Development of United-Atom Force Fields for Monomeric and Oligomeric Ionic Liquids through Regression-Guided Optimization of Electronic Continuum Correction Parameters

Abstract

Ionic liquids are salts that exist in the liquid state at room temperature and exhibit high viscosity because of their strong electrostatic interactions. It was difficult to reproduce their viscosity by molecular dynamics simulations with conventional nonpolarizable force fields; however, recent development of force fields implementing electronic continuum correction (ECC), which accounts for polarizability, has enabled accurate predictions. Here, we present a regression-guided strategy to optimize scaling factors for ECC charges and Lennard-Jones parameters for monomeric (BMIM⁺, MOEMIM⁺) and oligomeric (IL2²⁺, IL4⁴⁺) imidazolium-based cations paired with TFSI^– with united-atom models. The scaling factors were optimized to simultaneously reproduce experimental density and viscosity. To validate the strategy, we calculated the temperature dependence of density, diffusion coefficient, conductivity, and viscosity of BMIM–TFSI, achieving good agreements with experiments. Moreover, scaling factors optimized for MOEMIM⁺, which shares similar chemical structure and elemental compositions with IL2²⁺ and IL4⁴⁺, were found to be transferable to these compounds. Thus, this work not only provides a practical regression-guided workflow for selecting ECC-based united-atom force-field parameters but also suggests their transferability across chemically related ionic liquids.

1. Introduction

Ionic liquids (ILs) are a unique class of solvents that remain liquid at room temperature and exhibit distinctive physicochemical properties, including negligible vapor pressure, wide electrochemical windows, and high thermal stability. − Owing to these characteristics, ILs have emerged as promising candidates for applications in electrochemistry, lubrication, catalysis, and energy storage. − Among them, bis­(trifluoromethanesulfonyl)­imide (TFSI^–) is one of the most practically important and extensively studied room-temperature ionic liquids to date. − In particular, ILs composed of TFSI anions and imidazolium-based cations, such as 1-butyl-3-methylimidazolium (BMIM⁺), exhibit relatively low viscosity and high ionic conductivity compared to many other ILs, making them well-suited for use as electrolytes and extraction solvents. , Consequently, BMIM–TFSI has become a benchmark system and has been widely investigated through both experimental and computational approaches. − More recently, oligomeric ionic liquids such as imidazolium ion tetramers (IL4⁴⁺) and dimers (IL2²⁺) paired with TFSI^– have attracted increasing attention due to their multivalent structures, which enable stronger electrostatic interactions and unusually high electric double-layer capacitance.

Molecular simulations of ionic liquids have been extensively conducted using both all-atom (AA) and united-atom (UA) models, each offering different levels of molecular detail and computational cost. − Although AA models provide a more explicit representation of ionic liquids, their relatively high computational demand restricts the accessible time, particularly for systems exhibiting slow dynamics. In contrast, UA approaches, though they cannot account for hydrogen atoms, have been shown to reproduce the structural properties of imidazolium-based ILs with reasonable accuracy while enabling simulations of larger systems over longer time scales. − ,,, For both the AA and UA models, a persistent challenge in IL simulations is that conventional nonpolarizable force fields often fail to accurately reproduce transport properties such as conductivity and diffusion coefficients, because such force fields tend to overestimate Coulombic interactions by neglecting electronic polarization. , Accordingly, to simulate the diffusion coefficient at 300 K, it was necessary to unphysically increase the temperature to 400 K.

Recent studies have advanced the development of ionic liquid force fields in several directions, including refined classical force fields, polarizable models, automated or semiautomated parameter optimization, and machine-learning potentials. − Although some polarizable and machine-learning potentials can naturally reproduce viscosity, they require extensive computational costs than classical models. Accordingly, classical nonpolarizable force fields remain valuable for simulations of large ionic liquid systems, including electric double-layer structures and properties, as well as long-chain oligomeric ionic liquids.

The limitation of nonpolarizable force fields was first systematically addressed in studies of water and simple molecular liquids, where the electronic continuum correction (ECC) was introduced to account for fast electronic screening in a mean-field manner by uniformly scaling atomic charges according to the high-frequency dielectric constant. While this approach markedly improves the description of ion–solvent electrostatics and transport properties, charge scaling alone disrupts the balance of short-range interactions and can result in excessive local packing. ,− Consequently, a simultaneous adjustment of the Lennard-Jones size parameter is required to restore physically meaningful short-range repulsion. These insights from electrolyte solutions naturally extend to ionic liquids, where electrostatic interactions dominate in the absence of a neutral solvent. , Accordingly, the ECC framework, combined with Lennard-Jones size reparameterization, provides a practical strategy to incorporate missing electronic polarization effects and achieve better agreement with experimental structural and transport properties of ionic liquids. ,

To the best of our knowledge, no rigorous and universally accepted protocol currently exists for selecting the optimal combination of charge and Lennard-Jones size scaling factors that ensures quantitative agreement with the experimental data. In a previous study, these factors have been chosen empirically, based on visual inspection and qualitative comparison between simulated and experimental results. Within the limited parameter ranges typically explored, density and viscosity exhibit smooth and approximately linear dependence on the scaling factors for charge and Lennard-Jones parameters, which allows the use of a linear approximation to interpolate simulation results and determine optimal force-field parameters consistent with experimental data.

In this study, we developed UA models of BMIM⁺, 1-(2-methoxyethyl)-3-methylimidazolium (MOEMIM⁺), IL2²⁺, and IL4⁴⁺ using the electronic continuum correction (ECC) combined with Lennard-Jones size reparameterization through more than 0.4 ms molecular dynamics (MD) simulations in total (Figure a). These very long simulations were required to explore the parameter space, since IL2²⁺ and IL4⁴⁺ exhibit very slow dynamics. A comprehensive evaluation of structural and transport propertiesincluding density, diffusion coefficients, radial distribution functions, ionic conductivity, viscosity, and dielectric constantswas performed for the BMIM–TFSI system to validate the developed model. To enable a more objective and reproducible selection of the charge-scaling factor and Lennard-Jones size parameter, we propose a mathematical framework based on linear regression analysis. The advantages and limitations of constructing a common parameter-selection model using BMIM–TFSI alone are examined and compared with models incorporating IL2–TFSI and IL4–TFSI. Particular attention is given to the role of cation size and the number of charged units in determining the transferability of the optimized parameters. It was found that MOEMIM⁺, whose chemical structure and elemental composition are closer to IL2²⁺ and IL4⁴⁺ compared to BMIM⁺, has higher transferability than BMIM⁺. Furthermore, we analyze which physical propertiesparticularly density and viscosityshould be prioritized in identifying optimal parameter sets, thereby providing practical guidelines for force-field refinement of ionic liquids within the ECC framework.

1.

Chemical structure of developed ionic liquids and simulated systems. (a) Chemical structures of the monomeric BMIM⁺, TFSI^–, MOEMIM, dimeric IL2²⁺, and tetrameric IL4⁴⁺ used in this study. Scale bar: 10 Å. (b) Representative snapshots in MD simulations for BMIM–TFSI, MOEMIM–TFSI, IL2–TFSI, and IL4–TFSI systems. Cations and anions are shown in blue and red, respectively. Scale bar: 10 Å.

2. Methods

2.1. United-Atom Molecular Model Construction and Force-Field Parameterization

Our strategy to develop a UA model basically follows the Siqueira and Ribeiro’s paper. Specifically, molecular structures were constructed using the Avogadro molecular editor and preoptimized with a force-field-based method to eliminate unfavorable atomic contacts. The resulting geometries were subsequently used as input for quantum chemical calculations. Ab initio electronic structure calculations were performed using the Gaussian 09 package, and geometry optimizations were carried out at the second-order Møller–Plesset perturbation theory (MP2) level with the 6–311+G* basis set. Partial atomic charges were derived from the electrostatic potential computed on the optimized electron density.

Following the quantum chemical calculations, a UA representation was generated by summing the partial charges of hydrogen atoms into their corresponding bonded heavy atoms, after which explicit hydrogens were removed. For all cations, force field of bonds, angles, and dihedral parameters were adopted following OPLS. Lennard-Jones parameters of σ were first taken from OPLS but scaled as explained later.

2.2. Computational Setup

The simulated systems consisted of 1000 ion pairs for BMIM–TFSI and MOEMIM–TFSI, 500 IL2²⁺ and 1000 TFSI^– for IL2–TFSI, and 250 IL4⁴⁺ and 1000 TFSI^– for IL4–TFSI, corresponding to a total of 25000 particles in both BMIM–TFSI and MOEMIM–TFSI systems and 28000 particles in both IL2–TFSI and IL4–TFSI systems (Figure b).

MD simulations were performed using the AMBER 22 simulation package under periodic boundary conditions. Initially, the system was equilibrated in the isothermal–isobaric (NPT) ensemble at 300 K and 1 bar using the Langevin thermostat and the Berendsen barostat. Temperature coupling was applied with a collision frequency of 2 ps^–1, and isotropic pressure coupling was employed to maintain ambient pressure. The equilibrium density was estimated from the NPT simulations after the density reached a stable plateau (Figure S1). Then, the systems were further relaxed at constant volume. Subsequently, production simulations were carried out in both NVT and microcanonical (NVE) ensembles. The viscosity, self-diffusion coefficient, radial distribution functions (RDFs), and structure factor were calculated from trajectories in the NVT ensemble, while conductivity was evaluated from NVE simulations. A time step of 2 fs was used throughout all simulations. Nonbonded interactions were treated using a cutoff distance of 12 Å in real space, and long-range interactions were treated using the Particle Mesh Ewald (PME) method. The analysis was conducted using MDAnalysis and our in-house programs based on python. The simulation lengths and box sizes used in the parameter-scan and optimized parameter simulations are summarized in Tables S1 and S2.

2.3. Electronic Continuum Corrections

The electronic continuum correction (ECC) accounts for electronic polarization in a mean-field manner by treating electrons as a fast, uniform screening background, thereby incorporating electronic screening into nonpolarizable MD simulations without explicit polarization. This effect can be incorporated by scaling the atomic charges (q) using a constant factor, f = ε_el ^–1/2, yielding ECC-corrected charges (q ^ECC) as

$$q^{\mathrm{E}\mathrm{C}\mathrm{C}}=fq$$ 1

The pure high-frequency (electronic) dielectric constant is approximately ε_el ≈ 2.0, which corresponds to a charge-scaling factor of f ≈ 0.70. For ionic liquids, this value provides a physically motivated starting point to mimic fast electronic screening, significantly weakens excessive ion–ion over binding, and improves agreement with experimental structural and transport properties. However, while charge scaling effectively reduces electrostatic attraction, it simultaneously weakens short-range electrostatic repulsion. To restore a balanced description of short-range interactions and prevent unrealistically close ion contacts, the Lennard-Jones size parameters must therefore be scaled upward accordingly. ,,

To obtain inputs for the mathematical model to exploring the best charge-scaling factor (f) that best agrees with experiments, for BMIM–TFSI, f was varied from 0.70 to 1.00 in increments of 0.05, while the Lennard-Jones size scaling parameter (s) was increased from 1.00 to 1.10 in increments of 0.02. This parameter space resulted in a total of 42 distinct (f, s) combinations examined for BMIM–TFSI and MOEMIM–TFSI. For IL2–TFSI and IL4–TFSI, the charge-scaling factor was varied from 0.70 to 0.85 with a step size of 0.05, while the Lennard-Jones parameters were kept identical across all cases. Under these conditions, 24 parameter sets were evaluated for each of IL2–TFSI and IL4–TFSI. For every simulation set, the density and viscosity were calculated and compared against experimental values to identify the optimal parameter combination. Figure summarizes the dependence of density and viscosity of BMIM–TFSI on the charge and Lennard-Jones size scaling factors. The results for MOEMIM–TFSI, IL2–TFSI, and IL4–TFSI will be shown later. All the optimized charges and atom type in AMBER are described in Figures S2 and S3, respectively. The parameter files are also provided as electronic Supporting Information.

2.

Dependency of the (a) density and (b) viscosity on the charge-scaling factor (f) and the scaling factor for the Lennard-Jones parameters of σ (s) in the case of BMIM–TFSI.

2.4. Viscosity Calculation

In this work, viscosity was calculated from equilibrium MD simulations using the traditional Green–Kubo formalism, where the shear viscosity, η, is calculated by integrating the time autocorrelation function of the second-order pressure tensor (P _s ) as

$$\eta =\frac{V}{k_{\mathrm{B}}T}{\int }_0^{\infty }⟨P_s(t_0+t)P_s(t_0)⟩dt$$ 2

where V is the system volume, k _B is the Boltzmann constant, T is the temperature, and t denotes time. The angular brackets represent an ensemble average. To calculate P _s using AMBER, we followed the method proposed by Ando: P _s corresponds to one of the shear components of the pressure tensor, specifically (P _xx – P _yy )/2, (P _yy – P _zz )/2, and (P _xx – P _zz )/2. This is because P _xy , P _yz , and P _xz , typically used for viscosity estimation, are not available in AMBER, whereas P _xx , P _yy , and P _zz are given in the output files. The autocorrelation functions were computed for each stress component from 1 μs simulation for BMIM–TFSI and MOEMIM–TFSI, 5 μs for IL2–TFSI, and 7 μs for IL4–TFSI and integrated numerically over time to obtain the corresponding cumulative viscosity contributions. The time integration was performed using a trapezoidal scheme, yielding a time-dependent viscosity η­(t). The viscosity was obtained from the plateau region of η­(t) between 10 and 20 ns, where the integral converges and statistical noise remains minimal. Figure S4 exhibits cumulative integrals of the pressure tensor autocorrelation functions at different temperatures, and the enlarged plateau region in panel (b) indicates the time window used to estimate viscosities. The final viscosity was calculated as the average of the viscosities obtained from three stress tensor components.

2.5. Conductivity Calculation

An appropriate method for calculating the static conductivity (κ) for an electrically neutral system is the Einstein–Helfand method shown in the paper by Zeindlhofer et al. In this approach, the mean-squared displacement of the collective translational dipole moment M _J (t)­

$$\kappa =\frac{1}{6V{k}_{\mathrm{B}}T}\underset{t\gg t_c}{\lim }\frac{d}{dt}⟨\mathrm{\Delta }{\mathit{M}}_D^2(t)⟩$$ 3

with t _c being an initial correlation time and M _D (t) being derived from the charges q _iβ and the center of mass r _i (t) of molecule i

$${\mathit{M}}_D(t)=\sum _i\sum _{\beta }{q}_{i\beta }({\mathit{r}}_{i\beta }(t)-{\mathit{r}}_i(t))$$ 4

where summation runs over all particles or interaction sites β in all molecules. This reference choice is necessary because both BMIM⁺ and TFSI^– possess net molecular charge. In this method, unfolded MD trajectories are used to construct collective translational dipole moments. A clear linear region appeared in the correlation function of M _D (t) in 1 ns (=t _c ) < t < 10 ns, so κ was estimated from the slope. Figure S5 shows the mean-squared displacement of the collective translational dipole moment for BMIM–TFSI at different temperatures, together with the time window for the linear fitting. The total conductivity was decomposed into molecular-species contributions following refs ,, to evaluate the roles of cationic, anionic, and cation–anion correlated motions.

2.6. Calculation of Radial Distribution Functions

RDFs were calculated from MD trajectories to characterize the local structural organization between the BMIM⁺ cation and the TFSI^– anion. The RDF, g(r), describes the probability of finding a particle at a distance r from a reference site relative to the number density and is defined as

$$g(r)=\frac{1}{4\pi r^2\rho }⟨\frac{dN(r)}{dr}⟩$$ 5

where ρ is the number density and dN(r) is the average number of particles within a spherical shell of thickness dr at distance r.

The RDFs were computed using the MDAnalysis package from equilibrated trajectories generated at 300 K. Atom groups were defined based on chemically relevant interaction sites: carbon atoms of the imidazolium ring (C _ring), nitrogen atoms of the imidazolium ring (N _ring), the center of carbon and nitrogen atoms of the imidazolium ring (I _ring), the central nitrogen atom of the TFSI^– anion (N _TFSI), and the oxygen atoms of TFSI^– (O _TFSI). RDFs were evaluated for the C _ring – O _TFSI, C _ring – N _TFSI, N _ring – N _TFSI, and I _ring – N _TFSI pairs.

2.7. Calculation of Structure Factors

The static X-ray scattering intensity was computed using a reciprocal-space formalism. For each reciprocal vector k , the scattering amplitude was defined as

$$A(\mathit{k})=\sum _{i=1}^N{f}_i(k)e^{-i\mathit{k}{\mathit{r}}_i}$$ 6

where r _i is the position of particle i and f _i (k) is the X-ray atomic form factor. The k-dependent atomic form factors were evaluated using the Cromer–Mann coefficients:

$$f(k)=\sum _{i=1}^4{a}_i\exp \{-b_i{\left(\frac{k}{4\pi }\right)}^2\}+c$$ 7

where a _i , b _i , and c are coefficients for each element. The scattering intensity was then calculated as

$$I(k)=\frac{1}{N}⟨{|A(\mathit{k})|}^2⟩$$ 8

where N is the number of particles in the system.

The structure factor of the UA model was estimated in two ways. First, hydrogen atoms were assumed to be located at their bonded carbon atoms using the following equation for the atomic form factor of the carbon atoms:

$$f_{\mathrm{C}\mathrm{H}n}(k)=f_{\mathrm{C}}(k)+n{f}_{\mathrm{H}}(k)$$ 9

where n is the number of hydrogen atoms associated with the carbon atoms. Second, we computed the structure factor without taking the hydrogen into account.

2.8. Dielectric Constant

The dielectric constant of the ionic liquid was evaluated using the permanent dipole time-correlation formalism

$$\epsilon (\omega )-{\epsilon }_{\infty }=\frac{4\pi }{3V{k}_{\mathrm{B}}T}{\int }_0^{\infty }-\left(\frac{\partial }{\partial t}⟨{\mathit{M}}_D(0)·{\mathit{M}}_D(t)⟩\right)e^{i\omega t}dt$$ 10

where ε_∞ is 1. The static dielectric constant was obtained from the zero-frequency limit (ω = 0).

2.9. Molar Enthalpy of Vaporization

The molar enthalpy of vaporization (Δh _vap) was estimated using the following equation:

$$\mathrm{\Delta }{h}_{\mathrm{v}\mathrm{a}\mathrm{p}}=⟨u_{\mathrm{g}\mathrm{a}\mathrm{s}}⟩+k_{\mathrm{B}}T-⟨u_{\mathrm{l}\mathrm{i}\mathrm{q}}⟩$$ 11

where u _gas and u _liq are the molar internal energy in gas and liquid phases, respectively. Separate gas-phase simulations for each type of cluster were conducted to estimate u _gas.

2.10. Mathematical Modeling

To identify the optimal charge-scaling factor (f) and Lennard-Jones size scaling parameter (s) for BMIM–TFSI, MOEMIM–TFSI, IL2–TFSI, and IL4–TFSI systems, a regression-guided mathematical modeling approach was employed based on the simulation results obtained across the explored parameter space.

For each system, MD simulations were performed for a set of predefined combinations (f, s). For every parameter set, the equilibrium density (ρ) and shear viscosity (η) were calculated. The dependence of these two properties on the force-field scaling parameters was approximated using a linear approximation model in the (f, s)­parameter space. Specifically, density and viscosity were independently fitted to planar functions of the form

$$\rho (f,s)=a_{1}f+b_{1}s+c_1$$ 12

$$\eta (f,s)=a_{2}f+b_{2}s+c_2$$ 13

where a _i , b _i , and c _i are regression coefficients obtained via least-squares fitting to the simulation data.

Once the regression models were established, the optimal scaling factors were determined by inverse mapping. The experimental density ρ_exp and viscosity η_exp were imposed as target values, and the corresponding values of f and s were obtained by solving the coupled linear equations

$$\rho (f,s)={\rho }_{\exp }\text{and}\eta (f,s)={\eta }_{\exp }$$ 14

These equations were solved analytically to yield explicit expressions for f(ρ, η) and s(ρ, η), allowing direct estimation of the optimal parameter pair consistent with the experimental thermodynamic properties.

3. Results and Discussion

3.1. Dependency of Density and Viscosity of BMIM–TFSI on the Scaling Factors for Charge and Lennard-Jones Parameters

To develop a mathematical model which simultaneously best fits the density and viscosity with experiments, the densities and viscosities were calculated at several scaling factors for charge (f) and the Lennard-Jones σ (s). Figure shows how they depend on f and s. The density decreases approximately linearly with increasing s, while it increases with higher f. As expected, this indicates enhanced molecular packing under smaller radius of the particles and/or stronger electrostatic interactions. In contrast, Figure b shows that viscosity is significantly more sensitive to f and sharply increases at higher values of f. This pronounced electrostatic dependence explains the difficulty in accurate simulation without the charge-scaling factor. From the density plot, six possible intersection points with the experimental value (dashed line) can be identified, while four such points are obtained from the viscosity plot. Based on a visual inspection of these discrete data points, the optimal parameter region can be inferred to lie around s ∼ 1.07 and f ∼ 0.8. However, a more rigorous and sophisticated approach is required to determine the optimal parameter set with higher accuracy.

3.2. Mathematical Model for Optimizing Scaling Factors of BMIM–TFSI

To precisely obtain a best pair of f and s, a mathematical model was built. Using the inverse linear modeling approach described in methodology, explicit expressions relating f and s to the density (ρ) and viscosity (η) were obtained for BMIM–TFSI:

$$f=0.388\rho +0.310\eta +0.246$$ 15

$$s=-0.433\rho +0.024\eta +1.690$$ 16

By substituting the experimental density (1.435 g/mL) and viscosity (0.045 Pa·s) into these expressions, the optimal parameter pair was estimated as f = 0.817 and s = 1.070. To confirm that these factors successfully reproduce the experimental density and viscosity of BMIM–TFSI within the target accuracy, MD simulations were performed using this parameter set. The simulated density was 1.43 g/mL, and viscosity was 0.049 Pa·s. This confirmed the validity of the proposed linear inverse modeling strategy for parameter optimization.

3.3. Validation of the BMIM–TFSI Model

MD simulation was performed with f = 0.817 and s = 1.070 for the BMIM–TFSI system to validate the developed model. The temperature dependencies of the density, the diffusion coefficient, viscosity, and conductivity were computed for comparison with experimental data reported by Tokuda et al. The f and s optimized at 300 K were used at different temperatures. RDFs at 300 K were computed and compared with those obtained by the AA models and ab initio force field, and the structure factor was computed for comparison with the data obtained in a recent small-angle X-ray scattering (SAXS) experiment. The dielectric constant and molar enthalpy of vaporization at 300 K were also computed and compared with experimental data. ,

3.3.1. Density

Figure presents the temperature dependence of the simulated density of BMIM–TFSI compared with the experimental data by Tokuda et al. The density decreases monotonically with increasing temperature from 280 to 320 K, which is consistent with the expected thermal expansion behavior of ionic liquids. Despite the use of f and s values optimized at 300 K, the simulation results show good agreement with the experimental data at different temperatures. It means that f and s do not have a strong dependence on temperature. The average densities obtained for all systems investigated in this study are summarized in Table S3.

3.

Comparison of the density of BMIM–TFSI at different temperatures (280–320 K) with experimental data (dashed black line) reported by Tokuda et al. The shaded region represents the standard deviation obtained from three independent simulation runs.

3.3.2. Diffusion Coefficient

The simulated and experimental self-diffusion coefficients of BMIM⁺ and TFSI^– as a function of temperature are shown in Figure and Table S4. The self-diffusion coefficients were obtained from the linear slope of the mean-squared displacement using the Einstein relation and were subsequently corrected using the Yeh-Hummer correction. Figure S6 shows the MSD curves of (a) BMIM⁺ and (b) TFSI^– at different temperatures. The Yeh-Hummer correction was small (less than 5% for BMIM⁺ and 8% for TFSI^–), since the system size was relatively large. In both cases, the diffusion coefficients increase with temperature, reflecting enhanced molecular mobility at higher thermal energy. For BMIM⁺ (Figure a), the simulated diffusion coefficients are slightly higher than the experimental values over the entire temperature range, but the overall trend and magnitude remain comparable. In contrast, for TFSI^– (Figure b), the simulation results closely match the experimental data over the entire temperature range. Thus, the diffusion coefficients were well reproduced in our approach using the ECC.

4.

Comparison of simulated and experimental diffusion coefficients of BMIM–TFSI as a function of temperature: (a) BMIM cation and (b) TFSI anion. Experimental data (dashed black lines) are taken from Tokuda et al. The shaded region represents the standard deviation obtained from three independent simulation runs.

3.3.3. Conductivity

Figure compares the calculated temperature-dependent electrical conductivities of BMIM–TFSI with experimental data reported by Tokuda et al. The conductivity values are provided in Table S5. The simulated conductivity monotonically increases with temperature, reflecting enhanced ionic mobility at higher thermal energy. The simulation results show good quantitative agreement with the experimental values across the studied temperature range.

5.

Electrical conductivity of BMIM–TFSI at different temperatures. The experimental data reported by Tokuda et al. is shown in the dashed black line. The shaded region represents the standard deviation obtained from three independent simulation runs.

To examine the molecular origin of the ionic conductivity, the total conductivity was decomposed into species-resolved contributions. The corresponding contributions to the mean-squared displacement of the collective translational dipole moment for BMIM-TFSI at 300 K are shown in Figure S7. The species-resolved conductivity contributions for all systems investigated in this study are also summarized in Table S6. Interestingly, the cation–anion cross-correlation term contributes positively to the total conductivity. This behavior is consistent with the discussion by Avula et al., who reported that the cation–anion contribution can be positive in pure ionic liquids and attributed this feature to momentum conservation, following the work of Kashyap et al.

3.3.4. Viscosity

As explained in the Methods, viscosity was computed using the Green–Kubo formalism for independent validation, instead of using the Stokes–Einstein relation with the diffusion coefficients and assumed radius. Figure compares the viscosities obtained from the simulations at different temperatures with experimental values reported by Tokuda et al. Table S7 summarizes the viscosity values. The calculated viscosities exhibited a monotonic decrease with an increase in temperature, consistent with the expected temperature dependence of ionic liquids. The simulation results showed good quantitative agreement with the experimental data over the entire temperature range examined.

6.

Comparison of the simulated viscosity of BMIM–TFSI at different temperatures with experimental data (dashed black line) reported by Tokuda et al. The shaded region represents the standard deviation obtained from three independent simulation runs.

3.3.5. Radial Distribution Functions

Figures a–d show the radial distribution functions between selected atomic sites of the BMIM⁺ cation and the TFSI^– anion. All RDFs exhibit a pronounced first peak at distances of approximately 4–6 Å, reflecting strong short-range correlations between the imidazolium ring and the anion. This distance roughly corresponds to the molecular size of BMIM⁺ and TFSI^–. Beyond the first coordination shell, all RDFs gradually approach unity with weak oscillations, consistent with a liquid-like structure lacking long-range order.

7.

Radial distribution functions (RDFs) and structure factor. (a) RDF from N _ring (nitrogen atoms in the imidazole) to N _TFSI (N atom in TFSI). (b) RDF from C _ring (carbon atoms in the imidazole) to N _TFSI. (c) RDF from C _ring to O _TFSI (O atoms in TFSI). (d) RDF from I _ring (geometrical center of 5 atoms in imidazole) to N _TFSI. (e) Structure factor calculated using only heavy atoms. (f) Structure factor calculated by assuming that hydrogen atoms are located at their bonded carbon atoms.

The RDFs of N _ring – N _TFSI (Figure a), C _ring – N _TFSI (Figure b), and C _ring – O _TFSI (Figure c) were very close to those of TFSI^– in the transoid conformation obtained by the ab initio simulation. However, the ab initio study showed that TFSI^– in liquids takes 70% cisoid and 20% transoid conformation, and an additional small peak appears at ∼3.3 Å in the RDFs of N _ring – N _TFSI and C _ring – N _TFSI in the cisoid conformation. Given that the difference between the cisoid and transoid conformation is the orientation of the CF₃, tune-up of the dihedral angle parameter of TFSI^– is considered to improve the RDFs. In addition, a small peak at ∼3.7 Å seen in the RDF of I _ring to N _TFSI in the previous simulation using an AA model was not present in our model (Figure d). This discrepancy is presumably attributed to the absence of hydrogen atoms in our UA model, since the bond, angle, and dihedral parameters were shared. On the other hand, the RDF of N _ring – N _TFSI simulated using another AA model showed no such minor peak at ∼3.7 Å, which is consistent with our results.

3.3.6. Structure Factor

The overall shape of the computed structure factor of BMIM–TFSI was in agreement with the recent SAXS experiment. The structure factor was computed either by assuming that hydrogen atoms are located at their bonded carbon atoms or considering only heavy atoms as shown in Figures e and f, respectively. In both cases, the intensity monotonically increased up to the first peak at k = 0.87 Å^–1, followed by the second peak at 1.43 Å^–1, and then gradually decreased.

When hydrogen atoms are assumed to be located at their bonded carbon atoms, the intensity of the second peak is higher than that of the first peak. This trend is consistent with the experimental results, but the second peak is overestimated compared to those of experiments, suggesting that this assumption may not be appropriate. On the other hand, when only heavy atoms are used to calculate the structure factor, the intensity of the first peak is slightly higher than the second opposite to the experiments, but the ratio of peak intensity is comparable to that in experiments. Although these results highlight the difficulty of accurately calculating structure factors using UA models, neglecting hydrogen atoms is a better approximation since its overall shape is closer to the experimental result (Figure S8).

3.3.7. Dielectric Constant

The static dielectric constant of BMIM–TFSI at 300 K was calculated using the dipole-fluctuation formalism described in eqs and . The calculated dielectric constant was found to be ε = 5.8, which is lower than the experimental value of approximately 11.6 reported for BMIM–TFSI at comparable conditions. This underestimation of the dielectric constant, which sometimes happen in the MD simulation of ionic liquids, , indicates that the present nonpolarizable and charge-scaled force field does not fully capture the dielectric response of the ionic liquid.

3.3.8. Molar Enthalpy of Vaporization

The calculated molar enthalpy of vaporization was 133.7 kJ/mol. It is very close to the experimental value of 134 kJ/mol at 298 K. The vaporization enthalpies and relative errors for all monomeric and oligomeric ionic liquid systems investigated in this study are summarized in Table S8.

3.4. Modeling of IL2, IL4, and MOEMIM

The same inverse linear modeling procedure as used for the BMIM-TFSI system was applied to the IL2–TFSI and IL4–TFSI systems to determine the optimal charge-scaling factor (f) and the scaling factor for the Lennard-Jones σ parameter (s). First, we simulated the densities and viscosities of the IL2–TFSI and IL4–TFSI systems with varying f and s (Figure ). Then, the obtained results were used to make mathematical models.

8.

Dependence of density ((a) IL2–TFSI, (c) IL4–TFSI) and viscosity ((b) IL2–TFSI, (d) IL4–TFSI) on the charge-scaling factor (f) and the Lennard-Jones scaling parameter σ­(s).

For IL2–TFSI (Figure a,b), the fitted relations between the scaling factors and the thermodynamic properties were obtained as

$$f=0.289\rho +0.156\eta +0.299$$ 17

$$s=-0.444\rho +0.014\eta +1.707$$ 18

Substitution of the experimental density (1.436 g/mL) and viscosity (0.27 Pa·s) values into these equations yielded an optimal parameter set of f = 0.756 and s = 1.073. The simulated density and viscosity using these parameters were 1.43 g/mL and 0.30 Pa·s, respectively.

For IL4–TFSI (Figure c,d), the corresponding linear models were determined as

$$f=0.366\rho +0.058\eta +0.144$$ 19

$$s=-0.420\rho +0.003\eta +1.692$$ 20

Using experimental input values (1.438 g/mL and 1.2 Pa·s), the optimal parameters were f = 0.740 and s = 1.092. The simulated density and viscosity using these parameters were 1.43 g/mL and 1.12 Pa·s, respectively.

It is interesting to understand how ρ contributes to f and s. The contribution was quantified by the ratio of sum of the first and third terms in eqs – to f or s. For BMIM⁺, both f and s are determined almost entirely by ρ: accounting for 98.3% of f and 99.9% of s. For IL2²⁺, 94.4% of f and 99.6% of s arise from ρ, and 90.6% of f and 99.6% of s are from ρ for IL4⁴⁺. As the molecular length is elongated, the contribution of ρ to f decreases, since the longer molecules exhibit higher viscosity and η starts to contribute to the interactions. In contrast, more than 99% of s is determined by ρ irrespective of the molecular length. This is reasonable, since s is involved in determining the particle size of the molecules.

It was investigated if f and s obtained for a small molecule can be used for larger molecules. Substituting f and s optimized for BMIM–TFSI (f = 0.817 and s = 1.070) into the mathematical models for IL2–TFSI and IL4–TFSI (eqs –) enabled the density and viscosity prediction. The predicted viscosities of IL2–TFSI and IL4–TFSI were 0.626 and 2.16 Pa·s, respectively, which are approximately twice as the experimental values. These slower dynamics are reasonable since the f value is larger than those optimized for IL2–TFSI and IL4–TFSI, resulting in stronger electrostatic interactions. This also highlights that even slight differences in f can lead to overestimation or underestimation of the dynamics. The predicted densities of IL2–TFSI and IL4–TFSI were 1.45 and 1.50 g/mL, respectively. The density of IL4–TFSI was slightly higher than the experimental value, again indicating that the stronger interaction leads to an overestimation of the density.

From a chemical perspective, a notable difference between BMIM⁺ and the oligomer (IL2²⁺ and IL4⁴⁺) likely originates from the presence of oxygen atoms in the oligomers, which are absent in BMIM⁺. The inclusion of oxygen atoms changes the electronic state, which in turn alters the charge distribution (Figure S1). As such, we made a mathematical model for MOEMIM⁺ (Figure ), in which one carbon in the long chain of BMIM⁺ was replaced with an oxygen atom (Figure a). The model was

$$f=0.407\rho +0.187\eta +0.164$$ 21

$$s=-0.434\rho +0.018\eta +1.709$$ 22

With the experimental values (1.45 g/mL and 0.042 Pa•s), f = 0.763 and s = 1.081. Note that they are close to the optimized f and s for IL2²⁺ (f = 0.756 and s = 1.073) and those for IL4⁴⁺ (f = 0.740 and s = 1.092). Using the optimized f and s for MOEMIM⁺, the predicted densities of IL2–TFSI and IL4–TFSI using their models were, respectively, 1.42 and 1.46 g/mL, and the predicted viscosities of IL2–TFSI and IL4–TFSI were, respectively, 0.342 and 1.43 Pa•s. They are close to the experimental values. These results indicate that optimal scaling factors derived from a monomer or a small molecule can be transferred to its oligomers or larger molecules, if the chemical structures, including elemental compositions, are similar. This approach shortcuts the computational costs of estimating density and viscosity of oligomers using MD simulations, which otherwise required approximately 0.3 ms simulation to generate input data for the mathematical modeling of IL2–TFSI and IL4–TFSI.

9.

Dependency of the (a) density and (b) viscosity on the charge-scaling factor (f) and the scaling factor for the Lennard-Jones parameters of σ (s) in the case of MOEMIM–TFSI.

The equations for the scaling factors of the Lennard-Jones parameter (eqs , , , and ), which account for geometric packing effects, share a similar functional form, exhibiting similar variable dependencies and comparable coefficient magnitudes across the different systems. In contrast, the charge-scaling-factor equations (eqs , , , and ) are considerably more difficult to predict, suggesting a more complex and nonlinear dependence on the underlying electrostatic interactions. The difficulty in predicting the charge-scaling factor f, which is intended to approximate many-body polarization and local electrostatic screening, is presumably related to the nonadditive nature of electronic polarization.

In Table , the optimized factors of f and s and all computed thermodynamic quantities for all molecules are listed.

1. Optimized Scaling Factors (f and s) and Thermodynamic Quantities at 300 K.

	BMIM–TFSI	IL2–TFSI	IL4–TFSI	MOEMIM–TFSI
F	0.817	0.756	0.740	0.763
S	1.070	1.073	1.092	1.081
density (g/mL)	1.4338 ± 0.009	1.4331 ± 0.021	1.4308 ± 0.018	1.4486 ± 0.009
viscosity (Pa·s)	0.049 ± 0.0025	0.300 ± 0.038	1.12 ± 0.18	0.048 ± 0.0028
conductivity (S/m)	0.37 ± 0.02	0.051 ± 0.014	0.0087 ± 0.0031	0.39 ± 0.028
dielectric constant	5.8 ± 2.8	3.6 ± 1.8	2.8 ± 1.9	8.6 ± 2.6
molar enthalpy of vaporization (kJ/mol)	133.7 ± 5.6	187.4 ± 11.2	299.1 ± 24.8	132.9 ± 4.3

4. Conclusions

In this study, we developed a regression-guided framework for optimizing scaling factors for charge and the Lennard-Jones σ parameters within the electronic continuum correction (ECC) approach for BMIM⁺, MOEMIM⁺, and oligomers (IL2²⁺ and IL4⁴⁺) sharing a similar chemical structure and elemental composition with MOEMIM⁺. We demonstrated that a linear inverse modeling strategy based on density and viscosity can identify optimal force-field scaling factors without relying on empirical or trial-and-error procedures. Furthermore, based on more than 0.4 ms MD simulations, it was also found that scaling factors for charges and the Lennard-Jones size parameters optimized for small molecules are potentially transferable to their oligomer molecules with similar chemical structures and elements.

It is noteworthy that both scaling factors are mostly determined by the density; however, it is evident that the optimum scaling factors cannot be uniquely determined from the density alone. Density and viscosity reflect different aspects of intermolecular potential: density primarily reflects equilibrium packing determined by the free energy minimum, whereas viscosity reflects time correlations of stress and many-body friction or crossing of the free energy barriers. Accordingly, fitting them to a single property may reproduce that property while distorting others, because scaling factors for charge and the Lennard-Jones parameters influence structure and dynamics in different ways. Hence, reliable parameter optimization requires the simultaneous matching of multiple thermodynamic properties rather than relying on a single observable.

Using BMIM-TFSI as a benchmark system, it was revealed that the experimental density, diffusion coefficients, electrical conductivity, and shear viscosity across a range of temperatures were successfully reproduced with the optimized scaling factors. Thus, the proposed methodology provides clear guidelines for parameter selection and can be readily applied to other ionic liquids and multivalent electrolytes. On the other hand, comparison of the short-range RDF shows that the minor peak observed in the AA model and the ab initio simulation is absent in both our results and the other AA model, while the calculated structure factor remains in reasonable agreement with experimental observations. This discrepancy indicates that there is still room to improve the parameters. Addressing this issue remains as a future work, as the primary focus of this study is to develop a mathematical framework for constructing cation models using ECC and to clarify the possibility of transferability of the scaling factors.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.6c04969.

• Data S1: Amber library and parameter file (TXT)

• Simulation time and box dimensions; density equilibration analysis; ECC-scaled atomic charges; AMBER atom types; Green–Kubo viscosity analysis; mean-squared displacement and conductivity analyses; and structure factor comparison (PDF)

The authors declare no competing financial interest.