DSpace at VNU: Stages of melting of graphene model in two-dimensional space

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

Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tphm20

ISSN: 1478-6435 (Print) 1478-6443 (Online) Journal homepage: http://www.tandfonline.com/loi/tphm20

Stages of melting of graphene model in two-dimensional 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

© 2016 informa UK limited, trading as Taylor & Francis group

Stages of melting of graphene model in two-dimensional space

Vo Van Hoanga, Le Thi Cam Tuyenb and To Quy Dongc

a comp Phys lab, hochiMinh city Univ of Technology, Vietnam national University-hochiMinh city, hochiMinh city, Vietnam; b Department 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

10 4 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

multi-membered rings Unlike that found for melting of a 2D crystal with

an isotropic bonding, these defects do not occur homogeneously

throughout the system, they have a tendency to aggregate into a

region and liquid phase initiates/grows from this region via

tearing-like or crack-propagation-tearing-like mechanism Spontaneous melting point

of our graphene model occurs at T m = 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.

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 ref-erences 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 inter-atomic potential (a new version of LCBOP proposed in 2005), melting of graphene starts

KEYWORDS

spontaneous melting of graphene model; 2D melting; two-phase melting

ARTICLE HISTORY

Received 7 January 2016 accepted 27 april 2016

CONTACT Vo Van hoang vvhoang2002@yahoo.com

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 clus-ters containing from 2 to 55 atoms is studied using density-functional tight-binding and classical MD [11] Melting point is estimated using the bond energy, the Lindemann cri-terion 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 do 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 two-fold: (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 bound-ary 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:

(1)

Eb= 1 2

N

∑

i,j

Vijtot= 1 2

N

∑

i,j

(

fc,ijVijSR+SijVijLR

)

where the total pair interaction (Vtot

ij ) is a sum of a short-range part (fc,ijVSR

ij ) and a long-range one (SijVijLR), 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 do 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 calcu-lations 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 = N

S =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 1 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)

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

spon-taneous 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 liq-uid →  string-like 2D liquid transition (atoms in the latter mostly aggregate into strings)

Heat capacity at constant zero pressure is calculated approximately via relation: C P  = 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 T m = 5,800 K found for graphene by extrapolation to infinite- radius 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 C P 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 2 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 Z ≈ 1.9̄ 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 do 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

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]

(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

More details of structure evolution upon heating to molten state can be seen via temper-ature dependence of coordination number or ring distributions including mean value of these quantities (Figures 4–6) Mean ring size is equal to 6 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

Figure 4. Temperature dependence of mean ring size and mean coordination number (inset)

Figure 5. (colour online) coordination number distributions at various temperatures 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 5 and 6) Note that we do not employ PBCs for calculating coordination number to see what

hap-pens 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 which is related to the edge atoms (see Figures 2, 5(a)) However, fraction of atoms

with Z = 2 increases significantly at 7,700 K (Figure 5(b)) and becomes dominant at temper-ature just above a melting point (at 7,800 K, see Figure 5(c)) It is related to the formation of rings/strings in the system due to spontaneous melting of graphene Further heating leads

to increase in fraction of atoms not only with Z = 2 but those with Z ≤ 1 (Figure 5(d), at

8,000 K) while fraction of atoms with Z = 3 is very small This indicates an enhancement of

ring formation and starting of breaking of rings The latter is enhanced with further heating leading to the second phase transition, i.e transition from ring-like 2D liquid carbon into

a string-like 2D liquid one at around T2 = 8,100 K

Similarly, at T ≤ 7,600 K rings mostly are six-fold In particular, at 7,500 K about 97%

rings are six-fold meaning that system is still in the solid state with a honeycomb structure Small fraction of non-six-fold rings indicates a small distortion of honeycomb structure due

to heating (Figure 6(a)) Further heating increases fraction of non-six-fold rings including the dominant increase in five-fold and seven-fold rings while fraction of six-fold rings decreases (Figure 6(b)) Dominant increase in five-fold and seven-fold rings shows that spontaneous melting of a perfect graphene model is mediated by the spontaneous forma-tion SW defects After melting point, fracforma-tion of five-fold and seven-fold rings dominates

in the system Occurrence of rings with various sizes ranged from three-fold to thirty-fold ones is related to the formation of rings of various sizes including very large rings in the system (at 7,800 K, Figure 6(c)) At 8,000 K, such broad distribution of rings becomes more homogeneous meaning the full formation of ring-like 2D liquid carbon (Figure 6(d)) Above 8,000 K, fraction of large rings decreases while fraction of small rings (smaller than six-fold

Figure 6. (colour online) Ring distributions at various temperatures at below and above melting point

(T1 = 7750 K)

one) increases indicated a breaking of large rings It is related to the second phase transition like that discussed above It is found that octagons (eight-fold rings, R8) are precursors for the formation of larger rings leading to melting of graphene, i.e number of R8 increases when melting starts and it decreases when larger rings are formed [10] It may be not true Figure 6 shows that number of R8 increases up to 8,000 K at which a full ring-like 2D liquid carbon is formed (see Figure 4 and related discussion), i.e it increases up to temperature far above melting point of 7,750 K

In order to clarify structural characteristics of models obtained by heating from a perfect graphene model to a molten state, we show interatomic distance distributions in models obtained at various temperatures As shown in Figure 7, at a relatively low temperature (at 550 K) interatomic distance distribution is narrow and centred at around the value 1.425 Å which is almost equal to that of a perfect graphene Further heating broadens the distribution and the peak shifts toward a larger value of interatomic distance (see the curves for 7,500 and 7,750 K) However, unusual change has been found for the curve of liquid model at 8,100 K, i.e the distribution is broader and shifts toward smaller value The peak

is located at around 1.35 Å It corresponds to that found for evolution of position of the first peak in RDF (Figure 3) Shifting of distribution of interatomic distance of liquid model at 8,100 K is related to the phase transition from ring-like 2D liquid carbon into string-like one Maybe it is due to the formation of shorter double bonds in the string-like 2D liquid carbon compared to that in the ring-like one As discussed above, shifting of the position

of the first peak in RDF of liquid graphene compared to that of solid one is explained by the formation of shorter bonds in the rings/strings [10]

3.2 Structural defects occurred during heating to molten state

It is of great interest to discuss about occurrence/behaviour of defects occurred in the sys-tem upon heating to a molten state since it is related to the mechanism of melting First, temperature dependence of fraction of atoms with various coordination numbers can be seen in Figure 8 Atoms with Z ≠ 3 can be considered as structural defects (or liquid-like

atoms), this approach is an approximate way since some defects such as SW ones still keep

three-fold coordination and we find no atoms with Z > 3 in the whole temperature range

Figure 7. (colour online) interatomic distance distributions in models obtained at various temperatures upon heating to molten state

studied This indicates that there are no overcoordinated defects in graphene models sim-ulated 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

maxi-mum 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 mul-ti-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

Figure 8. Temperature dependence of fraction of atoms with various coordination numbers