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Analytical Carbon-Oxygen Reactive Potential ͒ A

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Analytical Carbon-Oxygen Reactive Potential ͒ A THE JOURNAL OF CHEMICAL PHYSICS 128, 234706 ͑2008͒ Analytical carbon-oxygen reactive potential ͒ A. Kutana and K. P. Giapisa Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA ͑Received 22 January 2008; accepted 9 May 2008; published online 19 June 2008͒ We present a reactive empirical potential with environment-dependent bond strengths for the carbon-oxygen ͑CO͒ system. The distinct feature of the potential is the use of three adjustable parameters characterizing the bond: the strength, length, and force constant, rather than a single bond order parameter, as often employed in these types of potentials. The values of the parameters are calculated by fitting results obtained from density functional theory. The potential is tested in a simulation of oxidative unzipping of graphene sheets and carbon nanotubes. Previous higher-level theoretical predictions of graphene unzipping by adsorbed oxygen atoms are confirmed. Moreover, nanotubes with externally placed oxygen atoms are found to unzip much faster than flat graphene sheets. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2940329͔ I. INTRODUCTION has led to many electronic structure and potential energy surface calculations for oxygen-hydrocarbon interactions Empirical interatomic potentials offer substantial advan- using ab initio methods.9–12 Yet, fast analytical interatomic tages in calculating the properties of large statistical entities potentials capable of capturing the chemical interactions as compared to more rigorous ab initio or tight binding between these elements have been scarce. In fact, there exists methods, which typically produce more accurate results al- currently only one such potential6 to our best knowledge. In beit for much smaller systems. In particular, reactive empiri- order to fill this gap, a carbon-oxygen potential based on an cal bond order ͑REBO͒ potentials have been very successful extensive fitting database of ab initio calculations is pre- in simulating covalently bonded materials such as Si ͑Ref. 1͒ sented here. The motivation for this work has also been an and hydrocarbons.2,3 They have also facilitated the descrip- attempt to avoid any predefined fixed relationships between tion of chemical reactions and barriers in large systems4 and bond energy, length, and force constants present in established molecular dynamics simulations as an important traditional bond order potentials. tool in designing new materials, estimating their properties, In Sec. II, we describe the formalism used to represent and understanding materials processing. This success moti- the binding energy as a function of the environment of a vated the development of REBO potentials for chemically chemical bond. In Sec. III, the parameters of the formulas complex systems of different atoms.5–7 given in Sec. II are fitted to results of ab initio calculations. The REBO approach is based on a fixed relationship In Sec. IV, the CuO potential developed is applied to the between bond energy and bond order, which is a function of problem of unzipping of graphene sheets and carbon the coordination numbers of atom pairs forming a bond. This nanotubes by adsorbed oxygen. Conclusions are given in empirical relationship is only approximately satisfied in real Sec. V. molecules and may yield imprecise results especially in sys- tems encompassing two or more chemically distinct atoms. Each new atom considered introduces a new set of param- eters, which must be optimized by fitting the equilibrium II. REACTIVE POTENTIAL distance and energy of bonds formed between the new atom and the previous set of atoms. This fitting strategy is not One of the most successful approaches to modeling re- specific to the REBO potential. It can be implemented to active chemistry in atomistic simulations has been the bond other potentials with their own sets of parameters, provided order method.1 It introduces environment-dependent correc- they are complex enough to capture an increasing number of tions to the strengths of chemical bonds based on the local bonding configurations. coordination of atoms. Unlike multibody potential expan- We present here a new approach to generating empirical sions, where each successive term depends on one extra reactive potentials for covalent systems. We focus on the atomic coordinate, the bond order formalism retains the form carbon-oxygen system as a model. Carbon-oxygen chemical of a pair potential; instead, all the corrections are included reactions are ubiquitous, from combustion to metabolic reac- into the coefficients of the potential function. Previous reac- tions in cells.8 The importance of carbon-oxygen chemistry tive potentials for silicon1 and hydrocarbons ͑C+H system͒2 ͑ ഛ ͒ used a bond order parameter bij bij 1 to account for the ͒ a Author to whom correspondence should be addressed. Electronic mail: environment-dependent bond order between atoms i and j as [email protected]. follows: 0021-9606/2008/128͑23͒/234706/8/$23.00128, 234706-1 © 2008 American Institute of Physics Downloaded 20 Jun 2008 to 131.215.225.9. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp 234706-2 A. Kutana and K. P. Giapis J. Chem. Phys. 128, 234706 ͑2008͒ is followed only approximately: the CO3 and CO4 molecules are more stable than what would follow from a linear fit for the CO and CO2 molecules. Usually, a best fit to the Pauling rule is performed in order to minimize the errors over differ- ent coordinations. The fitting could also be improved by in- troducing corrections to the angular energy of the molecule, at the cost of a more complex expression for it. As an alter- native to these approaches, we remove the requirement that the approximate Pauling rule be satisfied, and allow all three parameters A, B, and ␣ in Eqs. ͑1͒ and ͑2͒ to vary indepen- dently. Using the parameters 2 / ͑ /␣͒ ͑ / ͒ U0 =B 4A and re = 1 ln 2A B , which are the poten- tial well depth and equilibrium interatomic distance, respec- tively, instead of A and B in Eq. ͑2͒, the expression for the bond energy becomes FIG. 1. The Pauling plot for the potential well depth and equilibrium bond length of the CO bond represented by a Morse potential with variable coefficients as a function of the number of the O atoms connected to a C ͑ ͒ ͑ ͑ ␣͑ ͒͒ ͑ ␣͑ ͒͒͒ atom. The molecular energies are obtained using GAMESS ͑DFT, 6-311G*, Uij = fc rij U0 exp −2 rij − re − 2 exp − rij − re , and B3LYP density functional͒. ͑6͒ ͑ ͒͑ ͑ ͒ ͑ ͒͒ ͑ ͒ Uij = fc rij VR rij − bijVA rij . 1 ␣ where U0, re, and are functions of the numbers of neigh- ͑ ͒ bors on both sides of the bond between atoms i and j as Here, rij is the distance between atoms i and j, fc rij is the cutoff function used for smooth potential truncation, and follows: ͑ ͒ ͑ ͒ VR rij and VA rij are the repulsive and attractive interactions usually given by Morse-type expressions ͑ ͒ U0 = U0 NCi,NOi,NCj,NOj , ͑ ͒ ͑ ␣ ͒ VR r = A exp − 1r , ͑2͒ V ͑r͒ = B exp͑− ␣ r͒, ͑ ͒ ͑ ͒ A 2 re = re NCi,NOi,NCj,NOj , 7 ␣ ␣ ͑␣ ␣ ␣ where A, B, 1, and 2 are all positive 1 =2 2 = in the original Morse potential͒. The total bonding energy is ␣ ␣͑ ͒ obtained by summing Uij over all bonds as follows: = NCi,NOi,NCj,NOj . ͑ ͒ Ebond = ͚ Uij. 3 jϾi Here, NCi is the number of carbon atoms connected to atom i of the ij bond, N is the number of oxygen atoms con- In a more recent REBO potential for hydrocarbons,3 a Oi nected to this atom, while N and N are similar quantities repulsive term of the form Cj Oj for atom j. Each parameter is a function of 2K variables, V ͑r͒ = ͑1+Q/r͒A exp͑− ␣ r͒͑4͒ where K is the total number of the atomic species in the R 1 ␣ system. The coefficients U0, re, and are calculated for was introduced, where Q is a fitting parameter. The modified stable molecules where bonds are assigned integers NC,Oi and pre-exponential factor 1+Q/r accounts for Coulombic repul- NC,Oj. Use of fractional N values permits smooth switching sion, thus rectifying the unphysical behavior of the Morse through different coordinations and allows chemical reac- potential at small r. tions to occur. Morse-type functions are well suited for capturing the The particular form of the smooth interpolating universal energetics of covalent and metallic bonds.13,14 The functions used here for fitting the parameters in Eq. ͑7͒ as bond order parameter bij depends on the atomic coordination functions of the number of neighbors is a multivariate in a molecule or solid and is central to the potential’s ability polynomial of m variables of the order n as follows: to reproduce correct binding energies. The form of Eqs. ͑1͒ ͑ ͒ ͑ ␣ ␣ ␣͒ and 2 with 1 =2 2 = requires that bond energies and bond distances follow the Pauling rule,15 stating that the n n−i n−i−j n−i−j−k−... ͑ ͒ P ͑x ,x , ...,x ͒ = ͚ xi ͚ xj ͚ xk ͚ a xl , logarithm of the bond order bond strength U0 is a linear n 1 2 m 1 2 3 ¯ ijk...l m i=0 j=0 k=0 l=0 function of the equilibrium bond distance re as follows: ͑8͒ ␣ ͑ ͒ log U0 = log A −2 re. 5 Figure 1 shows the energies of COn n=1,4 molecules calcu- where xi are the variables corresponding to NCi ,...,NOj in lated using density functional theory ͑DFT͒. The Pauling rule Eq.
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