# User:Ivan Shmakov/Potopt 2014

Potopt is a software package to perform (potential) w:energy minimization of a multiple-particle system.

## Morse potential

The potential at the 𝑖th particle 𝑟𝑖 is 𝑉𝑖 and is the sum of potentials 𝑉𝑖𝑗 created by selected (𝐷e𝑖𝑗 ≠ 0) other individual particles. (Where 𝐷e𝑖𝑗, 𝜌e𝑖𝑗, and 𝑎e𝑖𝑗 are per-pair parameters.)

{\displaystyle {\begin{aligned}V_{i}&=\sum _{j\neq i}V_{ij};\\V_{ij}&=D_{\mathrm {e} ij}\left[1-\exp(-a_{ij}(\rho _{ij}-\rho _{\mathrm {e} ij}))\right]^{2};\\\rho _{ij}&={\sqrt {({\bar {r}}_{i}-{\bar {r}}_{j})^{2}}}.\\\end{aligned}}}

In order to apply the w:gradient descent method, we derive ∇𝑉𝑖𝑗 as follows.

{\displaystyle {\begin{aligned}\nabla V_{ijk}&=D_{\mathrm {e} ij}{\frac {\partial \left[1-\exp(-a_{ij}(\rho _{ij}-\rho _{\mathrm {e} ij}))\right]^{2}}{\partial x_{ik}}}\\&=2a_{ij}D_{\mathrm {e} ij}\left[1-\exp(-a_{ij}(\rho _{ij}-\rho _{\mathrm {e} ij}))\right]^{2}\exp(-a_{ij}(\rho _{ij}-\rho _{\mathrm {e} ij})){\frac {x_{ik}-x_{jk}}{\rho _{ij}}}\\&=2a_{ij}D_{\mathrm {e} ij}(1-\kappa _{ij})\kappa _{ij}{\frac {x_{ik}-x_{jk}}{\rho _{ij}}};\\\nabla {\bar {V}}_{ij}&=2a_{ij}D_{\mathrm {e} ij}(1-\kappa _{ij})\kappa _{ij}{\frac {{\bar {\rho }}_{ij}}{\rho _{ij}}};\\{\bar {\rho }}_{ij}&={\bar {r}}_{i}-{\bar {r}}_{j}.\\\end{aligned}}}

### Parameters

The parameters for the C─C w:single bond are as follows.[1]

Bond 𝐷e, 10⁻¹⁹ N ⋅ m 𝜌e, 10⁻⁹ m 𝑎e, 10¹⁰ m⁻¹
C─C 6.03105 0.1421 2.625

## References

1. Avinash Parashar; Pierre Mertiny (2012-10-26). "Multiscale model to investigate the effect of graphene on the fracture characteristics of graphene/polymer nanocomposites". Nanoscale Research Letters 7 (1): 595. doi:10.1186/1556-276X-7-595. Retrieved 2014-02-08.