Parameterization of Lennard-Jones Equation in the qAIREBO Model via Tetrafluoromethane

MINH H. TRAN, STEVEN J. STUART *

Department of Chemistry, Clemson University, Clemson, South Carolina 29634

Abstract: The objective of this project is to determine a pair of Lennard-Jones (LJ) parameter values, σ and ε, that describe van der Waals interactions between two non-bonded atoms in tetrafluoromethane (CF4). Recently developed as an improvement of the Reactive Empirical Bond-Order (REBO) model,1 the AIREBO model simulates chemical reactions more realistically by introducing torsion, dispersion and non-bonded repulsion terms into the original potential function.2

The AIREBO model has been parameterized for use with systems of carbon and hydrocarbon. In 2008, the qAIREBO, which is under development, is promising to be the most adaptive model by adding electrostatic interaction. With this new advantage, this project extends the application of the qAIREBO to fluorocarbon systems by parameterization. Molecular dynamics simulations were performed on models of CF4 gas and CF4 liquid, composed of 128 molecules. The LJ parameters were obtained by fitting them to reproduce CF4 liquid with experimental properties of 90.792 Å3 / molecule in density and 0.12415 eV in heat of vaporization at the boiling temperature of 145.19 K and standard pressure of 1 bar. If successful, these parameter values will be validated with larger systems of 256 and 512 CF 4

molecules.

Introduction

Computational molecular modeling has a long history of development. To simulate molecular interactions on computer, chemists have, at first, applied molecular dynamics and Monte Carlo method. This classical molecular simulation made use of Newton’s second law to describe motion through time of each individual atom in a chemical system. However, this technique successfully simulates physical processes, but reflects no chemical reactions such as bond breaking and bond formation, which are considered quantum mechanical processes. Not until 1988, Tersoff introduced a new model, which modified the classical potential in order to empirically simulate the formation and dissociation of saturated and unsaturated covalent bonds in chemical systems such as silicon and carbon. The most successful and well-known example of Tersoff potential is the reactive empirical bond-order (REBO) which was significantly modified by Brenner in 1990 to include radical and conjugated hydrocarbon bonds.3 One feature that gives the REBO model an advantage over non-reactive molecular dynamics is the contribution of both the repulsive and attractive potentials to the interacting potential between two bonded atoms which is expressed as

Vij(r) = (r) + bij (r) (1)

where i, j represent two bonded atoms, r is the distance between them, and b is the bond order whose terms vary as bonding environment changes. As a result, an atom is no longer covalently constrained to its original neighbor or a specific hybridization state, but able to dissociate and make a new bond with other atoms in an adaptive way during a chemical reaction. This REBO model was originally parameterized to simulate properties of diamond. New development has extended the application of the REBO potential to the simulation of the energetic, elastic and vibrational properties of solid carbon and small hydrocarbons.4 The REBO model, however, still has some drawbacks. With the absence of intermolecular (van der Waals) and torsional interaction potentials, the REBO model poorly reflects chemical properties of condensed phase molecular systems or large hydrocarbons. The need for a more complete model led to a new generation of the REBO’s family, called adaptive intermolecular REBO (AIREBO), proposed by Stuart in 2000.5 There are still some disadvantages that make this model less attractive. The Lennard-Jones (LJ) equation that describes the van der Waals interaction can adapt its strength to the change in local chemical environment, but not the shape of the interaction. In other words, the parameter values do not vary as the bonding environment changes. Another characteristic that is included in classical molecular dynamics, but absent in the REBO potentials is the electrostatic interaction. Therefore, the current model, called qAIREBO, is under development, and expected to address these limitations. Table 1 provides a brief comparison of different models mentioned above.

Classical Molecular Dynamics REBO AIREBO qAIREBO

Covalent Bonding Bond Stretching Bond Angle Bending

Torsion van der Waals Electrostatics Table 1. Brief comparison of molecular modeling potentials

With more complete terms in the potential expression, the qAIREBO model is intended to not only simulate hydrocarbon systems more accurately, but also simulate systems other than hydrocarbon such as FC, NH, CHNO, which is the motivation of this paper. Experimental energetic property of tetrafluoromethane (CF4) is used as a target to fit the LJ parameters. The pair of parameter values obtained then can be used for all F atoms in any chemical environment. Table 2 lists the family of bond-order potentials and their applications.

Model Applications

Tersoff Si and C systems

REBO Si, C and small HC systems

AIREBO C, HC and condensed phase molecular systems

qAIREBO AIREBO’s applications and, possibly, FC, NH, CHNO systems

Table 2. REBO potential family and their applications

The application of FC potential can be widely appreciated, including the fields of materials science and biological research. One example is the simulation of C2F6 etching of silicon. Due to its favorable kinetic properties and high selectivity, C2F6 plasmas are used in the etching industry. A potential model will simulate the surface reaction mechanisms so that a future plan will be developed to optimize the etching process. In the semiconductor manufacturing, a similar application is the simulation of monatomic fluorine, removing the photoresist from an etched wafer during the plasma ashing process. In pharmacy, many pharmaceutical products contain C-F bonds such as 5-fluorouracil, an anticancer drug. As a result, the capability of simulating fluorocarbon systems proves itself useful in experimenting new medical products. The list is long and an effort to improve the qAIREBO model to handle more types of chemical systems is, therefore, a demand.

Model

A simplified equation for the qAIREBO potential has the form:

V = VREBO + VLJ + Vtors (2)

in which the Lennard-Jones potential, VLJ, is the expression that will be parameterized for the purpose of this paper. VLJ is a function of the distance rij between two non-bonded atoms i and j. Its dependence on two parameters σij and εij for the description of the intermolecular interactions is expressed as the following

(rij) = 4εij [( )12

_

( )6

] (3)

With its shortness and simpler terms, this 12-6 form proves itself to be more computationally efficient than other functional forms, which may be more accurate, yet too complicated. The sum of the intermolecular interactions acting on atoms i and j is then expressed as

Eij = α (rij) (4)

where the many-body term α varies according to changes in bonding environment.

There are three factors that determine whether long range interactions should take effect during a simulation. These factors are covalent-bonding cutoff distance and bond order between atoms i and j, and the relative positions of all atoms bonded to i and j. As a result, if two atoms are located beyond a cutoff distance and not 1-4 neighbors on the same molecule, and have a low value of bond order, a full LJ potential will exist between them. The strength of the LJ interactions, therefore, will vary smoothly with different bonding environment and totally disappear if all three conditions above are not met. This adaptability is where the REBO and the AIREBO models compromise. Two atoms, when approaching close enough, will react or bond to each other and the REBO model is valid. In this case, the r ij

-12 term in equation (3) does not contribute. However, its contribution is taken into account when atoms are chemically saturated.

The LJ potential is formulated around the variability of the parameters σ and ε. In the simulation of soft, molecular systems, these parameters have an important role in determining material properties such as cohesive energy, boiling point and density. The variability of σ and ε reflects the change in the polarization and the radius of the electronic distribution on an atom, which invokes intermolecular interactions. In the AIREBO model, each atom is assigned a pair of σ and ε values regardless of any hybridization that occurs during a chemical reaction. As a result, the AIREBO model simulates a chemical system that was used in the fitting process with greater accuracy than systems that were not. It is the case of saturated and unsaturated hydrocarbons in which the AIREBO model is better suited to saturated hydrocarbons. In addition, a single pair of σ and ε is used when an atom is bonded to other different atoms. The type of covalent bonds, therefore, should not be ignored since different bond length and polarity result in different intermolecular interactions. The most recent version of the AIREBO model, however, resolves these problems. By considering the number of each type of neighbor that is bonded to the target atom, the LJ parameters appear in the AIREBO model as functions, such as σF(NC, NF) and εF(NC, NF), rather than fixed numbers. Again, the advantage of the AIREBO model over its counterparts lies in the simplification of its functions to tackle several problems that, otherwise, require complicated mathematical expressions. One limitation that may become a goal for future development is that the AIREBO model uses the first-neighbor method to count the number of atoms bonded to the target atom. When simulating many different chemical structures, it is sometimes necessary to consider second or third neighbors. This functional form is applied to both the reactive and nonreactive, bond-order potential. The difference is that in the latter, the values of N are integers whereas in the former, the values of N are non-integers.

As mentioned in the previous section, fitting the values of σ and ε using CF4 is the topic of this paper. However, due to the scope of this paper, only σ and ε values of F will be determined. Based on the functional form that was developed for the LJ parameters, the mathematical expressions for σ and ε are σF(1, 0) and εF(4, 0). This single pair of parameters is adequate for F since all F atoms in CF 4 are bonded to C in the same manner and none is bonded to each other.

Fitting Process

The objective of this fitting process was to find a pair of parameter values σ F and εF that best reproduce the energetic property of CF4 liquid, enthalpy of vaporization (∆Hvap) in this case, under the experimental conditions. For consistency with the setting of the qAIREBO model, the units of the

experimental density and ∆Hvap are converted to Å3/molecule and eV/molecule, respectively. Therefore, ∆Hvap of CF4, which was taken from a published paper,6 was recorded as 0.12425 eV/molecule. At the boiling point of 145.19K, the density of liquid CF4 was taken from the same source as 90.792 Å3/molecule under the pressure of 1 bar.

According to the definition of the enthalpy,

∆Hvap = ∆Uvap - ∆(PV) (5)

It is assumed that CF4 vapor behaves as an ideal gas and the molar volume of the liquid is negligible compared to that of the gas. Then, equation (5) becomes

∆Hvap = ∆Uvap – (∆ngas)RT (6)

Since ∆ngas = 1, equation (6) is reduced to

∆Hvap = ∆Uvap – RT (7)

The change in internal energy ∆Uvap is calculated from a simulation by taking the difference in internal energy of a gas-phase CF4 molecule and a CF4 liquid molecule at the boiling point.

Two separate simulations were run on two CF4 systems. The gas phase system included only one single CF4 molecule under the non periodic boundary condition while the liquid phase included 128 CF4

molecules under periodic boundary conditions. The CF4 gas molecule was heated up from an initial temperature of 2 K to a final temperature of 145.19 K with periodic velocity reassignment thermostat for a period of 80 ps (Figure 1). Then the molecule was equilibrated over 10 ps from which its average internal energy was obtained (Figure 2). Unlike the CF4 gas simulation, the simulation of CF4 liquid required much more labor. In order to obtain a periodic liquid cell with each side length greater than 20.4 Å or twice of the LJ cutoff of 3σCC, a minimum number of 128 CF4 molecules were required. The side length was calculated and adjusted based on the experimental liquid density of 90.792 Å3/molecule so that the total volume of the cell was exactly equal to that of the physical liquid. Figure 3 shows a 3-D view of a CF4 liquid cell. After the cell was built, its internal energy at the boiling point was measured in a similar manner as done with CF4 gas. The cell was heated up from 2 K to its boiling point of 145.19 K with Langevin thermostat for a period of 20 ps (Figure 4). For molecular interactions beyond the LJ cutoff, long range corrections were applied assuming a uniform density over that distance. The equations of motion were integrated with the velocity Verlet integrator. Then the cell was equilibrated over 200 ps from which its average internal energy and external pressure were measured (Figures 5 & 6). The average internal energy of a single CF4 liquid molecule was calculated by dividing the cell’s average energy by the number of CF4 molecules contained in the cell. The difference in the internal energy between two phases was then substituted into equation (7) to calculate ∆Hvap at the boiling

temperature and with the universal gas constant R of 0.000086 . This result was compared

with the experimental ∆Hvap of 0.12425 eV / molecule. Also, the average external pressure measured from the simulation graph was compared with the experimental pressure of 1 bar. If all values match, the values of σ and ε, which were set to run the simulations, will be the LJ parameters for F. If they don’t, the simulation for CF4 liquid cell must be performed again from the beginning with a new pair of parameter values until the cell reaches the experimental energetic property under the experimental conditions.

Figure 1. Internal energy of CF4 gas molecule from Figure 2. Internal energy of CF4 gas molecule2 K to 145.19 K over 80 ps. during the equilibrium state over 10 ps. The line shows the average internal energy

Figure 3. A 3-D view of CF4 liquid with 128 Figure 4. Internal energy of CF4 liquid from 2 K molecules to 145.19 K over 20 ps

Figure 5. Internal energy of CF4 liquid Figure 6. Pressure on CF4 liquid during the during the equilibrium state over 200 ps. equilibrium state over 200 ps. The line The line shows the average internal energy shows the average pressure.

Results and Discussions

Due to the substantial amount of time required to fit a pair of values for the LJ parameters, the current work has not reached its goal yet. So far, the best pair of parameter values is σ = 2.13 Å and ε = 50.717 K. These values will reproduce the enthalpy of vaporization of CF4 liquid under the pressure of -225 bar ± 63 bar. In figure 7, at σ = 2.13, each value of ε provides a ∆H vap, which is then compared to the experimental ∆Hvap to give the difference. As seen from the graph, at ε = 50.717 K, ∆Hvap from the simulation is equal to that from the experiment. Figure 8 shows a graph of pressure corresponding to each value of σ with the value of ε that reproduces the ∆Hvap from the experiment. According to this graph, the pressure is getting closer to 1 bar as the value of σ decreases.

Figure 7. The difference between ∆Hvap from simulations and ∆Hvap from experiment corresponding to each value of ε at σ = 2.13 Å

Figure 8. The value of pressure corresponding to each value of σ with the value of ε* that reproduces the ∆Hvap from the experiment.

More research is needed to determine the final pair of parameter values that can reproduce ∆H vap

of CF4 liquid under 1 bar pressure. As extrapolated from the graph in figure 8, a pressure close to 1 bar

can be achieved at σ = 1.36 Å. However, this approximation should be confirmed by simulations when available.

When the size of the simulation is larger with more molecules, the values of σ and ε may require adjustment. Only the pair of values that do not vary with larger models will be accepted. In other words, a satisfied pair of parameters will provide a consistent internal energy for each individual CF4 liquid molecule (U/N) over different large CF4 liquid systems. As a result, after successfully obtaining proper values of σ and ε for CF4 liquid corresponding to 128 CF4 molecules, a series of simulations will be run for 256 and 512 CF4 molecules with those same parameter values. If these systems continue to reproduce exact energetic property of CF4 liquid under the experimental conditions, the parameter values obtained are valid. If not, a new quest for new parameter values will resume. As explained in the previous sections, with the adaptability of the LJ parameters that gives the current model an advantage, it is necessary to direct future project to the transferability of the LJ parameters to other fluorocarbon systems.

References

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