# Molecular dynamic simulation

The motion of each molecule i in the system is described by Newton’s second law, i.e.,

$\sum\limits_{j = 1(j \ne i)}^N {{\mathbf{F}}_{ij} } = m_i \frac{{d^2 {\mathbf{r}}_i }} {{dt^2 }}, i = 1,2,3...n$ $i = 1,2,3...n\qquad \qquad(1)$

where mi and ${{\mathbf{r}}_i}$ are the mass and position of the ith molecule in the system. In arriving at eq. (1), it is assumed that the molecules are monatomic and have only three degrees of freedom of motion. For molecules with more than one atom, it is also necessary to consider the effect of rotation. As the simplest case, the intermolcular potential between the ith and jth molecules is obtained by the Lennard-Jones potential:

${\phi _{ij}} = 4\varepsilon \left[ {{{\left( {\frac{{{r_0}}}{{{r_{ij}}}}} \right)}^{12}} - {{\left( {\frac{{{r_0}}}{{{r_{ij}}}}} \right)}^6}} \right]\qquad \qquad(2)$

The intermolecular force between the ith and jth molecules can be obtained from

${{\mathbf{F}}_{ij}} = - \nabla {\phi _{ij}} = \frac{{24\varepsilon }}{{{r_o}}}\left[ {{{\left( {\frac{{{r_o}}}{{{r_{ij}}}}} \right)}^{13}} - {{\left( {\frac{{{r_o}}}{{{r_{ij}}}}} \right)}^7}} \right]\frac{{{{\mathbf{r}}_{ij}}}}{{{r_{ij}}}}\qquad \qquad(3)$

where rij is the distance between the ith and jth molecules.

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.