Philosophical Magazine ISSN: 1478-6435 (Print) 1478-6443 (Online) Journal homepage: http://www.tandfonline.com/loi/tphm20 Stages of melting of graphene model in twodimensional space Vo Van Hoang, Le Thi Cam Tuyen & To Quy Dong To cite this article: Vo Van Hoang, Le Thi Cam Tuyen & To Quy Dong (2016): Stages of melting of graphene model in two-dimensional space, Philosophical Magazine, DOI: 10.1080/14786435.2016.1185183 To link to this article: http://dx.doi.org/10.1080/14786435.2016.1185183 Published online: 25 May 2016 Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tphm20 Download by: [University of Sussex Library] Date: 26 May 2016, At: 16:36 Philosophical Magazine, 2016 http://dx.doi.org/10.1080/14786435.2016.1185183 Stages of melting of graphene model in two-dimensional space Vo Van Hoanga, Le Thi Cam Tuyenb and To Quy Dongc a Downloaded by [University of Sussex Library] at 16:36 26 May 2016 Comp Phys Lab, HochiMinh City Univ of Technology, Vietnam National University-HochiMinh City, HochiMinh City, Vietnam; bDepartment of Physics, College of Natural Sci., Can Tho University, Can Tho, Vietnam; c Laboratoire de Modélisation et Simulation Multi Echelle Equipe Transferts de Chaleur et de Matière, Université de Marne-la-Vallée, Marne-la-Vallée, France ABSTRACT Spontaneous melting of a perfect crystalline graphene model in 2D space is studied via molecular dynamics simulation Model containing 104 atoms interacted via long-range bond-order potential (LCBOP) is heated up from 50 to 8,450 K in order to see evolution of various thermodynamic quantities, structural characteristics and occurrence of various structural defects We find that spontaneous melting of our graphene model in 2D space exhibits a first-order behaviour of the transition from solid 2D graphene sheet into a ring-like structure 2D liquid Occurrence and clustering of Stone–Wales defects are the first step of melting process followed by breaking of C–C bonds, occurrence/growth of various types of vacancies and multimembered rings Unlike that found for melting of a 2D crystal with an isotropic bonding, these defects not occur homogeneously throughout the system, they have a tendency to aggregate into a region and liquid phase initiates/grows from this region via tearinglike or crack-propagation-like mechanism Spontaneous melting point of our graphene model occurs at Tm = 7,750 K The validity of classical nucleation theory and Berezinsky–Kosterlitz–Thouless–Nelson– Halperin–Young (BKTNHY) one for the spontaneous melting of our graphene model in strictly 2D space is discussed ARTICLE HISTORY Received January 2016 Accepted 27 April 2016 KEYWORDS Spontaneous melting of graphene model; 2D melting; two-phase melting 1. Introduction Graphene research has become one of the hottest research directions of the modern science and technology for more than one decade since it was discovered in 2004 (see [1–7] and references therein), although much attention has been turned out recently on its counterpart – silicene (see [8,9] and references therein) So far, our understanding of graphene’s properties including melting point and thermodynamics is still poor Melting of graphene has not been widely studied, maybe due to difficulty to carry out traditional calorimetric experiments for studying of melting of single atomic layer materials, one can find only some computer simulation works [10–12] Indeed, via Monte-Carlo (MC) simulation with LCBOP-II interatomic potential (a new version of LCBOP proposed in 2005), melting of graphene starts CONTACT Vo Van Hoang vvhoang2002@yahoo.com © 2016 Informa UK Limited, trading as Taylor & Francis Group Downloaded by [University of Sussex Library] at 16:36 26 May 2016 2 V V Hoang et al via clustering of Stone–Wales (SW) defects and formation of octagons After that, final molten state of 3D network of entangled carbon chains is formed [10] The spontaneous melting of graphene model containing up to 16,128 atoms, estimated by the Lindemann criterion, occurs at around 4,900 K This value constitutes an upper limit for the melting temperature However, the data in [10] are obtained only at some selected temperatures Therefore, information related to melting of graphene is limited Melting of graphene clusters containing from to 55 atoms is studied using density-functional tight-binding and classical MD [11] Melting point is estimated using the bond energy, the Lindemann criterion and specific heat It is found that the edges of graphene clusters proceed through different metastable states during heating before melting occurs [11] Recently, the melting of a graphene model with LCBOP-II interatomic potential has been determined by means of classical nucleation theory (CNT) for the first-order phase transition from 2D solid to 3D liquid via intermediate quasi-2D liquid [12] The melting temperature is found to be around 4,510 K, also in agreement with the asymptotic results of melting simulations for finite discs and ribbons of graphene with the same interatomic potential [12] The melting temperature of graphene is higher than that of graphite, carbon nanotubes or fullerenes [10] Note that MC method has been used for a rather small model containing 1,008 atoms [12] So far, details of melting of graphene in 2D space have not been investigated yet, i.e a clearer atomic mechanism of melting should be clarified via detailed analysis of evolution of structure and spatio-temporal arrangements of occurrence/growth of liquid-like atoms in the melting region (below and above a melting point) For describing melting of 2D crystals, BKTHNY theory is commonly accepted, i.e it is considered that melting occurs via unbinding of topological defects [13–19] The validity of this theory has been checked via studying of melting of 2D monatomic Lennard-Jones (LJ) crystals and other systems (see [20] and references therein) The validity of the BKTHNY theory has been under debate since results of simulations and experiments not agree with each other (see [20–27] and references therein) In particular, for melting of graphene it is found that splitting of a SW defect into two 57 defects and their subsequent diffusion is very unfavorable, i.e it does not support BKTHNY theory [12] Our main aim here is twofold: (i) comprehensive study of atomic mechanism of the spontaneous melting of graphene in strictly 2D space which has not been studied well yet; (ii) checking the validity of some models of melting of 2D crystals proposed in the past 2. Calculations We study the spontaneous melting of a perfect 2D graphene model with a bond length of 1.42 Å by MD simulations using zero pressure NPT ensemble in 2D space Periodic boundary conditions (PBCs) are applied only in the x and y directions while coordination of atoms in the z direction is fixed at z = 0.0 We use LAMMPS software for MD simulation with the Verlet algorithm and MD time step is 01 fs [28] Model contains 104 atoms interacted via LCBOP potential proposed in 2003 and denoted as LCBOP-I [29] According to LCBOP-I, total binding energy is a sum of two-pair terms: Eb = N N ( ) ∑ tot ∑ fc,ij VijSR + Sij VijLR Vij = i,j i,j (1) Downloaded by [University of Sussex Library] at 16:36 26 May 2016 Philosophical Magazine 3 where the total pair interaction (Vijtot) is a sum of a short-range part ( fc,ij VijSR) and a longrange one (Sij VijLR), see more details in [29] LCBOP-I is an approximately parameterised mix of a short-range Brenner-like bond order potential and a long-range radial one LCBOP-I gives good elastic constants for diamond and graphite, a reasonable description of the reaction path for the bulk diamond to graphite transition and a good description of the interlayer interaction energy in graphite [29] Moreover, LCBOP-I has been implemented in LAMMPS and it is convenient for users to carry out simulations in the field Note that an improved version of LCBOP (it is called LCBOP-II) was proposed in 2005 [30] However, it has not been implemented in LAMMPS yet Note that various interatomic potentials for carbon have been proposed (see [29] in details) and we not pause here for more discussions We employ ISAACS software for calculating ring statistics [31] We use VMD software for 2D visualisation of atomic configurations [32] We use the cutoff radius of 2.04 Å in order to calculate coordination number and interatomic distance distributions in the system This cutoff radius is equal to the position of the first minimum after the first peak in radial distribution function (RDF) of molten model obtained at 8,100 K For calculations of rings, the ‘shortest path’ Guttman’s rule is applied [31] Initial 2D configuration of a perfect graphene has a size of S = 213.000 × 122.976 Å2 and density 𝜌 = NS = 0.382 with a bond length of 1.42 Å Initial configuration has been relaxed at 50 K for 106 MD steps before heating up to 8,450 K at the heating rate of 1011 K/s Temperature of the system is increased via velocity rescaling at every time step as follows: T = T0 + γt Here, T0 = 50 K and γ is a heating rate, t is a time required for heating Model obtained at each temperature has been relaxed at a given temperature for 5,000 MD steps before analysing structural characteristics or 2D visualisation Note that we use Nose-Hoover barostat implemented in LAMMPS for controlling pressure [28] 3. Results and discussions 3.1. Thermodynamics of melting and evolution of structure upon heating to molten state Temperature dependence of total energy per atom and heat capacity under constant zero pressure upon heating from 50 to 8,450 K can be seen in Figure A long linear part of total energy from 50 K up to around 7,600 K is related to the solid state of graphene, where atoms Figure 1. Temperature dependence of total energy per atom and heat capacity (inset) 4 V V Hoang et al Downloaded by [University of Sussex Library] at 16:36 26 May 2016 mostly perform vibrations around their equilibrium positions However, above 7,600 K total energy has a sharp increase exhibiting a first order-like phase transition, i.e spontaneous melting occurs around T1 = 7,750 K On the other hand, heat capacity exhibits a sharp peak at T1 = 7,750 K and a smaller peak at around T2 = 8,100 K A first sharp peak at around T1 = 7,750 K exhibits a first-order behaviour of the 2D crystal → ring-like 2D liquid transition (atoms in the latter mostly aggregate into rings, see our discussion given below) The second phase transition occurred at around T2 = 8,100 K is ring-like 2D liquid → string-like 2D liquid transition (atoms in the latter mostly aggregate into strings) Heat capacity at constant zero pressure is calculated approximately via relation: CP = ΔE/ ΔT Some important points can be drawn here as follows: (1) Spontaneous melting of our graphene model occurs at rather high temperature compared to that found via MC simulation with LCBOP II in [10] (7,750 K vs 4,900 K) due to a strictly 2D space used in our MD simulation It is also higher than the value Tm = 5,800 K found for graphene by extrapolation to infiniteradius nanotubes in [33] Note that much lower melting point of around 4,000 and 4,800 K is found for fullerenes and carbon nanotubes, respectively [34,35] Note again that all simulations in [10,34,35] have been done in 3D space In the present work, we focus attention on the mechanism of spontaneous melting in 2D space of graphene which has not been studied well in the past (2) Individual two peaks in CP reflect the latent heat of transition between different phases of the system Similarly, appearance of additional smaller peaks after a first sharp peak in temperature dependence of the heat capacity of fullerenes (for C60 and C240) has been found previously [34] More details of the phase transitions related to the spontaneous melting of our graphene model in 2D space can be seen in Figure It is clear that spontaneous melting of our graphene model initiates/ grows by the occurrence of various defects (Figure 2(a)) Coexistence of crystalline and melted regions in atomic configuration at temperature around T1 = 7,750 K indicates a first order behaviour of the transition (Figure 2(b)) (3) We find that system completely melted at around 7,900–8,000 K Molten state contains mainly rings of various sizes of liquid-like atoms and mean coordination number at 8,000 K is Z̄ ≈ 1.9 In contrast, at around T2 = 8,100 K rings are mostly broken into strings indicating ring-like 2D liquid → string-like 2D one transition The spontaneous melting of our graphene model is also reflected via evolution of RDF upon heating to molten state (Figure 3) Crystalline structure of models at low temperature is reflected by very sharp peaks in RDF, with increasing temperature these peaks start to broaden and their intensity decreases As temperature increases up to around 7,600 K, there are still peaks with similar location At around a melting point (T1 = 7,750 K), the peaks at intermediate and far distances not completely disappear indicating the existence of a significant crystalline order in the system (see Figures 2(b) and 3) However, at T2 = 8,100 K RDF is rather smooth and the peaks at intermediate/far distances completely disappear This means that system transforms into a relatively homogeneous liquid like that discussed above On the other hand, position of the first peak in RDF is shifted to smaller distance with increasing temperature indicating the ring formation and crystalline structure is washed out like that found in [10] Location of the first sharp peak in RDF at the distance smaller than interatomic distance in solid graphene may be due to the formation of shorter bonds Philosophical Magazine 5 Downloaded by [University of Sussex Library] at 16:36 26 May 2016 (a) (b) Figure 2. (colour online) 2D visualisation of atomic configurations obtained at various temperatures upon heating to molten state: atoms with Z = 3 are coloured by the blue one and atoms with Z ≠ 3 are coloured by the red one Figure 3. (colour online) Temperature dependence of RDF upon heating in the rings/strings like that suggested in [10] However, negative thermal expansion of graphene found by experiments (see [36] and references therein) has not been detected in the present work due to strictly 2D space used in simulation Note that in simple systems, the first peak shifts to a larger distance when it is going from crystal into liquid [37] Downloaded by [University of Sussex Library] at 16:36 26 May 2016 6 V V Hoang et al Figure 4. Temperature dependence of mean ring size and mean coordination number (inset) (a) (b) (c) (d) Figure 5. (colour online) Coordination number distributions at various temperatures below and above melting point (T1 = 7750 K) More details of structure evolution upon heating to molten state can be seen via temperature dependence of coordination number or ring distributions including mean value of these quantities (Figures 4–6) Mean ring size is equal to up to around 7,600 K indicates that model still remains in a solid state with a honeycomb structure After that, it strongly increases due to occurrence of melting This is related to the formation of large rings Mean ring size reaches a maximum value at around 7,900–8,000 K, which is related to the full formation of ring-like liquid At T > 8,000 K, mean ring size starts to decrease which is related to the breaking of rings and indeed ring-like liquid → string-like liquid transition occurs at around T2 = 8,100 K (see Figure 4) In contrast, mean coordination number remains at around value Z = 3 up to around 7,600 K It strongly decreases at T1 = 7,750 K due to occurrence of melting of graphene (see the inset of Figure 4) The decrease changes slope at around T2 = 8,100 K indicating a second phase transition at around this point Downloaded by [University of Sussex Library] at 16:36 26 May 2016 Philosophical Magazine (a) (b) (c) (d) 7 Figure 6. (colour online) Ring distributions at various temperatures at below and above melting point (T1 = 7750 K) More details of evolution of structure upon heating to a molten state can be seen via coordination number and ring distributions at various temperatures (see Figures and 6) Note that we not employ PBCs for calculating coordination number to see what happens at the edges of graphene We find that at T ≤ 7,600 K atoms mostly have coordination number Z = 3 (i.e at 7,500 K, around 96% atoms have Z = 3) indicating that model still remains in the solid state with a honeycomb structure Only small fraction of atoms has Z 3 in the whole temperature range Downloaded by [University of Sussex Library] at 16:36 26 May 2016 Philosophical Magazine 9 Figure 8. Temperature dependence of fraction of atoms with various coordination numbers studied This indicates that there are no overcoordinated defects in graphene models simulated in 2D space As shown in Figure 8, fraction of atoms with Z = 3 remains constant at high value (~97%) up to around 7,600 K and then it has a sudden drop at the melting point (at T1 = 7,750 K) In contrast, fraction of atoms with Z = 1 and Z = 2 remains constant at low value (~3%) up to around 7,600 K, then it strongly increases in the melting region (7,700–7,900 K) Existence of atoms with Z = 1 and Z = 2 in the solid state (at T ≤ 7,600 K) is due to non-employment of PBCs in the calculation of coordination number and they are the edge atoms (see Figure 2) However, in the melting region atoms with Z = 1 and Z = 2 occur inside model and they can be considered as liquid-like ones (Figure 2) Note that fraction of atoms with Z = 2 does not monotonously increase with temperature in the high-temperature region unlike that found for atoms with Z = 1, i.e it passes over a maximum at around 7,900 K and then it decreases similarly to that found for fraction of six-fold rings Such evolution of fraction of atoms with Z = 2 is related to the competition between formation and breaking of rings in the molten state like that discussed above In contrast, monotonous increase in fraction of atoms with Z = 1 is related to the breaking of rings and formation of the dangling bonds At very high temperatures, we also find occurrence of atoms with Z = 0 Existence of atoms with Z = 1 and Z = 0 in the high temperature region with a significant amount may be related to the vaporisation of the system (not shown) Structural defects play an important role in melting of graphene Various structural defects and their role for graphene have been reviewed [5] The most popular defects found in graphene can be listed as follows: SW defects, single vacancies (SVs), multiple vacancies including di-vacancies (DVs) and adatoms In the present work, we also find the formation of various structural defects during heating of graphene model (see Table 1) We find that breaking bond occurs leading to the formation of the simplest SVs located inside multi-membered ring at temperature around 6,500 K or higher while rotation of C–C bond of six hexagons leading to the formation of single SW (55-77) defects These defects initiate at much lower temperature of around 5000 K due to their smallest formation energy It is found that the SW (55-77) defect has a formation energy of around 5 eV [38,39], whereas a lower value of 4.6 eV is found using LCBOP II [10] In addition, further heating leads to the formation of double SW defects or pair of SWs SW defects may play a role of 10 V V Hoang et al Table 1. Occurrence of various defects in graphene model upon heating to temperature below melting point T (K) Downloaded by [University of Sussex Library] at 16:36 26 May 2016 Type of defect 5,000–6,900 SW(55-77), simple SV 7,000–7,490 SW(55-77), double SW(55-77) 7,500–7,600 SV(5-9), DV(5-8-5), DV(555-6-777), DV(5555-6-7777), DV(5555-666-7777) 7,650–7,750 Multi-membered rings precursors for the formation of larger rings upon further heating Note that single SWs and pair of SWs have been found previously in [10] However, isolated pentagons, heptagons or 5-7 dislocations have not been found during melting process of graphene model with LCBOP II [10,12] The same situation is found for our model in the temperature region up to 7,600 K On the other hand, we find the existence of octagons in the region rich of 5-7 clusters, octagons may be the precursors for the formation of larger rings like that found in [10] The simplest defect in any material is missing atom of lattice leading to formation of SVs and we find that SVs also occur at temperature much lower melting point Note that SVs have been found by the experiment via using TEM [40,41] and STM techniques [42] The formation energy of SV in graphene is rather high because of the presence of undercoordinated C atoms It is calculated that formation energy for SV in graphene is around 7.5 eV which is much higher than that found in other materials while their migration barrier is rather low of around 1.3 eV (see [5] and references therein) indicating their relatively high mobility in graphene On the other hand, DVs should be observed in graphene DVs can be formed by coalescence of two SVs or by removing two neighbouring atoms in graphene lattice Formation of various types of DVs in our graphene models has been found (Table 1) Simulations show that the formation energy of DVs is of the same order as for SV, i.e about 8 eV [43,44] Structure reconstruction of DV(5-8-5) can lead to the formation of various DVs In particular, rotation of one of the bonds in the octagon of DV(5-8-5) transforms defect into DV(555-777) like that found in [45] Total formation energy of this defect is about 1 eV which is lower than that for DV(5-8-5), existence of this defect was found by electron microscopy [5] By rotating another bond, DV(555-777) transforms into DV(5555-6-7777) defect, the latter is also found by experiment [5] Indeed, DV(5555-6-7777) is also found in our model (see Table 1) The formation energy of DV(5555-6-7777) is between those of DV(5-8-5) and DV(555-777) [38] Moreover, we find the existence of new DVs such as DV(555-6-777) and DV(5555-666-7777) Note that similar complex DVs, e.g DV(5555-667777) for example, have been found recently by experiment [46] Formation of complex DVs such as DV(2 × 555-66-777) or DV(2 × 5555-666-7777) plus one additional pentagon is also found via TEM technique and then it is interpreted by DFT calculation [47] In addition, we also find the existence of SV-like defect with a strange form containing dangling bonds inside (not shown) New defects found in the present work give a deeper understanding of structural defects in graphene In general, DVs are immobile in graphene due to very high migration barrier compared to that of SVs [5] It is essential to note that the formation energy of various structural defects listed above is calculated in 3D space and we present here only for discussion since our simulation is carried out in 2D space In general, missing of two or more atoms and further reconstruction of structure may lead to the formation of the more complicated defective configurations It is found that missing Downloaded by [University of Sussex Library] at 16:36 26 May 2016 Philosophical Magazine 11 of even number of atoms allows a full reconstruction, i.e complete saturation of dangling bonds, and such vacancies are energetically favored over structure with an odd number of missing atoms where an open bond remains [48] One interesting defect, adatom, has not been found in our graphene model over whole temperature range studied due to strictly 2D space used in simulation Moreover, other type of defect such as multi-membered rings (R9, R10, R11, etc.) has been found in our model (not shown) These large rings may act as attractive centres for foreign atoms or precursors for melting Additional defect is 1D-like defect (or line defect) which may occur in our model Indeed, upon heating to high temperature many defects including vacancies, SW defects, multi-membered rings etc have a tendency to concentrate together to form a typical 1D-like defect in model The latter acts as precursor for melting (Figure 2(a) and (b)) Various types of defects occur in graphene model in different temperature regions upon heating due to the difference in the formation energy (Table 1) One can see in Table that SW defects occur first in the early stage of heating process due to the smallest formation energy like that found in [5], i.e we find that SW(55-77) occur at around 5,000–6,900 K and further heating increases their concentration leading to the formation of double SW(55-77) in the range of 7,000–7,490 K In addition, at around 6500 K breaking of C–C bond occurs leading to the formation of the simplest SV and their number increases with increasing in temperature In the high temperature region of 7,500–7,600 K, various SVs including SV(5-9) occur and their number increases with heating and leading to the formation of multiple vacancies (see Table 1) Finally, in the vicinity of melting region other type of defects, i.e large rings, occurs leading to melting of graphene lattice This means that the spontaneous melting of a perfect graphene lattice starts by the appearance and growth of various structural defects leading to the formation of rings of carbon atoms 3.3. Atomic mechanism of melting We pause here for more details of atomic mechanism of the spontaneous melting of our graphene model based on analysis of the occurrence/growth of structural defects in model upon heating to a molten state Since atoms in a perfect graphene have the same coordination number Z = 3, therefore, atoms with Z ≠ 3 can be considered as structural defects Note that it is an approximation approach since SW defects still keep this three-fold coordination One can see in Figure that fraction of structural defects is small and remains constant up to around 7,600 K Note that we employ non-PBCs for calculation of coordination number, therefore, atoms at the edge of graphene model are under-coordinated and small fraction of structural defects found in the temperature range up to 7,600 K is related to the edge atoms (see Figure 2(a)) However, small part of them also occurs inside model in the hightemperature region and they can be considered as liquid-like ones (Figure 2(a)) Further heating increases number of liquid-like atoms leading to the melting process (Figures 2(b), 9) Sudden growth in number of liquid-like atoms in the melting region exhibits a first-order behaviour of the spontaneous melting of our graphene model At a melting point, fraction of liquid-like atoms reaches about 35% It is not much smaller than that commonly found for melting/freezing of 2D or 3D materials [49–51] This means that total melting should occur at much higher temperature, i.e fraction of liquid-like atoms reaches 98% at around T2 = 8,100 K (Figure 9) However, some specific points of the spontaneous melting of our graphene model can be listed as follows Liquid-like atoms occur locally but they Downloaded by [University of Sussex Library] at 16:36 26 May 2016 12 V V Hoang et al Figure 9. Temperature dependence of fraction of atoms with Z ≠ 3 (NL/N) and size of their maximal cluster (Smax ∕N) Here, NL is total number of atoms with Z ≠ 3, while Smax is size of their maximal cluster and N is total number of atoms in model not homogeneously distribute throughout 2D model They have a tendency to aggregate into a region and melting is mediated/grown from this region via tearing-like or crackpropagation-like mechanism (Figure 2(b)) It is unlike that found for melting of simple 2D crystals with an isotropic interaction (see [52] and references therein) Due to strong C–C bond of 2D graphene sheet, tearing-like mechanism of melting grown from the defect-concentrated region is the energy-lowest cost one In addition, such an unusual spontaneous melting of our graphene model may be size dependent and it should be checked again It is found that melting regions are surrounded by pentagons and heptagons [10] and our simulation confirms this point Moreover, isolated pentagons and heptagons have not been found in graphene model with LCBOP II [10,12] We also find that isolated pentagons and heptagons not occur in the rigid solid state up to 7,600 K It is commonly found that melting initiates homogeneously and finishes via percolation threshold of liquid-like atoms throughout the model (see [49,52] and references therein) However, it is not true for the spontaneous melting of our 2D graphene model We find that although liquid-like atoms have a tendency to form clusters (Figures and 9), the largest liquid-like cluster does not contain all liquid-like atoms in the system for the whole temperature range studied (Figure 9) In the high temperature region, size of the largest cluster of liquid-like atoms increases with temperature reaching a maximum value at around 8,000 K, then it deceases (Figure 9) Unusual behaviour of the largest cluster of liquid-like atoms is related to the specific features of molten state of 2D graphene, i.e melting of graphene leads to the formation of a ring-like 2D liquid carbon (not simple liquid) In the ring-like 2D liquid carbon, atoms have a tendency to form rings and not scatter homogeneously throughout model It is unlike that found in simple systems with an isotropic interaction [52] We consider that if the distance between two atoms is less than 2.04 Å, i.e the radius of the first coordination sphere, they should belong to the same cluster We have checked a nature of the spontaneous melting of our graphene model heated at the higher heating rate of 5 × 1011 K/s and found that there is only one first-order phase transition occurred at around 8,200 K (not shown) This means melting of 2D materials maybe heating rate dependent However, data obtained at lower heating rate are more practical since it is closer to that carried out in practice In addition, melting of partially fluorinated Philosophical Magazine 13 graphene has been studied recently via MD simulation with reactive force field and found that melting of fluorographene is very unusual depending strongly on the degree of fluorination [53] Note that we study the spontaneous melting of the pristine graphene under PBCs and any edge effect on melting is out of scope of the paper On the other hand, similar scenario of melting showing the coexistence of the 2D solid and liquid phase during melting is found for melting of other 2D material, MoS2 [54] However, MoS2 transforms quickly to a phase with MoSx clusters without the appearance of random coils unlike graphene and graphite [54] Other problems such as self-healing of vacancies, atomic structure and energetics of large vacancies in graphene are also out of scope of the present work [55,56] Downloaded by [University of Sussex Library] at 16:36 26 May 2016 4. Conclusions We have carried out a comprehensive MD simulation of the spontaneous melting of graphene model with a realistic LCBOP I potential in 2D space and found important points as follows: (1) Spontaneous melting of our graphene model in 2D space starts by the formation of SW defects followed by occurrence of other defects such as SVs, multiple vacancies and large rings This means that spontaneous melting of graphene model in 2D space also exhibits the same behaviour like those found for the spontaneous melting of graphene model in 3D one [10] However, we found that defective/ liquid-like atoms not occur homogenously throughout the model Instead, they have a tendency to aggregate into a region and melting initiates/grows from this region into crystalline part via tearing-like or crack-propagation-like mechanism It is new and may be size dependent (2) We find that spontaneous melting of our graphene model in 2D space proceeds not via percolation of liquid-like atoms unlike that commonly found for melting of 2D or 3D crystals with an isotropic interaction in the past At a melting point, only one third of atoms become liquid-like which aggregate into single part of model while remaining part is still crystalline solid Melting completely occurs at temperature much higher than a melting point (3) It is found that melting of graphene model exhibits a first-order behaviour and can be described by the CNT [12] Our simulation results support this point Moreover, it is also found that melting of graphene does not occur via unbinding of topological defects, i.e splitting of a SW defect into two 57 defects and their subsequent diffusion have not been found [12] The same situation is found in the present work for the spontaneous melting of our graphene model in 2D space This means that melting scenario is not in agreement with the BKTHNY theory Disclosure statement No potential conflict of interest was reported by the authors Funding This work was supported by Vietnam 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