^{1}, Abdul N. A. Samadh

^{1}, Lionel M. Raff

^{2,a)}, Martin T. Hagan

^{3}, Satish T. Bukkapatnam

^{4}and Ranga Komanduri

^{5}

### Abstract

A new approach involving neural networks combined with molecular dynamics has been used for the determination of reaction probabilities as a function of various input parameters for the reactions associated with the chemical-vapor deposition of carbon dimers on a diamond (100) surface. The data generated by the simulations have been used to train and test neural networks. The probabilities of chemisorption, scattering, and desorption as a function of input parameters, such as rotational energy, translational energy, and direction of the incident velocity vector of the carbon dimer, have been considered. The very good agreement obtained between the predictions of neural networks and those provided by molecular dynamics and the fact that, after training the network, the determination of the interpolated probabilities as a function of various input parameters involves only the evaluation of simple analytical expressions rather than computationally intensive algorithms show that neural networks are extremely powerful tools for interpolating the probabilities and rates of chemical reactions. We also find that a neural network fits the underlying trends in the data rather than the statistical variations present in the molecular-dynamics results. Consequently, neural networks can also provide a computationally convenient means of averaging the statistical variations inherent in molecular-dynamics calculations. In the present case the application of this method is found to reduce the statistical uncertainty in the molecular-dynamics results by about a factor of 3.5.

This project is funded by grants from the National Science Foundation (DMI-0200327) and (DMI-0457663). We thank Dr. W. DeVries, Dr. G. Hazelrigg, Dr. J. Cao, and Dr. D. Durham of the Division of Design, Manufacturing, and Industrial Innovation, Dr. B. M. Kramer, Engineering Centers Division, and Dr. J. Larsen Basse, Tribology and Surface Engineering program for their interest and support of this work. This project was also sponsored by a DEPSCoR grant on the Multiscale Modeling and Simulation of Material Processing (F49620-03-1-0281). The authors thank Dr. Craig S. Hartley and Dr. J. Tiley of the AFOSR for their interest in and support of this work. One of the authors (R.K.) also thanks, A. H. Nelson, Jr. Endowed Chair in Engineering for additional support.

I. INTRODUCTION

II. MOLECULAR-DYNAMICS SIMULATIONS

III. NEURAL NETWORKS

IV. RESULTS AND DISCUSSION

A. Various reactions

B. Training and testing of the neural networks

1. Training by a single set of data

2. Ensemble (committee) of networks

3. Dependence of reaction probabilities on input parameters

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Artificial neural networks
- 17.0
- Chemical reactions
- 15.0
- Molecular dynamics
- 13.0
- Carbon
- 11.0
- Diamond
- 10.0

## Figures

(a) The projection of the top three layers of carbon atoms of diamond (100) lattice on plane: first layer (엯), second layer (◻), and third layer (*). The central atom denotes the radical site. (b) A schematic diagram to depict parameters and ; denotes the radical site and represents the point of intersection of the initial velocity vector and the diamond surface.

(a) The projection of the top three layers of carbon atoms of diamond (100) lattice on plane: first layer (엯), second layer (◻), and third layer (*). The central atom denotes the radical site. (b) A schematic diagram to depict parameters and ; denotes the radical site and represents the point of intersection of the initial velocity vector and the diamond surface.

A schematic illustration of the two-layer neural network used in the present study.

A schematic illustration of the two-layer neural network used in the present study.

(a) The potential energy of the system (in eV), (b) coordinate of the center of mass of carbon dimer (in Å), and (c) the internuclear distance between the carbon atoms of the dimer (in Å), as a function of the integration time step for a typical trajectory leading to chemisorption. One time step corresponds to 0.5 fs.

(a) The potential energy of the system (in eV), (b) coordinate of the center of mass of carbon dimer (in Å), and (c) the internuclear distance between the carbon atoms of the dimer (in Å), as a function of the integration time step for a typical trajectory leading to chemisorption. One time step corresponds to 0.5 fs.

(a) The potential energy of the system (in eV) and (b) coordinate of the center of mass of carbon dimer (in Å), as a function of the integration time step for a typical trajectory leading to reaction . One time step corresponds to 0.5 fs.

(a) The potential energy of the system (in eV) and (b) coordinate of the center of mass of carbon dimer (in Å), as a function of the integration time step for a typical trajectory leading to reaction . One time step corresponds to 0.5 fs.

The probabilities of an event (chemisorption/scattering/desorption) given by neural network vs the corresponding probabilities given by molecular dynamics for training and testing data sets: [(a) and (b)] chemisorption, [(c) and (d)] scattering, and [(e) and (f)] desorption.

The probabilities of an event (chemisorption/scattering/desorption) given by neural network vs the corresponding probabilities given by molecular dynamics for training and testing data sets: [(a) and (b)] chemisorption, [(c) and (d)] scattering, and [(e) and (f)] desorption.

The expected statistical spread of the probabilities as given by Eq. (6) vs the true probabilities .

The expected statistical spread of the probabilities as given by Eq. (6) vs the true probabilities .

The probabilities of an event (chemisorption/scattering/desorption) given by neural networks vs the corresponding probabilities given by molecular dynamics using sets of 500 trajectories: (a) chemisorption, (b) scattering, and (c) desorption.

The probabilities of an event (chemisorption/scattering/desorption) given by neural networks vs the corresponding probabilities given by molecular dynamics using sets of 500 trajectories: (a) chemisorption, (b) scattering, and (c) desorption.

The variation of chemisorption probability , scattering probability , and desorption probability as a function of impact parameter (*b*), for , , , and . The triangles (▴) joined by the line denote the NN ensemble predictions and the circles (●) represent the MD data. The error bars associated with the MD points correspond to one-sigma limit.

The variation of chemisorption probability , scattering probability , and desorption probability as a function of impact parameter (*b*), for , , , and . The triangles (▴) joined by the line denote the NN ensemble predictions and the circles (●) represent the MD data. The error bars associated with the MD points correspond to one-sigma limit.

Same as Fig. 8 except that these curves show variation with for , , , and (left curves), and that with for , , , and (right curves).

Same as Fig. 8 except that these curves show variation with for , , , and (left curves), and that with for , , , and (right curves).

Same as Fig. 8 except that these curves show variation with for , , , and (left curves), and that with for , , , and (right curves).

The variation of chemisorption probability , scattering probability , and desorption probability predicted by the NN ensemble as a function of angle for different values of impact parameter: 1.0 (*), 1.5 (●), 2.0 (▴), and 2.5 Å (▪), for , , and .

The variation of chemisorption probability , scattering probability , and desorption probability predicted by the NN ensemble as a function of angle for different values of impact parameter: 1.0 (*), 1.5 (●), 2.0 (▴), and 2.5 Å (▪), for , , and .

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