater than Na- and Ca-MMT under the same water content. However, there is almost no difference between the latter two. This is mainly due to the fact that Cs\({}^{+}\) ions are a heavy metal ion and their ion radius is about 1.7 times that of Na\({}^{+}\) and Ca\({}^{2+}\) ions, while the latter two's ion radii are almost the same. The simulated results can well penetrate the distribution range of other available measurements. It has been shown that the energy contribution to free energy dominates clay swelling and the entropy term only played a minor role. The local minima of the swelling energy curve correspond to the stable hydration state. Hence, the relative stabilities of different states can be determined by immersion energy and hydration energy. The immersion energy is defined as in Equation (14):  
\[Q=\left\langle E(N)\right\rangle-\left\langle E(N^{0})\right\rangle-\left(N-N^{ 0}\right)E_{bulk} \tag{14}\]  
Here \(<\)\(E(N)\)\(>\) is the average potential energy of the hydrate with water content \(N\), and \(<\)\(E(N^{0})\)\(>\) is the average energy of the referential hydration state (the state with the highest water content, 49%, was selected). \(E_{bulk}\) is the mean interaction potential of the bulk flexible SPC water; \(E_{bulk}=-41.5392\) kJ/mol. The immersion energy \(Q\) is the energy released when MMT with water content \(N\) transforms into the referenced state with water content \(N^{0}\) by adsorbing water from bulk water. The hydration energy is defined as in Equation (15)  
\[\Delta E=\frac{\left\langle E(N)\right\rangle-\left\langle E(0)\right\rangle}{N} \tag{15}\]  
Here \(<\)\(E(0)\)\(>\) is the average potential energy of the dry MMT. The hydration energy \(\Delta E\) evaluates the energy change associated with water uptake by the dry clay. The simulated hydration energy and immersion energy curves of MMT with different compensation cations are presented in Figure 3. For Na- and Cs-MMT the local minima occur when the interlayer water molecular number is 40 and 80 in the hydration energy curves, while Ca-MMT has only a global minimum at low water content. Correspondingly, in the immersion energy curves of Na- and Cs-MMT, local minima occur when the interlayer water molecular numbers are also 40 and 80. The global minima of Cs-MMT is found at \(n=80\), while Na-MMT global minima is located at \(n=160\), as shown by the enlarged illustration in Figure 3. Meanwhile, the variation of the Ca-MMT immersion energy curve decreases monotonically with the increase of water content, and the global minima occurs also at \(n=160\).