Molecular dynamic study of the mechanicalproperties of two-dimensional titaniumcarbides Tin+1Cn (MXenes)

Vadym N Borysiuk1,2, Vadym N Mochalin1 and Yury Gogotsi1

1Department of Materials Science and Engineering and A J Drexel Nanomaterials Institute, DrexelUniversity, Philadelphia, Pennsylvania 19104, USA2 Sumy State University, 2 Rimsky-Korsakov Street, 40007 Sumy, Ukraine

E-mail: gogotsi@drexel.edu

Received 18 March 2015, revised 12 May 2015Accepted for publication 15 May 2015Published 11 June 2015

AbstractTwo-dimensional materials beyond graphene are attracting much attention. Recently discovered2D carbides and nitrides (MXenes) have shown very attractive electrical and electrochemicalproperties, but their mechanical properties have not been characterized yet. There are neitherexperimental measurements reported in the literature nor predictions of strength or fracturemodes for single-layer MXenes. The mechanical properties of two-dimensional titanium carbideswere investigated in this study using classical molecular dynamics. Young’s modulus wascalculated from the linear part of strain–stress curves obtained under tensile deformation of thesamples. Strain-rate effects were observed for all Tin+1Cn samples. From the radial distributionfunction, it is found that the structure of the simulated samples is preserved during thedeformation process. Calculated values of the elastic constants are in good agreement withpublished DFT data.

Keywords: MXenes, 2D materials, molecular dynamics, Young’s modulus

(Some figures may appear in colour only in the online journal)

1. Introduction

A new, large family of two-dimensional transition metalcarbides and nitrides, known as MXenes, was recently dis-covered [1]. Due to their atomically thin structure [2],investigation of the mechanical properties of two-dimensional(2D) materials experimentally is much more challengingcompared to 3D materials. Despite that, Lee et al [3] havemeasured the elastic properties of monolayer graphene byatomic force microscopy (AFM) indentation. Implementationof such technique to MXenes presents an even harder chal-lenge because of the smaller lateral size of sheets that arecurrently produced [4]. Besides that, basal planes of allMXenes produced so far are terminated by oxygen-containingfunctional groups, which make the experimental investigationof elastic properties of pure non-terminated MXene sheetsimpossible, at least for now. Therefore, all current

information on elastic properties of pristine MXene sheetswas obtained from theoretical (modeling) studies.

Important structural characteristics such as lattice para-meters, and interatomic distances, as well as elastics con-stants, band structure and electronic density of states of theMXenes were obtained using the first principles techniques[5–13]. Although ab initio techniques are precise and pow-erful, usually only a small number of atoms, i.e. a few unitcells, can be modeled. Thus, some important collective effectsrequiring larger statistics may be overlooked.

The large-scale computer simulation by classical mole-cular dynamics (MD) is an alternative approach that is widelyused in investigating mechanical properties of 2D materials[14–17].

In classical MD, the trajectory of a system is obtained byintegration of Newton equations of motion containinginteratomic forces for all particles involved in simulation [18].

Nanotechnology

Nanotechnology 26 (2015) 265705 (10pp) doi:10.1088/0957-4484/26/26/265705

0957-4484/15/265705+10$33.00 © 2015 IOP Publishing Ltd Printed in the UK1

However, due to the recent discovery of MXenes, the exactexpressions for interatomic potentials directly fitted to theinteraction energy in two-dimensional Tin+1Cn crystals are notyet established. There are, however, several examples of MDsimulations involving Ti and C atoms [19, 20] in other mate-rials. Modified embedded atom method (MEAM) described in[19] was fitted for a 3D NaCl-type Ti-C binary system, but itcannot reproduce a 2D Tin+1Cn structure without majoradjustments. Moreover, the MEAM contains a large number ofparameters that need to be revised to fit the potential energy ofTi–C interaction in MXene. Empirical potential energy function(PEF) [20] was fitted to a layered-type Ti–C interaction intitanium covered carbon nanotube, which has atomic config-uration similar to Tin+1Cn MXene. The analytical expressionsfor potentials in a PEF approach have only three empiricalparameters; therefore it may be easily adjusted to represent Ti–C interaction in 2D Tin+1Cn carbides.

In this paper, we report the first classical MD study of themechanical properties of 2D Tin+1Cn MXenes. The aim of thiswork is to calculate the structural and elastic properties of the2D Tin+1Cn carbides by a large-scale MD simulation andcompare them with those obtained from DFT data on singlecrystal cells [9]. In the absence of a specific force field forMXenes, we seek to develop a simple and accurate approachthat can be used in future investigations of the structural andmechanical properties of the MXenes. Using this approach, wealso aim to obtain new insights into fracture modes of MXenes.

2. Model

We consider Tin+1Cn sheets with n 1, 2, 3,= located in aCartesian coordinate box as shown in figure 1 (all MD

snapshots in this paper were produced using Visual MolecularDynamics software [21]). Ti2C and Ti3C2 are among the firstMXenes reported [1, 4], while Ti4C3 has only been studiedcomputationally [12]. The titanium and carbon atoms wereplaced at initial positions according to the MXene lattice [4]. Inthis initial study we did not consider any surface terminations ofMXene, which are present when wet chemical synthesis is used[4]. It should be noted, however, that at least some bareMXenes are expected to be stable, because the calculatedcohesion energy of the appropriate MAX phases Ti3C2 andTi4C3 is lower than the energy of the appropriate MAX phasesTi3AlC2 and Ti4AlC3, respectively [22]. Just like silicene,germanene or electrides, bare MXenes must be produced in anoxygen-free environment by bottom-up approach. The intera-tomic distances within the five boundary atomic layers on eachside in the x direction− are held fixed during the simulation(figure 1), while periodic and free boundary conditions wereapplied in the y− and z directions− respectively. The tensileloading in the MD simulations of 2D materials can be imple-mented in a few ways [23, 24]; in our simulations we used atensile model as in [23]. During the tensile deformation, the fivelayers of atoms with fixed interatomic distance on one side ofthe sample were pulled in the x direction− at a constant strainrate of 0.0004 ps−1, while the corresponding five layers ofatoms on the opposite side of the MXene sheet remained fixed.The lateral (x- and y-) dimensions of the samples considered inour simulations vary from about 19.50 16.84 nm× to38.71 33.47 nm,× where the total number of atoms involvedin simulation varies from 12 288 to 87 808, depending on theMXene and system dimensions.

Interatomic bonding in MAX phases is known tobe a combination of metallic, covalent, and ionic [4, 25].Therefore, various types of potential energy functions were

Figure 1. Top (a) and perspective (b) view of the initial configuration of the Ti2C with a size of 38.71 33.47 nm× consisting of 49 152atoms. Five lateral layers of atoms, whose interatomic distances were fixed during simulations, are shown in yellow on both sides.

2

Nanotechnology 26 (2015) 265705 V N Borysiuk et al

used to simulate interactions between different types ofatoms.

As shown before [26], Tin+1Cn are metallic conductors,so the metallic type of bonding is supposed to prevail withinthe Ti layers of the material. With this assumption, theembedded atom method (EAM [27]) was chosen to describethe interactions between titanium atoms in the Tin+1Cn sheets.The generalized EAM potential is widely used in MD simu-lations of metal alloys and is well fitted to reproduce the basicmaterial properties [28, 29].

Forces between the titanium and carbon atoms arederived from the empirical potential energy function (PEF)parameterized for C–Ti compounds [20]. Within thisapproach, the total potential energy of the system can bepresented as a truncated sum of two- and three-body poten-tials, where the two-body term is approximated by the Len-nard–Jones (LJ) potential [30]

Ur

r

r

r2 , (1)ij

ij ij0

012

06⎡

⎣⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤

⎦⎥⎥ε= −

with r ,ij r ,0 and 0ε being the distance between atoms i and j,equilibrium distance, and the minimum energy, respectively;

and the three-body term by Axilrod–Teller (AT) potential [31]

( )W

Z

r r r

1 3 cos cos cos

( ), (2)ijk

i j k

ij ik jk3

θ θ θ=

+

Figure 2. Evolution of the potential energy (a) and probability distribution of the velocities (b) for the Ti2C sample (inset shows the systemtemperature during simulations).

Figure 3. Snapshots of the Tin+1Cn samples after equilibration at 300 K: (a) Ti2C; (b) Ti3C2; (c) Ti4C3.

Figure 4. Strain–stress curves obtained for the Tin+1Cn samplesduring tensile loading. Dashed lines are extrapolated from the initiallinear regions of strain–stress curves.

3

Nanotechnology 26 (2015) 265705 V N Borysiuk et al

where r ,ij rik and rjk are the distances between atom pairs; Z isa three-body intensity parameter; ,iθ jθ and kθ are angles ofthe triangle formed by r ,ij r ,jk and r .ik Parameters of theLJ +AT model for C–Ti dimer and trimer were estimatedfrom DFT data and used in a simulation of the titaniumcoating of single-wall carbon nanotubes [20]. However, in oursimulations, we adjusted the equilibrium interatomic distancer0 in pair potential (1) according to the Tin+1Cn structure [4].The carbon atoms in Tin+1Cn MXenes are not chemicallybonded with each other, therefore C–C interactions in oursimulations are described only by a Lennard–Jones potentialas in [32]. The cutoff distance of r 0.25nmc

AT = was used inAT potential (2), while r 0.90 nmc

EAM = was set for long-range EAM and LJ potentials.

In the absence of a force field for MD simulations ofTin+1Cn, our approach based on combination of the existingdata for Ti and C interactions is the first effort aimed at a

qualitatively correct description of the system and may evenproduce quantitative results.

In order to investigate the elastic properties of theTin+1Cn carbides, the tensile deformation was applied to thesamples and the stress during the deformation was computedaccording to the virial theorem [33, 34] as

Vr f m v v

1 1

2, (3)ij

N Ni j

i j

1 1

⎛⎝⎜⎜

⎞⎠⎟⎟∑ ∑σ = −

α βαβ αβ

α α α

= =

where ij are the Cartesian coordinates, α and β are atomindexes, ri

αβ f jαβ are corresponding components of the distance

and force between atoms α and ,β m is the atom mass andV isthe volume of the simulation box. The stress values wereestimated from equation (3) with 0.005Å increments in straincomponents and averaging over the 2000 simulation steps.

Figure 5. Snapshots of the atomistic configuration of the 2D Ti2C carbide at different strains: (a) 0.025; (b) 0.05; (c) 0.065; (d) 0.08.

4

Nanotechnology 26 (2015) 265705 V N Borysiuk et al

Young’s moduli were obtained from the initial slope of thesimulated stress–strain curves.

The computational code is implemented using massiveparallelism provided by the NVIDIA CUDA [35, 36] platform.Algorithms from the LAMMPS software package [37] werepartially used for interatomic forces calculation. Equations ofmotion were integrated with a time step td 0.2= fs.

3. Results

In the beginning of the simulation, all atoms were placed intheir initial positions to form an ideal Tin+1Cn monolayer andthe system was equilibrated over 2 105⋅ simulation steps. Thetemperature was maintained at 300 K, using Berendsen ther-mostat [38]. During the simulations, the potential energy ofthe particles, system temperature and the velocities wererecorded. Typical evolution charts for potential energy and

temperature, as well as velocity distribution for the Ti2Csample, are presented in figure 2.

As seen in figure 2, the potential energy of the systemafter several major oscillations in the beginning of the runreaches an equilibrium value with anticipated statisticalfluctuations around it. The distribution of the velocities hasthe Maxwell–Boltzman profile, while the system temperatureis maintained at 300 K. The snapshots of the simulatedMXenes after equilibration are shown in figure 3.

After the equilibration, the tensile load was applied to thesamples, during which the overall stress was calculated byequation (3). Figure 4 shows the stress–strain curves obtainedfor the Ti2C, Ti3C2, and Ti4C3 samples during their tensiledeformation.

All obtained strain–stress curves have a similar shapewith an initial linear region, related to elastic deformation. Ata higher strain, stress continues to increase up to the thresholdpoint of a yield stress, followed by a sharp drop associatedwith sample fracture.

Figure 6. Snapshots of the atomistic configuration of the Ti3C2 2D carbide at different strains: (a) 0.04; (b) 0.06; (c) 0.075; (d) 0.1.

5

Nanotechnology 26 (2015) 265705 V N Borysiuk et al

The Young’s moduli obtained by linear fit of the strain–stress curves at the strain 0.01ε < are 597, 502 and534 GPa for Ti2C, Ti3C2, and Ti4C3, respectively, with aninterpolation error within 10%. These results are close to thepublished data estimated from first principles calculations [9].As expected, the highest Young’s modulus was observed forTi2C MXene sheet and decreased with increasing thickness ofMXene layer. Ti4C3, which is still to be produced experi-mentally, showed the highest strain to failure.

Under critical stress all the Tin+1Cn samples exhibitsimilar behavior during elastic deformation; no significantdefects are formed in the structure and the overall strain of thesample is distributed over the interatomic bonds, resulting intheir tension. With further increase in stress, the evolution ofthe Tin+1Cn samples is different.

For the thinnest Ti2C MXene, the bond breakage beginsin the outermost regions of the sample at the points of thehighest local stress, propagating deeper to the center along the

shifting atomic layers. Cracking of the material is accom-panied by crumpling and folding of the Ti2C sheet near thegrowing gap.

After the sample is completely torn apart on its right-sideedge, it continues to fold and roll up due to the high surfaceenergy and free boundary conditions in the z direction.However, the radial distribution function (RDF) calculated forthe Ti2C sample after its fracture at the end of simulationshows that the atoms locally preserve their Ti2C structurewithin the crumpled sheet, like in a rolled-up graphene sheet[39]. Snapshots of the simulated Ti2C sample at several strainvalues are shown in figure 5.

A Ti3C2 sample behaves similarly to a thinner Ti2C,except showing less tendency to wrap up during the fractureunder tensile strain, which is probably due to its thickness, i.e.five atomic layers compared to three in Ti2C. As figure 6shows, only the edges along the crack folded, while the mainbody of the sample largely retained the initial sheet shape.

Figure 7. Snapshots of the atomistic configuration of the Ti4C3 2D carbide at different strains: (a) 0.025; (b) 0.035; (c) 0.06; (d) 0.1.

6

Nanotechnology 26 (2015) 265705 V N Borysiuk et al

In a striking contrast to Ti2C and Ti3C2, cracks in theTi4C3 MXene appear first in the central part of the sheet. At ahigher stress, these cracks grow in size, leading to the com-plete fracture of the sample with formation of several Ti4C3

fragments (figure 7). It is important to mention that Ti4AlC3

MAX phase or Ti4C3 MXene have not been producedexperimentally yet.

To detect the changes in the structure of MXenesheets, the 2D radial distribution functions [18] (RDF) g r( )

were calculated at the end of the strain–stress simulation(figure 8).

As can be seen from the figure, the short-range orderrelated to the local MXene structure is preserved in the atomicconfiguration for all samples.

To validate the obtained results, the strain–stress simu-lations were performed at different strain rates for all threeTin+1Cn. Three strain rate values of 0.0002 ps−1 0.0004 ps−1

and 0.001ps−1 were used in these simulations. The results of

Figure 8. Radial distribution functions (left panels) and corresponding atomistic configurations (right panels) of the 2D Tin+1Cn samples afterfracture under tensile deformation: (a), (d) Ti2C; (b), (e) Ti3C2; (c), (f) Ti4C3.

7

Nanotechnology 26 (2015) 265705 V N Borysiuk et al

these simulations are shown in figure 9. As expected, all thestrain–stress curves obtained at different strain rates have alinear region of elastic deformation, with the slope in thisregion being independent of the strain rate.

The shift of the yield point to larger strain values withincreasing strain rate was observed in all Tin+1Cn simulations.This phenomenon, known as strain rate effect [40], has beenobserved experimentally for metals and has several theoreticalexplanations that have been discussed in the literature [41].

4. Conclusions

Using large-scale classical molecular dynamics simulationswe have studied the mechanical properties of the 2D titaniumcarbides (MXenes) under tensile loading. Calculated strain–stress curves have linear regions at small strains ( 1%).⩽Young’s moduli estimated from these initial regions of strain–stress curves by a linear fit are close to previously publisheddata [9] obtained by DFT. These results were reproduced atdifferent strain rates and were shown to be insensitive to the

strain rate in the range 0.0002 ps−1–0.001 ps−1. Mechanicalproperties and evolution of the atomic structure of 2D Tin+1Cn

under tensile loading depend on the number of atomic layersn in MXenes. This may indicate different mechanisms ofmechanical failure for Tin+1Cn with different n (layer thick-ness). The highest Young’s modulus was obtained for thethinnest Ti2C carbide.

As soon as the free boundary edge appears in the sample,the MXene begins to roll-up onto itself due to a high surfaceenergy in vacuum. However, in the Ti4C3 sample, smallerfragments, which are formed during fracture, largely retaintheir original flat sheet-like appearance, as the surface tensionis not enough to overcome the high rigidity of the thickest ofthe considered MXenes. The calculated RDFs show thatoverall, the local Tin+1Cn structure remains intact in all thesamples after fracture. With increasing rate, a slight growth ofyield stress and critical strain was observed, which indicatesthat Tin+1Cn MXenes may possess a mechanism of defectformation and plastic deformation typical for metals [40].

The examined MXenes are almost twice weaker thanatomically thin graphene with Young’s modulus of

Figure 9. Strain–stress curves for the 2D Tin+1Cn samples at different strain rates: (a) Ti2C; (b) Ti3C2; (c) Ti4C3. Dashed lines indicate theregion of linear strain–stress relation. Vertical arrows show the yield points for each curve.

8

Nanotechnology 26 (2015) 265705 V N Borysiuk et al

E 1.0 0.1 TPa= ± [3]. At the same time, the elastic con-stant, obtained for Ti2C is almost twice higher compared toE 0.33 0.07 TPa= ± [42] of MoS2, another 2D crystal witha similar structure. Thus, as potential materials for nanode-vices with high mechanical property requirements or asreinforcement for composites, MXenes can be the preferredchoice among other 2D materials beyond graphene.

Although the calculated Young’s moduli are close tothose previously obtained by DFT, in the absence of experi-mental data we cannot claim that other mechanical properties,such as yield stress and related strain, are reproduced cor-rectly by classical MD. Nevertheless, besides the goodagreement with DFT data, our simulations have reproducedstrain-rate effects that are also observed in experiments onmetals and graphene [43]. Overall, we conclude that theemployed classical MD approach correctly describesmechanical behavior of the two-dimensional titanium car-bides and may be used in further investigations of the struc-tural and mechanical properties of other bare and surface-terminated MXenes. Further efforts in this area should befocused on creating and parameterizing a reliable force field(ReaxFF [44] or similar), which could be used for classicalMD simulations of different MXenes.

Acknowledgments

The work reported in this paper was performed at DrexelUniversity. This material is based upon work supported by theUS Army Research Laboratory and the US Army ResearchOffice under grant number W911NF-14-1-0568. VNB wassupported by the internship program of the Ministry ofEducation and Science of Ukraine.

References

[1] Naguib M, Kurtoglu M, Presser V, Lu J, Niu J J, Heon M,Hultman L, Gogotsi Y and Barsoum M W 2011 Two-dimensional nanocrystals produced by exfoliation ofTi3AlC2 Adv. Mater. 23 4248–53

[2] Novoselov K S, Jiang D, Schedin F, Booth T J,Khotkevich V V, Morozov S V and Geim A K 2005 Two-dimensional atomic crystals Proc. Natl Acad. Sci. USA 10210451–3

[3] Lee C, Wei X D, Kysar J W and Hone J 2008 Measurement ofthe elastic properties and intrinsic strength of monolayergraphene Science 321 385–8

[4] Naguib M, Mochalin V N, Barsoum M W and Gogotsi Y 201425th anniversary article: MXenes: a new family of two-dimensional materials Adv. Mater. 26 992–1005

[5] Er D, Li J, Naguib M, Gogotsi Y and Shenoy V B 2014 Ti3C2

MXene as a high capacity electrode material for metal (Li, Na,K, Ca) ion batteries ACS Appl. Mater. Interfaces 6 11173–9

[6] Gan L-Y, Zhao Y-J, Huang D and Schwingenschlögl U 2013First-principles analysis of MoS2/Ti2C and MoS2/Ti2CY2

(Y=F and OH) all-2D semiconductor/metal contacts Phys.Rev. B 87 245307

[7] Hu J, Xu B, Ouyang C, Yang S A and Yao Y 2014Investigations on V2C and V2CX2 (X=F, OH) monolayer as

a promising anode material for Li ion batteries from first-principles calculations J. Phys. Chem. C 118 24274–81

[8] Khazaei M, Arai M, Sasaki T, Chung C-Y,Venkataramanan N S, Estili M, Sakka Y and Kawazoe Y2013 Novel electronic and magnetic properties of two-dimensional transition metal carbides and nitrides Adv.Funct. Mater. 23 2185–92

[9] Kurtoglu M, Naguib M, Gogotsi Y and Barsoum M W 2012First principles study of two-dimensional early transitionmetal carbides MRS Commun. 2 133–7

[10] Lane N J, Barsoum M W and Rondinelli J M 2013 Correlationeffects and spin–orbit interactions in two-dimensionalhexagonal 5d transition metal carbides, Tan+1Cn (n = 1, 2, 3)Europhys. Lett. 101 57004

[11] Shein I R and Ivanovskii A L 2012 Planar nano-blockstructures Tin+1Al0.5Cn and Tin+1Cn (n= 1, and 2) fromMAX phases: Structural, electronic properties and relativestability from first principles calculations SuperlatticesMicrostruct. 52 147–57

[12] Xie Y and Kent P R C 2013 Hybrid density functional study ofstructural and electronic properties of functionalized Tin+1Xn

(X =C, N) monolayers Phys. Rev. B 87 235441[13] Xie Y, Naguib M, Mochalin V N, Barsoum M W, Gogotsi Y,

Yu X Q, Nam K W, Yang X Q, Kolesnikov A I andKent P R C 2014 Role of surface structure on Li-ion energystorage capacity of two-dimensional transition-metalcarbides J. Am. Chem. Soc. 136 6385–94

[14] Cranford S W and Buehler M J 2011 Mechanical properties ofgraphyne Carbon 49 4111–21

[15] Pei Q X, Zhang Y W and Shenoy V B 2010 A moleculardynamics study of the mechanical properties of hydrogenfunctionalized graphene Carbon 48 898–904

[16] Zheng Q, Li Z, Geng Y, Wang S and Kim J-K 2010 Moleculardynamics study of the effect of chemical functionalizationon the elastic properties of graphene sheets J. Nanosci.Nanotechnol. 10 7070–4

[17] Zhao Y, Smalley R E and Yakobson B I 2002 Coalescence offullerene cages: Topology, energetics, and moleculardynamics simulation Phys. Rev. B 66 195409

[18] Rapaport D C 2004 The Art of Molecular Dynamics Simulation(Cambridge: Cambridge University Press)

[19] Kim Y M and Lee B J 2008 Modified embedded-atom methodinteratomic potentials for the Ti-C and Ti-N binary systemsActa Mater. 56 3481–9

[20] Oymak H and Erkoc F 2004 Titanium coverage on a single-wall carbon nanotube: molecular dynamics simulationsChem. Phys. 300 277–83

[21] Humphrey W, Dalke A and Schulten K 1996 VMD: visualmolecular dynamics J. Mol. Graphics Modelling 14 33–8

[22] Shein I R and Ivanovskii A L 2012 Graphene-like titaniumcarbides and nitrides Tin+1Cn, Tin+1Nn (n= 1, 2, and 3) fromde-intercalated MAX phases: First-principles probing oftheir structural, electronic properties and relative stabilityComput. Mater. Sci. 65 104–14

[23] Bu H, Chen Y, Zou M, Yi H, Bi K and Ni Z 2009 Atomisticsimulations of mechanical properties of graphenenanoribbons Phys. Lett. A 373 3359–62

[24] Grantab R, Shenoy V B and Ruoff R S 2010 Anomalousstrength characteristics of tilt grain boundaries in grapheneScience 330 946–8

[25] Barsoum M W and Radovic M 2011 Elastic and mechanicalproperties of the MAX phases Annu. Rev. Mater. Res. 41195–227

[26] Tang Q, Zhou Z and Shen P W 2012 Are MXenes promisinganode materials for Li ion batteries? Computational studieson electronic properties and Li storage capability of Ti3C2

and Ti3C2X2 (X=F, OH) monolayer J. Am. Chem. Soc. 13416909–16

9

Nanotechnology 26 (2015) 265705 V N Borysiuk et al

[27] Zhou X W et al 2001 Atomic scale structure of sputtered metalmultilayers Acta Mater. 49 4005–15

[28] Francis M F, Neurock M N, Zhou X W, Quan J J,Wadley H N G and Webb E B 2008 Atomic assembly ofCu/Ta multilayers: surface roughness, grain structure,misfit dislocations, and amorphization J. Appl. Phys. 104034310

[29] Zou W, Wadley H N G, Zhou X W, Ghosal S, Kosut R andBrownell D 2001 Growth of giant magnetoresistancemultilayers: Effects of processing conditions during radio-frequency diode deposition J. Vac. Sci. Technol. A 192414–24

[30] Jones J E 1924 On the determination of molecular fields II.From the equation of state of a gas Proc. R. Soc. Lond. A106 463–77

[31] Axilrod B M and Teller E 1943 Interaction of the van derWaals type between three atoms J. Chem. Phys. 11 299

[32] Sasaki N, Kobayashi K and Tsukada M 1996 Atomic-scalefriction image of graphite in atomic-force microscopy Phys.Rev. B 54 2138–49

[33] Hadizadeh Kheirkhah A, Saeivar Iranizad E, Raeisi M andRajabpour A 2014 Mechanical properties of hydrogenfunctionalized graphene under shear deformation: Amolecular dynamics study Solid State Commun. 17798–102

[34] Tsai D H 1979 The virial theorem and stress calculation inmolecular dynamics J. Chem. Phys. 70 1375–82

[35] Anderson J A, Lorenz C D and Travesset A 2008 Generalpurpose molecular dynamics simulations fully implementedon graphics processing units J. Comput. Phys. 227 5342–59

[36] van Meel J A, Arnold A, Frenkel D, Zwart S F P andBelleman R G 2008 Harvesting graphics power for MDsimulations Mol. Simul. 34 259–66

[37] Plimpton S 1995 Fast parallel algorithms for short-rangemolecular dynamics J. Comput. Phys. 117 1–19

[38] Berendsen H J C, Postma J P M, Vangunsteren W F,Dinola A and Haak J R 1984 Molecular-dynamics withcoupling to an external bath J. Chem. Phys. 81 3684–90

[39] Cendula P, Kiravittaya S, Monch I, Schumann J andSchmidt O G 2011 Directional roll-up of nanomembranesmediated by wrinkling Nano Lett. 11 236–40

[40] Meyers M A 1994 Dynamic Behavior of Materials (New York:Wiley)

[41] Kocks U F, Argon A S and Ashby M F 1975 Thermodynamicsand Kinetics of Slip (Oxford: Pergamon)

[42] Castellanos-Gomez A, Poot M, Steele G A, van der Zant H S J,Agraït N and Rubio-Bollinger G 2012 Elastic properties offreely suspended MoS2 nanosheets Adv. Mater. 24 772–5

[43] Lee J H, Loya P E, Lou J and Thomas E L 2014 Dynamicmechanical behavior of multilayer graphene via supersonicprojectile penetration Science 346 1092–6

[44] van Duin A C T, Dasgupta S, Lorant F and Goddard W A 2001ReaxFF: a reactive force field for hydrocarbons J. Phys.Chem. A 105 9396–409

10

Nanotechnology 26 (2015) 265705 V N Borysiuk et al