143. Cooling rate effects on structure of amorphous graphene

143. Cooling rate effects on structure of amorphous graphene tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bà...

Author's Accepted Manuscript Cooling rate effects on structure of amorphous graphene Vo Van Hoang www.elsevier.com/locate/physb PII: DOI: Reference: S0921-4526(14)00686-3 http://dx.doi.org/10.1016/j.physb.2014.08.020 PHYSB308609 To appear in: Physica B Received date: 29 May 2014 Revised date: 20 July 2014 Accepted date: 16 August 2014 Cite this article as: Vo Van Hoang, Cooling rate effects on structure of amorphous graphene, Physica B, http://dx.doi.org/10.1016/j.physb.2014.08.020 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain 1    Cooling rate effects on structure of amorphous graphene Vo Van Hoang Comp Phys Lab, Institute of Technology, Vietnam National University- HochiMinh City, 268 Ly Thuong Kiet Street, District 10, HochiMinh City-Vietnam Email: vvhoang2002@yahoo.com  Tel: 84 08 38647256 Fax: 84.08 38656295 Abstract Simple monatomic amorphous 2D models with Honeycomb structure are obtained from 2D simple monatomic liquids with Honeycomb interaction potential [M.C Rechtsman et al., Phys Rev Lett 95, 228301 (2005)] via molecular dynamics (MD) simulations Models are observed by cooling from the melt at various cooling rates Temperature dependence of thermodynamic and structural properties including total energy, mean ring size, mean coordination number is studied in order to show evolution of structure and thermodynamics upon cooling from the melt Structural properties of the amorphous Honeycomb structures are studied via radial distribution function (RDF), coordination number and ring distributions together with 2D visualization of the atomic configurations Amorphous Honeycomb structures contain a large amount of structural defects including new ones which have not been previously reported yet Cooling rate dependence of structural properties of the obtained amorphous Honeycomb structures is analyzed Although amorphous graphene has been proposed theoretically and/or recently obtained by the experiments, our understanding of structural properties of the system is 2    still poor Therefore, our simulations highlight the situation and give deeper understanding of structure and thermodynamics of the glassy state of this novel 2D material PACS numbers: 81.05.ue, 64.70.Nd, 61.43 ± j, 64.70.Pf Keywords: Cooling rate effects, Amorphous graphene, Honeycomb structure, MD simulation, Simple 2D system, 2D glass Introduction Amorphous graphene (a-graphene), 2D material with amorphous Honeycomb structure, is of great interest due to theoretical and technological importance A-graphene is often considered as 2D fully amorphous networks composed of sp hybridized C atoms, i.e the configurations contain various rings other than 6-fold ones in a disordered arrangement which has not been well understood up to date [1-7] However, while crystalline graphene (c-graphene) has become an experimental reality in 2004 and research related to c-graphene becomes the hottest one in condensed matter physics and materials science (see [3-7] and references therein), less attention has been paid on the amorphous counterparts Experimentally, indirect evidence of the formation of a-graphene is obtained using Raman spectroscopy in samples subject to electron-beam irradiation, ozone exposure and ion irradiation [8-10] A-graphene is obtained directly in electron-beam irradiation experiments and is visualized by high-resolution electron transmission microscopy [11,12], and recently large-scale a-graphene has been obtained by chemical vapor deposition [13] However, detailed information of structure of a-graphene at the atomic level can be obtained only by computer simulation and limited number of works in this 3    direction can be listed as follows [14-20] Simple a-graphene 800 atoms model containing oddmembered rings (5 and 7-fold ones) and no coordination defects has been proposed via inserting Stone-Wales defects into a perfect graphene [14] It is found that odd-membered rings increase the electronic density of states at the Fermi level of a-graphene relatively to the crystalline counterpart [14] Subsequently, using DFT method, it is found that carbon pentagons (5-fold rings) induce local curvature - breaking the initial planar symmetry and puckering of the agraphene is studied [15,20] It is found that upon relaxation of initially flat a-graphene model, puckering occurs, and may be due to existence of pentagons in analogy with the situation for fullerenes [20] Disorder of structure may lead to strong change in transport properties of agraphene Indeed, a transition to metallicity when a sufficient amount of disorder is induced in agraphene is predicted by using first principles stochastic quench method [16] In contrast, it is predicted that a-graphene should be a strong Andersen insulator using the Kubo transport methodology and Lanczos method [19] Note that a-graphene model used in [19] is obtained by inserting Stone-Wales defects into a perfect graphene which may be quite different from the sample models used in [16] It may be an origin of the discrepancy between results obtained in the two works More details of microstructure of a-graphene have been found [17] A-graphene samples are obtained by two different methods The first one, Monte-Carlo bond switching algorithm is employed in order to amorphize a crystalline graphene sheet The second one, MD method is used to quench the sample from the liquid state Two approaches lead to similar results via analyzing RDF and diffraction pattern, it is found that a-graphene is a realization of 2D Zachariasen glass proposed 80 years ago [17] Details of structure of a-graphene including ring statistics and angular distortion are sensitive to the preparation method like that found in practice [17] Similarly, amorphous structural model for graphene oxides has been proposed via first 4    principle calculations, i.e geometric structures, thermodynamic stabilities and electron density of states of the model are presented [19] Moreover, recent MD simulations show that good insights of structure and thermodynamics of the formation of graphene can be obtained by using 2D simple monatomic liquids with Honeycomb interaction potential [21,22] Models are observed by cooling from the melt at various cooling rates [22] Thermodynamics and atomic mechanism of the transition from 2D liquid state into crystalline Honeycomb structure (i.e c-graphene) is analyzed in details Cgraphene obtained by cooling from the melt contains new structural defects which have not been previously reported in literature Similar situation can be proposed for a-graphene, however, it has not been studied yet although it is found that depending on the cooling rate used in simulations amorphous or crystalline graphene can be obtained [22] Therefore, clarification of the structure and thermodynamics of the formation of a-graphene is our main aim in the present work Calculations Glass formation in 2D supercooled liquids containing 16428 atoms interacted via Honeycomb pair potential has been studied by MD simulations The Honeycomb pair potential used in the present work has the form as give below [21]: U ( R) ε = 5.89 − 10 + 17.9 exp( −2.49 R ) − 0.4 exp[ −40( R − 1.823) ] 12 R R (1) 5    This potential has two minima, the shallow one is located at R1 = 1.07 in the positive region of potential and the deeper minimum is located at R2 = 1.84 in the negative region of potential Initially, atomic configurations of 2D simple square lattice structure of the size of S = 203.004σ × 117.282σ and at the density ρ = N N = = 0.69 have been relaxed at V S temperature as high as T = 1.0 for 106 MD steps in order to get a good equilibrium liquid state We use ‘ NAT ’ ensemble simulation and periodic boundary conditions (PBCs), i.e simulation with fixed number of atoms ( N ), area ( A ) and temperature ( T ) under PBCs is carried out We adopt the density ρ = 0.69 in order to get a final honeycomb structure like that proposed in [23] Here, R = r σ and r is the distance between atoms and σ is a diameter of atom We employ Lennard-Jones reduced units in the present work as follows: the length in units of σ , temperature T in units of ε / k B , and time in units of τ = σ m / ε where k B is the Boltzmann constant and m is an atomic mass The Verlet algorithm is employed and MD time step is dt = 0.001τ We take the same cutoff for potential at ro = 3σ like that done in [21,23] Then the system is cooled down at the ‘constant area’ while temperature is decreased linearly with time as T = T0 − γ × n via the simple atomic velocity rescaling until reaching T = 0.01 Here, γ is a cooling rate and n is the number of MD steps In order to get amorphous one we use two different cooling rates as high as γ = 9.9 × 10−6 and 9.9 × 10−5 per MD step In order to improve the statistics, we average results over five independent runs In order to calculate coordination number and ring distributions, the cutoff radius RC = 1.45σ is taken which is equal to the position of the first minimum after the first peak in RDF of model obtained at low temperature We use VMD software [24] for 2D visualization of atomic configurations in the paper and 6    LAMMPS for MD simulations [25] Since the Honeycomb interatomic potential presented in Eq (1) has not been implemented in LAMMPS yet, and therefore, it has been given in the form of table of values stored in one of the input files We employ ISAACS software for calculating ring statistics following the “shortest-path” criterion for determination of rings [26] Results and discussions 3.1 Thermodynamics and evolution of structure upon cooling from the melt Temperature dependence of total energy per atom can be seen in Fig 1, the change of total energy upon cooling from the melt is rather smooth indicating a glass transition in the system and glass transition temperature can be considered as the point of starting of deviation from the linearity of a low temperature region (see the inset), i.e it is found that Tg1 = 0.14 and Tg = 0.10 for the cooling rate of 9.9 × 10 −5 and 9.9 × 10 −6 per MD step, respectively Note that if a lower cooling rate is used, crystallization occurs in the system and temperature dependence of total energy exhibits a first-order behavior at the transition point [22] On the other hand, potential of Eq.(1) has two wells [21], the first one is located at the position of positive value of potential and atoms in our models mainly concentrate in this well yielding positive value for total energy presented in Fig RDF of the amorphous models can be seen in Fig Note that a quite similar RDF of a-graphene models obtained by both MC and MD simulations with a realistic Tersoff-II potential for carbon atoms is presented in [17] RDF of amorphous models is rather smooth while for crystalline model additional peaks of crystalline Honeycomb structure occur (see the inset of Fig 2) The first peak of RDF is located at around R = 1.05 which is close to the 7    shallow minimum of potential at R1 = 1.07 The second peak of RDF is located at around R = 1.83 which is close to the deeper minimum of potential at R2 = 1.84 A small peak at around R ~ 2.10 corresponds to twice the nearest-neighbor bond length in our model and hence it is related to the spatial correlations across 6-membered rings like that found and discussed for model of graphene in [17] This small peak is strongly enhanced with lowering cooling rate toward the highest value in the crystalline models (see the inset of Fig 2) indicating a much higher fraction of 6-membered rings in crystalline models compared to that in the amorphous ones (see our discussions given below) Note that RDF of models at T = 1.0 is rather smooth, without the peak at R ~ 2.10 and height of its peaks is small exhibiting clearly a liquid-like behavior (not shown) More details about evolution of structure of the system upon cooling from the melt can be seen in Fig One can see that in the liquid state, mean coordination number is equal to around 3.8 - 3.9 and it decreases down to around 3.20 and 3.45 for the cooling rates of 9.9 × 10 −6 and 9.9 × 10 −5 per MD step at T = 0.01 , respectively This means that the lower cooling rate used the lower coordination is toward the formation of a low-coordinated crystal with coordination number Z = 3.0 like that found for an ideal Honeycomb structure Evolution of mean coordination number in the models is rather smooth indicating liquid-glass behavior of the transition (Fig 3) Mean coordination number equaled to 3.20 and 3.45 for the glassy state obtained at T = 0.01 indicates a strong distorted Honeycomb structure with a large amount of structural defects Moreover, decrease in mean coordination number in the Honeycomb structure-forming system upon cooling from the melt may be related to the negative thermal expansion coefficient (TEC) of our amorphous 2D Honeycomb models like that found by both experiments and computer simulations for c-graphene [27-30] We have no experimental data 8    related to TEC of a-graphene in order to compare and discuss, however, one can take the data for c-graphene for discussion That is, Monte-Carlo simulations show that graphene models have a negative TEC in the range – 300 K and after that it becomes positive [27] In contrast, TEC of graphene as function of temperature is estimated by using a first-principle calculation and find that it is negative for the whole temperature range studied up to 2500 K (see Ref 28) However, it is found experimentally that TEC of graphene is negative just for the temperature range 300 350 K (Ref 29) or 200 - 400 K (Ref 30) The reason of the discrepancy between calculations and experimental data may be related to the uncertainties in the accuracy of the experimental measurements or to the limitation of the computer simulations Decrease of mean coordination number of our Honeycomb structure-forming 2D system for the whole temperature range studied, ranged from the normal liquid state to amorphous solid, is anomalous and unlike that commonly found for 3D systems It is suggested in Ref 28 that the anomaly, i.e a negative TEC of graphene, is due to a low-lying bending phonon branch [31] Based on our simulations, TEC should be also obtained for a-graphene like that found for c-graphene Concerning on the rings, we find that rings almost have not been formed in the high temperature liquid state and significant amount of rings occurs at temperature just above the freezing one (see the inset of Fig 3) However, further cooling leads to the strong increase of the mean size of rings At the cooling rate of 9.9 × 10 −5 and 9.9 × 10 −6 per MD step, mean size of rings reaches the value around 4.60 and 5.10, respectively It is smaller than the 6-fold ring of the ideal Honeycomb structure This means that the Euler’s theorem is violated since the average ring size in amorphous-graphene-like models obtained by cooling from the melt is not six Note that the Euler’theorem is followed well for c-graphene-like models obtained by cooling from the melt [22] A significant deviation from the 6-fold ring indicates the existence of a large amount 9    of structural defects - small membered rings in the amorphous Honeycomb models compared to that of the ideal Honeycomb structure It is also clear that the lower cooling rate the larger mean ring size is toward the value of an ideal Honeycomb structure like that found for coordination number 3.2 Structural properties of amorphous Honeycomb structure Amorphous models obtained at T = 0.01 and at two different cooling rates of 9.9 × 10 −6 and 9.9 × 10 −5 per MD step have been relaxed for 10 MD steps before analyzing structural properties As shown in Fig 4, although atoms with coordination number Z = dominate in the models fraction of atoms with Z ≠ is significant which are related to the structural defects Concerning coordination number distributions in amorphous models presented in Fig 4, some points can be drawn: (i) Coordination number distributions are broad indicated an inhomogeneous structure of the models, (ii) The lower cooling rate the higher fraction of atoms with Z = is More details of coordination distribution in a-models can be seen in Table compared to that of crystalline model (c-model) obtained at much lower cooling rate of 9.9 × 10 −9 per MD step For a perfect Honeycomb structure (i.e perfect crystalline graphene), all atoms should have coordination number Z = Indeed, although c-model has small fraction of atoms with Z = and (structural defects), about 98% atoms have Z = indicating a nearly perfect Honeycomb structure However, fraction of atoms in a-models with Z = is much lower than that of c-model and coordination number distribution is much broader ranged from Z = to Z = indicating a strong distorted Honeycomb structure like that discussed above Note that a similar broad coordination number distribution (from Z = to Z = ) has been found in a-model 15    than that in the latter The lower cooling rate the larger amount of structural defects is A moderate cooling rate is recommended for preparation of 2D amorphous Honeycomb structure materials in order to maintain a domination of 6-fold rings in the system ii) Vacancies of various sizes and shapes have been found in amorphous Honeycomb structures which are ‘naturally’ formed upon cooling from the melt unlike the simple ones proposed in literature for both amorphous and crystalline graphene Since vacancies play an important role in various structural and chemico-physical performances of 2D materials, their formation, dynamics and stability are of great interest and further investigation in this direction is needed iii) We find a large amount of adatoms in amorphous Honeycomb structures, the existence of adatoms in graphene has been ignored due to their high formation energy in the 2D solid state However, we find that adatoms are formed in the liquid state prior freezing Energy required for the formation of adatoms in the liquid state should be small since it is just related to the rearrangement of atoms in the liquid state Existence of the adatoms in graphene-like materials should be reconsidered iv) We find a large amount of 3-fold rings (triangles) in amorphous Honeycomb structures in good agreement with amorphous models obtained by a first-principle stochastic quench method Existence of 3-fold rings in graphene has not been found by experiments yet v) We find the existence of dangling atoms in amorphous Honeycomb structures, together with other structural defects such as vacancies and small membered rings, dangling atoms may serve as reactive sites for chemical reactivity of amorphous 16    graphene-like materials Due to containing a larger amount of structural defects compared to those of crystalline counterparts, amorphous graphene-like materials should be more reactive with the environment compared to that of the crystalline counterparts vi) A-graphene cannot be simply considered as 2D fully amorphous networks composed of sp hybridized C atoms based on our 2D visualization since it should contain various bondings like that found in the present work Other words, our simulations not support the idea that a-graphene is a realization of Zachariasen’s 2D glass since it is too simple model Acknowledgments This research is funded by Vietnam National University - HoChiMinh City (VNU-HCM) under grant number B2014-20-01 References [1] D.S.L Abergel, V Apalkov, J Berashevich, K Ziegler, and T Chakraborty, Adv Phys 59 (2010) 261 [2] S Roche, N Leconte, F Ortmann, A Lherbier, D 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[39] T Mizuguchi and T Odagaki, Phys Rev E 79 (2009) 051501 20    Table Table Coordination number distributions: models obtained at the cooling rate of 9.9 × 10 −5 and 9.9 × 10 −6 per MD step are denoted as a-models A and B compared to that of crystalline one (c-model) obtained at cooling rate of 9.9 × 10 −9 per MD step at T = 0.01 Z a-model A 0.004 0.083 0.457 0.356 0.091 0.009 a-model B 0.002 0.065 0.689 0.208 0.033 0.004 c-model [22] 0.000 0.014 0.978 0.007 0.000 0.001 21    Table Table Ring distributions: models obtained at the cooling rate of 9.9 × 10 −5 and 9.9 × 10 −6 per MD step are denoted as a-models A and B compared to that of crystalline one (c-model) obtained at cooling rate of 9.9 × 10 −9 per MD step at T = 0.01 We also present the data of amodels obtained by inserting Stone-Wales defects and further annealing [15,19] Ring size a-model A 0.393 0.039 0.284 0.175 0.064 0.024 0.012 a-model B 0.237 0.020 0.306 0.352 0.059 0.014 0.007 a-model [15] a-model [19] 0 0 0.335 0.240 0.380 0.520 0.240 0.240 0.045 0 c-model [22] 0 0.029 0.949 0.006 0.002 22    Fig Fig Temperature dependence of total energy of the system upon cooling from the melt at two different cooling rates The inset: determination of glass transition temperature, straight lines serve as a guide for eyes 23    Fig.2 Fig Radial distribution function at T = 0.01 for models obtained at two different cooling rates compared to that of crystalline one (see the inset) 24    Fig Fig Evolution of mean coordination number and mean ring size (the inset) upon cooling from the melt at two different cooling rates 25    Fig Fig (Color online) Coordination number distribution in models obtained at T = 0.01 and at two different cooling rates of 9.9 × 10 −6 and 9.9 × 10 −5 per MD step 26    Fig Fig (Color online) Ring statistics in models obtained at T = 0.01 and at two different cooling rates of 9.9 × 10 −6 and 9.9 × 10 −5 per MD step 27    Fig Fig (Color online) A full 2D visualization of the well-relaxed atomic configuration of the amorphous models obtained at the cooling rate of 9.9 × 10 −6 per MD step and at T = 0.01 28    Fig Fig (Color online) 2D visualization of the enlarged part of well-relaxed atomic configuration of the amorphous models obtained at cooling rate of 9.9 × 10 −6 per MD step and at T = 0.01 Vacancies of various sizes and shapes, various structural defects including adatoms, dangling bonds, 3-fold, 4-fold, 5-fold, 7-fold, 8-fold, 9-fold rings etc can be seen clearly in the visualization 29    Fig (a) (b) (c) (d) Fig (Color online) 2D visualization of atoms with some specific coordination numbers Z ≠ in model obtained at the cooling rate of 9.9 × 10 −6 per MD step and at T = 0.01 : (a) Central atom of hexagon marked by the red color has bonds with neighbors ( Z = ), (b) Central atom of the region marked by the red color has bonds with neighbors ( Z = ), (c) Central atom of the region marked by the red color has bonds with neighbors ( Z = ), (d) One atom of pentagon marked by the red color has only bonds with neighbors ( Z = ) ... distribution function (RDF), coordination number and ring distributions together with 2D visualization of the atomic configurations Amorphous Honeycomb structures contain a large amount of structural... depending on the cooling rate used in simulations amorphous or crystalline graphene can be obtained [22] Therefore, clarification of the structure and thermodynamics of the formation of a -graphene. .. conclusions can be given: i) We find that depending on the cooling rate used in simulation, amorphous or crystalline Honeycomb structures can be obtained Amorphous Honeycomb structures contain