Bonded interactions are configured by the espressomd.interactions.BondedInteractions class, which is a member of espressomd.system.System. Generally, one may use the following syntax to activate and assign a bonded interaction:
system.bonded_inter.add(bond)
system.part[pid1].add_bond((bond, pid2...))In general, one instantiates an interaction object bond and subsequently passes it to espressomd.interactions.BondedInteractions.add. This will enable the bonded interaction and allows the user to assign bonds between particle ids pidX. Bonded interactions are identified by either their bondid or their appropriate object.
Defining a bond between two particles always involves three steps: defining the interaction, adding it to the system and applying it to the particles. To illustrate this, assume that three particles with ids 42, 43 and 12 already exist. One could for example create FENE bonds (more information about the FENE bond is provided in subsection FENE bond) between them using:
fene = FeneBond(k=1, d_r_max=1)
system.bonded_inter.add(fene)
system.part[42].add_bond((fene, 43), (fene, 12))
system.part[12].add_bond((fene, 43))This will set up a FENE bond between particles 42 and 43, 42 and 12, and 12 and 43. Note that the fene object specifies the type of bond and its parameters, the specific bonds are stored within the particles. you can find more information regarding particle properties in Setting up particles.
A FENE (finite extension nonlinear elastic) bond can be instantiated via espressomd.interactions.FeneBond:
from espressomd.interactions import FeneBond
fene = FeneBond(k=<float>, d_r_max=<float>, r_0=<float>)This command creates a bond type identifier with a FENE interaction. The FENE potential
models a rubber-band-like, symmetric interaction between two particles with magnitude K, maximal stretching length Δr0 and equilibrium bond length r0. The bond potential diverges at a particle distance r = r0 − Δrmax and r = r0 + Δrmax.
A harmonic bond can be instantiated via espressomd.interactions.HarmonicBond:
from espressomd.interactions import HarmonicBond
hb = HarmonicBond(k=<float>, r_0=<float>, r_cut=<float>)This creates a bond type identifier with a classical harmonic potential. It is a symmetric interaction between two particles. With the equilibrium length r0 and the magnitude k. It is given by
The third, optional parameter defines a cutoff radius. Whenever a harmonic bond gets longer than rcut, the bond will be reported as broken, and a background error will be raised.
Note
Requires ROTATION feature.
A harmonic Dumbbell bond can be instantiated via espressomd.interactions.HarmonicDumbbellBond:
from espressomd.interactions import HarmonicDumbbellBond
hdb = HarmonicDumbbellBond(k1=<float>, k2=<float>, r_0=<float>, r_cut=<float>)This bond is similar to the normal harmonic bond in such a way that it sets up a harmonic potential, i.e. a spring, between the two particles. Additionally the orientation of the first particle in the bond will be aligned along the distance vector between both particles. This alignment can be controlled by the second harmonic constant k2. Keep in mind that orientation will oscillate around the distance vector and some kind of friction needs to be present for it to relax.
The roles of the parameters k1, r0, rcut are exactly the same as for the harmonic bond.
Note
Requires ELECTROSTATICS feature.
A pairwise Coulomb interaction can be instantiated via espressomd.interactions.BondedCoulomb:
bonded_coulomb = espressomd.interactions.BondedCoulomb(prefactor=1.0)
system.bonded_inter.add(bonded_coulomb)
system.part[0].add_bond((bonded_coulomb, 1))This creates a bond with a Coulomb pair potential between particles 0 and 1. It is given by
where q1 and q2 are the charges of the bound particles and α is the Coulomb prefactor. This interaction has no cutoff and acts independently of other Coulomb interactions.
Note
Requires the P3M feature.
This bond can be instantiated via espressomd.interactions.BondedCoulombP3MSRBond:
from espressomd.interactions import BondedCoulombP3MSRBond
subtr_p3m_sr = BondedCoulombP3MSRBond(q1q2=<float>)The parameter q1q2 sets the charge factor of the short-range P3M interaction. It can differ from the actual particle charges. This specialized bond can be used to cancel or add only the short-range electrostatic part of the P3M solver. A use case is described in Particle polarizability with
thermalized cold Drude oscillators.
espressomd.interactions.SubtLJ can be used to exclude certain particle pairs from the non-bonded Lennard-Jones interaction:
subtLJ = espressomd.interactions.SubtLJ()
system.bonded_inter.add(subtLJ)
system.part[0].add_bond((subt, 1))This bond subtracts the type-pair specific Lennard-Jones interaction between the involved particles. This interaction is useful when using other bond potentials which already include the short-ranged repulsion. This often the case for force fields or in general tabulated potentials.
Note
Requires BOND_CONSTRAINT feature.
A rigid bond can be instantiated via espressomd.interactions.RigidBond:
from espressomd.interactions import RigidBond
rig = RigidBond(r=<float>, ptol=<float>, vtol=<float> )To simulate rigid bonds, uses the Rattle Shake algorithm which satisfies internal constraints for molecular models with internal constraints, using Lagrange multipliers.andersen83a The constrained bond distance is named r, the positional tolerance is named ptol and the velocity tolerance is named vtol.
Note
Requires TABULATED feature.
A tabulated bond can be instantiated via espressomd.interactions.Tabulated:
from espressomd.interactions import Tabulated
tab = Tabulated(type=<str>, min=<min>, max=<max>,
energy=<energy>, force=<force>)This creates a bond type identifier with a two-body bond length, three-body angle or four-body dihedral tabulated potential. For details of the interpolation, see Tabulated interaction.
The bonded interaction can be based on a distance, a bond angle or a dihedral angle. This is determined by the type argument, which can be one of the strings distance, angle or dihedral.
The potential is calculated as follows:
type="distance": is a two body interaction depending on the distance of two particles. The force acts in the direction of the connecting vector between the particles. The bond breaks above the tabulated range, but for distances smaller than the tabulated range, a linear extrapolation based on the first two tabulated force values is used.type="angle": is a three-body angle interaction similar to the bond angle potential. It is assumed that the potential is tabulated for all angles between 0 and π, where 0 corresponds to a stretched polymer, and just as for the tabulated pair potential, the forces are scaled with the inverse length of the connecting vectors. The force on the extremities acts perpendicular to the connecting vector between the corresponding particle and the center particle, in the plane defined by the three particles. The force on the center particle p2 balances the other two forces.type="dihedral": tabulates a torsional dihedral angle potential. It is assumed that the potential is tabulated for all angles between 0 and 2π. This potential is not tested yet! Use on own risk, and please report your findings and eventually necessary fixes.
A virtual bond can be instantiated via espressomd.interactions.Virtual:
from espressomd.interactions import Virtual
tab = Virtual()This creates a virtual bond type identifier for a pair bond without associated potential or force. It can be used to specify topologies and for some analysis that rely on bonds, or for bonds that should be displayed in the visualization.
Please cite Cimrak2014 when using the interactions in this section in order to simulate extended objects embedded in a LB fluid. For more details we encourage the dedicated OIF documentation available at http://cell-in-fluid.fri.uniza.sk/en/content/oif-espresso.
The following interactions are implemented in order to mimic the mechanics of elastic or rigid objects immersed in the LB fluid flow. Their mathematical formulations were inspired by dupin07. Details on how the bonds can be used for modeling objects are described in section Object-in-fluid.
Note
Requires OIF_GLOBAL_FORCES feature.
OIF local forces are available through the espressomd.interactions.OifLocalForces class.
This type of interaction is available for closed 3D immersed objects flowing in the LB flow.
This interaction comprises three different concepts. The local elasticity of biological membranes can be captured by three different elastic moduli. Stretching of the membrane, bending of the membrane and local preservation of the surface area. Parameters LAB0, ks, kslin define the stretching, parameters ϕ, kb define the bending, and A1, A2, kal define the preservation of local area. They can be used all together, or, by setting any of ks, kslin, kb, kal to zero, the corresponding modulus can be turned off.
For each edge of the mesh, LAB is the current distance between point A and point B. LAB0 is the distance between these points in the relaxed state, that is if the current edge has the length exactly , then no forces are added. ΔLAB is the deviation from the relaxed state, that is ΔLAB = LAB − LAB0. The stretching force between A and B is calculated using
Fs(A, B) = (ksκ(λAB) + ks, lin)ΔLABnAB.
Here, nAB is the unit vector pointing from A to B, ks is the constant for nonlinear stretching, ks, lin is the constant for linear stretching, λAB = LAB/LAB0, and κ is a nonlinear function that resembles neo-Hookean behavior
Typically, one wants either nonlinear or linear behavior and therefore one of ks, ks, lin is zero. Nonetheless the interaction will work if both constants are non-zero.
The tendency of an elastic object to maintain the resting shape is achieved by prescribing the preferred angles between neighboring triangles of the mesh.
Denote the angle between two triangles in the resting shape by θ0. For closed immersed objects, one always has to set the inner angle. The deviation of this angle Δθ = θ − θ0 defines two bending forces for two triangles A1BC and A2BC
Fbi(AiBC) = kbΔθnAiBC
Here, nAiBC is the unit normal vector to the triangle AiBC. The force Fbi(AiBC) is assigned to the vertex not belonging to the common edge. The opposite force divided by two is assigned to the two vertices lying on the common edge. This procedure is done twice, for i = 1 and for i = 2.
This interaction conserves the area of the triangles in the triangulation. The area constraint assigns the following shrinking/expanding force to vertex A
where ΔS△ is the difference between current S△ and area S0 of the triangle in relaxed state, T is the centroid of the triangle, and ta, tb, tc are the lengths of segments AT, BT, CT, respectively. Similarly the analogical forces are assigned to B and C.
OIF local force is asymmetric. After creating the interaction
local_inter = OifLocalForces(r0=1.0, ks=0.5, kslin=0.0, phi0=1.7, kb=0.6, A01=0.2,
A02=0.3, kal=1.1, kvisc=0.7)it is important how the bond is created. Particles need to be mentioned in the correct order. Command
p1.add_bond((local_inter, p0.part_id, p2.part_id, p3.part_id))creates a bond related to the triangles 012 and 123. The particle 0 corresponds to point A1, particle 1 to C, particle 2 to B and particle 3 to A2. There are two rules that need to be fulfilled:
- there has to be an edge between particles 1 and 2
- orientation of the triangle 012, that is the normal vector defined as a vector product 01 × 02, must point to the inside of the immersed object.
Then the stretching force is applied to particles 1 and 2, with the relaxed length being 1.0. The bending force is applied to preserve the angle between triangles 012 and 123 with relaxed angle 1.7 and finally, local area force is applied to both triangles 012 and 123 with relaxed area of triangle 012 being 0.2 and relaxed area of triangle 123 being 0.3.
Notice that also concave objects can be defined. If θ0 is larger than π, then the inner angle is concave.
Note
Requires OIF_GLOBAL_FORCES feature.
OIF global forces are available through the espressomd.interactions.OifGlobalForces class.
This type of interaction is available solely for closed 3D immersed objects.
It comprises two concepts: preservation of global surface and of volume of the object. The parameters S0, kag define preservation of the surface while parameters V0, kv define volume preservation. They can be used together, or, by setting either kag or kv to zero, the corresponding modulus can be turned off.
The global area conservation force is defined as
where Sc denotes the current surface of the immersed object, S0c the surface in the relaxed state, SABC is the surface of the triangle, T is the centroid of the triangle, and ta, tb, tc are the lengths of segments AT, BT, CT, respectively.
The deviation of the objects volume V is computed from the volume in the resting shape ΔV = V − V0. For each triangle the following force is computed
where SABC is the area of triangle ABC, nABC is the normal unit vector of the plane spanned by ABC, and kv is the volume constraint coefficient. The volume of one immersed object is computed from
V = ∑ABCSABC nABC ⋅ hABC,
where the sum is computed over all triangles of the mesh and hABC is the normal vector from the centroid of triangle ABC to any plane which does not cross the cell. The force Fv(ABC) is equally distributed to all three vertices A, B, C.
This interaction is symmetric. After the definition of the interaction by
global_force_interaction = OifGlobalForces(A0_g=65.3, ka_g=3.0, V0=57.0, kv=2.0)the order of vertices is crucial. By the following command the bonds are defined
p0.add_bond((global_force_interaction, p1.part_id, p2.part_id))Triangle 012 must have correct orientation, that is the normal vector defined by a vector product 01 × 02. The orientation must point inside the immersed object.
OIF out direction is available through the espressomd.interactions.OifOutDirection class.
This type of interaction is primarily for closed 3D immersed objects to compute the input for membrane collision. After creating the interaction
out_direction_interaction = OifOutDirection()it is important how the bond is created. Particles need to be mentioned in the correct order. Command
p0.add_bond((out_direction_interaction, p1.part_id, p2.part_id, p3.part_id))calculates the outward normal vector of triangle defined by particles 1, 2, 3 (these should be selected in such a way that particle 0 lies approximately at its centroid - for OIF objects, this is automatically handled by oif_create_template command, see Section [ssec:oif-create-template]). In order for the direction to be outward with respect to the underlying object, the triangle 123 needs to be properly oriented (as explained in the section on volume in oif_global_forces interaction).
Note
Feature BOND_ANGLE required.
Bond-angle interactions involve three particles forming the angle ϕ, as shown in the schematic below.
This allows for a bond type having an angle-dependent potential. This potential is defined between three particles. The particle for which the bond is created, is the central particle, and the angle ϕ between the vectors from this particle to the two others determines the interaction.
Similar to other bonded interactions, these are defined for every particle triplet and must be added to a particle (see espressomd.particle_data.ParticleHandle.bonds). For example, for the schematic with particles id=0, 1 and 2 the bond was defined using :
>>> system.part[1].add_bond((bond_angle, 0, 2))The parameter bond_angle is a bond type identifier of three possible bond-angle classes, described below.
espressomd.interactions.AngleHarmonic
A classical harmonic potential of the form:
$$V(\phi) = \frac{K}{2} \left(\phi - \phi_0\right)^2.$$ K is the bending constant, and the optional parameter ϕ0 is the equilibrium bond angle in radians ranging from 0 to π.
If this parameter is not given, it defaults to ϕ0 = π, which corresponds to a stretched conformation.
Unlike the two other variants, this potential has a kink at ϕ = ϕ0 + π and accordingly a discontinuity in the force, and should therefore be used with caution.
Example:
>>> angle_harmonic = AngleHarmonic(bend=1.0, phi0=np.pi) >>> system.bonded_inter.add(angle_harmonic) >>> system.part[1].add_bond((angle_harmonic, 0, 2))
espressomd.interactions.AngleCosine
Cosine bond angle potential of the form:
V(ϕ) = K[1−cos(ϕ−ϕ0)]K is the bending constant, and the optional parameter ϕ0 is the equilibrium bond angle in radians ranging from 0 to π.
If this parameter is not given, it defaults to ϕ0 = π, which corresponds to a stretched conformation.
Around ϕ0, this potential is close to a harmonic one (both are 1/2(ϕ − ϕ0)2 in leading order), but it is periodic and smooth for all angles ϕ.
Example:
>>> angle_cosine = AngleCosine(bend=1.0, phi0=np.pi) >>> system.bonded_inter.add(angle_cosine) >>> system.part[1].add_bond((angle_cosine, 0, 2))
espressomd.interactions.AngleCossquare
Cosine square bond angle potential of the form:
$$V(\phi) = \frac{K}{2} \left[\cos(\phi) - \cos(\phi_0)\right]^2$$ This form is used for example in the GROMOS96 force field. The potential is 1/8(ϕ − ϕ0)4 around ϕ0, and therefore much flatter than the two potentials before.
Example:
>>> angle_cossquare = AngleCossquare(bend=1.0, phi0=np.pi) >>> system.bonded_inter.add(angle_cossquare) >>> system.part[1].add_bond((angle_cossquare, 0, 2))
Dihedral interactions are available through the espressomd.interactions.Dihedral class.
This creates a bond type with identifier with a dihedral potential, a four-body-potential. In the following, let the particle for which the bond is created be particle p2, and the other bond partners p1, p3, p4, in this order. Then, the dihedral potential is given by
V(ϕ) = K[1−cos(nϕ−p)],
where n is the multiplicity of the potential (number of minima) and can take any integer value (typically from 1 to 6), p is a phase parameter and K is the bending constant of the potential. ϕ is the dihedral angle between the particles defined by the particle quadruple p1, p2, p3 and p4, the angle between the planes defined by the particle triples p1, p2 and p3 and p2, p3 and p4:
Together with appropriate Lennard-Jones interactions, this potential can mimic a large number of atomic torsion potentials.
This bond can be used to apply Langevin thermalization on the centre of mass and the distance of a particle pair. Each thermostat can have its own temperature and friction coefficient.
The bond is configured with:
from espressomd.interactions import ThermalizedBond
thermalized_bond = ThermalizedBond(temp_com=<float>, gamma_com=<float>,
temp_distance=<float>, gamma_distance=<float>,
r_cut=<float>)
system.bonded_inter.add(thermalized_bond)The parameters are:
temp_com: Temperature of the Langevin thermostat for the COM of the particle pair.gamma_com: Friction coefficient of the Langevin thermostat for the COM of the particle pair.temp_distance: Temperature of the Langevin thermostat for the distance vector of the particle pair.gamma_distance: Friction coefficient of the Langevin thermostat for the distance vector of the particle pair.r_cut: Specifies maximum distance beyond which the bond is considered broken.
The bond is closely related to simulating Particle polarizability with
thermalized cold Drude oscillators.