# Multiresolution molecular mechanics: Surface effects in nanoscale materials

## Abstract

Surface effects have been observed to contribute significantly to the mechanical response of nanoscale structures. The newly proposed energy-based coarse-grained atomistic method Multiresolution Molecular Mechanics (MMM) (Yang, To (2015), ) is applied to capture surface effect for nanosized structures by designing a surface summation rule SR{sup S} within the framework of MMM. Combined with previously proposed bulk summation rule SR{sup B}, the MMM summation rule SR{sup MMM} is completed. SR{sup S} and SR{sup B} are consistently formed within SR{sup MMM} for general finite element shape functions. Analogous to quadrature rules in finite element method (FEM), the key idea to the good performance of SR{sup MMM} lies in that the order or distribution of energy for coarse-grained atomistic model is mathematically derived such that the number, position and weight of quadrature-type (sampling) atoms can be determined. Mathematically, the derived energy distribution of surface area is different from that of bulk region. Physically, the difference is due to the fact that surface atoms lack neighboring bonding. As such, SR{sup S} and SR{sup B} are employed for surface and bulk domains, respectively. Two- and three-dimensional numerical examples using the respective 4-node bilinear quadrilateral, 8-node quadratic quadrilateral and 8-node hexahedral meshes are employed tomore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22622287

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 336; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 77 NANOSCIENCE AND NANOTECHNOLOGY; ATOMS; BEAMS; BONDING; CAPTURE; COMPARATIVE EVALUATIONS; COMPUTERIZED SIMULATION; DEGREES OF FREEDOM; DISTRIBUTION; ENERGY SPECTRA; FINITE ELEMENT METHOD; NANOSTRUCTURES; PERFORMANCE; QUADRATURES; SAMPLING; SURFACE AREA; SURFACES; THREE-DIMENSIONAL LATTICES

### Citation Formats

```
Yang, Qingcheng, E-mail: qiy9@pitt.edu, and To, Albert C., E-mail: albertto@pitt.edu.
```*Multiresolution molecular mechanics: Surface effects in nanoscale materials*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.01.058.

```
Yang, Qingcheng, E-mail: qiy9@pitt.edu, & To, Albert C., E-mail: albertto@pitt.edu.
```*Multiresolution molecular mechanics: Surface effects in nanoscale materials*. United States. doi:10.1016/J.JCP.2017.01.058.

```
Yang, Qingcheng, E-mail: qiy9@pitt.edu, and To, Albert C., E-mail: albertto@pitt.edu. Mon .
"Multiresolution molecular mechanics: Surface effects in nanoscale materials". United States.
doi:10.1016/J.JCP.2017.01.058.
```

```
@article{osti_22622287,
```

title = {Multiresolution molecular mechanics: Surface effects in nanoscale materials},

author = {Yang, Qingcheng, E-mail: qiy9@pitt.edu and To, Albert C., E-mail: albertto@pitt.edu},

abstractNote = {Surface effects have been observed to contribute significantly to the mechanical response of nanoscale structures. The newly proposed energy-based coarse-grained atomistic method Multiresolution Molecular Mechanics (MMM) (Yang, To (2015), ) is applied to capture surface effect for nanosized structures by designing a surface summation rule SR{sup S} within the framework of MMM. Combined with previously proposed bulk summation rule SR{sup B}, the MMM summation rule SR{sup MMM} is completed. SR{sup S} and SR{sup B} are consistently formed within SR{sup MMM} for general finite element shape functions. Analogous to quadrature rules in finite element method (FEM), the key idea to the good performance of SR{sup MMM} lies in that the order or distribution of energy for coarse-grained atomistic model is mathematically derived such that the number, position and weight of quadrature-type (sampling) atoms can be determined. Mathematically, the derived energy distribution of surface area is different from that of bulk region. Physically, the difference is due to the fact that surface atoms lack neighboring bonding. As such, SR{sup S} and SR{sup B} are employed for surface and bulk domains, respectively. Two- and three-dimensional numerical examples using the respective 4-node bilinear quadrilateral, 8-node quadratic quadrilateral and 8-node hexahedral meshes are employed to verify and validate the proposed approach. It is shown that MMM with SR{sup MMM} accurately captures corner, edge and surface effects with less 0.3% degrees of freedom of the original atomistic system, compared against full atomistic simulation. The effectiveness of SR{sup MMM} with respect to high order element is also demonstrated by employing the 8-node quadratic quadrilateral to solve a beam bending problem considering surface effect. In addition, the introduced sampling error with SR{sup MMM} that is analogous to numerical integration error with quadrature rule in FEM is very small. - Highlights: • Surface effect captured by Multiresolution Molecular Mechanics (MMM) is presented. • A novel surface summation rule within the framework of MMM is proposed. • Surface, corner and edges effects are accuterly captured in two and three dimension. • MMM with less 0.3% degrees of freedom of atomistics reproduces atomistic results.},

doi = {10.1016/J.JCP.2017.01.058},

journal = {Journal of Computational Physics},

number = ,

volume = 336,

place = {United States},

year = {Mon May 01 00:00:00 EDT 2017},

month = {Mon May 01 00:00:00 EDT 2017}

}