Particle-based simulation of charge transport in discrete-charge nano-scale systems: the electrostatic problem
© Berti et al; licensee Springer. 2012
Received: 18 November 2011
Accepted: 16 February 2012
Published: 16 February 2012
The fast and accurate computation of the electric forces that drive the motion of charged particles at the nanometer scale represents a computational challenge. For this kind of system, where the discrete nature of the charges cannot be neglected, boundary element methods (BEM) represent a better approach than finite differences/finite elements methods. In this article, we compare two different BEM approaches to a canonical electrostatic problem in a three-dimensional space with inhomogeneous dielectrics, emphasizing their suitability for particle-based simulations: the iterative method proposed by Hoyles et al. and the Induced Charge Computation introduced by Boda et al.
The investigation of the properties of a large variety of physical systems requires accurate computation of the electrostatic interactions among discrete fixed or mobile charges. This problem has been faced in a large number of cases including the analysis of electronic and optoelectronic devices [1–6], the investigation of fluid properties and the simulation of ion transport through membrane pores [7–12].
In nano-scale physical systems, some of the properties of the interacting bodies are strongly localized and can be approximated as Dirac delta functions. For example, the properties of spatially homogeneous ionic solutions have been investigated using the so called primitive model for the ions, considering them as hard spheres with finite radii and discrete point charges placed at the center of the spheres. On the other hand, in relatively large physical systems, the charge can be described by a continuous function of the spatial coordinates representing volume (or surface) charge density. The electrostatic problem requires the solution of Poisson's differential equation and boundary conditions. The numerical solution of Poisson's equation requires the discretization of the partial differential equation into a system of algebraic equations on a discretization grid of the simulation domain. In the case of nano-scale systems, the discrete nature of charges cannot be neglected. If the primitive model is adopted for the charges, the electrostatic interactions among them can be computed from Coulomb's law. If the system of interest includes different dielectric regions (or phases) characterized by different values of the permittivity and separated by abrupt boundaries, Coulomb's law is not enough. The charges on the boundary are not discrete. Indeed, every charge can interact significantly with every other charge through the boundary condition and a two-body treatment of electric forces (like Coulomb's law) is incomplete, and in that sense incorrect. In the boundary element method (BEM), the polarization effects associated with the discontinuity of permittivity at boundaries is accounted for by adding to the system the polarization charge induced over the boundary surface. This approach does not require the discretization of the whole simulation domain. It requires the discretization of boundary surfaces where discrete polarization charges are associated with discrete surface elements.
In this article, we compare the accuracy and computational speed of two different BEM approaches for the simulation of charged particles.
2 The electrostatic problem
The evaluation of the induced charges in Equation 3 and of the potential in Equation 4 requires the evaluation of surface integrals over the dielectric boundaries. This task can be accomplished by N-point quadrature, dividing the dielectric boundary into a number N of surface elements that we call "tiles". Therefore, the solution of Poisson's equation is converted into the solution of the linear system of equations, A h = b, where A is a N × N matrix describing the mutual electrostatic interaction among the N surface elements, the vector b is the electric field impinging on each surface element and the vector h contains the polarization charges induced at the discretization tiles of the dielectric boundaries.
3 Boundary elements methods
In this study, two different BEM implementations are compared: the iterative method (ITER) introduced by Hoyles et al.  based on the study of Levitt  and the induced-charge computation (ICC) of Boda et al. .
ITER and ICC differ in the way they evaluate the induced charges on the dielectric boundary. ITER approach does not solve a linear system of equations, but evaluates the polarization charges using an algorithmic iterative procedure: at each step, the charge induced by each source/induced charge on the surface elements is evaluated; the procedure is recursively repeated in order to take into account the mutual electrostatic interaction among the polarization charges induced at the surface elements, resulting in a progressive refinement of the estimated-induced polarization charges until a convergence criterion is met. ICC solution is obtained by simultaneously solving the coupled equations derived by the discretization of Equation 3 over the discretized boundary. In both cases, the evaluation of the electrostatic potential at a given position (Equation 4) is performed via Coulomb's law and superposition with an integral form of Coulomb's law for the induced charge on the boundaries.
Figure 1 shows the relative error, with respect to the analytic solution, for the electrostatic potential due to the surface-boundary induced polarization charges only, evaluated along the sphere diameter passing through the point charge. Both ITER and ICC yield very accurate results, limiting the relative error along the whole diameter to approximately 7 · 10-3.
It is worth noting that the advantage of ICC is even more significant in numerical simulations in which the dielectric boundaries do not change during a simulation run. In such a case, the matrix must be inverted only once at the beginning of the simulation and then, at each time step, the electric forces can be evaluated on the basis of the charge distribution in the system through a N × N-matrix by N- vector multiplication. On the other hand, the large computation time needed by ITER makes the computation of electrostatics at run time difficult or even unfeasible.
To avoid long computation times, the electric forces in an ITER calculation are often stored in a number of look-up tables for different (possibly all) ion configurations in the system [18–20]. This approach has two major drawbacks: (i) the loss in accuracy due to the need to interpolate between look-up table entries; (ii) the practical impossibility to determine potential for asymmetric charge distributions due to extremely large memory requirements. ICC is not subject to these kind of problems since it can rapidly solve Poisson's equation for any charge distribution without lookup tables. Due to its accuracy and computation speed, ICC appears well suited for the simulation of nano-scale discrete-charge physical systems.
Finally, we tested ICC solutions in systems with asymmetrical charge distributions. We added two rings of 20 dipoles in the membrane to mimic ion channel charged groups. The dipoles are oriented such that their negative charges are placed 2 Å far from water-protein boundary, and the positive charges are placed 2 Å further inside the membrane. The orientation of the dipoles alter the potential profile in proximity of and inside the channel. No ions are present in the system.
We compared ICC and ITER BEMs for the calculation of the electrostatics in discrete-charge systems. Our results show that both ICC and ITER feature high accuracy, but ICC is remarkably faster. This enables the evaluation of the electrostatics at run time during simulations. One can avoid the lookup tables to retrieve pre-calculated values that in our experience do not work well. ICC is therefore much more well suited for the simulation of nano-scale charged-particle systems such as ion channels or electronic devices.
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