Additional documentation for the RE-squared ellipsoidal

potential as implemented in LAMMPS

Mike Brown, Sandia National Labs, October 2007

Let the shape matrices S_{i} = diag(a_{i}, b_{i}, c_{i}) be given by the ellipsoid radii. Let the relative energy matrices E_{i} = diag(ϵ_{ia}, ϵ_{ib}, ϵ_{ic}) be given by the relative well depths (dimensionless energy scales inversely proportional to the well-depths of the respective orthogonal configurations of the interacting molecules). Let A_{1} and A_{2} be the transformation matrices from the simulation box frame to the body frame and r be the center to center vector between the particles. Let A_{12} be the Hamaker constant for the interaction given in LJ units by A_{12} = 4π^{2}ϵ_{LJ}(ρσ^{3})^{2}.

The RE-squared anisotropic interaction between pairs of ellipsoidal particles is given by

U = U_{A} + U_{R},

U_{α} = \frac{A_{12}}{m_{α}} (\frac{σ}{h})^{n_{α}}(1 + o_{α}ηχ\frac{σ}{h}) × ∏_{i} \frac{a_{i}b_{i}c_{i}}{(a_{i} + h/p_{α})(b_{i} + h/p_{α})(c_{i} + h/p_{α})},

m_{A} = −36, n_{A} = 0, o_{A} = 3, p_{A} = 2,
m_{R} = 2025, n_{R} = 6, o_{R} = 45/56, p_{R} = 60^{1/3},
χ = 2r̂^{T} B^{−1}r̂,
r̂ = r/|r|,
B = A_{1}^{T}E_{1}A_{1} + A_{2}^{T}E_{2}A_{2}

η = \frac{det[S_{1}]/σ_{1}^{2} + det[S_{2}]/σ_{2}^{2}}{[det[H]/(σ_{1} + σ_{2})]^{1/2}} ,

σ_{i} = (r̂^{T} A_{i}^{T}S_{i}^{−2}A_{i}r̂)^{−1/2},

H = \frac{1}{σ_{1}} A_{1}^{T}S_{1}^{2}A_{1} + \frac{1}{σ_{2}} A_{2}^{T}S_{2}^{2}A_{2}

Here, we use the distance of closest approach approximation given by the Perram reference, namely

h = |r| − σ_{12},

σ_{12} = [\frac{1}{2}r̂^{T} G^{−1}r̂]^{−1/2},

and

G = A_{1}^{T}S_{1}^{2}A_{1} + A_{2}^{T}S_{2}^{2}A_{2}

The RE-squared anisotropic interaction between a ellipsoidal particle and a Lennard-Jones sphere is defined as the lim_{a_{2}−>0} U under the constraints that a_{2} = b_{2} = c_{2} and \frac{4}{3}πa_{2}^{3}ρ = 1:

U_{elj} = U_{A_{elj}} + U_{R_{elj}},

U_{α_{elj}} = (\frac{3σ^{3}c_{α}^{3}}{4πh_{elj}^{3}} )\frac{A_{12_{elj}}}{m_{α}} (\frac{σ}{h_{elj}} )^{n_{α}}(1+o_{α}χ_{elj} \frac{σ}{h_{elj}} )×\frac{a_{1}b_{1}c_{1}}{(a_{1} + h_{elj}/p_{α})(b_{1} + h_{elj}/p_{α})(c_{1} + h_{elj}/p_{α})},

A_{12_{elj}} = 4π^{2}ϵ_{LJ}(ρσ^{3}),

with h_{elj} and χ_{elj} calculated as above by replacing B with B_{elj} and G with G_{elj}:

B_{elj} = A_{1}^{T}E_{1}A_{1} + I,

G_{elj} = A_{1}^{T}S_{1}^{2}A_{1}.

The interaction between two LJ spheres is calculated as:

U_{lj} = 4ϵ [(\frac{σ}{|r|})^{12} − (\frac{σ}{|r|})^{6}]

The analytic derivatives are used for all force and torque calculation.