Additional documentation for the RE-squared ellipsoidal ... - Lammps

Additional documentation for the RE-squared ellipsoidal
potential
as implemented in LAMMPS
Mike Brown, Sandia National Labs, October 2007
Let the shape matrices Si = diag(ai, bi, ci) be given by the ellipsoid
radii. Let the relative energy matrices Ei = diag(ɛia, ɛib, ɛic) 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 A1 and A2 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 A12 be the Hamaker constant for the interaction
given in LJ units by A12 = 4π 2 ɛLJ(ρσ 3 ) 2 .
The RE-squared anisotropic interaction between pairs of ellipsoidal particles
is given by
Uα = A12
mα
U = UA + UR,
( σ
h )nα (1 + oαηχ σ
aibici
) ×
h (ai + h/pα)(bi + h/pα)(ci + h/pα) ,
i
mA = −36, nA = 0, oA = 3, pA = 2,
mR = 2025, nR = 6, oR = 45/56, pR = 60 1/3 ,
χ = 2ˆr T B −1 ˆr,
ˆr = r/|r|,
B = A T 1 E1A1 + A T 2 E2A2
1

η = det[S1]/σ 2 1 + det[S2]/σ 2 2
[det[H]/(σ1 + σ2)] 1/2 ,
σi = (ˆr T A T i S −2
i Aiˆr) −1/2 ,
H = 1
A
σ1
T 1 S 2 1A1 + 1
A
σ2
T 2 S 2 2A2
Here, we use the distance of closest approach approximation given by the
Perram reference, namely
and
h = |r| − σ12,
σ12 = [ 1
2 ˆrT G −1 ˆr] −1/2 ,
G = A T 1 S 2 1A1 + A T 2 S 2 2A2
The RE-squared anisotropic interaction between a ellipsoidal particle and
a Lennard-Jones sphere is defined as the lima2−>0 U under the constraints
that a2 = b2 = c2 and 4
3 πa3 2ρ = 1:
Uαelj = (3σ3 c 3 α
4πh 3 elj
) A12elj
(
mα
σ
helj
Uelj = UAelj + URelj ,
) nα (1+oαχelj
σ
helj
A12elj = 4π2 ɛLJ(ρσ 3 ),
a1b1c1
)×
(a1 + helj/pα)(b1 + helj/pα)(c1 + helj/pα) ,
with helj and χelj calculated as above by replacing B with Belj and G with
Gelj:
Belj = A T 1 E1A1 + I,
2

Gelj = A T 1 S 2 1A1.
The interaction between two LJ spheres is calculated as:
⎡
σ
Ulj = 4ɛ ⎣
|r|
12
−
⎤
6
σ
⎦
|r|
The analytic derivatives are used for all force and torque calculation.
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