Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 56, 3, pp. 601-614, Warsaw 2018 DOI: 10.15632/jtam-pl.56.3.601 COARSE-GRAINING MODELS FOR MOLECULAR DYNAMICS SIMULATIONS OF FCC METALS Pourya Delafrouz, Hossein Nejat Pishkenari Department of Mechanical Engineering, Sharif University of Technology, Tehran, Islamic Republic of Iran e-mail: nejat@sharif.edu In this paper, four coarse-graining (CG)models are proposed to acceleratemolecular dyna- mics simulations of FCC metals. To this aim, at first, a proper map between beads of the CGmodels and atoms of the all-atom (AA) system is assigned, afterwardsmass of the beads and the parameters of the CGmodels are determined in amanner that the CGmodels and the original all-atom model have the same physical properties. To evaluate and compare precision of these four CGmodels, different static and dynamic simulations are conducted. The results show that these CGmodels are at least 4 times faster than theAAmodel, while their errors are less than 1 percent. Keywords: accelerated molecular dynamics, coarse-grain models, FCC metals, EAM po- tential 1. Introduction One of the important challenges of working at the nanoscale is to analyze materials properties. Because experimental researches in characterization and analysis of materials characteristics at nano-scale are significantly expensive, improvement of theoretical modeling methods is an in-progress research field. Also, there are some important subjects which experimental setups and techniques cannot capture. On the other hand, it seems necessary to assess the system behavior and amend the experimental setup based on the results of modeling and simulation techniques before any practical attempt. Consequently, computational modelling techniques, with their great potential to model and predict atomic features of nano-structures, entice the interest of many researchers in nanotechnology. The necessity for this predictive capability has empoweredcomputationalmodelingandsimulationmethods tobecomean important component for inspecting phenomena in different nano-scale engineering applications. The most popular computational methods used in simulations atmicro and nano scales are classic and non-classic continuum theories (Chandramouli, 2014), molecular dynamics (MD) (Leach, 2001; Muc, 2011) and Monte-Carlo (Berg, 2005; Samani and Pourtakdoust, 2014). Molecular dynamics (MD) is a powerful technique in the computational study of nano-scale structures and processes, which can provide data on the time dependent performance of a molecular system. Augmented by quantum mechanical computations of interatomic potentials that explain the interaction of atoms, molecular dynamics can model behavior of a variety of materials with all-atom detail. However, the all-atom (AA) molecular dynamics method is not able to model many phenomena because it is practically limited to system sizes less than a few tens of nanometers and simulation times less than a few hundreds of nanoseconds (Chen et al., 2011). These restrictions have attracted a lot of attention in the development of techniques to accelerate molecular dynamics simulations. To accelerate long-time molecular simulations, differentmethods including increasing the integration time step, reducing the computational cost per step, or combinations of thesemethodologies have been developed (Poursina andAnderson, 2014). These improvements may be achieved through clustering atoms to make a larger rigid 602 P. Delafrouz, H.N. Pishkenari bead or by employing continuum theories in combinationwith atomisticmodeling.Totally, there are two main schemes for accelerating molecular dynamics simulations: multi-scale modeling (Cranford and Buchler, 2010; Park andKlein, 2007; Jovanovic and Filipovic, 2006) and coarse- -graining (CG)methods (Ouldridge, 2012; Yang andOu, 2014; Hedrich and Hedrich, 2010). As noted by Yang and To (2015), numerous multiscale techniques have been proposed to achieve the accuracy of full atomistic simulationswith a reduced computational cost (Tadmor et al., 2013; Burbery et al., 2017). These methods couple atomistic and continuum models to link different spatial and time scales from nanometer and femtosecond to meter and second (Paćko and Uhl, 2011; Karma and Tourret, 2016). FCCmetals are commonlyused inMEMs/NEMs systemsas amainmaterial.Although there exists a lot of researches employing themolecular dynamics technique for the modeling of FCC metals (Pishkenari and Meghdari, 2010; Lao et al., 2013; Pishkenari, 2015), faster techniques with enough accuracy are highly demanded for modeling these metals (Park and Klein, 2007; Oren et al., 2016). In this paper, we have proposed four effective CGmodels based on the first technique intro- duced in the previous paragraph, for simulation of FCCmetals. To this aim, at first, the proper atomic structure is defined and based on this new atomic structure, mass of the beads is deter- mined. Then, the potential parameters are determined in amanner that the original model and the coarse-grained models have the same physical properties like cohesive energy, bulkmodulus and elastic constants. After developing CG models, the bulk properties of an FCC metal such as elastic constants, bulkmodulus and potential energy are determined and compared with the results of the AA model. As a case-study, to show the speed and accuracy of the developed coarse-graining models in prediction of mechanical properties of nano-scale systems, we have studied longitudinal and transversal vibrations of nanowires. The results demonstrate that the error of the CGmodels is reduced when the surface effects decrease. Themaingoal of thispaper is introductionof anovel coarse-grainmodel inorder toaccelerate the molecular dynamics simulations. The way of implementing these methods for modeling of nanowires are as same as using the molecular dynamics method with few changes. Owing to decreasing the number of beads in thesemethods, nanowires of larger size can bemodeled with better accuracy compared to the Finite Elementmodel. As an example, the error of calculating thenatural transversal frequencyamong theall-atommodel andoneof theCGmodel is less than 0.6percent for ananowirewith 65.28nmlength and13.056nmthicknesswhile it is 4 times faster. Hence, these CG models are faster than the Molecular Dynamics method and more accurate than theFinite Elementmethod. The rest of the paper is organized as follows. In Section 2, four CGmodels for analyzing FCCmetals are introduced. In that Section, propermapping, mass of beads and potential parameters are developed. In Section 3, size-dependency of the proposed models is investigated by various static and dynamic simulations. The last Section is devoted to concluding remarks. 2. Proposed CG method The term “coarse graining” is used for methods based on replacing some atoms with one bead in order to reduce the number of degrees of freedom. Since the number of degrees of freedom in a CG model is significantly lower than in the main atomic system, simulation of the CG system needs considerably less computational resources; consequently, the simulation time and the system size can be substantially decreased (Cascella and Dal Peraro, 2009; Marrink et al., 2007). To develop a CG model for analyzing FCC metals, three steps must be done. At first, the location of the beads must be assigned. After choosing a proper atomic structure, the mass of the beads is specified and, finally, inter-atomic potential parameters are determined. Coarse-graining models for molecular dynamics simulations of FCC metals 603 2.1. Mapping One way to arrange the locations of beads is to choose an FCC crystalline structure as the CGmodel having a different lattice constant in comparison with the main atomic structure. In this way, a new FCC metal is made of the representative beads. Herein, the lattice constant of the CG model is set as twice of the lattice constant of the AA model, i.e. aCG = 2aAA, where aAA and aCG are respectively the lattice constants of the AA and CG models. Figure 1 shows this mapping for a cubic structure with an edge of 2aAA. Fig. 1. 3-D display of (a) the AAmodel and (b) the CGmodel The number of the beads in the resultant CG structure is about one-eighth of the number of atoms in the original structure. It should be mentioned that all of the four CG models have the same arrangement of the beads. After introducing the proposed mapping, the mass of the beads must be specified. 2.2. Mass of beads By calculating contributions of each atom to the assigned beads, the mass of the beads can be determined. At first, we describe how the mass of the internal beads, named as bulk beads, can be calculated. Based on the structure of the CG and AA models, there is an atom that is in the same place where the bead is and its entire mass contributes to the corresponding bead. There are also 12 atoms surrounding the bead with a distance ( √ 2/2)aAA from the bead and being shared with another bead equally. Also, there are 6 atoms with distance aAA from the bulk bead each of which are shared among 6 beads. Consequently, the mass of the bulk bead can be calculated as below Mbulk = ( 1+12 · 1 2 +6 · 1 6 ) matom =8matom (2.1) wherematom is the atomicmass of the FCCmaterial. In addition to bulk beads, there are other new beads including surface beads, edge beads and corner beads. For the surface beads, there is an atom at the same place where the bead is. There are 8 atoms at the distance ( √ 2/2)aAA from the surface bead each of which is shared between two beads. There are also 4 atoms on the surface with the distance aAA from the surface bead, and they are shared among 5 beads. Also there exists one atom of the systemwith the distance aAA from the surface bead that is shared among 6 beads. So the mass of the surface bead can be calculated as follows Msurface = ( 1+8 · 1 2 + 1 6 +4 · 1 5 ) matom = 179 30 matom ≈ 5.97matom (2.2) The number of atoms with the distance ( √ 2/2)aAA from the edge bead is 5, and each of them is shared between two beads.There are 4 atomswith the distance aAA from the edge bead. 604 P. Delafrouz, H.N. Pishkenari Two of them are on the surface and are shared among 4 beads while the remaining atoms are shared among 5 beads are at the edge. By noticing one atom at the same place where the edge bead is, the mass of the bead is calculated as follows Medge = ( 1+5 · 1 2 +2 · 1 4 +2 · 1 5 ) matom =4.4matom (2.3) Themass of the corner bead can be calculated as follows. There are 3 atoms that are shared between two beads with the distance ( √ 2/2)aAA from each atom. There are also 3 atoms with the distance aAA from the corner bead, and each of them are shared among 4 beads. Hence, the mass of the corner bead is obtained as follows Mcorner = ( 1+3 · 1 2 +3 · 1 4 ) matom =3.25matom (2.4) In addition to the four aforementioned beads, there are somenewbeadswithdifferent atomic neighborhood.To determine themass of these newbeads, a system is createdwith bothAAand CGstructures.Thenall of atomswith thedistancewhich is equal or less thanone lattice constant from each bead are determined. Also, the number of beads, inheriting from each atom, must be calculated. With this data, mass allocation for each bead can be computed and used inMD simulations. LAMMPS software (Plimpton, 1995) is used as themain solver in our simulations. Fig. 2. Eight types of beads in the last CGmodel: (a) one corner of a nanowire which can be zoomed in to display different bead types, (b) a cubic box on the corner of the nanowire with the edge of 2aCG (all types of beads are shown in this part of figure simultaneously), (c) external layer of the beads of the corner, (d) the first internal layer of the beads of the corner, (e) the second internal layer of the beads of the corner By using thementioned algorithm, 8 types of beads are obtained (Fig. 2). Four of them are previouslymentioned as the bulk, surface, edge and corner beads. The rest of them is as follows: • Beads shared between the surface and corner: these beads are on the surface with the distance aAA from the corner and their mass isMSC =6.0667matom. • Beads shared between the surface and edge: there are beads with mass of MSE = 6.0167matom. These beads are on the surface with the distance aAA from the edge. Coarse-graining models for molecular dynamics simulations of FCC metals 605 • Beads shared between the bulk (internalmaterial) and edge: these internal beads have the distance aAA from the edge and their mass is MBE =8.0667matom. • Beads shared between the bulk and surface: these internal beads have the distance aAA from the surface and their mass is MBS =8.033matom. Currently, we can introduce 4 CG models based on the mass assignment. Table 1 lists the specifications of these CGmodels. Although the second and third coarse-grainingmethods have the samemass assignments, their potential parameters are different (see the next Section). Table 1. Four different CGmodels CGmodel Mass assignment CG1 One bead type:Mbulk =8matom CG2 Four bead types:Mbulk =8matom,Msurface =5.9667matom, Medge =4.4matom,Mcorner =3.25matom CG3 Four bead types:Mbulk =8matom,Msurface =5.9667matom, Medge =4.4matom,Mcorner =3.25matom CG4 Eight bead types:Mbulk =8matom,Msurface =5.9667matom, Medge =4.4matom,Mcorner =3.25matom,MSC =6.0667matom, MSE =6.0167matom,MBE =8.0667matom,MBS =8.033matom In this Section, we have introduced 8 types of beadswhichwas developed for a cubic system. In the next Section, the potential parameters are determined for the CGmodels. 2.3. Potential parameters Themost widely used potential for themodeling of FCCmetals is EAM (Zhou et al., 2004). In this potential, the total potential energy can be expressed as E = 1 2 ∑ i,j,i6=j ϕij(rij)+ ∑ i Fi(ρi) (2.5) This potential model has two parts. The first part is ϕij that is known as pair energy between atoms i and j, and the second part is embedding energy Fi(ρi) that describes the effect of electron density ρi. The electron density can be obtained as below ρi = ∑ i,j 6=i fi(rij) (2.6) where fi(rij) is the electron density produced by atom j at the position of atom i. Herein, at first, the potential parameters for the first model (CG1) is introduced. For the first model, each bead is representative of eight atoms, hence the cohesive energy per bead in the CG model should be 8 times bigger than the cohesive energy per atom in the AA model. Consequently, to have the same potential energy for both CG1 andAAmodels at any arbitrary point, the potential energy obtained from Eq. (2.5) for the first CG model, must be multiplied by 8 to reproduce the correct potential energy. On the other hand, since the lattice constant of the CGmodels is aCG =2aAA, the distance between beads is also twice the distance between atoms. Thus, every part of the potential which is a function of distance, must be modified to be a function of distance divided by 2. Totally, the potential function for the first CGmodel, can be represented as follows ϕCG1(rij)= 8ϕAA (rij 2 ) ρCG1(rij)= ρAA (rij 2 ) FCG1(ρi)= 8FAA(ρi) (2.7) 606 P. Delafrouz, H.N. Pishkenari For the second CG model, the mass assignment for the beads is different from the first CG model; however, the potential parameters for these models are considered to be the same. For the third and fourth CGmodels, the potential parameters are modified based on the mass proportions. This means that each bead is not necessary representative of 8 atoms, and only bulk atoms are representative of eight atoms and other beads are representative ofMbead/matom atoms. Therefore, it is reasonable to replace number 8 in Eq. (2.7) withMbead/matom as follows ϕ(CG1)(rij)= Mbead matom ϕAA (rij 2 ) ρCG1(rij)= ρAA (rij 2 ) FCG1(ρi)= Mbead matom FAA(ρi) (2.8) For this model, a different pairwise potential is required for calculating the interaction be- tween two different beads. So, the EAM potential relation for alloys can be used (Zhou et al., 2004), which has the following relation ϕab(r)= 1 2 [ρb(r) ρa(r) ϕaa(r)+ ρa(r) ρb(r) ϕbb(r) ] (2.9) Equation (2.9) shows the pairwise potential function for alloys a and b. Since electron density is equal for all beads, the average of the two pairwise potential function should be considered for the interaction between two beads. Thus, in this paper, four different CG models which have the following specifications are considered: 1. CG1: one type of bead, 2. CG2: four types of beads having four different mass values while using the same potential parameters, 3. CG3: four types of beads having four differentmass values and potential parameters tuned based on themass of the beads, 4. CG4: eight types of beads having eight different mass values and potential parameters tuned based on the mass of the beads. After introducing the CGmodels, the validity of themmust be checked. For this purpose, bulk simulations such as calculating elastic properties are performed firstly. In our simulations, gold is studied as one of the FCCmetals. To simulate a bulkmaterial, periodic boundary conditions are used in three dimensions to remove the effect of boundaries. A cubic central box with an edge length of 10aAA is considered. Our goal is to compare the total potential energy, bulk modulus and elastic properties obtained from the CGmodels with those obtained from the AA method and experiment (Simmons and Wang, 1971). For evaluating the elastic constants, a specified strain is applied to the system and stress changes caused by the strain are measured. By dividing the stress changes by strain, different elastic constants are calculated. Generally, a material has 36 elastic constants; however, for FCC metals, due to symmetry, there are only three different elastic constants. As can be expected, our simulation results demonstrate that all CGmodels predict the bulk properties exactly as same as the AAmodel. In fact, all of the CG models behave completely similar, since for the bulk materials there exists only one bead type. Therefore, all CGmodels are appropriate for the modeling of the bulk FCCmaterials. In the following Section, some static and dynamic simulations are performed so as to check the size-dependency of these models. To this aim, in different nanowires, the natural frequency of longitudinal and transversal vibrations by employing these models is calculated. Then the results are compared with the results of the AAmodel. Investigation of how accurate these CG models can predict the mechanical properties of an FCC metal having free surfaces is studied in the next Section. Coarse-graining models for molecular dynamics simulations of FCC metals 607 3. Case study In thisSection, theaccuracyof theCGmodels in thepredictionofmechanical propertiesof small- -size nanowires is examined. To this aim, Young’s modulus in static longitudinal deformation and the natural frequency in longitudinal and transversal vibrations of nanowires are studied. Figure 3 depicts a gold nanowire with clampled-free boundary conditions. Fig. 3. A gold nanowire. Three zones are recognizable at each setup: the boundary zone having a length 2aAA (shown by black color), moving zone having a length aAA (shown by yellow color), and the excitation zone displayed in red color 3.1. Longitudinal deformation Herein, Young’s modulus through static analysis is calculated and the size effect on the accuracy of 4 CG models is investigated. Then, longitudinal vibrations of the nanowire are studied. 3.1.1. Calculating Young’s modulus Young’s modulus is calculated based on the potential energy changes followed by strain changes. For this purpose, five different strains are applied to the nanowire, and variations of the potential energy per unit volume are calculated for these strains as follows ∆u= 1 2 Eε2 (3.1) where ∆u stands for potential energy changes per volume, E is Young’s modulus and ε is the strain. By fitting a quadratic curve to the data, Young’s modulus is calculated. Figure 4 shows the results for a gold nanowire with 56aAA in length and 14aAA in thickness. According to the relative error of the CG models, the most accurate model is the fourth model, which is predictable due to more corrections applied to bead masses and potential pa- rameters in this model. Furthermore, as it is expected, the errors of the first and second CG models are the same. In fact, in static simulations, the mass distribution plays no significant role, and the difference between the first two models and two other models is only because of the distinct potential parameter of the beads. Next, Young’s modulus is calculated for different sizes of nanowires and the accuracy of these 4 CGmodels is examined. Nanowires with thickness of 14aAA andwith various lengths are considered for checking the effect of length. The length of these nanowires varies from 4 to 10 times larger than their thick- nesses. The results of these simulations show that the error of the CGmodels is approximately constant for different lengths. Since there is no change in the surface effects by changing length of the nanowire, it is expected that the error of the models does not change significantly with variations of the length. 608 P. Delafrouz, H.N. Pishkenari Fig. 4. Variations of the potential energy per volume with respect to strain for a gold nanowire with length of 56aAA and thickness of 14aAA Thickness is another size parameter influencing the error of themodels. Thenanowire length is set to be 8 times larger than its thickness. The results depict that the error of the CGmodels in the prediction of Young’s modulus decreases as the nanowire thickness increases. The reason for this behavior is that the surface effects decrease as the nanowire thickness increases. The first and secondmodels use the samepotential parameters.Nevertheless, theirmass assignments are different. Therefore, it is expected that their behavior in the static and dynamic simulations are similar and different, respectively. Since in this Section a static simulation is performed, the results of these twomethods are the same. 3.1.2. Calculating the natural frequency of longitudinal vibration Here we aim at the investigation of the size effect on the exactness of the proposed CG models in the estimation of the dynamic behavior of the nanowires in longitudinal vibrations. In this regard, the effect of length, thickness, and the scaling parameter on the first longitudinal vibrational frequency of the clamped-free gold nanowire is studied. The details of simulation setup are explained as follows: • Afterminimization, two lattices (one lattice in theCGmodels) at one end of the nanowire are fixed and temperature of the nanowire is maintained at 0.01K employing the Nosé- -Hoover thermostat (Nosé, 1984; Hoover, 1985). It should be mentioned that the higher the temperature of the system is, the greater its kinetic energy is, thatmay yield incorrect results. Therefore, the nanowire temperature is controlled at 0.01K. • The other end of the nanowire is displaced with a constant speed. This displacement is about 1% of the nanowire length to avoid nonlinearity and plastic deformation. • Then the excited atoms are fixed (at a new position), and temperature of the nanowire is controlled at 0.01K. • At the last stage, the end of the nanowire is released to vibrate freely. The corresponding frequency of longitudinal vibrations can be calculated by fitting a sinusoidal function to its end-point displacement (Pishkenari et al., 2015, 2016). Figure 5 shows the x position of the nanowire free end for a nanowire with length of 80aAA and thickness of 20aAA. According to Fig. 5, all CG methods similar to the AA model pre- Coarse-graining models for molecular dynamics simulations of FCC metals 609 dict oscillating behavior for motion of the nanowire free end. The natural frequencies that are calculated from Fig. 5 are listed in Table 2. Fig. 5. Position x of the nanowire free end in the fourth step of simulation. The nanowire length and thickness are 80aAA and 20aAA, respectively Table 2. Natural frequency of longitudinal vibrations for the proposed CG models and the relative error in the estimation of the natural frequency in addition to computation time AA CG1 CG2 CG3 CG4 Longitudinal natural 75.4 74.8 77.1 75.1 75.1 frequency [GHz] Relative error [%] — 0.80 2.3 0.40 0.40 CPU time [s] 15356 2021 3666 3832 3917 Based on the relative error of the CG models, the most accurate models are the third and fourth ones. Despite the fact that we expect the error of the second model to be smaller than the first one, the first model is more accurate in this simulation setup. It should be mentioned that the error of the CGmodels highly depends on the nanowire thickness, and the dependency is different in four CG models. It is worthy to note that functionality of the first, third and fourth CG models on the size parameters of the nanowire is nearly similar (where always the third and fourth models are more precise than the first one), but behavior of the second model is completely distinct. In fact, for a considerable range of nanowire thickness, the secondmodel may give less exact results with respect to the first model (follow the next Sections). Further simulations are done to investigate the effect of the size parameters on the error of the CGmodels. Length, thickness and scale parameter are the three size parameters that their effects are studied in this Section. Length effect: In the simulations, the nanowire length is changed from 48aAA to 120aAA while its thickness is set as 12aAA. The results show that the nanowire length does not influence the accuracy of four CGmodels. This behavior is due to the fact that the surface effects do not significantly change with the nanowire length. A striking point observed in this figure is that at this nanowire thickness the second model is the most precise one. 610 P. Delafrouz, H.N. Pishkenari Thickness effect: Figure 6 illustrates the effect of thickness on theprecision of theCGmodels. InFig. 6a, thenanowire thickness is varied from8aAA to 20aAA and its length is set tobe10 times larger than its thickness. In Fig. 6b, the nanowire length is set to be 100aAA and its thickness is varied from 8aAA to 20aAA. Fig. 6. Error of the estimated longitudinal first natural frequency of the gold nanowire for different CG models: (a) length of the nanowires is 10 times larger than its thickness, (b) nanowire length is set as 100aAA Based on Fig. 6, the error of the CG models in the prediction of the longitudinal natural frequency decreases to zero by increasing thickness of the nanowire. The reason for this behavior is that the surface effects reduce as nanowire thickness increases. It should be noted that the secondmodel cannotproperlypredict thenatural frequency for nanowire thicknesses over 14aAA. In this model, the exterior beads have differentmasses with respect to the interior beads, while the potential parameters of all beads are the same.Due to limitations in computational facilities, simulations of larger nanowires are not possible in this work. However, for larger thickness, it is expected that the error of the secondmodel converges to zero because all coarse-grainedmodels behave similar to bulkmaterials. Scale parameter effect: To examine another effect of nanowire size on the accuracy of the CG models, a nanowire with length of 40aAA and thickness of 8aAA is considered as the base nanowire, and all of its dimensions aremultiplied bya scale parameter.The results show that the frequency error decays to zero for three of the CGmodels. It should be noted that the surface- -to-volume ratio reduces as the nanowire size increases, hence the surface effects should vanish for larger values of the scale parameter. For the second CG model, at first the frequency error descends from positive values to negative values and then it ascends gradually. It is expected that the error of the second model eventually converges to zero for larger values of the scale parameter because the bulk simulations proved that all of the CG models behave as same as theAAmodel. It should bementioned that our computational facilities limit our simulations to scale parameter 4. 3.2. Transversal vibrations In this Section, transversal vibration of the nanowire is investigated. The nanowire free end is displaced and released in the z direction. By fitting a sinusoidal function on motion of this point in the direction of excitation, the transversal natural frequency ismeasured (Pishkenari et al., 2016). The initial displacement of the beam is equal to 1% of the total nanowire length to avoid large-displacement nonlinear effects. Herein, wewill examine the effect of nanowire length and thickness as well as the scale parameter on the accuracy of four CGmodels. Coarse-graining models for molecular dynamics simulations of FCC metals 611 Length effect: for investigating this effect, a nanowire with thickness of 16aAA is conside- red and its length is varied from 64aAA to 160aAA. As previously discussed, the surface effect does not change with variation of the nanowire length, hence the error of the models remains approximately constant as the nanowire length is altered in these simulations. Thickness effect: in this Section, the effect of nanowire thickness is studied. The thickness of the nanowire varies from 8aAA to 20aAA. The length of the nanowire for simulations in Figs. 7a and 7b are set to be 10 times larger than the nanowire thickness and 100aAA, respectively. Fig. 7. The error of 4 CGmodels in estimating the transversal natural frequency as a function of nanowire thickness: (a) length of the nanowire is 10 times larger than its thickness, (b) nanowire length is set to be 100aAA Increasing the thickness decreases the surface effect and, therefore, the error of the CG models decays to 0. It should be noted that the error of the first and second CG models does not converge to zero at this thickness interval. However, based on the results of bulkmaterials, it is expected that the error of thesemodels eventually converges to zero for larger values of the nanowire length whose simulation is beyond capability of our computational resources. The scale parameter effect: the effect of the scale parameter on the accuracy of the CG models is investigated here. The base nanowire in these simulations has length of 40aAA and thickness of 8aAA. According to the results, the error of the CG models at first descends from positive values to negative values and then slowly ascends. As the nanowire size increases, its properties become closer to behavior of the bulk material. Consequently, the error of the CG models should eventually converge to zero for larger values of the scale parameter. Totally, the third and fourth models have the least error. In this part, the natural frequency of a nanowire with 160aAA in length and 32aAA in thickness is evaluated. The AA model for the nanowi- re has more than 670000 atoms, however, the CG model of the nanowire has less than 90000 beads. According to the dimension of the nanowire, approximately 3 million steps are needed to calculate the natural transversal frequency of the nanowire. It causes a seventeen-day si- mulation for the AA model. Our model can decrease the time to about 4 days while its error is less than 0.6%. When nanowire size rises, not only the number of the beads increases, but also steps of simulations go up. Hence, evaluating the natural frequency for a larger-size na- nowire by the AA model is very time-consuming and it is not possible for us to compare our results for larger-size nanowires. Nevertheless, the CG models are capable of modeling these nanowires. 612 P. Delafrouz, H.N. Pishkenari 4. Conclusions In this paper, 4 different CGmodels are introduced: 1. CGmodel with one type of bead and one type of potential parameter. 2. CGmodel with four types of bead and one type of potential parameter. 3. CGmodel with four types of bead and four types of potential parameter. 4. CGmodel with eight types of bead and eight types of potential parameter. To reveal the ability of the CG models for reproduction of results of the original atomic system, we have conducted different simulations. At first, it is shown that these models can predict behavior of the bulk material, including elastic constants, bulk modulus and potential energy, exactly as same as the AA model. Then we have applied longitudinal and transversal displacements to the free end of a nanowire and comparedYoung’smodulus aswell as the longi- tudinal and transversal vibrational frequency of theCGmodels with theAAmodel. The results exhibit that the CG models can properly estimate the results of the AA model. Nevertheless, the accuracy of the proposed CG models in calculating mechanical properties of the nanowire highly depends on the nanowire size. Totally, among four proposed CG models, the error of the second model highly and nonli- nearly depends on the nanowire size. Also the error of the second model at a considerable size interval is relatively larger than in other models. The simulation results of the third and fourth models are approximately the same but the third method is slightly faster. 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