## 2 Model and Methods
### Methods for Calculating the Elastic Constants
\begin{array}{ccc}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{xy}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{xz}&\sigma_{yz}&\sigma_{zz}\end{array} \right]\Rightarrow\left[ \begin{array}{ccc}\sigma_{1}&\tau_{6}&\tau_{5}\\ \sigma_{2}&\tau_{4}\\ &\sigma_{3}\end{array} \right] \tag{8}\]  
The strain is represented as in Equation (9):  
\[\left[ \begin{array}{ccc}\varepsilon_{xx}&\varepsilon_{xy}&\varepsilon_{xz} \\ \varepsilon_{xy}&\varepsilon_{yy}&\varepsilon_{yz}\\ \varepsilon_{xz}&\varepsilon_{yz}&\varepsilon_{zz}\end{array} \right] \Rightarrow\left[ \begin{array}{ccc}\varepsilon_{1}&\gamma_{6}&\gamma_{5} \\ \varepsilon_{2}&\gamma_{4}\\ &\varepsilon_{3}\end{array} \right] \tag{9}\]  
The generalized [PERSON]'s law is thus often written as in Equation (10):  
\[\left[ \begin{array}{c}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \tau_{4}\\ \tau_{5}\\ \tau_{6}\end{array} \right]=\left[ \begin{array}{cccccc}C_{11}&C_{12}&C_{13}&C_{ 14}&C_{15}&C_{16}\\ &C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ &&C_{33}&C_{34}&C_{35}&C_{36}\\ &&&C_{44}&C_{45}&C_{46}\\ &&&C_{55}&C_{56}\\ &&&C_{66}\end{array} \right]\left[ \begin{array}{c}\varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \tau_{4}\\ \tau_{5}\\ \tau_{6}\end{array} \right] \tag{10}\]  
alternatively:  
\[\sigma_{i}=C_{ij}\varepsilon_{j} \tag{11}\]  
where \(C_{ij}\) denotes the stiffness tensor and is a symmetric matrix of \(6\times 6\). The number of independent elastic constants for any system is 21. The mineral of MMT crystals belongs to the monoclinic system (\(a\
eq b\
eq c,\;\alpha=\beta=90^{\circ}\
eq\gamma\)) and has C2/M symmetry. It, then, only needs 13 independent parameters to describe its stiffness tensor. However, according to the study in [23], the MMT crystal can be considered approximately as orthotropic symmetry, thus further reducing the stiffness tensor parameters to 9 to characterize the sheet. These 9 independent elastic constants include 3 [PERSON]'s moduli (\(E_{1}\), \(E_{2}\), \(E_{3}\)), 3 shear moduli (\(G_{23}\), \(G_{31}\) and \(G_{12}\)), and 3 [PERSON]'s ratios (\(\mu_{12}\), \(\mu_{13}\) and \(\mu_{23}\)). In terms of these parameters, the stiffness tensor \(C_{ij}\) can be further expressed as in Equation (12):  
\[\left[ \begin{array}{cccccc}C_{11}&C_{12}&C_{13}&0&0&0\\ C_{21}&C_{22}&C_{23}&0&0&0\\ C_{31}&C_{32}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{55}&0\\ 0&0&0&0&C_{66}\end{array} \right]= \tag{12}\] \[\left[ \begin{array}{cccccc}\frac{1-\mu_{12}\mu_{32}}{E_{21}E_{3 }}&\frac{\mu_{12}+\mu_{13}\mu_{23}}{E_{21}E_{3}}&\frac{\mu_{31}+\mu_{12}\mu_{32 }}{E_{21}E_{3}}&0&0&0\\ \frac{\mu_{12}+\mu_{13}\mu_{23}}{E_{21}E_{3}}&\frac{1-\mu_{13}\mu_{13}}{E_{21} E_{3}}&\frac{\mu_{32}+\mu_{13}\mu_{12}}{E_{21}E_{3}}&0&0&0\\ \frac{\mu_{13}+\mu_{12}\mu_{23}}{E_{1}E_{2}}&\frac{\mu_{23}+\mu_{13}\mu_{23}}{ E_{1}E_{2}\Delta}&\frac{1-\mu_{12}\mu_{21}}{E_{1}E_{2}\Delta}&0&0\\ 0&0&0&2G_{23}&0&0\\ 0&0&0&0&2G_{31}&0\\ 0&0&0&0&0&2G_{12}\end{array} \right]\]  
where \(\Delta=\frac{1-\mu_{12}\mu_{21}-\mu_{23}\mu_{32}-\mu_{31}\mu_{13}-2\mu_{12}\mu _{23}\mu_{31}}{E_{1}E_{2}E_{3}}\). Therefore, the key to characterizing the elastic mechanical properties of the MMT-water-ion system at the nanoscale level is to obtain the parameters of the stiffness tensor or the compliance tensor.