#
Optimization of Monolayer MoS_{2} with Prescribed Mechanical Properties

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}nanostructures. The numerical results show that the proposed EA and the use of optimization method allowed accurately obtaining nanostructures with predefined mechanical material properties by introducing elliptical voids in the 2D MoS

_{2}nanosheets.

## 1. Introduction

_{2}nanomaterial have attracted many researchers in this field. Transition metal dichalcogenide (TMD) flat MoS

_{2}is a triple layer of molybdenum and sulfur atoms arranged in a hexagonal crystal lattice. It has excellent mechanical [1] electrical and chemical properties [2,3]. Consequently, MoS

_{2}has been the focus of substantial research in recent years, ensuring that the material can be used in a wide range of emerging technologies and future applications, including, for example, nanoporous MoS

_{2}membranes for efficient reverse osmosis desalination and gas separation, membrane separation, desalination of water [4], DNA sequencing [5], and power generation [6,7]. Therefore, the properties and behavior of monolayer MoS

_{2}must be understood and accurately predicted under various conditions to introduce MoS

_{2}to novel applications. However, fabricated MoS

_{2}sheets typically contain a variety of defects, including nanopores [8,9]. Thus, microstructural voids can profoundly impact the properties of MoS

_{2}, which helps design the nanomaterial, ultimately influencing the performance of MoS

_{2}-based devices. Likewise, during the growth and processing of MoS

_{2}, different topological defects (such as vacancies, inclusions, dislocation, and grain boundaries) and other sizable defects (such as nano-holes and nano-cracks) are inevitable, which can compromise the expected performance in the preparation and handling of MoS

_{2}-based nanodevices [10]. In addition, such defects can also occur due to the conditions in which MoS

_{2}-based devices are used [11]. Functional materials in nanosystems are often based on well-known materials but with tuned material properties. The 2D MoS

_{2}sheet can be functionalized by introducing voids of appropriate sizes. The sizes of the voids determine the mechanical properties of the material. In recent years, researchers have carried out investigations on the mechanical performance of monolayer MoS

_{2}related to aspects such as defects, inclusions, strength, damage, debonding, and failure that greatly influence its properties [1,12].

_{2}with various mechanical properties can be computed by molecular statistics or MD methods with a set of numerical tests comprising uniaxial tension, compression, and shear [1,13,14,15]. The well-known MD code LAMMPS [16] was used in this study for direct problem solving. In recent years, inverse methods have been widely applied to predict the properties of structures and materials. These methods allow solving problems regarding parameters using optimization techniques and a set of direct problem solutions. Inverse methods were used for mechanical and thermomechanical problems in which the properties of materials and their structures were searched [17]. Inverse problems can be solved using direct problem formulations computed using numerical methods such as the finite element method (FEM), boundary element method (BEM), and MD. The objective function of optimization algorithms in inverse methods is, in most cases, multimodal; thus, global optimization techniques are often used during the problem-solving process. Applications of new approaches of evolutionary algorithms coupled with BEM computation in optimization and identification for cracked structures and internal void defects under thermomechanical and dynamical loading were shown in [17,18,19]. Sigmund [20] used the inverse homogenization method to tune the elastic properties of the material for periodic truss, frame, and continuum structures, as well as design microstructures with prescribed elastic properties and negative Poisson’s ratios. An in-house implementation of EA was used to search for new stable molecular graphene-like 2D materials [21,22,23].

_{2}with single and multiple random defects. The computational results from our previous work showed the significant influence of defects on the elastic material properties of MoS

_{2}nanosheets. In this work, we aimed to identify the void size for the prescribed elastic properties defined by the user. We intuitively define the prescribed elastic properties, assuming that the sheet contains a void, which should be lower than the material without a void. Thus, we use the tools discussed earlier in the implementation of EA coupled with LAMMPS. As a result, we obtained the size of the void for the prescribed elastic properties.

_{2}with voids and the evolutionary identification of voids with prescribed properties by minimizing the objective function. Numerical identification examples proving the ability of this method in solving the intended optimization problem iteratively are also provided.

## 2. Materials and Methods

#### 2.1. Optimization Problem Formulation

_{2}structures with predefined material properties. The mechanical stiffness is taken into account in this paper; however, the optimization problem can also be solved for thermal, optical, or other properties of the microstructure. The objective function depends on the prescribed material properties and the actual properties computed for each design of the microstructure. The design vector

**ch**may define the size, shape, and topology of the microstructure. The MoS

_{2}structure considered in this paper was modified by introducing voids with the properties described using design variables ${g}_{i}$. The optimization problem can be formulated as

**P**denotes the nanostructure properties obtained for design vector

**ch**, and ${P}_{ref}$ describes the prescribed reference properties of the nanostructure. The goal of minimizing the difference between current and reference material properties is set as a function of design variables leading to an objective function equal to zero (i.e., identical reference and obtained properties). Optimization with a small difference between the reference and obtained properties is also acceptable. The paper is devoted to the optimization of the nanostructure taking into account mechanical properties (in our case, $P$ and ${P}_{ref}$

**)**depending on the stiffness of the nanostructure with introduced voids. The stress–strain relationship for small strains can be expressed with Voigh notation as follows:

**ch**by introducing an elliptic void, as shown in Figure 1.

_{2}. The atomic model consisted of about 10,000 atoms in the nanosheet for a domain size of 175 Å

^{2}. The Stillinger–Weber (SW) [24,25] potential was applied to describe interatomic interactions between atoms. Before tensile deformation, the model was relaxed at 300 K and 0 bar pressure through an isothermal–isobaric ensemble (NPT) for 30 ps (picoseconds). All MoS

_{2}tensile deformations were carried out at a constant temperature of 300 K. The Nose–Hoover thermostat was used to maintain the temperature of the simulation system. The position and velocity of all atoms were updated by the Verlet integration algorithm. The LAMMPS software package and the Open Visualization Tool (OVITO) [26] were used for molecular dynamics simulation, visualization, and output data analysis. Uniaxial tensile deformation at a constant strain rate of 0.0001 ps

^{−1}was applied to estimate the stiffness of the structure. The stress tensor components [27] were calculated as follows:

_{ij}is the force acting on atom i due to another atom j, V, m

_{i}, and u

_{i}are the volume, mass, and velocity of atom i, and N is the number of atoms. Figure 3 shows examples of two simulation results with a stiffness of 163 GPa and 158 GPa.

#### 2.2. Evolutionary Optimization

**ch**, with vectors of genes representing design variables g

_{i}. Genes may contain coded design variables; however, in our approach, we used floating point genes. Hence, additional coding was not needed.

## 3. Results

_{ref11}= P

_{ref22}= 160 GPa are shown in Figure 5.

_{2}nanosheet with different prescribed material properties. The void (elliptical void) was induced at the center of the sheet. Table 1 contains the values of the best obtained solutions for the numerical tests. The obtained material properties (P

_{11}, P

_{22}) for the ellipse radius (g

_{1}, g

_{2}) were not identical to those prescribed (

**P**

_{ref}), as denoted by the errors eP

_{11}and eP

_{22}. The obtained properties were very close to the prescribed ones.

_{2}nanostructure was not symmetric in the x (g

_{1}) and y (g

_{2}) directions; thus, we did not expect the same ellipse radii for the second case in the x- and y-directions. The results agreed with the intuitive approach and gave exact values of void size. The method can be used for any prescribed stiffness values; however, it is of course limited by the size of the void and maximum stiffness of the MoS

_{2}sheet.

## 4. Conclusions

_{2}can weaken its mechanical properties, such as fracture strength and Young’s modulus. However, such defects have potential in novel applications of MoS

_{2}, such as graded materials or nanosystems. We used the optimization method to tune the material properties for a periodic monolayer MoS

_{2}with a void as a defect. The tailoring problem was formulated as an optimization problem to identify the simplest possible nanostructure with constraints given by the prescribed elastic constants. Numerical examples proved that the proposed method could find the void size with prescribed mechanical properties. The present study shows the potential of molecular simulations for 2D nanostructures. It is revealed as an efficient method to design nanostructures with prescribed properties. The method is generally applicable and can be used to take into account the thermal or optical properties of the nanostructure by modifying the components of the objective function and using an adequate direct problem method.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Akhter, M.J.; Kuś, W.; Mrozek, A.; Burczyński, T. Mechanical Properties of Monolayer MoS
_{2}with Randomly Distributed Defects. Materials**2020**, 13, 1307. [Google Scholar] [CrossRef] [PubMed][Green Version] - Khan, M.F.; Nazir, G.; Lermolenko, V.M.; Eom, J. Electrical and photo-electrical properties of MoS
_{2}nanosheets with and without an Al_{2}O_{3}capping layer under various environmental conditions. Sci. Technol. Adv. Mater.**2016**, 17, 166–176. [Google Scholar] [CrossRef] [PubMed][Green Version] - Li, X.; Zhu, H. Two-dimensional MoS
_{2}: Properties, preparation, and applications. J. Mater.**2015**, 1, 33–44. [Google Scholar] [CrossRef][Green Version] - Heiranian, M.; Farimani, A.B.; Aluru, N.R. Water desalination with a single-layer MoS
_{2}nanopore. Nat. Commun.**2015**, 6, 8616. [Google Scholar] [CrossRef][Green Version] - Feng, J.; Liu, K.; Bulushev, R.D.; Khlybov, S.; Dumcenco, D.; Kis, A.; Radenovic, A. Identification of single nucleotides in MoS
_{2}nanopores. Nat. Nanotechnol.**2015**, 10, 1070–1076. [Google Scholar] [CrossRef] - Yang, T.; Lin, H.; Zheng, X.; Loh, K.P.; Jia, B. Tailoring pores in graphene-based materials: From generations to applications. J. Mater. Chem. A
**2017**, 5, 16537–16558. [Google Scholar] [CrossRef][Green Version] - Macha, M.; Marion, S.; Nandigana, V.V.; Radenovic, A. 2D materials as an emerging platform for nanopore-based power generation. Nat. Rev. Mater.
**2019**, 4, 588–605. [Google Scholar] [CrossRef] - Su, S.; Wang, X.; Xue, J. Nanopores in two-dimensional materials: Accurate fabrication. Mater. Horiz.
**2021**, 8, 1390–1408. [Google Scholar] [CrossRef] - Graf, M.; Lihter, M.; Thakur, M.; Georgiou, V.; Topolancik, J.; Ilic, B.R.; Liu, K.; Feng, J.; Astier, Y.; Radenovic, A. Fabrication and practical applications of molybdenum disulfide nanopores. Nat. Protoc.
**2019**, 14, 1130–1168. [Google Scholar] [CrossRef] - Hong, J.; Hu, Z.; Probert, M.; Li, K.; Lv, D.; Yang, X.; Gu, L.; Mao, N.; Feng, Q.; Xie, L.; et al. Exploring atomic defects in molybdenum disulphide monolayers. Nat. Commun.
**2015**, 6, 6293. [Google Scholar] [CrossRef][Green Version] - Vancsó, P.; Magda, G.Z.; Pető, J.; Noh, J.; Kim, Y.; Hwang, C.; Biró, L.P.; Tapasztó, L. The intrinsic defect structure of exfoliated MoS
_{2}single layers revealed by Scanning Tunneling Microscopy. Sci. Rep.**2016**, 6, 29726. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wang, S.; Robertson, A.; Warner, J.H. Atomic structure of defects and dopants in 2D layered transition metal dichalcogenides. Chem. Soc. Rev.
**2018**, 47, 6764. [Google Scholar] [CrossRef] [PubMed] - Mahata, A.; Jiang, J.; Mahapatra, D.R.; Rabczuk, T. Effect of intrinsic structural defects on mechanical properties of single layer MoS
_{2}. Nano-Struct. Nano-Objects**2019**, 18, 100247. [Google Scholar] [CrossRef] - Li, Y.; Chen, P.; Zhang, C.; Peng, J.; Gao, F.; Liu, H. Molecular dynamics simulation on the buckling of single-layer MoS
_{2}sheet with defects under uniaxial compression. Comput. Mater. Sci.**2019**, 162, 116–123. [Google Scholar] [CrossRef] - Li, M.; Wan, Y.; Tu, L.; Yang, Y.; Lou, J. The Effect of VMoS
_{3}Point Defect on the Elastic Properties of Monolayer MoS_{2}with REBO Potentials. Nanoscale Res. Lett.**2016**, 11, 155. [Google Scholar] [CrossRef][Green Version] - Thompson, A.P.; Aktulga, H.M.; Berger, R.; Bolintineanu, D.S.; Brown, W.M.; Crozier, P.S.; in’t Veld, P.J.; Kohlmeyer, A.; Moore, S.G.; Nguyen, T.D.; et al. LAMMPS-a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. Comp. Phys. Comm.
**2022**, 271, 10817. [Google Scholar] [CrossRef] - Burczyński, T.; Kuś, W.; Beluch, W.; Długosz, A.; Poteralski, A.; Szczepanik, M. Intelligent Computing in Optimal Design. In Solid Mechanics and Its Applications; Springer: Berlin/Heidelberg, Germany, 2020. [Google Scholar]
- Kokot, G.; Burczyński, T. Evolutionary Algorithms and Boundary Element Method in Generalized Shape Optimization. J. Theor. Appl. Mech.
**2003**, 41, 341–364. [Google Scholar] - Kokot, G.; Orantek, P. The Topology Optimization Using Evolutionary Algorithms. In IUTAM Symposium on Evolutionary Methods in Mechanics. Solid Mechanics and Its Applications 117; Burczyński, T., Osyczka, A., Eds.; Springer: Berlin/Heidelberg, Germany, 2004; pp. 173–186. [Google Scholar] [CrossRef]
- Sigmund, O. Tailoring materials with prescribed elastic properties. Mech. Mater.
**1995**, 20, 351–368. [Google Scholar] [CrossRef] - Mrozek, A.; Kuś, W.; Burczyński, T. Searching of stable configurations of nanostructures using computational intelligence methods. Tech. Sci.
**2010**, 20, 85–97. [Google Scholar] - Mrozek, A.; Kuś, W.; Burczyński, T. Nano level optimization of graphene allotropes by means of a hybrid parallel evolutionary algorithm. Comput. Mater. Sci.
**2015**, 106, 161–169. [Google Scholar] [CrossRef] - Mrozek, A.; Kuś, W.; Burczyński, T. Method For Determining Structures of New Carbon-based 2D Materials with Predefined Mechanical Properties. Int. J. Multiscale Comput. Eng.
**2017**, 15, 379–394. [Google Scholar] [CrossRef] - Kandemir, A.; Yapicioglu, H.; Kinaci, A.; CaÇin, T.; Sevik, C. Thermal transport properties of MoS
_{2}and MoSe_{2}monolayers. Nanotechnology**2016**, 27, 55703. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wen, M.; Shirodkar, S.N.; Plech, P.; Kaxiras, E.; Elliott, R.S.; Tadmor, E.B. A force-matching Stillinger-Weber potential for MoS
_{2}: Parameterization and Fisher information theory based sensitivity analysis. J. Appl. Phys.**2017**, 122, 244301. [Google Scholar] [CrossRef] - Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO—the Open Visualization Tool. Simul. Mater. Sci. Eng.
**2010**, 18, 015012. [Google Scholar] [CrossRef] - Thompson, A.P.; Plimpton, S.J.; Mattson, W. General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions. J. Chem. Phys.
**2009**, 131, 154107. [Google Scholar] [CrossRef][Green Version] - Michalewicz, Z. Genetic Algorithms + Data Structures = Evolutionary Algorithms; Springer: Berlin, Germany, 1996. [Google Scholar]
- Kennedy, J.; Eberhart, R.C.; Shi, Y. Swarm Intelligence; Morgan Kaufmann Publishers: San Francisco, CA, USA, 2001. [Google Scholar]

**Figure 2.**Determination of nanostructure stiffness in two directions. An ellipse is introduced into the pristine nanostructure by removing atoms in black area. Next, the structure is relaxed, and two analyses of microstructure stretching are performed in two different directions. Then, the stiffness is computed on the basis of the MD results.

**Figure 5.**Progress of the convergence of the objective function indicating the evolution of the void during iterative generations for prescribed elastic properties P

_{ref11}= 160 GPa, P

_{ref22}= 160 GPa. The corresponding void dimensions obtained during generations were (i) g

_{1}= 46.06 Å and g

_{2}= 2.95 Å, (ii) g

_{1}= 32.18 Å and g

_{2}= 6.07 Å, (iii) g

_{1}= 33.92 Å and g

_{2}= 8.86 Å, (iv) g

_{1}= 21.36 Å and g

_{2}= 14.88 Å, (v) g

_{1}= 18.41 Å and g

_{2}= 19.38 Å, and (vi) g

_{1}= 28.12 Å and g

_{2}= 23.15 Å.

**Figure 6.**Monolayer MoS

_{2}nanosheet with void identified by optimization: (

**a**) P

_{ref11}= 180, P

_{ref22}= 150 GPa (g

_{1}= 15.35 Å and g

_{2}= 33.44 Å); (

**b**) P

_{ref11}= 160, P

_{ref22}= 160 GPa (g

_{1}= 28.12 Å and g

_{2}= 23.15 Å); (

**c**) P

_{ref11}= 150, P

_{ref22}= 180 GPa (g

_{1}= 36.35 Å and g

_{2}= 12.19 Å).

Case | P_{ref11}(GPa) | P_{11}(GPa) | P_{ref22}(GPa) | P_{22}(GPa) | g_{1}(Å) | g_{2}(Å) | eP_{11} (%) | eP_{22} (%) |
---|---|---|---|---|---|---|---|---|

1 | 150.0 | 149.2 | 180.0 | 179.5 | 36.35 | 12.19 | 0.5 | 0.3 |

2 | 160.0 | 162.6 | 160.0 | 158.0 | 28.12 | 23.15 | 1.6 | 1.3 |

3 | 180.0 | 179.5 | 150.0 | 148.0 | 15.35 | 33.44 | 0.3 | 1.3 |

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

© 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/).

## Share and Cite

**MDPI and ACS Style**

Kuś, W.; Akhter, M.J.; Burczyński, T. Optimization of Monolayer MoS_{2} with Prescribed Mechanical Properties. *Materials* **2022**, *15*, 2812.
https://doi.org/10.3390/ma15082812

**AMA Style**

Kuś W, Akhter MJ, Burczyński T. Optimization of Monolayer MoS_{2} with Prescribed Mechanical Properties. *Materials*. 2022; 15(8):2812.
https://doi.org/10.3390/ma15082812

**Chicago/Turabian Style**

Kuś, Wacław, Mohammed Javeed Akhter, and Tadeusz Burczyński. 2022. "Optimization of Monolayer MoS_{2} with Prescribed Mechanical Properties" *Materials* 15, no. 8: 2812.
https://doi.org/10.3390/ma15082812