Polyelectrolyte solutions have been proposed as a method to improve the efficiency of lithium-ion batteries by increasing the cation transference number because the polymer self-diffusion coefficient is much lower than that of the counterion. However, this is not necessarily true for the polymer mobility. In some cases, negative transference numbers have been reported, which implies that the lithium ions are transporting to the same electrode as the anion, behavior that is often attributed to a binding of counterions to the polyion. We use a simple model where we bind some counterions to the polymer via harmonic springs to investigate this phenomenon. We find that both the number of bound counterions and the strength of their binding alter the transference number, and, in some cases, the transference number is negative. We also investigate how the transference number depends on the Manning parameter, the ratio of the Bjerrum length to charge separation along the chain. By altering the Manning parameter, the transference number can almost be doubled, which suggests that charge spacing could be a way to increase the transference number of polyelectrolyte solutions.

## I. INTRODUCTION

Lithium-ion batteries are ubiquitous in modern electronic technology. A number of metrics are used to categorize battery performance. One of these metrics is known as the cation transference number, which is a measure of the fraction of current carried by the cation.^{1} A large transference number allows the charging rate of a battery to increase^{2} and decreases the formation of dendrites. However, common battery electrolytes usually have transference numbers less than 0.5,^{3} which means that the anion contributes more to the conductivity than the cation.

One method proposed to increase this transference number is using polyelectrolyte solutions as the battery electrolyte.^{4,5} The rationale for this proposed method is that if the mobility of the lithium ion is not substantially decreased, but the mobility of the polyelectrolyte chain is smaller than the mobility of a molecular anion, the transference number will be large.^{4,6} An additional advantage of polyelectrolyte solutions as battery electrolytes is the low barrier for utilization in battery technologies.^{7}

Polyelectrolytes synthesized for use in battery electrolytes usually have sulfonate groups,^{8,9} but polyelectrolytes based on bis(nonenylmalonato)borate (BNMB) anions have also been synthesized.^{10} The measured transference numbers of polyz(LiBNMB) were much larger than those of the sulfonate-based polyelectrolytes,^{8–10} which have been attributed to the formation of a gel-like structure in the poly(LiBNMB) system, making the anion mobility very low. In this work, we focus on mobile cations and anions and use a simple coarse-grained model for the polyelectrolyte solution.

The chemical identity of the polyelectrolyte charged group has a large impact on the behavior of the polyelectrolyte solution. Compared to BNMB, sulfonate groups usually have a poorer electrochemical stability.^{10–12} These sulfonated polyelectrolytes typically have a low solubility in the carbonate solvents employed in lithium-ion batteries compared to BNMB-based polyelectrolytes.^{10,13} In addition, lithium ions tend to interact strongly with sulfonate groups due to the presence of three sulfonyl-oxygens in the sulfonate group.^{14–17} Since BNMB has demonstrated reasonable performance in single-ion conductors^{18} and gel polymer electrolytes,^{19} it is likely that lithium ions and BNMB do not interact as strongly as lithium ions and sulfonate groups.

An important parameter for polyelectrolytes is the charge spacing along the polyelectrolyte. One way to characterize polyelectrolyte charge spacing is the Manning parameter, *ξ*, which is defined as the ratio of the Bjerrum length, *l*_{B}, to the distance between charges.^{20–28} In the original formulation of counterion condensation, an infinitely thin and long charged rod will condense counterions when the Manning parameter is greater than 1.^{20} However, examination of systems with finite length,^{21} explicit counterions and coions,^{22} or flexible chains^{26,27} has shown that the original understanding of counterion condensation is not completely accurate, but increasing the Manning parameter does increase the interaction between polyelectrolytes and counterions. The sulfonated polymers usually have charged groups closer together than poly(LiBNMB),^{8–10} so it is expected that the sulfonated polymers have stronger interactions with their counterions than poly(LiBNMB). This work seeks to examine the effect of counterion condensation on the transference number of the polyelectrolyte solution.

Measurements of the transference number have revealed negative transference numbers in a number of systems.^{29–34} These negative transference numbers have been attributed to the presence of strong ion pairing, which causes the charged species to move toward the unexpected electrode.^{29–35} However, experimental measurements of transference numbers have proven to be method dependent.^{9,30,31,33,36–38} Due to this method dependence, quantitatively comparing transference numbers measured via different experimental techniques is difficult.^{9,35,39} Thus, molecular simulation may be a useful approach to better understand transference numbers in polyelectrolyte solutions.

Determining transference numbers from molecular simulations can largely be divided into three categories of techniques. Defining the transference number as $t+=(z+u+c+)/(\u2211iziuici)$, where *z*_{i}, *u*_{i}, and *c*_{i} are the charge, electrophoretic mobility, and number density of species *i*. Perhaps the simplest method to calculate the transference number is using the self-diffusion coefficient.^{9,40} This approach relates the electrophoretic mobility to the self-diffusion coefficient via the Nernst–Einstein equation.^{9} However, this method does not capture correlations between ions due to bonded potentials or strong ion pairing.^{9,39} An alternative approach is simulating the system in an electric field because *u*_{i} = ⟨*v*_{x}⟩/*E*, where ⟨*v*_{x}⟩ is the drift velocity in the *x*-direction and *E* is the electric field strength.^{41,42} However, the relationship between the mobility and the electric field strength is only valid in the linear response regime, which means that simulations have to be performed at a number of *E* values to determine the relevant electric field strength.^{41} The final approach is using an Onsager coefficient framework,^{39} which relies on the expression $ui=\u2211jLij,tzjF/ci$,^{35,39} where *F* is Faraday’s constant and *L*_{ij,t} is the total Onsager coefficient for species *i* and *j*, which is calculated via an Einstein expression^{35,39} (similar to how the self-diffusion coefficient is calculated from the mean-square displacement). This has the advantage of only needing a single simulation to calculate the electrophoretic mobility and being capable of capturing all correlations between ionic species.^{39}

In this article, we use molecular dynamics simulations to analyze how explicitly binding counterions to a polyelectrolyte chain affects the transference number. This is accomplished using a harmonic bond potential to explicitly bind counterions to the polyelectrolyte chain and allows us to independently alter the number of bound counterions and the strength of that binding. While this is not necessarily a real system, it models the effect of strong ion binding and allows us to analyze if the transference number indeed becomes negative with sufficiently high ion binding. In Sec. II A, we discuss the details of our simulations, and in Sec. II B, we explain the Onsager coefficient framework. The presence of the explicit solvent and chain length effects are investigated in Secs. III B and III C, respectively. We then analyze how the transference number depends on the number of bound counterions in Sec. III D. These results are then used to analyze the effect of the Manning parameter on the transference number and conductivity in Sec. III E.

## II. METHODS

### A. Simulation details

Simulations are performed for salt-free solutions of one polyion in dilute solution with the polarizable, coarse-grained Martini water (POL) model^{43} as solvent, degree of polymerization, *N*, and sufficient counterions to neutralize the system. Adjacent monomers, *i* and *j*, are connected via a harmonic bond,

where the separation between *i* and *j* is *r*_{ij}, the strength of the bonding potential is $kbM$ (=8000 kJ mol^{−1} nm^{−2}), and the equilibrium bond length is *b*_{ij} (=0.51 nm). A number of counterions, $nCB$, are bound to the polyelectrolyte via a harmonic bond potential given by

where $kbC$ is the bond strength between the monomer and the counterion. The number of unbound counterions is $nCU=nCT\u2212nCB$, where $nCT$ is the total number of counterions. This is illustrated schematically in Fig. 1. An equal number of monomers, $nMB=nCB$, have a counterion bound to them, and the number of bare monomers is $nMU=nMT\u2212nMB=N\u2212nMB$, where $nMT$ is the total number of monomers. These monomers will be referred to as bound and unbound, respectively, but this does not indicate they are not part of the polymer chain. We vary the value of $nCB$ and $kbC$ to analyze how both the number and bond strength of bound counterions affect dynamic properties of these solutions.

Non-electrostatic interactions between beads are included via a cut and shifted Lennard-Jones potential,

where the well depth is *ɛ*_{ij}, the collision diameter is *σ*_{ij} (=0.47 nm), and the constants that ensure the continuity of the potential and force are given by

for *α* = 6 or 12. With this potential form, the force decays smoothly to 0 between the distances *r*_{1} and *r*_{c}, the potential-cutoff distance. The POl water model^{43} sets *r*_{1} = 0.9 nm and *r*_{c} = 1.2 nm. *ɛ*_{WW} = *ɛ*_{CW} = *ɛ*_{CC} = *ɛ*_{MC} = *ɛ*_{MM} = 4.0 kJ mol^{−1}, and *ɛ*_{MW} = 4.2 kJ mol^{−1}, where *W* refers to water, *C* is a counterion, and *M* is a monomer. These epsilon values ensure that an equivalent neutral polymer exhibits good solvent scaling for this solvent model (see Sec. S2 for details).

The POL water model is a coarse-grained water model, which maps four water molecules to each coarse-grained molecule.^{43} Orientation polarizability is captured by modeling each coarse-grained water molecule with three sites: one positive site denoted WP, one negative site denoted WN, and one neutral site denoted W, as shown in Fig. 1.^{43} Sites WP and WN are bound to site W with a linear constraint solver (LINCS)-constrained bond length of 0.14 nm, and these three sites include an angle potential with strength 4.2 kJ mol^{−1} rad^{−2} and equilibrium angle 0 rad.^{43} ±0.46*e* charges are assigned to sites WP and WM, respectively.^{43} For POL water, the relative dielectric constant, *ɛ*_{r}, throughout the simulation box is 2.5, which gives a bulk dielectric constant, $\epsilon rB$, of 75.6 at 300 K.^{43} Modeling the solvent polarizability in this manner allows ions to transition from regions with little water and low dielectric constant to mostly aqueous regions with large dielectric constants.^{43}

All simulations are performed at 300 K using GROMACS 2019.3.^{44−51} We note that bonded interactions generated due to Eq. (1) are of type “1” and those generated due to Eq. (2) are of type “6.” A type “6” bonded potential allows a harmonic bond to exist, which does not generate exclusions.^{52} By using this bonded potential type for counterions bound to monomers, the full Lennard-Jones and electrostatic interactions will be included for these pairs. Steepest descent minimization minimizes the energy of the initial configuration. Energy minimized configurations are equilibrated using the leapfrog integrator in the NPT ensemble with the Berendsen thermostat and barostat utilizing 1 ps temperature coupling and 4 ps pressure coupling for 20 ns of simulation time. Production simulations are performed in the NVT ensemble using the Nose–Hoover thermostat with 4 ps temperature coupling for 1 *µ*s, and every 10 ps trajectories are saved. For *N* = 128, the simulation is only run for 503 ns due to the large size of the simulation (see Table S5). The particle mesh Ewald (PME) technique calculated electrostatic interactions using a 0.2 nm spacing and a 1.5 nm real space cutoff distance, and POL water has a dielectric constant of *ɛ*_{r} = 2.5. Monomers and counterions have a −1*e* and 1*e* charge, respectively. Final box sizes and the number of solvent molecules for each simulation are given in Tables S5 and S6 of the supplementary material. We also consider implicit solvent (IMP) simulations in Secs. III A and III B. Simulation parameters and procedures for IMP are discussed in detail in Sec. S1 of the supplementary material.

### B. Onsager coefficient analysis

A total Onsager coefficient for species *i* and *j* can be defined as

where $\Delta ri\alpha =ri\alpha (t)\u2212rCM(t)\u2212ri\alpha (0)+rCM(0)$, $ri\alpha (t)$ is the position of molecule *α* of species *i* at time *t*, *r*_{CM}(*t*) is the position of the system center of mass at time *t*, *k*_{B} is the Boltzmann constant, *T* is the temperature, and *V* is the volume. Superscript *TT* denotes that contributions from both bound and unbound members of species *i* and *j* are included in the total Onsager coefficient calculation. Subscript *t* denotes a total Onsager coefficient. This can be separated into a series of total Onsager coefficient between bound and unbound species as

where superscripts *B* and *U* denote the bound or unbound molecules of species *i* and *j*, where the first superscript letter corresponds to species *i* and the second superscript letter corresponds to species *j*. Total Onsager coefficients for *i* = *j* allow self- and distinct Onsager coefficient to be defined as

where subscripts *s* and *d* refer to the self-Onsager coefficients and distinct Onsager coefficients, respectively. While we could use these summations over mean squared displacement-like (msd) to calculate distinct Onsager coefficients, we choose to calculate them via $Lii,djj=Lii,tjj\u2212Lii,sjj$. These Onsager coefficients are used to calculate electrophoretic mobilities.

A representative plot of the values in ⟨⋯⟩ in Eqs. (8) and (10) is shown in Fig. S5 for *N* = 32 and $nCB=16$. The averages in the equations are calculated via a fast Fourier transform method.^{53,54} If we define ⟨⋯⟩ ∝ *t*^{β}, the exponent *β* allows linearity to be defined as 0.94 ≤ *β* ≤ 1.05,^{35,55} which is the criterion used in this article. Statistical uncertainty is calculated for all quantities using block averaging. The linear regime is separated into one decade blocks of time, and an average slope is obtained for each block length. Slopes for each block are used to calculate the standard error. This block averaging is used to estimate the statistical uncertainty in the Onsager coefficients, and, subsequently, the electrophoretic mobilities and transference numbers.

Total Onsager coefficients are used to define electrophoretic mobilities for bound, unbound, and all molecules of species *i* as

where *F* is Faraday’s constant. These electrophoretic mobilities are used to calculate transference numbers as

The conductivity, *κ*, is given by

where *F* is Faraday’s constant, the summations over *i* and *j* are over the bound and unbound states, and summations over *k* and *l* are over counterions and polyelectrolytes.

### C. Finite size effects

Finite size effects can be important with long-ranged electrostatic interactions.^{56} Self-diffusion coefficients can be corrected by applying a finite size correction according to^{56}

where $Dj,pbci$ is the uncorrected self-diffusion coefficient calculated from the simulation. *A* is a constant that corrects the diffusion coefficient, and *L* is the length of 1 edge of the simulation box. To determine the constant *A*, a number of simulations with differing box sizes are performed, and $Dj,pbci$ is plotted as a function of 1/*L*. Application of a linear fit to the data gives $Dji$ as the intercept and *A* as the negative slope. This procedure can be applied to Onsager coefficients with slight modifications because the Onsager coefficients depend on the system volume and particle number.

We modify Eq. (19) for total Onsager coefficients as

and self-Onsager coefficients as

where $Ljk,t,pbci$ and $Ljj,s,pbci$ are the uncorrected total and self-Onsager coefficients, respectively. Both self- and total Onsager coefficients are normalized by the number concentration of species *j*, *c*_{j} = *n*_{j}/*V*. Total Onsager coefficients are also divided by the number of molecules of species *k*, *n*_{k}. This removes the explicit concentration dependence from the Onsager coefficients and allows only finite size effects occurring due to periodic boundary conditions to be captured. Since this correction is proportional to the solution viscosity^{56} and the polyelectrolyte concentration is small (≈4 mM monomer concentration), we assume that the finite size effects are largely due to the viscosity of POL water and independent of chain length. *B* and *C* are the correction factors analogous to *A* and are determined in the same manner as *A*.

We examine finite size effects using *N* = 16 with a bound counterion fraction, *f*^{B} = 0.5, and a number of different box sizes (simulation parameters given in Table S2). Fits to the raw Onsager coefficient data are shown in Fig. S3, and correction values are given in Tables S3 and S4. These finite size corrections cause changes in the observed transference number values (Fig. S4) and are applied to all results presented in this article.

## III. RESULTS AND DISCUSSION

### A. Polyelectrolyte radius of gyration scaling with chain length

A fully charged chain without bound counterions is expanded in both the POL and implicit solvent. Figure 2 depicts *R*_{g} as a function of *N* for implicit and explicit solvents. Defining the scaling of *R*_{g} with *N* as *R*_{g} ∼ *N*^{ν} gives scaling exponents of *ν* ≈ 0.934 and ≈0.896 for POL and IMP, respectively. One expects that *R*_{g} ∼ *N*, i.e., rod-like behavior^{57} and deviations from this may be attributed to finite size of the chains.

### B. Implicit solvent

Onsager coefficients [Fig. S6(a)] for IMP show that the counterion and polyelectrolyte total Onsager coefficients are equal, and the counterion-polyelectrolyte Onsager coefficient has the same magnitude and the opposite sign of the counterion or polyelectrolyte total Onsager coefficient. For an ionic system, the number of independent Onsager coefficients is given as *n*(*n* − 1)/2, where n is the number of components in the system, and $\u2211i,kLkl,tij=0$, assuming the masses of the counterion and monomer are equal.^{35,39} An implicit solvent system has only two components, the counterion and polyelectrolyte, so $LCC,tUU+LCP,tUU=0$ and $LPP,tUU+LCP,tUU=0$. Since a Brownian dynamics integrator is used for implicit solvent systems, all ionic species experience a frictional force due to the implicit solvent.^{58} Even though this frictional force affects the polyelctrolyte and counterion momentum, no momentum transfer occurs from the ionic species to the solvent, so only one independent Onsager coefficient is measured. This means that the counterion and polyelectrolyte mobilities have equal magnitudes [Fig. S6(b)], so the transference number is 0.5, regardless of the chain length [Fig. S6(c)].

### C. Chain length dependence

For polyelectrolytes in POL water with no bound counterions, Fig. 3(a) shows that the polyelectrolyte distinct Onsager coefficient is much larger than the polyelectrolyte self-Onsager coefficient, which we attribute to the monomer connectivity. For short chain lengths (*N* < 64), the counterion self-Onsager coefficient is the largest contribution to the counterion total Onsager coefficient, as shown in Fig. 3(b). However, for longer chain lengths (*N* > 32), the counterion self- and distinct Onsager coefficients have similar magnitudes. Counterion–polyelectrolyte Onsager coefficients increase slightly with the chain length, but the statistical uncertainty makes this assignment uncertain.

The polyion electrophoretic mobility has a larger magnitude than the counterion electrophoretic mobility, as shown in Fig. 4. Scaling analysis of $uPU$ $(uPU\u223cN\alpha )$ reveals *α* = 0.14 ± 0.06. However, theory predicts *α* = 0.^{59} This theoretical prediction is based on a rod-like chain, but flexible chains are not expected to be rod-like even if the *R*_{g} scaling exponent is 1.^{59} Thus, we attribute the discrepancy between our simulations and the theoretical prediction to the flexible chains used in these simulations. The counterion mobility, $uCU$, exhibits similar scaling as the polyelectrolyte chain.

Transference numbers are largely independent of the chain length, as shown in Fig. 5. This is in contrast to the work of Fong *et al.*^{35} who found a decreasing transference number with increasing chain length. We attribute this to the presence of multiple chains and larger concentrations in the work of Fong *et al.*^{35} In contrast, this work contains only a single chain, so we are in the infinitely dilute limit. Further investigation is necessary to explicitly analyze the concentration dependence of transference numbers, but this is beyond the scope of this work.

### D. Fraction of monomers with bound counterions

Self-Onsager coefficients for all species and chain lengths are largely independent of the fraction of monomers with bound counterions, *f*^{B}, as shown in Figs. S12(a)–S12(d) for $kbC=kbM$ = 8000 kJ mol^{−1} nm^{−2}. For poleyelctrolytes, the distinct Onsager coefficient is the largest contribution to the total Onsager coefficient for both bound and unbound monomers and is strongly dependent on *f*^{B}. In addition, the bound–bound polyelectrolyte increase with *f*^{B} increases with *N*. The bound–bound counterion and bound–bound counterion–polyelectrolyte Onsager coefficients are similar to the polyelectrolyte Onsager coefficients. Unbound–unbound counterion Onsager coefficients are largely independent of *f*^{B} and *N*. The bound–unbound polyelectrolyte Onsager coefficient displays a nonmonotonic dependence on *f*^{B}. At *f*^{B} < 0.5, the number of unbound monomers is greater than the number of bound monomers, so the bound–unbound and unbound–unbound polyelectrolyte Onsager coefficients are greater than the bound–bound polyelectrolyte Onsager coefficient. For *f*^{B} > 0.5, this trend is inverted due to a larger number of bound monomers, so the bound–bound polyelectrolyte Onsager coefficient is the largest of these three Onsager coefficients. At *f*^{B} = 0.5, these three polyelectrolyte Onsager coefficients are nearly equal because the number of bound and unbound monomers is equal. These trends hold for the three chain lengths investigated.

The total counterion electrophoretic mobility decreases with increasing *f*^{B} until it becomes negative at *f*_{B} = 0.75, as shown in Fig. 6. As *f*^{B} becomes 1, this mobility is expected to become 0 because the bound–bound counterion and bound–bound polyelectrolyte Onsager coefficients become equal and, according to Eq. (14), $uiT$ = 0. These simulations do not exactly give $uiT$ = 0 due to statistical error. Electrophoretic mobility is a measure of how charged species would move due to an applied electric field. When *f*^{B} is less than 0.5, a majority of counterions do not have a bound monomer and would move in the opposite direction of the polyelectrolyte under an applied electric field. At *f*^{B} equal to 0.5, exactly half of the counterions have a bound monomer, which means half of the counterions would move with the polyelectrolyte and half in the opposite direction of polyelectrolyte under an applied electric field. This results in 0 mobility. At *f*^{B} greater than 0.5, a majority of counterions are bound to the polyelectrolyte and will move with the polyelectrolyte due to an applied electric field. For *f*^{B} = 1, the counterions are completely neutralized due to all being bound and the mobility becomes 0 because the electric field would no longer be able to move the neutral counterion–polyelectrolyte complex. This is consistent for all three chain lengths.

Unbound counterion mobility is independent of *f*^{B} and *N*, as shown in Fig. 6. The unbound counterions are always able to move opposite to the polyelectrolyte if an electric field is applied. However, the bound counterion mobility is always negative (until *f*^{B} = 1) because these counterions always move with the polyelectrolyte. The magnitude of this mobility decreases with increasing *f*^{B} because as more counterions are bound to the polyelectrolyte, the net charge on the polyelectrolyte–counterion complex becomes smaller. A smaller overall charge means that an applied electric field would not interact as strongly with this species, and the species would exhibit less mobility.

Concomitantly, all polyelectrolyte mobilities increase with increasing *f*^{B} because as the number of counterions bound to the chain increases, the total charge on the chain decreases and, thus, becomes less susceptible to movement due to an applied electric field. The magnitude of the mobility decreases as the charge on the polymer (plus bound counterions) decreases due to the reasons discussed for the bound counterion mobility. However, the bound, unbound, and total polyelectrolyte mobilities are all similar regardless of *f*^{B} because the monomers are bound together and cannot move independently of each other. The magnitude of the polyelectrolyte does increase slightly with *N* for all *f*^{B} except when *f*^{B} = 1.

At large enough *f*^{B} (>0.5), the counterion transference number becomes negative for *N* = 32 and 64 but not for 16. This is a consequence of the counterion mobility becoming negative for large *f*_{B} values, and for *N* = 16, the mobilities of both counterion and polyelectrolyte are too small to give a negative transference number. As shown in Fig. 7, the transference numbers of the unbound counterions are always positive, and those of the bound counterions are always negative. However, the magnitude of the bound transference number increases with increasing *f*^{B}, while the unbound transference number is largely independent of *f*^{B}. The transference numbers of bound counterions are negative because these counterions are completely correlated with the polyelectrolyte chain. Our results suggest that if over half of the counterions are bound to the polyelectrolyte chain, the transference number will be small and possibly negative. Note that *f*^{B} = 1 is not shown because the mobilities are 0, and the transference number is ill-defined. In addition to decreasing the fraction of bound counterions, decreasing the bonded interaction strength between the counterions and polyelectrolyte, $kbC$, increases the counterion transference number, as shown in Fig. S11 of the supplementary material.

### E. The Manning parameter

As the Manning parameter, *ξ*, is decreased, the transference number increases. In Fig. 8, the transference number is plotted as a function of *ξ* for *N* = 16 with no bound counterions. Even though the range of *b*_{ij} values is reasonably small (0.25–1.25 nm), the transference number almost doubles in value. In the context of the previously discussed explicitly bound counterion data, decreasing the Manning parameter should decrease both the number of counterions closely interacting with the chain and their strength of interaction. The counterion–polyelectrolyte Onsager coefficient (Fig. S13) is indeed decreasing with decreasing *ξ*. Decreasing the counterion–polyelectrolyte correlations allows the counterion mobility to increase while the polyelectrolyte mobility magnitude decreases (Fig. S14). These effects cause the transference number to almost double. This Manning parameter allows the transference number to increase with minimal impact to the conductivity, as shown in Fig. 9. This suggests that the spacing of charges along the polyelectrolyte chain is an important design parameter.

## IV. CONCLUSIONS

We present molecular dynamics simulations of infinitely dilute polyelectrolyte solutions to investigate the effects of the explicit solvent, chain length, number and strength of bound counterions, and the Manning parameter. Our results suggest that an explicit solvent is necessary to calculate Onsager coefficients and dynamic quantities calculated from Onsager coefficients. Transference numbers are largely insensitive to the chain length. However, the number of bound counterions has a profound effect on the transference number, and when the fraction of bound counterions is greater than 0.5, the transference becomes small and sometimes negative.

Analysis of the Manning parameter showed that the transference number can be almost doubled without altering the conductivity. This suggests that the Manning parameter is an important design parameter for increasing the transference number in polyelectrolyte solutions. More broadly, these results highlight the importance of ion binding when considering transference numbers. If enough ion pairs are formed, the transference number can indeed be negative.

## SUPPLEMENTARY MATERIAL

See the supplementary material for details regarding the implicit solvent simulations, results for dynamics with an implicit solvent, characterization of the neutral polymer solution, finite size corrections, results from varying the bond strength of the bound counterions, examples of mean square displacements for calculation of dynamic properties and Onsager coefficients, effect of simulation duration on Onsager coefficients, and effect of the Manning parameter on Onsager coefficients.

## ACKNOWLEDGMENTS

This work was supported by the National Science Foundation (Grant No. CHE-1856595). All simulations presented here were performed in computational resources provided by the UW-Madison Department of Chemistry HPC Cluster under NSF Grant No. CHE-0840494 and the UW-Madison Center for High Throughput Computing (CHTC).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors declare no conflicts of interes.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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