Interaction Potentials

CustomPotential

Implements a potential that evaluates a user defined function.

Expects values f, and p, which respecively are a callable and a list of parameters. The function should be called f(r::Number, p) and it should produce either a Number, in the case of a single-component system, or an SMatrix, in the case of a multicomponent system.

Example:

f = (r, p) -> 4*p[1]*((p[2]/r)^12 -  (p[2]/r)^6)
potential = CustomPotential(f, (1.0, 1.0))

GaussianCore

Gaussian core (ultrasoft) pair interaction

u(r) = ϵ * exp(-(r/σ)^2)

Constructors:

• GaussianCore(ϵ::Number, σ::Number) — single-component
• GaussianCore(ϵ::AbstractVector, σ::AbstractVector) — mixture with σij = (σi + σj)/2 (additive), ϵij = √(ϵi ϵj) (geometric mean)
• GaussianCore(ϵij::AbstractMatrix, σij::AbstractMatrix) — explicit pair tables

Example:

potential = GaussianCore(1.0, 1.5)

Example (mixture with mixing rules):

eps = [1.0, 0.8]
sig = [1.0, 1.2]
potential = GaussianCore(eps, sig)

HardSpheres

Implements the hard-sphere pair interaction for single component systems $ u(r) = \infty$ for $r < 1$ and $u(r) = 0$ for $r > 1$, or $u_{ij}(r) = \infty$ for $r < D_{ij}$ and $u_{ij}(r) = 0$ for $r > D_{ij}$ for mixtures.

For mixtures expects either a vector $D_i$ of diameters for each of the species in which case an additive mixing rule is used $\left(D_{ij} = (D_{i}+D_{j})/2\right)$ or a matrix $D_ij$ of pair diameters.

Example:

potential = HardSpheres(1.0)

Example:

potential = HardSpheres([0.8, 0.9, 1.0])

Dij = rand(3,3)
potential = HardSpheres(Dij)

LennardJones

Implements the Lennard-Jones pair interaction $u(r) = 4\epsilon [(\sigma/r)^{12} - (\sigma/r)^6]$.

Expects values ϵ and σ, which respecively are the strength of the potential and particle size.

Example:

potential = LennardJones(1.0, 2.0)

Morse

Morse potential:

u(r) = ϵ * (exp(-2α(r-σ)) - 2 exp(-α(r-σ)))

Constructors:

• Morse(ϵ::Number, σ::Number, α::Number) — single-component
• Morse(ϵ::AbstractVector, σ::AbstractVector, α::AbstractVector) — mixture with σij = (σi + σj)/2, ϵij = √(ϵi ϵj), αij = (αi + α_j)/2
• Morse(ϵij::AbstractMatrix, σij::AbstractMatrix, αij::AbstractMatrix) — explicit pair tables

Example:

potential = Morse(1.0, 1.0, 2.0)

Example (mixture with mixing rules):

eps = [1.0, 0.8]
sig = [1.0, 1.2]
alp = [2.0, 1.5]
potential = Morse(eps, sig, alp)

PowerLaw

Implements the power law pair interaction $u(r) = \epsilon (\sigma/r)^{n}$.

Expects values ϵ, σ, and n, which respecively are the strength of the potential and particle size.

Example:

potential = PowerLaw(1.0, 2.0, 8)

SquareWell

Square-well pair interaction:

u(r) = ∞                     if r < σ
     = -ϵ                    if σ ≤ r ≤ λσ
     = 0                     if r > λσ

Constructors:

• SquareWell(σ::Number, ϵ::Number, λ::Number) — single-component
• SquareWell(σ::AbstractVector, ϵ::AbstractVector, λ::Number) — mixture with σij = (σi + σj)/2, ϵij = √(ϵi ϵj)
• SquareWell(σij::AbstractMatrix, ϵij::AbstractMatrix, λ::Number) — explicit pair tables

Example:

potential = SquareWell(1.0, 1.0, 1.5)

Example (mixture with mixing rules):

sig = [0.9, 1.1, 1.0]
eps = [1.0, 0.8, 1.2]
potential = SquareWell(sig, eps, 1.5)

TabulatedPotential

Piecewise-linear potential defined on a sorted grid r_grid with values u_grid.

• Linear interpolation within tabulated range.
• Extrapolation behavior controlled by extrapolation:
  • :error (default) — throw if r is outside [rmin, rmax]
  • :flat — clamp to end values
  • :linear — extend linearly using end slope

Example:

r = range(0.8, 6.0; length=500) |> collect
u = @. 4.0*((1.0/r)^12 - (1.0/r)^6)  # LJ shape as a table
pot = TabulatedPotential(r, u, :flat)

WCADivision{P<:Potential, T, UC}

fields

• potential::Potential
• cutoff : ($r_{c}$)
• U_c : $U(r=r_c)$

Splits the potential at the cutoff point using the WCA splitting rule. This means

\[u(r) = u_{SR}(r) + U_{LR}(r),\]

where $U_{LR}(r) = u(r)$ for $r > r_{c}$, and $U(r_{c})$ for $r < r_{c}$, and $USR(r) = 0$ for $r > r_{c}$, and $U(r) - U(r_{c})$ for $r < r_{c}$.

Yukawa

Screened-Coulomb / Yukawa pair interaction

u(r) = A * exp(-κ r) / r

Constructors:

• Yukawa(A::Number, κ::Number) — single-component
• Yukawa(q::AbstractVector, κ::Number) — mixture from "charges": Aij = qi q_j
• Yukawa(Aij::AbstractMatrix, κ::Number) — mixture with explicit pair amplitudes

Example (single component):

potential = Yukawa(1.0, 2.0)  # A=1, κ=2

Example (mixture from charges):

q  = [1.0, -1.0, 2.0]
pot = Yukawa(q, 1.5)

Example (mixture with explicit A_ij):

Aij = [1.0 0.2; 0.2 0.5]
pot = Yukawa(Aij, 1.0)