9. Electrostatics

The Coulomb (or electrostatic) interaction is defined as follows. For a pair of particles at distance $r$ with charges $q_1$ and $q_2$, the interaction is given by

$$U_C(r)=C\cdot \frac{q_1{q}_2}{r}$$

where

(1)$$C=\frac{1}{4\pi {\epsilon }_0{\epsilon }_r}$$

is a prefactor which can be set by the user. The commonly used Bjerrum length $l_B=e^2/(4\pi {\epsilon }_0{\epsilon }_r{k}_{B}T)$ is the length at which the Coulomb energy between two unit charges is equal to the thermal energy $k_{B}T$. Based on this length, the prefactor is given by $C=l_B{k}_{B}T/e^2$.

Computing electrostatic interactions is computationally very expensive. ESPResSo features some state-of-the-art algorithms to deal with these interactions as efficiently as possible, but almost all of them require some knowledge to use them properly. Uneducated use can result in completely unphysical simulations.

Coulomb interactions have to be attached to the system object to become active. Only one electrostatics method can be active at any time.

Note that using the electrostatic interaction also requires assigning charges to the particles via the particle property q.

All solvers need a prefactor and a set of other required parameters. This example shows the general usage of the electrostatic method P3M (often stylized as P³M). An instance of the solver is created and attached to the system, at which point it will be automatically activated. This activation will internally call a tuning function to achieve the requested accuracy:

import espressomd
import espressomd.electrostatics

system = espressomd.System(box_l=[10, 10, 10])
system.time_step = 0.01
system.part.add(pos=[[0, 0, 0], [1, 1, 1]], q=[-1, 1])
solver = espressomd.electrostatics.P3M(prefactor=2., accuracy=1e-3)
system.electrostatics.solver = solver

where the prefactor is defined as $C$ in Eqn. (1).

The solver can be detached with either:

system.electrostatics.solver = None

or:

system.electrostatics.clear()

9.1. Coulomb P3M

espressomd.electrostatics.P3M

For this feature to work, you need to have the fftw3 library installed on your system. In ESPResSo, you can check if it is compiled in by checking for the feature FFTW with espressomd.features. P3M requires full periodicity (True, True, True). When using a non-metallic dielectric constant (epsilon != 0.0), the box must be cubic. Make sure that you know the relevance of the P3M parameters before using P3M! If you are not sure, read the following references: [Cerdà et al., 2008, Deserno, 2000, Deserno and Holm, 1998, Deserno and Holm, 1998, Deserno et al., 2000, Ewald, 1921, Hockney and Eastwood, 1988, Kolafa and Perram, 1992].

9.1.1. Tuning Coulomb P3M

It is not easy to calculate the various parameters of the P3M method such that the method provides the desired accuracy at maximum speed. To simplify this, it provides a function to automatically tune the algorithm. Note that for this function to work properly, your system should already contain an initial configuration of charges and the correct initial box size. The tuning method is called when the handle of the Coulomb P3M is added to the actor list. Some parameters can be fixed (r_cut, cao, mesh) to speed up the tuning if the parameters are already known.

Please note that the provided tuning algorithms works very well on homogeneous charge distributions, but might not achieve the requested precision for highly inhomogeneous or symmetric systems. For example, because of the nature of the P3M algorithm, systems are problematic where most charges are placed in one plane, one small region, or on a regular grid.

The function employs the analytical expression of the error estimate for the P3M method [Hockney and Eastwood, 1988] and its real space error [Kolafa and Perram, 1992] to obtain sets of parameters that yield the desired accuracy, then it measures how long it takes to compute the Coulomb interaction using these parameter sets and chooses the set with the shortest run time.

During tuning, the algorithm reports the tested parameter sets, the corresponding k-space and real-space errors and the timings needed for force calculations. In the output, the timings are given in units of milliseconds, length scales are in units of inverse box lengths.

9.1.2. Coulomb P3M on GPU

espressomd.electrostatics.P3MGPU

The GPU implementation of P3M calculates the far field contribution to the forces on the GPU. The near-field contribution to the forces, as well as the near- and far-field contributions to the energies are calculated on the CPU. It uses the same parameters and interface functionality as the CPU version of the solver. It should be noted that this does not always provide significant increase in performance. Furthermore it computes the far field interactions with only single precision which limits the maximum precision. The algorithm does not work in combination with the electrostatic extension Dielectric interfaces with the ICC* algorithm.

The algorithm doesn’t have kernels to compute energies and pressures and therefore uses the respective CPU kernels with the parameters tuned for the GPU force kernel.

9.2. Debye-Hückel potential

espressomd.electrostatics.DH

The Debye-Hückel electrostatic potential is defined by

$$U^{C-DH}=C\cdot \frac{q_1{q}_2\exp (-\kappa r)}{r}\mathrm{for}r<r_{\mathrm{cut}}$$

where $C$ is defined as in Eqn. (1) and $\kappa$ is the inverse Debye screening length. The Debye-Hückel potential is an approximate method for calculating electrostatic interactions, but technically it is treated as other short-ranged non-bonding potentials. For $r>r_{\text{cut}}$ it is set to zero which introduces a step in energy. Therefore, it introduces fluctuations in energy.

For $\kappa =0$, this corresponds to the plain Coulomb potential.

9.3. Reaction Field method

espressomd.electrostatics.ReactionField

The Reaction Field electrostatic potential is defined by

$$U^{C-RF}=C\cdot q_1{q}_2[\frac{1}{r}-\frac{B{r}^2}{2r_{\mathrm{cut}}^3}-\frac{1-B/2}{r_{\mathrm{cut}}}]\mathrm{for}r<r_{\mathrm{cut}}$$

where $C$ is defined as in Eqn. (1) and $B$ is defined as:

$$B=\frac{2({\epsilon }_1-{\epsilon }_2)(1+\kappa r_{\mathrm{cut}})-{\epsilon }_2(\kappa r_{\mathrm{cut}}{)}^2}{({\epsilon }_1+2{\epsilon }_2)(1+\kappa r_{\mathrm{cut}})+{\epsilon }_2(\kappa r_{\mathrm{cut}}{)}^2}$$

with $\kappa$ the inverse Debye screening length, ${\epsilon }_1$ the dielectric constant inside the cavity and ${\epsilon }_2$ the dielectric constant outside the cavity [Tironi et al., 1995].

The term in $1-B/2$ is a correction to make the potential continuous at $r=r_{\mathrm{cut}}$.

9.4. Dielectric interfaces with the ICC$\star$ algorithm

espressomd.electrostatic_extensions.ICC

The ICC$\star$ algorithm allows to take into account arbitrarily shaped dielectric interfaces and dynamic charge induction. For instance, it can be used to simulate a curved metallic boundary. This is done by iterating the charge on a set of spatially fixed ICC particles until they correctly represent the influence of the dielectric discontinuity. All ICC particles need a certain area, normal vector and dielectric constant to fully specify the surface. ICC relies on a Coulomb solver that is already initialized. So far, it is implemented and well tested with the Coulomb solver P3M. ICC is an ESPResSo actor and can be activated via:

import espressomd.electrostatics
import espressomd.electrostatic_extensions
p3m = espressomd.electrostatics.P3M(...)
icc = espressomd.electrostatic_extensions.ICC(...)
system.electrostatics.solver = p3m
system.electrostatics.extension = icc

The ICC particles are setup as normal ESPResSo particles. Note that they should be fixed in space and need an initial non-zero charge. The following example sets up parallel metallic plates and activates ICC:

# Set the ICC line density and calculate the number of
# ICC particles according to the box size
box_l = 9.
system.box_l = [box_l, box_l, 12.]
nicc = 3  # linear density
nicc_per_electrode = nicc**2  # surface density
nicc_tot = 2 * nicc_per_electrode
iccArea = box_l**2 / nicc_per_electrode
l = box_l / nicc

# Lists to collect required parameters
iccNormals = []
iccAreas = []
iccSigmas = []
iccEpsilons = []

# Add the fixed ICC particles:

# Left electrode (normal [0, 0, 1])
for xi in range(nicc):
    for yi in range(nicc):
        system.part.add(pos=[l * xi, l * yi, 0.], q=-0.0001,
                        type=icc_type, fix=[True, True, True])
iccNormals.extend([0, 0, 1] * nicc_per_electrode)

# Right electrode (normal [0, 0, -1])
for xi in range(nicc):
    for yi in range(nicc):
        system.part.add(pos=[l * xi, l * yi, box_l], q=0.0001,
                        type=icc_type, fix=[True, True, True])
iccNormals.extend([0, 0, -1] * nicc_per_electrode)

# Common area, sigma and metallic epsilon
iccAreas.extend([iccArea] * nicc_tot)
iccSigmas.extend([0] * nicc_tot)
iccEpsilons.extend([100000] * nicc_tot)

icc = espressomd.electrostatic_extensions.ICC(
    first_id=0,
    n_icc=nicc_tot,
    convergence=1e-4,
    relaxation=0.75,
    ext_field=[0, 0, 0],
    max_iterations=100,
    eps_out=1.0,
    normals=iccNormals,
    areas=iccAreas,
    sigmas=iccSigmas,
    epsilons=iccEpsilons)

system.electrostatics.extension = icc

With each iteration, ICC has to solve electrostatics which can severely slow down the integration. The performance can be improved by using multiple cores, a minimal set of ICC particles and convergence and relaxation parameters that result in a minimal number of iterations. Also please make sure to read the corresponding articles, mainly [Arnold et al., 2013, Kesselheim et al., 2011, Tyagi et al., 2010] before using it.

9.5. Electrostatic Layer Correction (ELC)

espressomd.electrostatics.ELC

ELC is an extension of the P3M electrostatics solver for explicit 2D periodic systems. It can account for different dielectric jumps on both sides of the non-periodic direction. In more detail, it is a special procedure that converts a 3D electrostatic method to a 2D method in computational order N. The periodicity has to be set to (True, True, True). ELC cancels the electrostatic contribution of the periodic replica in $z$-direction. Make sure that you read the papers on ELC ([Arnold et al., 2002, de Joannis et al., 2002, Tyagi et al., 2008]) before using it. See ELC theory for more details.

Usage notes:

• The non-periodic direction is always the $z$-direction.

• The method relies on a slab of the simulation box perpendicular to the $z$-direction not to contain particles. The size in $z$-direction of this slab is controlled by the gap_size parameter. The user has to ensure that no particles enter this region by means of constraints or by fixing the particles’ z-coordinate. When particles enter the slab of the specified size, an error will be thrown.

ELC is an ESPResSo actor and is used with:

import espressomd.electrostatics
p3m = espressomd.electrostatics.P3M(prefactor=1, accuracy=1e-4)
elc = espressomd.electrostatics.ELC(actor=p3m, gap_size=box_l * 0.2, maxPWerror=1e-3)
system.electrostatics.solver = elc

Although it is technically feasible to detach elc from the system and then to attach the p3m object, it is not recommended because the P3M parameters are mutated by ELC, e.g. the epsilon is made metallic. It is safer to instantiate a new P3M object instead of recycling one that has been adapted by ELC.

ELC can also be used to simulate 2D periodic systems with image charges, specified by dielectric contrasts on the non-periodic boundaries ([Tyagi et al., 2008]). This is achieved by setting the dielectric jump from the simulation region (middle) to bottom (at $z=0$) and from middle to top (at $z=L_z-h$), where $L_z$ denotes the box length in $z$-direction and $h$ the gap size. The corresponding expressions are ${\mathrm{\Delta }}_t=\frac{{\epsilon }_m-{\epsilon }_t}{{\epsilon }_m+{\epsilon }_t}$ and ${\mathrm{\Delta }}_b=\frac{{\epsilon }_m-{\epsilon }_b}{{\epsilon }_m+{\epsilon }_b}$:

elc = espressomd.electrostatics.ELC(actor=p3m, gap_size=box_l * 0.2, maxPWerror=1e-3,
                                    delta_mid_top=0.9, delta_mid_bot=0.1)

The fully metallic case ${\mathrm{\Delta }}_t={\mathrm{\Delta }}_b=-1$ would lead to divergence of the forces/energies in ELC and is therefore only possible with the const_pot option.

Toggle const_pot on to maintain a constant electric potential difference pot_diff between the xy-planes at $z=0$ and $z=L_z-h$:

elc = espressomd.electrostatics.ELC(actor=p3m, gap_size=box_l * 0.2, maxPWerror=1e-3,
                                    const_pot=True, delta_mid_bot=100.0)

This is done by countering the total dipole moment of the system with the electric field $E_{\text{induced}}$ and superposing a homogeneous electric field $E_{\text{applied}}=\frac{U}{L}$ to retain $U$. This mimics the induction of surface charges $\pm \sigma =E_{\text{induced}}\cdot {\epsilon }_0$ for planar electrodes at $z=0$ and $z=L_z-h$ in a capacitor connected to a battery with voltage pot_diff.

9.6. MMM1D

espressomd.electrostatics.MMM1D

Please cite [Arnold and Holm, 2005] when using MMM1D. See MMM1D theory for the details.

MMM1D is used with:

import espressomd.electrostatics
mmm1d = espressomd.electrostatics.MMM1D(prefactor=C, far_switch_radius=fr,
                                        maxPWerror=err, tune=False, bessel_cutoff=bc)
mmm1d = espressomd.electrostatics.MMM1D(prefactor=C, maxPWerror=err)

where the prefactor $C$ is defined in Eqn. (1). MMM1D requires for systems with periodicity (0 0 1) and the N-squared cell system (see section Cell systems). The first form sets parameters manually. The switch radius determines at which xy-distance the force calculation switches from the near to the far formula. The Bessel cutoff does not need to be specified as it is automatically determined from the particle distances and maximal pairwise error. The second tuning form just takes the maximal pairwise error and tries out a lot of switching radii to find out the fastest one. If this takes too long, you can change the value of the timings argument of the MMM1D class, which controls the number of test force calculations.

9.7. ScaFaCoS electrostatics

espressomd.electrostatics.Scafacos

ESPResSo can use the methods from the ScaFaCoS Scalable fast Coulomb solvers library. The specific methods available depend on the compile-time options of the library, and can be queried with espressomd.electrostatics.Scafacos.get_available_methods().

To use ScaFaCoS, create an instance of Scafacos and attach it to the system. Three parameters have to be specified: prefactor (as defined in (1)), method_name, method_params. The method-specific parameters are described in the ScaFaCoS manual. In addition, methods supporting tuning have a parameter tolerance_field which sets the desired root mean square accuracy for the electric field.

To use a specific electrostatics solver from ScaFaCoS for your system, e.g. ewald, set its cutoff to $1.5$ and tune the other parameters for an accuracy of ${10}^{-3}$:

import espressomd.electrostatics
scafacos = espressomd.electrostatics.Scafacos(
   prefactor=1, method_name="ewald",
   method_params={"ewald_r_cut": 1.5, "tolerance_field": 1e-3})
system.electrostatics.solver = scafacos

For details of the various methods and their parameters please refer to the ScaFaCoS manual. To use this feature, ScaFaCoS has to be built as a shared library.