*2.3. Methods for Calculating the Elastic Constants*

Acting on external forces, the hydrated MMT system will be in a state of stress. If the system is in equilibrium, the external forces must be exactly balanced by internal stress. In general, stress is a second rank tensor with nine components as in Equation (1):

$$
\overrightarrow{\sigma} = \begin{bmatrix} \sigma\_{\overline{i}\overline{j}} \end{bmatrix} = \begin{bmatrix} \sigma\_{xx} & \sigma\_{xy} & \sigma\_{xz} \\ \sigma\_{xy} & \sigma\_{yy} & \sigma\_{yz} \\ \sigma\_{xz} & \sigma\_{yz} & \sigma\_{zz} \end{bmatrix} \tag{1}
$$

In an atomistic calculation, the internal stress tensor can be obtained using the so-called virial expression as in Equation (2):

$$\stackrel{\rightarrow}{\sigma} = -\frac{1}{V\_0} \left( \sum\_{i=1}^{N} m\_i (\stackrel{\rightarrow}{\upsilon}\_i \stackrel{\rightarrow}{\upsilon}\_i^T) + \sum\_{i$$

where index *i* runs over all particles 1 through *N*; *mi*, - *<sup>v</sup> <sup>i</sup>*, and - *Fi* denote the mass, velocity, and force acting on particle *i*, respectively; *V*<sup>0</sup> denotes the initial system volume.

The application of stress to the hydrated MMT system results in a change in the relative positions of particles within the system expressed quantitatively via the strain tensor as in Equation (3):

$$
\overrightarrow{\varepsilon} = \begin{bmatrix} \varepsilon\_{\hat{i}\hat{j}} \end{bmatrix} = \begin{bmatrix} \varepsilon\_{xx} & \varepsilon\_{xy} & \varepsilon\_{xz} \\ \varepsilon\_{xy} & \varepsilon\_{yy} & \varepsilon\_{yz} \\ \varepsilon\_{xz} & \varepsilon\_{yz} & \varepsilon\_{zz} \end{bmatrix} \tag{3}
$$

For a parallelepiped (for example, a periodic simulation cell) characterized by the three column vectors [ - *a* 0 - *b* 0 - *c* <sup>0</sup>] in some reference state, and by the vectors [ - *a* - *<sup>b</sup>* - *c* ] in the deformed state, the strain tensor is given by Equation (4):

$$\stackrel{\rightarrow}{\varepsilon} = \frac{1}{2} \left[ \stackrel{\rightarrow}{h}\_0^T \right]^{-1} \stackrel{\rightarrow}{G} \stackrel{\rightarrow}{h}\_0^{-1} - 1 \tag{4}$$

where - *h* <sup>0</sup> denotes the matrix formed from the three column vectors [ - *a* 0 - *b* 0 - *c* <sup>0</sup>], - *h* denotes the corresponding matrix forms from [ - *a* - *<sup>b</sup>* - *<sup>c</sup>* ], *<sup>T</sup>* denotes the matrix transpose, and - *G* denotes the metric tensor - *h T*- *h* .

Therefore, the elastic stiffness coefficients, relating the various components of stress and strain, are defined by Equation (5):

$$\stackrel{\rightarrow}{\mathbf{C}} = \left[ \mathbf{C}\_{ijkl} \right] = \left. \frac{\partial \sigma\_{ij}}{\partial \varepsilon\_{kl}} \right|\_{T, \varepsilon\_{kl}} = \left. \frac{1}{V\_0} \frac{\partial^2 A}{\partial \varepsilon\_{ij} \partial \varepsilon\_{kl}} \right|\_{T, \varepsilon\_{ij}, \varepsilon\_{kl}} \tag{5}$$

where *A* denotes the Helmholtz free energy, and *T* denotes temperature.

For small deformations, the relationship between the stresses and strains may be expressed in terms of a generalized Hooke's law:

$$
\sigma\_{ij} = \mathbb{C}\_{ijkl}\varepsilon\_{kl} \tag{6}
$$

and, alternatively:

$$
\varepsilon\_{ij} = S\_{ijkl}\sigma\_{kl} \tag{7}
$$

where *Cijkl* and *Sijkl* denote stiffness tensor and compliance tensor, respectively.

Since both the stress and strain tensors are symmetric, it is often convenient to simplify these expressions by making use of Voigt vector notation. The stress is represented as in Equation (8):

$$
\begin{bmatrix}
\sigma\_{xx} & \sigma\_{xy} & \sigma\_{xz} \\
\sigma\_{xy} & \sigma\_{yy} & \sigma\_{yz} \\
\sigma\_{xz} & \sigma\_{yz} & \sigma\_{zz}
\end{bmatrix}
\Rightarrow
\begin{bmatrix}
\sigma\_1 & \tau\_6 & \tau\_5 \\
& \sigma\_2 & \tau\_4 \\
& & \sigma\_3
\end{bmatrix}
\tag{8}
$$

The strain is represented as in Equation (9):

$$
\begin{bmatrix}
\varepsilon\_{xx} & \varepsilon\_{xy} & \varepsilon\_{xz} \\
\varepsilon\_{xy} & \varepsilon\_{yy} & \varepsilon\_{yz} \\
\varepsilon\_{xz} & \varepsilon\_{yz} & \varepsilon\_{zz}
\end{bmatrix}
\Rightarrow
\begin{bmatrix}
\varepsilon\_1 & \gamma\_6 & \gamma\_5 \\
& \varepsilon\_2 & \gamma\_4 \\
& & \varepsilon\_3
\end{bmatrix}
\tag{9}
$$

The generalized Hooke's law is thus often written as in Equation (10):

$$
\begin{bmatrix}
\sigma\_{1} \\
\sigma\_{2} \\
\sigma\_{3} \\
\tau\_{4} \\
\tau\_{5} \\
\tau\_{6}
\end{bmatrix} = \begin{bmatrix}
\text{C}\_{11} & \text{C}\_{12} & \text{C}\_{13} & \text{C}\_{14} & \text{C}\_{15} & \text{C}\_{16} \\
& \text{C}\_{22} & \text{C}\_{23} & \text{C}\_{24} & \text{C}\_{25} & \text{C}\_{26} \\
& & \text{C}\_{33} & \text{C}\_{34} & \text{C}\_{35} & \text{C}\_{36} \\
& & & \text{C}\_{44} & \text{C}\_{45} & \text{C}\_{46} \\
& & & & \text{C}\_{55} & \text{C}\_{56} \\
& & & & & \text{C}\_{66}
\end{bmatrix} \begin{bmatrix}
\varepsilon\_{1} \\
\varepsilon\_{2} \\
\varepsilon\_{3} \\
\gamma\_{4} \\
\gamma\_{5} \\
\gamma\_{6}
\end{bmatrix} \tag{10}
$$

alternatively:

$$
\sigma\_{\hat{i}} = \mathbb{C}\_{\hat{i}\hat{j}} \varepsilon\_{\hat{j}} \tag{11}
$$

where *Cij* denotes the stiffness tensor and is a symmetric matrix of 6 × 6. The number of independent elastic constants for any system is 21. The mineral of MMT crystals belongs to the monoclinic system (*a* = *b* = *c*, *α* = *β* = 90◦ = *γ*) 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 Young's moduli (*E*1, *E*2, *E*3), 3 shear moduli (*G*23, *G*<sup>31</sup> and *G*12), and 3 Poisson's ratios (*μ*12, *μ*<sup>13</sup> and *μ*23). In terms of these parameters, the stiffness tensor *Cij* can be further expressed as in Equation (12):


where Δ = <sup>1</sup>−*μ*12*μ*21−*μ*23*μ*32−*μ*31*μ*13−2*μ*12*μ*23*μ*<sup>31</sup> *<sup>E</sup>*1*E*2*E*<sup>3</sup> .

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.
