md.bond

Overview

Bond

Base class bond force.

FENEWCA

FENE and WCA bond force.

Harmonic

Harmonic bond force.

Table

Tabulated bond force.

Tether

Tether bond force.

Details

Bond forces.

Bond force classes apply a force and virial on every particle in the simulation state commensurate with the potential energy:

\[U_\mathrm{bond} = \sum_{(j,k) \in \mathrm{bonds}} U_{jk}(r)\]

Each bond is defined by an ordered pair of particle tags in the hoomd.State member bond_group. HOOMD-blue does not construct bond groups, users must explicitly define bonds in the initial condition.

Definition of the bond between particles j and k.

In the bond group (j,k), \(r\) is the length of the bond between the particle positions \(r= |\mathrm{minimum\_image}(\vec{r}_k - \vec{r}_j)|\).

Bond force classes assign 1/2 of the potential energy to each of the particles in the bond group:

\[U_i = \frac{1}{2} \sum_{(j,k) \in \mathrm{bonds}} U_{jk}(r) [i=j \lor i=k]\]

and similarly for virials.

class hoomd.md.bond.Bond

Bases: Force

Base class bond force.

Bond is the base class for all bond forces.

Warning

This class should not be instantiated by users. The class can be used for isinstance or issubclass checks.

class hoomd.md.bond.FENEWCA

Bases: Bond

FENE and WCA bond force.

FENEWCA computes forces, virials, and energies on all bonds in the simulation state with:

\[V(r) = - \frac{1}{2} k r_0^2 \ln \left( 1 - \left( \frac{r - \Delta}{r_0} \right)^2 \right) + U_{\mathrm{WCA}}(r)\]

where

\[\begin{split}U_{\mathrm{WCA}}(r) = \begin{cases} 4 \varepsilon \left[ \left( \frac{\sigma}{r - \Delta} \right)^{12} - \left( \frac{\sigma}{r - \Delta} \right)^{6} \right] + \varepsilon & r-\Delta < 2^{\frac{1}{6}}\sigma\\ 0 & r-\Delta \ge 2^{\frac{1}{6}}\sigma \end{cases}\end{split}\]

, \(k\) is the attractive force strength, \(r_0\) is the size of the bond, \(\varepsilon\) is the repulsive interaction energy, and \(\sigma\) is the repulsive interaction width.

params

The parameter of the FENEWCA potential bonds. The dictionary has the following keys:

  • k (float, required) - attractive force strength \(k\) \([\mathrm{energy} \cdot \mathrm{length}^{-2}]\).

  • r0 (float, required) - size parameter \(r_0\) \([\mathrm{length}]\).

  • epsilon (float, required) - repulsive force strength \(\varepsilon\) \([\mathrm{energy}]\).

  • sigma (float, required) - repulsive force interaction width \(\sigma\) \([\mathrm{length}]\).

  • delta (float, required) - radial shift \(\Delta\) \([\mathrm{length}]\).

Type

TypeParameter[bond type, dict]

Examples:

fenewca = bond.FENEWCA()
fenewca.params['A-A'] = dict(k=3.0, r0=2.38, epsilon=1.0, sigma=1.0,
                             delta=0.0)
fenewca.params['A-B'] = dict(k=10.0, r0=1.0, epsilon=0.8, sigma=1.2,
                             delta=0.0)
class hoomd.md.bond.Harmonic

Bases: Bond

Harmonic bond force.

Harmonic computes forces, virials, and energies on all bonds in the simulation state with:

\[U(r) = \frac{1}{2} k \left( r - r_0 \right)^2\]
params

The parameter of the harmonic bonds for each particle type. The dictionary has the following keys:

  • k (float, required) - potential constant \([\mathrm{energy} \cdot \mathrm{length}^{-2}]\)

  • r0 (float, required) - rest length \([\mathrm{length}]\)

Type

TypeParameter[bond type, dict]

Examples:

harmonic = bond.Harmonic()
harmonic.params['A-A'] = dict(k=3.0, r0=2.38)
harmonic.params['A-B'] = dict(k=10.0, r0=1.0)
class hoomd.md.bond.Table(width)

Bases: Bond

Tabulated bond force.

Parameters

width (int) – Number of points in the table.

Table computes computes forces, virials, and energies on all bonds in the simulation given the user defined tables \(U\) and \(F\).

Note

For potentials that diverge near r=0, to set r_min to a non-zero value.

The force \(\vec{F}\) is:

\[\begin{split}\vec{F}(\vec{r}) = \begin{cases} 0 & r < r_{\mathrm{min}} \\ F_\mathrm{table}(r)\hat{r} & r_{\mathrm{min}} \le r < r_{\mathrm{max}} \\ 0 & r \ge r_{\mathrm{max}} \\ \end{cases}\end{split}\]

and the potential \(U(r)\) is:

\[\begin{split}U(r) = \begin{cases} 0 & r < r_{\mathrm{min}} \\ U_\mathrm{table}(r) & r_{\mathrm{min}} \le r < r_{\mathrm{max}} \\ 0 & r \ge r_{\mathrm{max}} \\ \end{cases}\end{split}\]

where \(\vec{r}\) is the vector pointing from one particle to the other in the bond.

Warning

Bonds that stretch to a length \(r \ge r_{\mathrm{max}}\) result in an error.

Provide \(F_\mathrm{table}(r)\) and \(U_\mathrm{table}(r)\) on evenly spaced grid points points between \(r_{\mathrm{min}}\) and \(r_{\mathrm{max}}\). Table linearly interpolates values when \(r\) lies between grid points and between the last grid point and \(r=r_{\mathrm{max}}\). The force must be specificed commensurate with the potential: \(F = -\frac{\partial U}{\partial r}\).

params

The potential parameters. The dictionary has the following keys:

  • r_min (float, required) - the minimum distance to apply the tabulated potential, corresponding to the first element of the energy and force arrays \([\mathrm{length}]\).

  • r_max (float, required) - the minimum distance to apply the tabulated potential, corresponding to the first element of the energy and force arrays \([\mathrm{length}]\).

  • U ((width,) numpy.ndarray of float, required) - the tabulated energy values \([\mathrm{energy}]\). Must have a size equal to width.

  • F ((width,) numpy.ndarray of float, required) - the tabulated force values \([\mathrm{force}]\). Must have a size equal to width.

Type

TypeParameter [bond type, dict]

width

Number of points in the table.

Type

int

class hoomd.md.bond.Tether

Bases: Bond

Tether bond force.

The tethered network is described in Refs. Gompper, G. & Kroll, D. M. Statistical Mechanics of Membranes and Surfaces 2nd edn (eds Nelson, D. R. et al.) 359-426 (World Scientific, 2004) and Noguchi, H. & Gompper, G., Phys. Rev. E 72 011901 (2005).

Tether computes forces, virials, and energies on all bonds in the simulation state with:

\[U(r) = U_{\mathrm{att}}(r) + U_{\mathrm{rep}}(r)\]

where \(r\) is the distance from one particle to the other in the bond.

\[\begin{split}U_{\mathrm{att}}(r) = \begin{cases} k_b \frac{\exp(1/(l_{c0}-r)}{l_{max}-r} & r > l_{c0} \\ 0 & r \leq l_{c0} \\ \end{cases}\end{split}\]
\[\begin{split}U_{\mathrm{rep}}(r) = \begin{cases} k_b \frac{\exp(1/(r-l_{c1})}{r-l_{min}} & r < l_{c1}\\ 0 & r \ge l_{c1} \end{cases}\end{split}\]
\[l_{min} < l_{c1} < l_{c0} < l_{max}\]
params

The parameter of the Tethering potential bonds. The dictionary has the following keys:

  • k_b (float, required) - bond stiffness \([\mathrm{energy}]\)

  • l_min (float, required) - minimum bond length \([\mathrm{length}]\)

  • l_c1 (float, required) - cutoff distance of repulsive part \([\mathrm{length}]\)

  • l_c0 (float, required) - cutoff distance of attractive part \([\mathrm{length}]\)

  • l_max (float, required) - maximum bond length \([\mathrm{length}]\)

Type

TypeParameter[bond type, dict]

Examples:

bond_potential = bond.Tether()
bond_potential.params['A-A'] = dict(k_b=10.0, l_min=0.9, l_c1=1.2,
                                       l_c0=1.8, l_max=2.1)