Importance of Varying 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 , 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 -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 permittivity ε 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. (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 implicitly through a con- the fluid significantly reduce the local dielectric constant stant background dielectric. Crucially, the solvent me- because water 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 . 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 are important for a detailed 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 ε = , (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 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 ∈ {1,..., 25} 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 = εE, 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 ε 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 ε depends on space and trolyte fixed in space. time: The influence of magnetic waves that are partially We start out with a constant permittivity of ε = 78.5 reflected in lattice cells with variable permittivity, and and obtain a counterion distribution from an equilibrium a force directly pointing in direction of the permittivity MD simulation. We then map the cylindrically symmet- gradient, following the Lorentz force FL = qD/ε+v×B. ric salt concentration to a local permittivity using equa- The MEMD grid spacing is aMEMD = 0.4 nm and the tion (1) and used it as a fixed permittivity for the sub- time step is set to ∆tMEMD = ∆tMD. The last param- sequent simulation run. A successive under-relaxation eter is an artificial mass fmass = 0.05m0. To take into scheme equally weighting the two preceding results con- account changes in the solvent polarization due to ions verges to the counterion distribution in Figure 3. We 3

weighted: exit if The physical basis of the extended initial increase in start converged the counterion concentration when the permittivity is set permittivity adapted locally is the permittivity gradient in the vicin- ity of the polyelectrolyte, where there is an increase from simulation ion distribution approximately ε = 60 at the polymer surface to ε = 78.5 in the bulk (see inset of Figure 3). The observed distribu- Figure 2. Iterative scheme: Starting with a constant back- tion is the combined result of the repulsion of counterions ground permittivity, the resulting ion concentrations of suc- by the permittivity gradient, the electrostatic attraction cessive simulations are mapped to a spatially varying but fixed of the ions to the polyelectrolyte backbone, and ther- permittivity distribution, until the scheme converges towards mal fluctuations. Interestingly, there is almost no visi- a stable equilibrium. ble difference in the counterion distribution around the flexible polyelectrolyte and the infinite stiff charged rod, demonstrating that the cell model is a good approxima- then apply the new time-dependent adaptive scheme, tion [63, 64]. seen in Figure 1, to calculate the local charge concentra- Our results resemble earlier observations for the coun- tion and permittivity to a fixed rod-like polyelectrolyte, terion distribution around a colloid with spatially varying as well as a fully flexible polyelectrolyte. Figure 3 dis- dielectric background [6, 7]. All simulations with varying plays all three counterion distributions. permittivity also agree qualitatively with atomistic sim- ulations of a similar system using a Kirkwood-Buff based 0.03 force field for NaCl [65] in combination with the extended iterative rod PB 0.025 adaptive rod dielectric jump simple point charge model (SPC/E) water model [66]. adaptive flexible permittivity The atomistic simulations displayed a similar depletion 0.02 of counterions very close to the polyelectrolyte backbone 80 and a subsequent rise of the distribution. The excellent 0.015 75

ɛ 70 agreement between our three simulation setups, and both 0.01 65 60 existing ion distributions for colloids and our atomistic 55 simulations, demonstrates that our method for dynami- 0.005 0 2 4 6 8 10

counterion distribution cally adapting the local permittivity produces physically 0 very reasonable results. 0 2 4 6 8 10 In Figure 3 we have also plotted simulation results (r-R) [nm] where there is a sharp dielectric interface (thin green line) at the surface of the polyelectrolyte, with a discrete rise Figure 3. (color online) Counterion distribution around the from 2 within the polyelectrolyte to 78.5 in the fluid. polyelectrolyte backbone with varying permittivity as a func- As observed in other studies [9, 13], the discrete change tion of the distance from the polyelectrolyte surface. The iter- in the dielectric interface also produces a thickening of ative scheme for a stiff rod (dashed line), the adaptive scheme for a stiff rod (dotted line), and the adaptive scheme for a flex- the Debye layer. However, the difference in the distri- ible polymer (red triangles) are almost identical. They differ bution compared to the Poisson-Boltzmann (thick black qualitatively from the analytical Poisson-Boltzmann (PB) so- line) result is significantly less than in our simulations lution (thick black line) for a uniform permittivity ε = 78.5. with a permittivity adapted to the local salt concentra- The corresponding permittivity ε (inset) in the iterative case tion (dotted line). This shows that one not only needs goes from 78.5 in the bulk to 60 close to the polyelectrolyte take into account the reduced permittivity within a poly- backbone, similar to what is observed near a charged sur- mer, colloid, or charged surface, but also the reduction face [37, 62]. The counterion distribution for a sharp rise in the permittivity from 2 within the rod to 78.5 in the fluid in the permittivity in the surrounding Debye layer, i. e. a (thin light green line) shows only minor deviation from the gradient in the permittivity. PB solution. The structural differences within the EDL in Figure 3 have little influence on many properties such as the ra- The non-monotonicity in the counterion distribution dius of gyration or the polymer diffusion coefficient [67]. is not predicted by Poisson-Boltzmann (PB) theory with However, we found that adapting the local dielectric con- a fixed constant dielectric constant. Note that in the stant significantly impacts the response of the system to case of a uniform background permittivity there is also an external electric field. This is because the electroki- a sharp increase in the counterion density near the poly- netic behavior of the system strongly depends on the hy- electrolyte backbone, however, the increase only ex- drodynamic and electrostatic friction between the poly- tends over a relatively short distance (approximately electrolyte backbone and its counterions, and is thus very 0.1σ = 0.03 nm, data not shown) compared to the simu- susceptible to changes within the EDL. lations with a varying dielectric constant (approximately We simulated the equivalent conductivity Λ (the con- 4σ = 1.2 nm). ductivity over the polyelectrolyte concentration) of a 4

Colby, Na Kwak, Na constant ɛ(C) in the fraction of condensed counterions (red spheres) Kwak, Cs Kwak, Li constant ɛ Kwak, K Lipar-Oštir varying ɛ at high concentrations for the simulations including di- electric variations (top right snapshot). The simulations 1.04 assuming a constant dielectric background (bottom right 1.02 snapshot) resulted in a larger number of condensed coun- 1 terions (blue spheres). To quantify the fraction of con- 0.98 densed counterions, we used the criterion suggested by 0 0.96 L. Belloni [69] and M. Deserno [70]. In Figure 5, we plot Λ / 0.94 the fraction of condensed counterions fcci as a function 0.92 of the monomer concentration C. 0.9 0.88 0.75 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 55 0.7 C1/2 [M1/2] 50 0.65 cci poly

Figure 4. (color online) The rescaled equivalent conductiv- f

√ ɛ ity Λ/Λ0 over monomer concentration C for simulations 0.6 45 with constant permittivity (green dashed, blue dotted) and permittivity locally varying (red solid) permittivity. Experimental data fcci constant ɛ 0.55 f varying ɛ (gray symbols) from Kwak and Hayes [28], Colby et al. [27], 40 cci and Lipar-Oˇstiret al. [26] is reproduced with locally vary- 0.5 ing permittivity, while we observe a qualitative difference for 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 a constant dielectric background. The reason for this is a C1/2 [M1/2] significant drop in the dielectric constant around the poly- electrolyte backbone, as seen in Figure 5, and the subsequent repulsion of counterions from the polymer. Figure 5. (color online) The local permittivity εpoly around the polymer backbone (black squares), and the fraction of condensed counterions fcci for constant dielectric background (blue circles) and varying local permittivity (red triangles) salt-free solution of polyelectrolytes as a function of the as a function of the monomer concentration. The maximum monomer concentration C as plotted in Figure 4 together in the fraction of condensed counterions is due to the sharp with experimental data. The equivalent conductivity is decrease in the permittivity with increasing monomer concen- simply: tration and occurs at the same concentration as the minimum in the equivalent conductivity in Figure 4. ~j Λ = , (2) EC While fcci monotonically increases for simulations with where ~j is the current density, E is the applied electric a constant dielectric background, we observe a maximum field, and C is the monomer concentration. The con- and subsequent decrease when we adapt the dielectric ductivities have been additionally normalized by an ex- constant to the local ion concentration. The maximum trapolated Λ0(C = 0), since the hydrodynamic radius of occurs at the same monomer concentration as the min- the specific ions strongly influences the diffusion and con- imum in conductivity in Figure 4. The decrease in fcci ductivity even at infinite dilution. We performed two sets results in a higher effective charge of the polyelectrolytes, of simulations with a constant background permittivity, which both increases their mobility and the net charge one with εr = 78.5 and one where the ion concentration they carry with them. In addition, there is an increase in in the simulation box was substituted into equation 1 the number of free counterions contributing to the overall to determine the background permittivity. Both sets of conductivity. This is why the maximum in the fraction simulations with a constant dielectric background show of condensed counterions results in a minimum in the a monotonic decrease in the equivalent conductivity, in equivalent conductivity of the solution. good agreement with scaling theories which ignore the We average the dielectric permittivity within 1.4 nm effect of dielectric contrast [24, 25, 27, 29, 30, 68]. More of the polyelectrolyte backbone and plot the results as importantly, our simulations that adapt the permittivity black squares in Figure 5. We find that the reason for to the local ion concentration quantitatively reproduce the drop in fcci is a decrease in dielectric permittivity and the unexpected rise in conductivity at high salt concen- therefore a steeper gradient in ε, resulting in a stronger trations. dielectric repulsion from the backbone. 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