**1. Introduction**

Montmorillonite (MMT) consists of well-defined layers separated by interlayer spaces and has great water absorption potential [1–4]. Thus, it is often used as clay barriers or adsorption material in environmental geotechnical engineering, for example, the backfill in the nuclear waste disposal and the cushion in the landfill [5–10]. It is very challenging to directly measure the fundamental mechanical properties of MMT [11,12]. Atomic force acoustic microscopy is used to determine the elastic properties of clay mineral aggregates by measuring adhesion forces [13–15]. Nanoindentation and ultrasonic pulse velocity technology (UPV) are also used to measure the elastic stiffness constants of clays and shale samples [16–18]. However, these techniques are powerless to measure the anisotropic mechanical properties of MMT, due to the complications in sample preparation and even the testing process.

In contrast to difficulty in experimental methods, molecular dynamics (MD) simulation is a supplemental way to model and understand the properties of hydrated clay minerals on an atomic scale [1,19–21]. With the development of polymer–clay nanocomposites

**Citation:** Kuang, L.; Zhu, Q.; Shang, X.; Zhao, X. Molecular Dynamics Simulation of Nanoscale Elastic Properties of Hydrated Na-, Cs-, and Ca-Montmorillonite. *Appl. Sci.* **2022**, *12*, 678. https://doi.org/10.3390/ app12020678

Academic Editor: Bing Bai

Received: 17 December 2021 Accepted: 10 January 2022 Published: 11 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

in recent years, several studies on the nanomechanical properties of MMT minerals or hydrated MMT systems have emerged. The authors in [18] reported for the first time the stress–strain response of rock-forming minerals by MD simulation. The authors in [22] performed MD simulations to study the flexibility and the mechanical behavior of a single clay layer in a completely exfoliated state and found that the critical stress of a clay layer bending fracture is 0.8 MPa in both in-plane directions. In [23], the authors presented results for the elastic properties of a single lamella of MMT by MD simulation and further pointed out that the elastic constants characteristic of a thin plate is closely related to the thickness of the nanoplate. Based on the analogous minerals and modulus-density relations, ref. [24] deduced a convergence of opinion in the range 178–265 GPa for the elastic moduli of smectite clay platelets. The authors in [25] systematically simulated the structure and the thermomechanical properties of an isolated clay nanoplate and MMT crystals intercalated by water or polyethylene oxide. In [26], the authors clarified the linear elastic properties including tensile moduli, shear moduli, and potential failure mechanisms as a function of cation density and stress for the minerals pyrophyllite, MMT, and mica. In [27], the authors examined the mechanism of bending, the stored energy, and the failure of several clay minerals. They revealed that molecular contributions to the bending energy include bond stretching and bending of bond angles in the mineral as well as rearrangements of alkali ions on the surface of the layers. The authors in [28] further simulated the elastic and structural properties of muscovite as a function of temperature, pressure, and strain. The results demonstrate that the elastic properties of muscovite depend on both temperature and pressure. MD with the CLAYFF force field was used in [29] to simulate isothermal isobaric water adsorption of interlayer MMT, and nanoscale elastic properties of the clayinterlayer water system are calculated from the potential energy of the model system. Similar simulations have also been carried out on MMT by implementing the elastic bath method [30]. The mechanical properties of Na-MMT were investigated in [31] from the view of water content by MD, and the hysteresis phenomena of elastic constant during the swelling and shrinking was found. In addition, researchers found that the mechanical behavior of MMT crystal exhibits a clear dissymmetry between compression and tension and an important dependency on mean stress [32].

The above simulation studies are generally based on the strain method, that is, the stress is calculated by changing the size of the simulation box. The stress method and the large-scale fluctuation method are also usually adopted. For example, using steered molecular dynamics (SMD) technology, the mechanical response of the interlayer of hydrated MMT was evaluated by [33]. Using grid-computing technology, ref. [34] simulated MMT clay large-scale systems containing up to approximately ten million atoms; large-scale systems exhibit emergent behavior with increasing size, and the material mechanical properties were calculated based on thermal bending fluctuations. From the surveyed status of nanoscale mechanical properties of clay minerals, it can be found that the stress–strain response of minerals can be revealed by different methods (constant strain, SMD, or the large-scale fluctuation method) based on different force fields and different platforms. However, the moisture content under most simulated hydration conditions is usually arbitrarily selected and generally restricts some movement directions of clay plates or even the interlayer water molecules which affect the mechanical response characteristics of the hydrated mineral system.

The goal of this study is to simulate the nanoscale mechanical properties of hydrated MMT minerals with three cations (Na, Ca, and Cs) at variable water contents. In the simulation, all degrees of freedom of atoms are released, and the water content in different hydration states is determined according to stable-state thermodynamic conditions. To the best of our knowledge, no such treatments have been reported to study the mechanical properties of the MMT mineral up to now. The quantitative values of the basal spacing for different compensation cationic MMTs are firstly studied. The relative stabilities of different states are determined by comparing the immersion energy and hydration energy between MMT systems, and, then, the corresponding moisture content of thermodynamics stable states is obtained. Finally, the nanoscale elastic properties of MMT are further simulated by the constant strain method under the stable state water content.
