Importance of Varying Permittivity on the Conductivity of Polyelectrolyte Solutions Florian Fahrenberger, Owen A. Hickey, Jens Smiatek, and Christian Holm∗ Institut f¨urComputerphysik, Universit¨atStuttgart, Allmandring 3, Stuttgart 70569, Germany (Dated: September 8, 2018) Dissolved ions can alter the local permittivity of water, nevertheless most theories and simulations ignore this fact. We present a novel algorithm for treating spatial and temporal variations in the permittivity and use it to measure the equivalent conductivity of a salt-free polyelectrolyte solution. Our new approach quantitatively reproduces experimental results unlike simulations with a constant permittivity that even qualitatively fail to describe the data. We can relate this success to a change in the ion distribution close to the polymer due to the built-up of a permittivity gradient. The dielectric permittivity " measures the polarizabil- grained approach is necessary due to the excessive sys- ity of a medium subjected to an electric field and is one of tem size. Molecular Dynamics (MD) simulations of only two fundamental constants in Maxwell's equations. charged systems typically work with the restricted prim- The relative permittivity of pure water at room temper- itive model by simulating the ions as hard spheres while ature is roughly 78.5, but charged objects dissolved in accounting for the solvent implicitly through a con- the fluid significantly reduce the local dielectric constant stant background dielectric. Crucially, the solvent me- because water dipoles align with the local electric field diates hydrodynamic interactions and reduces the elec- created by the object rather than the external field [1{3]. trostatic interactions due to its polarizability. Hydro- When ions accumulate in the vicinity of a charged object dynamic interactions significantly impact the electroki- they further reduce the local dielectric constant [2, 4, 5], netic properties and lead to qualitatively different be- causing a permittivity gradient that repels ions from the havior [35, 36]. Polarizability actually depends on the surface [6{9]. The screening of electrostatic forces be- local electric field [37, 38]: these local variations in the tween charged objects is therefore affected, and it also polarizability thicken the Debye layer by repelling the influences any properties that depend on the specific ion counterions [6{8, 13, 39, 40]. More detailed atomistic distribution. However, almost all computational and the- computer simulations and experiments have also indi- oretical work to date using an implicit water model as- cated pronounced deviations to standard theories for ion sumes a constant dielectric permittivity. This is partially distributions around charged objects if the continuum due to a lack of suitable numerical approaches, since only solvent approach is replaced by an explicit water envi- few electrostatic algorithms can include spatial changes ronment [4, 41{49]. Hence, these findings show that the in the dielectric permittivity [10{13]. molecular properties of water are important for a detailed Electrophoresis is the directed motion of an object in description of the EDL. However, the consequences for an aqueous solution subject to an external electric field. electrokinetic properties are largely unknown. The relative ease with which electric fields can be applied In this letter, we show that including spatially and experimentally has led to wide use of electrophoresis in temporally varying dielectric properties significantly in- the characterization of polymers [14{16], colloids [17{21], fluences the structure of the Debye layer and the dynamic and cells [22, 23], which tend to ionize in aqueous solu- properties of polyelectrolytes. By means of a novel algo- tions. Measuring the electrophoretic velocity of individ- rithm, we locally couple the permittivity to the local ion ual particles is often difficult from a technical standpoint. concentration. When applied to a polyelectrolyte solu- For this reason, polyelectrolyte solutions are often char- tion in an external electric field, the algorithm quantita- acterized by their conductivity [24{33]. tively reproduces experimental data for the conductivity, In both electrophoresis and conductivity, the distribu- while simulations assuming a constant dielectric back- tion of oppositely charged counterions around the object ground disagree qualitatively with experiment. We show determines the magnitude of the relative velocity between that the decreased permittivity in the vicinity of the poly- arXiv:1509.03814v1 [cond-mat.soft] 13 Sep 2015 the object and the fluid. The surrounding counterion electrolyte reduces the fraction of condensed counterions cloud is comprised of two separate layers: the Stern layer on the polyelectrolyte backbone, explaining the experi- and the Debye layer [34]. The Stern layer, often called mentally observed increase in the equivalent conductivity the stagnant layer, consists of strongly adsorbed ions ad- at high monomer concentrations. jacent to the charged object that reduce the effective sur- We performed standard coarse-grained simulations us- face charge. Beyond the Stern layer is the Debye layer, ing a Weeks-Chandler-Anderson (WCA) potential [50] also called the diffuse layer, which consists of ions that for steric interactions with an equilibrium distance of are free to move relative to the charged surface. The two σ = 0:3 nm between particles. Adjacent monomers layers are collectively referred to as the electric double are connected by nitely extensible nonlinear elastic layer (EDL). (FENE) bonds. Simulations looking at static ion dis- For the study of charged macromolecules a coarse- tributions were performed with a Langevin thermostat, 2 in the double layer [2, 4, 5], the local dielectric permit- tivity is calculated from the nearby ion density using the empirical function found by B. Hess et al. [4], 78:5 " = ; (1) 1 + 0:278 · C where C is the molar salt concentration in [mol/L] or [M]. For each lattice cell, the charge density is averaged in the surrounding 73 cells as shown in the top left of Figure 1, weighted by the inverse square of the depicted shell number, resulting in weights from 1 to 1=16. This Figure 1. (color online) Our system setup with polyelec- is because the electric field created by an ion decays as 2 trolytes (black spheres) and counterions (red spheres). The 1=r with the distance r. Assuming linear response, the scheme to calculate local charge concentrations is depicted polarizability should be proportional to this local electric by the lattice in the top left corner. The concentration of field. The weighting also guarantees that charges enter- a lattice cell is determined via a weighted summation of all ing or leaving the cube do not lead to sharp jumps in the 3 charged particles in the 7 surrounding MEMD lattice cells. dielectric constant, which cause unphysical behavior such Ions thus influence the local permittivity within a distance as jumps in the total electric field energy. The volume d = 1:4 nm, or two Bjerrum lengths. taken into account for the averaging is (2:8 nm)3, which for a Bjerrum length of lB ≈ 0:7 nm is roughly the extent over which the electrostatic interactions are significant while simulations in which dynamic quantities were mea- compared to thermal fluctuations. sured used the D3Q19 lattice-Boltzmann (LB) fluid in The temporal changes in permittivity lead to addi- ESPResSo [51{53] and the corresponding thermostat. A tional interactions which have been called spurious [59], detailed description of the simulation method can be but are indeed physical as pointed out by Rottler and found in the supplemental material[54]. Maggs [60] and Pasichnyk et al. [61]. In our simulations, The interactions between charges are calculated us- however, these effects are very small since the algorithm ing an extension of a local electrostatics algorithm that described above only allows smooth and slow changes in was first introduced by A. Maggs [55] and later inde- the local permittivity. pendently adopted for MD simulations by J. Rottler [56] Initially we examine the counterion distribution and I. Pasichnyk [57]. This method of calculating elec- around a single rod-like chain of N = 80 monomers sepa- trostatic interactions has been coined Maxwell Equations rated by 0:3 nm in a cubic box with a side length of 24 nm Molecular Dynamics (MEMD). This algorithm's locality with periodic boundary conditions. In our conductivity permits arbitrary changes in the dielectric constant [58]. simulations, we vary the monomer concentrations C by The motion of charges qivi is interpolated to a regu- placing M 2 f1;:::; 25g chains of length N = 30, 45, and lar lattice. The resulting electric current gives rise to 60 in a box of size (32 nm)3, depending on the desired a change in the magnetic field B and the displacement polyelectrolyte concentration. On a modern workstation field D = "E, following Maxwell's equations. The algo- (Intel i7-5820K CPU, Nvidia GTX 780 Ti GPU), the rithm, however, treats the propagation speed of the mag- simulation time for each curve in Figure 4 was around 46 netic field c as a tunable parameter. For a wide range of hours. This includes 16 points with 3 different polymer values of c, MEMD produces the correct particle dynam- lengths for each, adding up to a total of 48 simulation ics and statistic observables [55, 58]. The algorithm is runs during this time. therefore computationally efficient, since only two update We first verify the validity of our new approach, in steps for the electromagnetic fields are performed per MD which the dielectric constant dynamically adapts to the time step. The permittivity " is used twice: The coupling local ion concentration. To this end, we use the itera- of the displacement field to the magnetic field and vice tive scheme sketched in Figure 2 to calculate the coun- versa (Amp`ere'slaw and Faraday's law). Effectively, this terion distribution around an infinite rod-like polyelec- creates two new driving forces if " depends on space and trolyte fixed in space.