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Conductance of concentrated electrolytes: multivalency and the Wien effect

The electric conductivity of ionic solutions is well understood at low ionic concentrations of up to a few millimolar but becomes difficult to unravel at higher concentrations that are still common in nature and technological applications. A model for the conductivity at high concentrations was recently put forth for monovalent electrolytes at low electric fields. The model relies on applying a stochastic density-functional theory and using a modified electrostatic pair-potential that suppresses unphysical, short-range electrostatic interactions. Here, we extend the theory to multivalent ions as well as to high electric fields where a deviation from Ohm's law known as the Wien effect occurs. Our results are in good agreement with experiments and recent simulations.


Abstract
The electric conductivity of ionic solutions is well understood at low ionic concentrations of up to a few millimolar but becomes difficult to unravel at higher concentrations that are still common in nature and technological applications. A model for the conductivity at high concentrations was recently put forth for monovalent electrolytes at low electric fields. The model relies on applying a stochastic density-functional theory and using a modified electrostatic pair-potential that suppresses unphysical, short-range electrostatic interactions. Here, we extend the theory to multivalent ions as well as to high electric fields where a deviation from Ohm's law known as the Wien effect occurs. Our results are in good agreement with experiments and recent simulations.

I. INTRODUCTION
Understanding the electric conductance of concentrated electrolytes has posed a great theoretical challenge for over a century. The theory of electrolytic conductivity was pioneered by Debye and Hückel [1]. They used the notion of an ionic cloud, where each ion is assumed to be surrounded by a smeared ionic distribution of net opposite charge, which gets distorted upon movement of the central ion. Onsager detected a flaw in the Debye-Hückel account of the central ion diffusion [2] and a few years later corrected the theory, yielding the so-called Debye-Hückel-Onsager (DHO) equation (also known as the "Onsager limiting law") for the conductivity of electrolytes [3,4]. Due to its elegance and accurate predictions, the DHO equation is considered to be one of the cornerstones of electrolyte theory.
A few decades after it was established, the DHO equation was extended to arbitrary electric-field strengths by Onsager and Kim [5], who relied on the unpublished thesis of Wilson on binary electrolytes [6]. The modified theory (often called "Onsager-Wilson (OW) theory") captures the Wien effect [7][8][9] that is an increase in the conductivity with the electric-field strength, attributed to the destruction of the ionic cloud. A related phenomenon called "the second Wien effect" [10,11], occurs in weak electrolytes. Here, the conductivity increases with the electric field strength due to a modification in the dissociation kinetics of chemically bound pairs. While being a remarkable achievement, the DHO and OW theories can be applied only for very dilute electrolyte solutions. They break down when the ion concentration exceeds the threshold of a few millimolar for monovalent ions and an even lower threshold for multivalent ions [12,13]. Since its onset in the 1920s', there have been many attempts to extend the DHO theory to higher concentrations. Initially, by Onsager himself (the "Onsager-Fuoss" theory) [14,15], and later on by others [16][17][18][19][20][21]. However, previous works either used fit parameters that limit their predictive power or included very complicated results that are difficult to use and are not thoroughly transparent. Moreover, to the best of our knowledge, no previous work in the concentrated regime was generalized to finite electric-field strengths; Thus, not capturing the Wien effect.
In recent years, highly concentrated electrolytes have attracted a lot of attention [22][23][24][25] due to their numerous potential applications and surprising experimental observations [26][27][28]. At the same time, advances in nonequilibrium theories such as stochastic density functional theory (often referred to as the Kawasaki-Dean equation) [29][30][31][32], have led to a new way of calculating the ionic conductivity in the dilute limit [33][34][35], which is far simpler than the previous ionic-cloud-based approach. Relying on these theoretical advances and using a modified pair-potential to account for the finite ion size, we recently formulated [36] a new model for the conductivity of concentrated electrolytes. The model was shown to agree well with experimental data for different aqueous solutions at concentrations as high as 3 M but was limited to binary monovalent ions and low electric fields.
In the present work, we extend this model and apply it to multivalent ions and finite electric fields. We derive a general expression for the conductivity of binary electrolytes and then focus on two cases: (i) the weak-field limit with any z 1 :z 2 ionic valencies, and; (ii) the symmetric z:z electrolyte at finite field intensities, where we recover the Wien effect and provide new predictions for the high concentration regime. Our results compare favorably to experiments and recent simulations.
The outline of this paper is as follows: in Sec. II, we present the model system and derive the conductivity equations for ionic solutions with an arbitrary number of species, at high ionic concentrations. In Sec. III, we restrict ourselves to binary electrolytes and analyze the low electric-field limit as well as the case of symmetric ions with any finite electric field.
In Sec. IV, we compare our results to experiments and simulations. Finally, in Sec. V, we conclude and suggest future experiments to further test our predictions.

A. The Equations of Motion
We consider a homogeneous ionic solution composed of M ionic species of charge q α and average concentration n 0 α , where α = 1, ..., M . The ions are embedded in a solvent with dielectric permittivity ε and viscosity η at temperature T . The solution is subjected to a constant (static) external electric field E 0 pointing in a fixed direction.
The local ionic concentrations, denoted by n α (r, t), satisfy the continuity equation where j α (r, t) is the ionic flux of the α species, given by, The first and second terms on the right-hand-side of Eq. (2) are advection and diffusion terms, respectively, where u(r, t) is the solvent velocity field, and D α is the diffusion coefficient of the α species at infinite ionic dilution. The third term accounts for the motion due to the external field and inter-ionic forces. Here, µ α is the ion mobility at infinite ionic dilution, related to D α by the Einstein relation µ α = D α /k B T , with k B being the Boltzmann constant, and f α (r, t) is the force density given by where v αβ is the pair interaction energy between ions of species α and β. Note that for generality sake we do not specify v αβ until Sec. II C. The last term in Eq. (2) is a stochastic flux, where ζ α (r, t) is a 3D white-noise function, satisfying where n and m denote the cartesian components of the vector ζ α . Equations (1) and (2) can be derived by transforming the Langevin equation from individual particle representation to concentration fields using Ito calculus, and it is referred to as stochastic density-functional theory (SDFT) [30]. 1 1 The derivation of the stochastic density-functional theory in Ref. [30] was done without considering the advection by the solvent. However, advection can be easily incorporated into the formalism by adding the solvent velocity to the Langevin equation, yielding Eqs. (1) and (2) exactly.
The ion continuity equation is coupled to the Navier-Stokes equation for an incompressible fluid, where p(r, t) is the local pressure and ρ is the solvent density. The ρ(u · ∇)u term will disappear when we linearize the equations of motion around u = 0 in the next II B subsection. The ρ∂ t u term can also be neglected as long as the electric field is not too strong.
The typical time and length scales that characterize the ionic motion, T and L, satisfy Applying this rescaling we get that |ρ∂ t δu|/|η∇ 2 u| ∼ ρD α /η, which is the inverse of the Schmidt number [37] and is roughly ∼ 0.001 for standard electrolytes. Therefore, the resulting equation for the solvent velocity is the Stokes equation for an incompressible fluid,

B. Calculation of the conductivity
The conductivity of the ionic solution is defined by the ratio where ... is the thermodynamic ensemble average, and J (r, t) is the electric current density, which depends on the ionic fluxes, j α , The subscript " " in Eq. (7) denotes the vector projection on the external field direction, J = J ·Ê 0 . Substituting Eq. (2) in Eq. (8) and performing the average in Eq. (7), we obtain where κ 0 is the conductivity at infinite dilution, Note that κ 0 depends linearly on the concentrations as the ions do not interact in this limit.
In order to obtain Eq. (9), we invoke the system homogeneity and the independence between n α and ζ α at equal times.
Equation (9) implies that at finite concentrations the conductivity deviates from its dilutelimit behavior due to two effects. The first effect, incorporated in κ hyd , is a hydrodynamically mediated interaction between the ions, and it is traditionally referred to as the electrophoretic effect. We note that in its present form, the average u (r, t) n α (r, t) in κ hyd includes the ion interaction with its own induced velocity field, resulting in a self-interaction that should be subtracted, as will be done later on. The second effect, incorporated in κ el , is a direct ionic interaction that is mostly electrostatic but will include finite-size corrections as we explain in Sec. II C, below. This effect is often referred to as the relaxation effect.
In order to calculate the averages in Eq. (9) we need to solve the equations of motion, Eqs. (1), (2) and (6). An exact solution cannot be obtained. Instead, we linearize the equations by writing n α (r, t) = n 0 α + δn α (r, t), u(r, t) = δu(r, t) and p(r, t) = p 0 + δp(r, t), and keeping only terms up to linear order in δn α , δu, δp, and ζ α . The linearization is justified for small fluctuations around the mean-field values. In Fourier space, the linearized form of the ion equation of motion is wheref (k) = d 3 r f (r)e −ik·r is the Fourier transform of the function f (r). The matrices andζ α (k, t) is a white-noise scalar function, α = 1, ..., M , satisfying Note that we used the fact that k ·ζ α (k) = 3 α=1 k αζ i α (k), is a sum of three independent white noise functions with zero mean. Therefore, it can be replaced by a single white noise function, whose variance is the sum of the variances of the three functions, kζ α (k).
The linearized form of the Stokes equation in Fourier space is We use the Fourier transform of the incompressiblity condition (Eq. (6)), k · δũ(k) = 0, to eliminatep(k) in Eq. (14), and obtain δũ (k) in terms of the concentrations, Writing Eq. (9) in terms of the fluctuational variables in Fourier space and using Eq. (15), we obtain In steady state, the set of linear equations in Eq. (11) leads to where the correlation matrix, C(k), is given by the relation [38] A(k)C(k) where † is the Hermitian conjugate. In order to subtract the ion self-correlation, we define the following subtracted correlation matrix [39], and use C instead of C from here on.
Substituting the subtracted correlation matrix in Eq (16) and recalling that κ hyd , κ el and v αβ (k) are real functions, we obtain < l a t e x i t s h a 1 _ b a s e 6 4 = " s d 0 6 i + c 1 j s 2 g u W n 3 2 In summary, the correlation matrix is obtained by solving Eq. (18) and redefining a subtracted correlation matrix C in Eq. (19). By substituting the matrix C in Eq. (20) we can compute the conductivity.

C. The modified pair-potential
Up until now, we did not specify the pair potential, and the results were written in terms of a general v αβ (r) interaction. For point-like ions, the pair potential equals to the Coulomb interaction, v αβ (r) = q α q β /(4πε 0 εr), where ε 0 is the vacuum permittivity. However, the point-like approximation leads to an unphysical attraction between oppositely charged ions at distances smaller than the ion diameter (see Fig. 1). This becomes problematic at high concentrations, where the ions are more likely to get close to each, leading to an unphysical decrease in the conductivity due to enhanced inter-ionic correlations. This deficiency is present in the DHO theory that assumes point-like ions.
The problem can be remedied by including in v αβ (r) a hard-core potential, where r α is the ion radius, and r α + r β is the distance of closest approach between two ions. Unfortunately, such a diverging potential breaks down the perturbative approach introduced in Sec. II B. Instead, a viable modification is to introduce a low cutoff to the Coulomb interaction [24], where Θ(r) is the Heaviside function. In Appendix A, Eq. (22) is shown to approximate well the average distance between two ions interacting via the pair potential as in Eq. (21), in a confined system that corresponds to concentrated electrolytes (with short inter-ionic distances). It is also shown that the approximation becomes less accurate for multivalent ions at high concentrations.
We note that while the modified potential successfully suppresses the short-range electrostatic attraction between oppositely charged ions, it induces an unphysical attraction at very short distances between ions with the same electric charge sign. It might seem that this problem can be circumvented by keeping the standard Coulomb potential (which diverges at small distances) for ions with the same electric charge sign, or by assigning a finite yet large positive value to the potential at r < r α + r β . However, to be consistent within the perturbative approach we need to keep the potential small enough. Thus, in our approach we keep v αβ = 0 for r < r α + r β , for any type of ions, α and β.
Substituting the Fourier transform of Eq. (22),ṽ αβ (k) = q α q β cos (kr α + kr β ) /(ε 0 εk 2 ), in Eq. (12) and following the analysis of Sec. II B leads to a closed-form expression for the conductivity. It is presented in the next Section for binary electrolytes.

III. BINARY ELECTROLYTES
We consider hereafter a binary electrolyte z + :z − containing cations of charge q + = ez + and anions of charge q − = −ez − , where z ± are the valencies (in absolute value) and e is the elementary charge. The electro-neutrality condition implies z + n 0 + = z − n 0 − , where n 0 + and n 0 − are the average cation and anion concentrations. The experimentally controlled salt concentration, n salt , is n salt ≡ n 0 + /z − = n 0 − /z + . Note that in the case where the two valencies z ± have a common divisor (e.g., 2:4), it is more natural to define the salt concentration as n salt multiplied by the greatest common divisor. This is done later on for symmetric z:z salts.
We make a further simplification by replacing the species-dependent cutoff length in Eq. (22), r α + r β , by a single cutoff that equals the sum of the cation and anion radii, a ≡ r + + r − . This simplification is motivated by the fact that the primary role of the cutoff is to prevent attraction between oppositely charged ions. In Appendix B, we explore the difference between the conductivity when a single cutoff is used as opposed to three different cutoffs (r + +r − , 2r + and 2r − ), and show that the difference is negligible for standard electrolytes.
Under these simplifications, the conductivity is written as follows, where we used the fact that C αβ is hermitian. Equation (23) indicates that κ hyd depends on the difference between the spatial correlations of equal and opposite charges, while κ el depends on the spatial correlations between opposite charges only.
The components of the subtracted correlation matrix are where cos θ =k ·Ê 0 , the Debye screening length is and the electric field length is l E = k B T /(eE 0 ) (note that l E is inversely proportional to the electric-field intensity). We also defined the average valency,z ≡ (z + + z − )/2, the parameter γ, and the following functions for brevity For z + = z − , γ < 1 (γ > 1) if the ion with the larger valency has a smaller (larger) mobility.
Typically, multivalent ions have smaller mobilities than monovalent ions; thus, asymmetric salts commonly have γ < 1 (see Table II in Sec. IV). For symmetric salts with z + = z − ≡ z, γ = 1 and n 0 + = n 0 − ≡ n, and the correlation matrix reduces to We note that n = zn salt as explained at the beginning of Sec. III.
A visualization of the correlations can be obtained by plotting the pair-correlation function, where δn α (r, t) = n α (r, t) − n 0 α . The pair-correlation function for symmetric ions is shown in Figs. 2 and 3. The correlation function is rescaled by (2e 2 /k B T ε 0 ε) 3/2 √ n and depends on two parameters: zλ D /l E , that is the normalized electric field E 0 , and a/λ D , a rescaled cutoff length.
At equilibrium, zλ D /l E = 0, and h αβ (r) is spherically symmetric (Fig. 2). For a/λ D = 0 (point-like ion), the standard ionic atmosphere, h αβ (r) ∝ e −r/λ D /r, is obtained. Equal charges are depleted, opposite charges are more abundant around the test charge, and the correlation function diverges at r → 0. For small non-zero a/λ D , the pair-correlation function behaves similarly to the point-like case, except that it has a finite value at r → 0. The value is positive for h ++ (r) and h −− (r) and negative for h +− (r). For larger a/λ D values (1 a/λ D 2.8 for symmetric ions), the pair-correlation function decays in an oscillatory manner (for a full derivation, see Ref. [24]). Similar damped oscillations were shown to exist at highly concentrated solutions [40][41][42]. When a/λ D is very large ( 2.8 for symmetric ions, not shown in Fig. 2), the correlation function diverges with pure oscillatory modes leading to unphysical long-range order [24]. For such high a/λ D values, which are reached only at very high concentrations (as high as 9 M for NaCl in water at room temperature, which gets beyond crystallization), the use of the modified cutoff potential cannot be justified.
When an electric field is applied to the system (Fig. 3), the pair-correlation function maintains rotational symmetry around the direction of the electric field,Ê 0 . Whereas h ++ (r) where r s = 1/(6πημ) is a reduced Stokes radius withμ = (µ + +µ − )/2 and l B = e 2 /(4πk B ε 0 ε) is the Bjerrum length. We can see from Eq. (30) that the rescaled conductivities, κ hyd /κ 0 and κ el /κ 0 depend on the ratios between the length-scales: λ D , l B , r s , l E and a, on the valencies z ± , and the asymmetry parameter γ. Equation (30) is the main result of this paper. In the next sections, we will explore different limits and cases.
A. The conductivity in the weak E 0 limit The first case that we would like to examine is the limit λ D /l E → 0, i.e., E 0 k B T /eλ D . As an example, for aqueous solutions at room temperature with monovalent , which means that the λ D /l E → 0 limit occurs when E 0 10 −4 V/Å = 1 V/µm for n salt = 1 mM and E 0 10 −2 V/Å = 100 V/µm for n salt = 1 M. In this limit, Eq. (30) reduces to where we used the change of variables x = λ D k. Although the integrals in Eq. (31) cannot be performed analytically, they can be easily computed numerically.
The rescaled conductivity correction terms κ hyd /κ 0 and κ el /κ 0 , are shown in Fig. 4 on a semi-log plot as a function of n salt , the salt concentration, for monovalent salts, z ± = 1.
Both κ hyd /κ 0 and κ el /κ 0 approach zero in the infinite dilution limit (n salt → 0). One sees from the figure that κ hyd /κ 0 decreases as the concentration increases until a minimum is reached at ∼ 1 M. Then, κ hyd /κ 0 increases until it diverges at a finite concentration. The minimum occurs very close to (but not exactly at) the onset of damped oscillations in the pair-correlation function, discussed in the previous subsection, while the divergence occurs exactly when the correlation function diverges. The second correction term, κ el /κ 0 , shows a different behavior. It decreases as the salt concentration increases until it diverges to −∞ at the same concentration where the correlation function diverges. We note that for γ > 2, which is very uncommon for small inorganic ions, κ el diverges prior to the diverges threshold of the correlation function, due to the term of 1 2 γ cos(ax/λ D ) + x 2 in the denominator of the κ el /κ 0 expression in Eq. (31).
The two integrals of Eq. (31) can be approximated for small a by approximating cos(ax/λ D ) ≈ 1 in their denominators. The integrals can then be calculated analytically using the residue theorem, yielding The divergence that occurs in the exact result at high concentrations is not present in the analytical approximation. By taking a → 0 in Eq. (32), the DHO result for the conductivity is exactly recovered [44], where both κ hyd /κ 0 and κ el /κ 0 (second and third terms on the right-hand-side, respectively) are inversely proportional to λ D .
In Fig. 5, we explore the effect of multivalency on the conductivity by plotting the rescaled conductivity using our numerical results of Eq. (31), the analytical approximation (Eq. (32)), and the DHO result (Eq. (33)) of 1:1 monovalent electrolytes and 2:1 electrolytes (+2e and −e charges). We used a crude approximation that the mobility is inversely proportional to the valency (see Table I (34) where ξ ≡ zλ D /l E ∝ E 0 . In the a → 0 limit, the integrals can be performed analytically.
A convenient way to perform the integrals is to take the a → 0 limit already in the 3dimensional integral expressions of Eq. (23), and then perform the radial integration in k-space before the angular part. The Onsager-Wilson (OW) result is then recovered [5], In Fig. 6, we show the rescaled conductivity, κ/κ 0 = 1 − κ hyd /κ 0 − κ el /κ 0 , according to Additionally, κ/κ 0 saturates at high electric fields. Such high fields are often not accessible experimentally, as they introduce other effects such as Joule heating [45]. The relative increase in κ/κ 0 , induced by the electric field, is more pronounced at high concentrations as compared to low concentrations. Our results predict that the relative increase in κ/κ 0 is smaller than the increase predicted by the OW theory. This is due to the suppression of the electrostatic interactions at short distances, included in our theory. As in the low electric field case analyzed in the previous subsection (Sec. III A), the difference between our results for κ/κ 0 as compared to the a → 0 case (OW theory) increases with the ion concentration. We conclude that the Wien effect becomes more pronounced as the ion concentration increases, but to a lesser extent than the OW theory prediction.
By expanding Eq. (34) in powers of ξ = zλ D /l E at weak electric fields, we see that the conductivity grows quadratically with ξ ∝ E 0 , In the ξ → ∞ limit, κ el /κ 0 → 0, while κ hyd /κ 0 approaches a constant value, The system behavior at strong electric fields can be understood from the correlation matrix, Eq. (28). Taking the ξ → ∞ limit is equivalent to l E → 0, and yields C +− (k) → 0 and C ++ (k), C −− (k) → −n cos (ka) /[cos (ka) + 2k 2 λ 2 D ]. The external electric field "tears apart" pairs of oppositely charged ions and the correlation between such pairs vanishes in the ξ → ∞ limit. However, since E 0 drags equally charged ions in the same direction, it does not destroy their (anti-) correlations. Thus, C ++ and C −− remain finite. Since κ el /κ 0 is proportional to C +− , it vanishes in the ξ → ∞ limit. However, κ hyd /κ 0 depends on the difference between C ++ and C +− . Therefore, it reaches a constant value in this limit.

IV. COMPARISON WITH EXPERIMENTS AND SIMULATIONS
In Fig. 7, we compare our numerical results for weak electric fields (Sec. III A) to experimental data for different aqueous ionic solutions at T = 25 • C, taken from Refs. [46] and [47], where an extensive body of measurements is summarized. The molar conductivity, κ/n salt , which is commonly used in experiments, is plotted as a function of n salt for 1:1 electrolytes (NaCl and KBr), 2:1 electrolytes (BaCl 2 and CaCl 2 ), and 3:1 electrolyte (LaCl 3 ).
Note that the molar conductivity is proportional to the rescaled conductivity, κ/κ 0 , since κ 0 is linear in n salt . The electrolytes we consider have monovalent anions, z − = 1, and different cationic valencies z + . Therefore, n salt is the concentration of the cations, n 0 + . At T = 25 • C, water viscosity is η = 0.890 mPa · s [53] and the dielectric permittivity is ε = 78.3 [54], yielding a Bjerrum length of l B = e 2 /(4πk B ε 0 ε) = 7.15Å, and a screening length of λ D = 1/ 4πl B (z 2 + n 0 + + z 2 − n 0 − ) = 4.30 [Å]/ z + (z + + 1)n salt [M]. In Table I, we summarize the values of the radii and diffusion coefficients at infinite dilution for all the ions considered in Figs. 7. In Table II, we present the electrolytes asymmetry parameter γ, cutoff length, a reduced Stokes radius r s , and molar conductivity κ 0 /n salt at infinite dilution. They are all calculated from the parameters in Table I and [52] and diffusion coefficients for aqueous solutions at T = 25 • C at infinite dilution limit [46]. We use the "Effective ionic radii" by Shannon with six-coordinate. Other sets for the ionic radii give very similar results [52].  (26)), cutoff length a = r + + r − , reduced Stokes radius r s = k B T /3πη(D + + D − ), and molar conductivity at infinite dilution κ 0 /n salt (Eq. (10) with n salt = n 0 + ) where S is the Siemens electric conductance unit, calculated from the parameters in Table I  potential with hardcore for multivalent ions at high concentrations, as demonstrated in Appendix A (see Fig. 10(a)). Finally, the high charge density orders the liquid around the ions.
The ordering changes the dielectric constant ε and viscosity η, and these extra factors are not accounted for in our theory.  Recently, field-dependent ionic conductivities were calculated from molecular dynamics simulations, using generalized fluctuation-dissipation relations [48,49]. This method yields the differential conductivity, κ diff ≡ d J /dE 0 , related to the standard conductivity, Fig. 9, the simulation results of Ref. [48] for the molar conductivity are reproduced, where the differential conductivity is converted by integration to the standard conductivity. The simulations take into account the solvent only implicitly and do not account for the conductivity correction due to the counterflow of the  35), without the hydrodynamic correction term, κ hyd . While our numerical results deviate significantly from the simulations, they describe the same qualitative behavior and are in much better agreement with the simulations than the OW theory is. In particular, the simulations support our prediction that at high ionic concentrations, the relative increase of the conductivity due to the Wien effect is smaller as compared to the increase predicted by the OW theory. We note that for system parameters as in Fig. 7, the OW result does not make sense as it predicts negative conductivity for weak electric fields.

V. CONCLUSIONS
This paper presents a theory for conductivity at high electrolyte concentrations, applicable for multivalent ions and finite electric fields. We used a stochastic density functional theory (SDFT) and a modified electrostatic potential that suppresses the unphysical shortrange attraction between oppositely charged ions. At low electric fields, the theory is particularly accurate for monovalent salts, showing excellent agreement with experimental data at concentrations as high as a few molars with no fit parameters. Its range of applicability decreases for multivalent ions due to the strong electrostatic interactions that break the correction term, and compared to simulation data with implicit solvent, taken from Ref. [48], and to the infinite dilution limit.
perturbative approach, and the inaccuracy of the modified potential at high concentrations.
Nevertheless, the theory provides accurate predictions for 2:1 and 3:1 electrolytes up to concentrations of ∼ 0.1 M and ∼ 0.02 M, respectively, without any fit parameters. This is far beyond the applicability range of the well-known Debye-Hückel-Onsager (DHO) theory.
For strong electric fields, we recover the Wien effect and show that similarly to dilute solutions, the rescaled conductivity at high ionic concentrations displays a sigmoid-like behavior, where the conductivity will make a transition between two limiting values when the field strength increases. The relative increase in the rescaled conductivity at high concentrations is smaller than the increase predicted by the WO theory due to the suppression of the For a z + :1 electrolyte, n salt = n 0 + = n 0 − /z + , and thus The salt concentration, n * salt , is an approximation of the electrolyte salt concentration of the real system, n salt . Figure 10 shows the normalized deviation σ calculated from Eq. (A1) as a function of n * salt . Figure 10a shows the case of cation-anion interactions and Fig. 10(b) cation-cation interactions, for three electrolytes: 1:1, 2:1 and 3:1. For the cation-anion case, σ barely deviates from 0, even at 1 M concentration, indicating that the modified potential approximates well the Coulomb with hardcore interaction for opposite charges. For comparison, we note that a pure Coulomb interaction, v c (r) = q 1 q 2 /4πε 0 εr, leads to r c = 0 and a normalized deviation of ( r c − r hc c )/ r hc c = −1 at any concentration [39].   Fig. 7), as a function of the salt concentration n salt , shown for the single cutoff length approximation (full lines, Eqs. (31)) and when using three different cutoffs (dashed lines, Eq. (B1)).
The electrolyte physical parameters are specified in Sec. IV.
In Fig. 11, we show the molar conductivity, κ/n salt , for the same five electrolytes analyzed in Fig. 7, and compare the results of using three different cutoffs (Eq. (B1)), to the single cutoff length case a = a ± = a +− (Eq. (31)). For concentrations below ∼ 0.05 M, the difference is negligible for all considered electrolytes. Above ∼ 0.05 M, there is a small yet visible difference, but only for electrolytes with trivalent ions (LaCl 3 ) or electrolytes with large size asymmetry between the cation and anion (MgCl 2 , where r − ≈ 2.5r + , see Table I in Sec. IV). In these cases, the molar conductivity for the single cutoff case is slightly higher.
For monovalent ions or roughly symmetric electrolytes, no difference is seen in Fig. 11 even at very high concentrations. We note that the regime where a difference is seen is beyond the concentration regime analyzed in Fig. 7. Moreover, in this regime, our theory deviates from experimental results in the case of a single or three cutoffs, where the single cutoff approximation gives slightly better agreement to experiments.