Supplementary Material: Crumpling Damaged Graphene

Transcription

1 Sulementary Material: Crumling Damaged Grahene I.Giordanelli 1,*, M. Mendoza 1, J. S. Andrade, Jr. 1,, M. A. F. Gomes 3, and H. J. Herrmann 1, 1 ETH Zürich, Comutational Physics for Engineering Materials, Institute for Building Materials, Wolfgang-Pauli-Strasse 7, HIT, CH-8093 Zürich, Switzerland Universidade Federal do Ceará, Deartamento de Física, Camus do Pici, Fortaleza, Ceará, Brazil 3 Universidade Federal de Pernambuco, Deartamento de Física, Recife-PE, Brazil * gilario@ethz.ch ABSTRACT This sulementary material contains some additional details regarding the fractal dimension measure, the simulation set-u, the anisotroy and ashericity of crumled damaged grahene sheets, the contribution of newly formed covalent bonds and the derivation of the theoretical model resented in the aer. Fractal dimension In order to determine the fractal dimension systematically and consistently, in the case of the sand-box method, we have to determine first the ower-law range for each DGM searately. A least-squares fit to the data is then alied and the resulting exonents are averaged over all DGM realizations ln(m) ln(r) Figure S1. Tyical examle of the fitting used to obtain the fractal dimension with the sand-box method. Here, we adot ar g, with a = 0.19, and R g as aroriate values for the lower and uer threshold, resectively. The dashed green lines delineate the thresholds of the scaling region. The fractal dimension for this articular case is.71 ± 0.03 and corresonds to a grahene sheet with = 0.8. In Fig. S we show the largest shere used by the sand-box method. The structure is between the flat and the crumled state. We clearly see that the DGM is still flat with some riles and the sheres of the sand-box method contain regions where there are no atoms. On the contrary, in Fig. S3 we see that the shere is comletely contained in the DGM. 1

2 Figure S. DGM at = The green oints reresent carbon atoms, the blue oints hydrogen atoms and the red oint the center of mass. The radius of the shere corresonds to the radius of gyration. Figure S3. DGM at = 0.8. The green oints reresent carbon atoms, the blue oints hydrogen atoms and the red oint the center of mass. The radius of the shere corresonds to the radius of gyration and covers the isotroic core of the structure. /8

3 d F Figure S4. Fractal dimension comuted with Eq. (3) (ensemble method) for sheets of different initial sizes ranging from 4 to 60 with NCIs. Ensemble method We rovide further evidence that the crumled DGM is fractal using an additional method, which we will call the ensemble method. It is based on the relation between the radius of gyration R g and the mass M of the DGM as exressed by Eq. (3). Different system sizes are required for this method. Small system sizes have a very small isotroic core and are thus not aroriate for measuring the fractal dimension. In contrast, large system sizes would roduce accurate results but at a very high comutational cost. Therefore we focus on intermediate system sizes in the range of 4 60 Å. We observe that the fractal dimension increases in region II as it does with the sand-box method (see Fig. S4). However, the lateau obtained with the sand-box method (seen in region III of Fig. 4) is more ronounced than with the ensemble-method seen in Fig. S4. Cooling rocess and fractal dimension From Fig. S5 we see that as we decrease the temerature of the samles we converge to the fractal dimension d F. Comutational details All the arameters used in the simulation have been summarized on Table 1. The initial temerature T i has been chosen to be 800 K, because it is low enough to kee covalent bonds, avoiding that some atoms would detach from the DGM. However, it is sufficiently high for achieving a fast equilibration rocess and hels the DGM to exlore a broader art of the hase sace with configurations that require a high kinetic energy for overcoming some ossible local minima configurations. Note also that the initially constructed configuration is not an energetically favourable configuration and a high temerature (and thus high velocity of the atoms) hels to equilibrate faster towards a more natural conformation described by the otential. Subsequently, we cool down the structure gradually using a temerature ste size of 1 T in the temerature interval T 1 and T in the temerature interval T (see Figure S6). The higher the temerature stes size, the longer it takes for the simulation to achieve the new equilibrium. However, the smaller the temerature ste size, the more intervals between 800 K and 0 K we need. We found that 1 T = 5 K is a good comromise. We defined the second interval T in order to achieve a better recision in the last 5 K. The amount of time-stes required for equilibration strongly deends on the system size. Furthermore, for DGM, the equilibration rocess is dominated by the carbon atoms, due to their higher mass m c comared to the hydrogen atoms. The hydrogen atoms lay a minor role in the equilibration rocess. Therefore, we have chosen the equilibration time ( 1 t, t ) in each temerature interval to be roortional to the number of carbon atoms (N c ) contained in the DGM. The re-factors have been determined emirically based on the energy evolution. Once we achieve an equilibrium, the energy end temerature fluctuates around the desired value and the velocity of each atom is determined by a Maxwell-Boltzmann distribution. Note that also the free arameter Q in the Nosé-Hoover thermostat determines the couling between the heat reservoir and the DGM, thus influencing also the temerature fluctuations. This arameter is usually found emirically, based on the fact that a too high value of Q leads to a oor temerature control and for Q it even leads to the micro-canonical ensemble (NVE) without 3/8

4 .8 I II III d F K 500 K 400 K 300 K 00 K 100 K 0 K Figure S5. Convergence of the fractal dimension during the cooling rocess comuted with the sand-box method for different vacancy densities. The arrow is in direction of the cooling rocess towards 0 K. The dashed blue line indicates the fractal dimension after thermalisation, i.e. d F =.71 ± 0.0. any temerature control. On the other hand, too small values might induce high-frequency temerature oscillations. In our study we want to roduce a structure based on hydrocarbon molecules, where the underlying lattice is hexagonal. Therefore, a natural choice of the initial bond-distance b i cc would be close to a stable known configuration, in this case a structure close to the one of grahene. During the simulation, the bond-lengths adjust dynamically in order to minimize the total energy and are not fixed arameters of the simulations. Therefore, the recise value of the initial bond-length is not relevant, as long as it is in a suitable range where we have stable covalent bonds between the atoms. In order to be on the safe side, we choose them to coincide with the minimum energy distance redicted for a 100 Å x 100 Å grahene sheet, which is 1.40 ± 0.0 Å as shown in the Figure S7. Indeed, it is matching with the values obtained from many numerical and exerimental results? (the average bond-length in exerimentally grown grahene is found to be 1.4 Å). Note that this argument can be alied to the carbon-hydrogen bonds. We choose an initial bond-length of 1 Å, which is not far away from the equilibrium bond-length between carbon and hydrogen atoms. Note that we could have also started with a bond-length of 1.38 Å or 1.4 Å and this would not have made any difference after reaching thermal equilibrium (We relaxed a grahene sheet with initial bond-length of 1.35 Å and achieved the same equilibrium bond-length mentioned above). However, if we would start with a comletely different bond-length for a honeycomb lattice, let s say Å, then the initial configuration will recombine towards a comletely different structure or even to amorhous carbon. This rocess will only be initially dominated by van der Waals forces, since this distance is too large for the covalent bond interaction in carbon. Even though this set-u might lead to interesting results, it is not the scoe of this research. Anisotroy and ashericity The eigenvalues of the gyration tensor are also useful for comuting several shae descritors. Shae descritors of articular interest are the relative shae anisotroy κ and ashericity b (see Fig. S8). The former is a measure for anisotroy and can reach values between 0 and 1. The value 0 occurs when all articles are sherically symmetric and thus the structure is comletely isotroic, 0.5 occurs when the articles lie on a lane, and 1 occurs when all oints lie on a line. We observe that κ reaches a value of 0.5 for = 0 and decreases for higher values of, roviding evidence that the DGM is more isotroic for large values of. The ashericity b, which is the second shae descritor considered, is a quantity that is 0 for articles that are sherically symmetric. We observe that the ashericity is also decreasing as a function of with a turning oint in region II, which is the region where the DGM crumles. Contribution from new covalent bonds In this section we study the evolution of the covalent bonds. It is reasonable to believe that ossibly newly-formed covalent bonds increase the comactness of the structure. It might be also ossible that during the evolution, some new covalent bonds 4/8

5 T T Ste # Ste #10 6 Figure S6. Temerature evolution of a DGM with = 0.13 Å. The DGM has 1193 carbon atoms and 543 hydrogen atoms. We clearly see that after each change of temerature we leave enough time to equilibrate again. Parameter m c = 1 u m h = 1 u b i cc = 1.4 Å b i ch = 1.0 Å σ cc = 3.4 Å C t =.5σ cc N c Descrition Mass in atomic units for carbon atoms Mass in atomic units for hydrogen atoms Initial bond-length of carbon-carbon bonds Initial bond-length of carbon-hydrogen bonds Lennard Jones arameter for carbon carbon interaction Cutoff length for the otential Carbon atoms Time ste Initial temerature t = 0.1 fs T i = 800 K 800 K T 1 5K Temerature interval 1 5 K T 0K Temerature interval 1 T = 5 K Temerature ste difference in T 1 T = 5 K Temerature ste difference in T t 1 = 57N c Time stes for each interval T 1 t = 17N c Time stes for each interval in T Q = N c /65000 Free arameter for the Nosé-Hoover thermostat Table 1. Comlete table with the arameters required to reroduce the molecular dynamics simulation. 5/8

6 Distribution r Figure S7. Distribution of the covalent bonds for a relaxed grahene sheet of 100 x 100 Å size. The green line denotes the fit with mean 1.40 Å and standard deviation of 0.0 Å I II III b κ Figure S8. Relative shae anisotroy κ and ashericity b comuted with the normalized eigenvalues λ n /R g. 6/8

7 .8 I II III.6 w/o enalty d F.4 w/ enalty Figure S9. Fractal dimension of the modified DGM, where a enalty has been introduced to suress newly formed covalent bonds. are temorarily created. They could lay an imortant role in the crumling transition. To address these questions we modified the simulation such that, whenever new covalent bonds are formed, a force is roduced that, tries to searate the covalent bonds again. This does not allow the DGM to kee newly-formed covalent bonds. Figs. S9 and S10 clearly show that the imact of newly formed covalent bonds is negligible. There are some new bonds which are created, but they disaear again some time stes later. Theoretical model An effective theory can be formulated to exlain the fractal dimensions obtained from our simulation results. The masssize scaling M R.3 valid for DGM in absence of NCIs is reminiscent of many other situations including DLA growth,? aggregating roteins,? crumled aer,? ensemble average of large grous of roteins,? and DNA in chromosomes,?,? all of them satisfying M R.5, differing 8% in the scaling exonent. The recurrence of d F for these structures, in site of chemical and hysical eculiarities, suggests a common underlying origin. In fact, a mean-field henomenological model using an entroic elastic energy U S = AR,? and a two-body reulsive energy U SA = Bρ V can exlain this scaling. In the last term, ρ is the average density given by mass/volume= M/V, with V R 3, and consequently, U SA = BM /R 3. The entroic energy favours collased configurations of the system, but the second term, due to self-avoidance interactions, favours oen, more extended, configurations of the sheet. Physically, the refactor A in U S is exected to deend on the ercentage of vacancies as z, with z > 0; that is, the entroy of the sheet increases with the damage level. After minimisation of E = U S +U SA with resect to R, the mass-size relation M R.5 follows straightforwardly. If NCIs exressed through a modified Lennard-Jones otential are resent, their attractive art that decays as 1/R 6 tends to combine with the ositive U SA term leading to a reulsive otential that effectively behaves more singularly. Our candidate for this comosition of energies is a four-body energy which decays as USA NCI = Cρ4 V = CM 4 /R 9. According to this qualitative model, in order to exlain the crumling transition observed in DGM in the resence of NCI, we conjecture that the relevant energy E of the damaged sheet has the form E = AR +CM 4 /R 9, which after minimisation leads to M R.75, in conformity with Fig. 4, i.e. the attractive art of the non-covalent interaction favours more comact conformations with a greater fractal dimension, and the radius of gyration R g = (9C/A) 1/11 M 4/11 is exected to scale inversely with (as z/11 ), as suggested by Fig.. For comleteness, Fig. 4 shows that there is reversibility between the rocesses leading to the conformations with fractal dimensions d F =.71 ± 0.0 and d F =.30 ± 0.05, i.e. with and without NCIs, resectively. Thus, if the steady-state configuration is initially obtained with NCI, the sudden elimination of this interaction restores the less comact structure with d F =.30, and vice-versa. References 1. Fasolino, A., Los, J. H. & Katsnelson, M. I. Intrinsic riles in grahene. Nat. Mater. 6, (007). 7/8

8 <λ/r g >, <Λ/Rg > I II III <λ 1 <λ <λ 3 <Λ 1 <Λ <Λ Figure S10. Radius of gyration of the modified DGM, where a enalty has been introduced to suress newly formed covalent bonds. Blue and red symbols refer to simulations with all and without non-covalent interactions, resectively.. Meakin, P. Diffusion-controlled cluster formation in -6-dimensional sace. Phys. Rev. A 7, (1983). 3. Feder, J., Jøssang, T. & Rosenqvist, E. Scaling behavior and cluster fractal dimension determined by light scattering from aggregating roteins. Phys. Rev. Lett. 53, 1403 (1984). 4. Gomes, M. A. F. Paer crushes fractally. J. Phys. A: Math. Gen. 0, L83 (1987). 5. Enright, M. B. & Leitner, D. M. Mass fractal dimension and the comactness of roteins. Phys. Rev. E 71, (005). 6. Bancaud, A., Lavelle, C., Huet, S. & Ellenberg, J. A fractal model for nuclear organization: current evidence and biological imlications. Nucleic Acids Res. 40, 8783 (01). 7. Metze, K. Fractal dimension of chromatin: otential molecular diagnostic alications for cancer rognosis. Exert Rev. Mol. Diagn. 13, (013). 8. Keten, S. & Buehler, M. J. Strength limit of entroic elasticity in beta-sheet rotein domains. Phys. Rev. E 78, (008). 8/8