# Anomalous ion transport in saline solutions under electric fields: A molecular dynamic study

Marie Landeiro dos Reis, S. Touzain, A. El Hamidi

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Marie Landeiro dos Reis, S. Touzain, A. El Hamidi. Anomalous ion transport in saline solutions under electric fields: A molecular dynamic study. Physical Review E, 2026, 113 (2), pp.025414. 10.1103/c1sp-lnjp. hal-05507485

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# Anomalous ion transport in saline solutions under electric fields : A molecular dynamic study

M. Landeiro Dos Reis, S. Touzain, and A. El Hamidi

LaSIE UMR CNRS 7356, La Rochelle Université,
Av. Michel Crépeau, 17042, La Rochelle Cedex 1, France *

(Dated: January 15, 2026)

In this study, we employ molecular dynamics simulations to investigate the anomalous diffusion of $Na^+$ and $Cl^-$ ions in water at the microscopic scale. By carefully distinguishing the migration (drift) component from the diffusive motion, we analyze the influence of ion concentration and applied electric fields on ionic transport. Our results reveal clear deviations from classical Brownian dynamics under non-zero electric fields, highlighting the emergence of anomalous diffusion regimes. We further show that the degree of anomalous behavior strongly depends on the ionic chemical activity and the strength of Coulombic interactions between ions.

## I. INTRODUCTION

Classical diffusion models are based on the assumption that diffusing particles follow Brownian motion. This theoretical framework implies a linear growth of the mean squared displacement (MSD), $\langle r^2(t) \rangle \propto t$, as established by the foundational works of Louis Bachelier [1] and Albert Einstein in 1905 [2].

While this framework successfully describes diffusion in several systems, both experimental and theoretical studies have highlighted deviations from this linear behavior in complex environments such as biological media, porous materials, or systems under external fields [3–5]. These situations are referred to as anomalous diffusion, characterized by a fractional power-law scaling of the form $\langle r^\beta(t) \rangle \propto Dt^\alpha$, where the exponents $\alpha \neq 1$ and $\beta \neq 2$ indicate non-classical transport behaviors, and $D$ denotes a generalized diffusion coefficient. The nonlinearity may arise from time dependence ($\alpha \neq 1$) [3, 4, 6–10] or spatial dependence ($\beta \neq 2$) [4, 11, 12].

In this work, we investigate the ionic diffusion of saline solutions under the influence of an external electric field. A thorough understanding of this phenomenon is crucial for advancing electrochemical water desalination technologies [13–16], which are increasingly recognized for their energy efficiency and environmental sustainability [17]. Beyond desalination, electric-field-driven ionic transport plays a key role in biological and biochemical applications, such as electrophoresis used to separate charged biomolecules like proteins, DNA, and RNA [18] and in the study of ion mobility within biological environments, including cell membranes and tissues [19].

Several experimental studies, notably in electrochemical impedance, have revealed that ionic diffusion can deviate from this classical model, particularly in confined environments or under electric fields, and exhibit anomalous diffusion of the form $\langle r^2(t) \rangle \propto t^\alpha$ with $\alpha \neq 1$ [3, 10, 20, 21].

To investigate ionic transport phenomena, several modeling approaches are available, each with its own strengths and limitations: (i) The Poisson-Nernst-Planck (PNP) model [22–26] employs a mean-field approach by iteratively solving the Poisson and Nernst-Planck equations. Ion flow is represented as a continuous concentration profile, driven by electrostatic potential force and ion concentration gradient. Although computationally efficient, it approximates critical effects, such as the finite size of the ions and discrete charges. (ii) Brownian dynamics (BD) [24, 25, 27] explicitly simulating the motion of mobile ions while treating the surrounding water implicitly as a continuum. The motion of ions is governed by the Langevin equation, which incorporates forces such as frictional and random forces from the solvent and electrostatic interactions with other ions, calculated by solving the Poisson equation. (iii) Hybrid methods that combine the Poisson-Nernst-Planck (PNP) model with Monte Carlo simulations [28] or dynamic lattice Monte Carlo [29] have been developed to improve the description of ions that are treated explicitly. These approaches enhance the accuracy of ion transport modeling by incorporating molecular-level ion interactions within a continuum framework. (iv) Atomistic calculations, such as molecular dynamics (MD) simulations [30–35], provide a complete framework to study ion transport by explicitly modeling atomic-level interactions.

Despite growing interest, the microscopic origin of anomalous ion transport under external fields remains poorly understood. In this study, we employ molecular dynamics simulations combined with a anomalous diffusion framework to investigate the transport properties of $Na^+$ and $Cl^-$ ions in saline solutions. By extracting the anomalous diffusion exponent $\alpha$, we aim to quantify deviations from classical diffusion and to explore how these deviations depend on ion concentration and the strength of an externally applied electric field. This atomistic perspective provides new insights into the mechanisms driving anomalous transport in confined or driven ionic systems.

<footnote>* Contact author: mlandeir@univ-lr.fr</footnote>
## II. METHODOLOGY

Molecular dynamics simulations were conducted using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [36]. The atomic visualization were performed using OVITO [37]. The simulation cell consists of two electrodes with a saline solution placed between them. Periodic boundary conditions were applied in the x and y directions, while there was free surface in the z direction (see Fig. 1). The system contain N particles, comprising $n_{H_2O}$ water molecules, $n_{Cl^-}$ chloride ions and $n_{Na^+}$ sodium ions. All parameters for the size of the simulation cell is given Tab. I. The electrodes were modeled as a body-centered cubic conductive metal (M) (Tab. II), aimed solely to generate an electric field within the saltwater solution. To immobilize the electrodes, forces on all atoms in the electrodes were set to zero, effectively freezing their positions. We ensured that the system size was sufficient to accurately address diffusion-related phenomena [38].

<figure><img src="image_2.png" /><figcaption>FIG. 1. Simulation cell.</figcaption></figure>

<table><caption>TABLE I. Simulation cell parameters. The inter-electrode distance is denoted as $L$, while $d_x$, $d_y$, and $d_z$ represent the cell dimensions in the x, y, and z directions (Fig. 1). The number of water molecules is noted as $n_{H_2O}$, and $n_M$ refers to the number of atoms in the electrodes.</caption><thead><tr><th>L (Å)</th><th>d<sub>x</sub> (Å)</th><th>d<sub>y</sub> (Å)</th><th>d<sub>z</sub> (Å)</th><th>n<sub>H2O</sub></th><th>n<sub>M</sub></th></tr></thead><tbody><tr><td>90</td><td>79</td><td>79</td><td>95</td><td>≈18200</td><td>4800</td></tr></tbody></table>

The interatomic potential used is a combination of Coulombic interactions and a simple Lennard-Jones (LJ) potential. The potential energy, for the particle i writes:

$$V(\mathbf{r}_i) = \underbrace{\sum_{j \neq i} \frac{q_i q_j e^2}{r_{ij}}}_{\text{Coulomb}} + \underbrace{\sum_{j \neq i} 4\epsilon_{ij} \left[ \left( \frac{\sigma_{ij}}{r_{ij}} \right)^{12} - \left( \frac{\sigma_{ij}}{r_{ij}} \right)^{6} \right]}_{\text{LJ}} \quad (1)$$

where $\mathbf{r}_i$ is the position of the particle i, $q_i$ is the charge of the particle i, e is the elementary charge, $r_{ij}$ is the distance between the particles i and j, $\epsilon_{ij}$ and $\sigma_{ij}$ are respectively the potential depth and the distance parameter of the LJ contribution.

The parameters of the potential $V(\mathbf{r}_i)$ (Eq. 1) are provided in Table II. A rigid water model based on TIP4P/2005 [39] was used, while the parameters for Na$^+$ and Cl$^-$ ions were derived from the CHARMM-27 force field [40]. The long-range Coulombic interactions were accounted for using the FFT-based particle-particle/mesh (PPPM) method in LAMMPS [36].

<table><caption>TABLE II. LJ pair potential parameters used in this study [39, 40].</caption><thead><tr><th></th><th>σ<sub>ij</sub> (Å)</th><th>ε<sub>ij</sub> (kcal/mol)</th></tr></thead><tbody><tr><td>O-O</td><td>0.185199</td><td>3.1589</td></tr><tr><td>Na-Na</td><td>0.04690</td><td>2.4299</td></tr><tr><td>Cl-Cl</td><td>0.1500</td><td>4.04470</td></tr><tr><td>M-M</td><td>11.697</td><td>2.574</td></tr><tr><td>O-M</td><td>0.4</td><td>2.86645</td></tr></tbody></table>

System equilibration was achieved through a 1 ns NVE simulation at 300 K, i.e in the microcanonical ensemble, ensuring energy stabilization and convergence of the interelectrode distance. Following equilibration, a 1 ns NVT simulation was performed to study the diffusion of water, i.e in the canonical ensemble, with or without charge of the electrodes. For the NVT simulations we used a Nose Hoover thermostat. Hence, the equations of motion are those given by of Shinoda et al [41], combining the hydrostatic equations of Martyna et al [42] with the strain energy of Parrinello and Rahman [43]. The time integration schemes are based on Verlet [44].

We investigated the influence of ion concentration and electrode charge on ionic diffusion. To ensure robust statistical sampling, 50 molecular dynamics simulations were conducted for each combination of concentration (ranging from 4 g/L to 36 g/L of NaCl solution) and charge of the atoms within the electrode, quoted $q^M$. The charge $q^M$ was varied from 0e to 0.05e per metallic atom on the cathode and from 0e to -0.05e on the anode, corresponding to average electric field strengths between approximately $1 \times 10^5$ V/m ($q^M \approx 0.001e$) and $1 \times 10^7$ V/m ($q^M \approx 0.05e$).

## III. RESULTS

To characterize the diffusion processes at play, we directly extracted from our MD simulations the Mean Squared Displacement (MSD) for each type of ion without (Fig. 2(a)) and with electric field (Fig. 2(b)-(c)), using:

$$\langle \mathbf{r}_{\text{ion}}^2(t) \rangle = \frac{1}{N_{\text{ion}}} \sum_{i=1}^{N_{\text{ion}}} \| \mathbf{r}_{\text{i}}(t) - \mathbf{r}_{\text{i}}(0) \|^2 \quad (2)$$
<figure><img src="image_3.png" /><figcaption>FIG. 2. Comparison between $\langle \mathbf{r}_{ion}^2(t) \rangle - \langle \mathbf{r}_{ion}(t) \rangle^2$ obtained from molecular dynamics simulations and the predictions of Eq. 3. (a) Without external electric field at an ion concentration of 36 g/L. (b) Under an external electric field with $q^M = 0.025e$ of 5 g/L. (c) Under an external electric field with $q^M = 0.05e$ and an ion concentration of 36 g/L. In each case, the dashed black line corresponds to normal diffusion ($\alpha = 1$), while the purple line represents the best fit obtained by treating $\alpha$ as a free parameter.</figcaption></figure>

where $N_{ion}$ represents the number of ions in the solution and $||\mathbf{r}_i(t) - \mathbf{r}_i(0)||$ is the distance traveled by ion $i$, between instants 0, the beginning of the NVT simulation procedure, and $t$. The distinct behavior of $Na^+$ and $Cl^-$ ions will be carefully analyzed in Eq. 2.

To extract the diffusion coefficient and assess whether the diffusion is anomalous, we compared the MSD from MD simulations (Eq. 2) with the following diffusion model, where the MSD of mobile ions is described by a fractional time dependence:

$$ \langle \mathbf{r}_{ion}^2(t) \rangle - \langle \mathbf{r}_{ion}(t) \rangle^2 = \frac{6D_{ion}}{\Gamma(1+\alpha)} t^{\alpha} \quad (3) $$

where $D_{ion}$ is the diffusion coefficient, $0 < \alpha \le 2$ is a real number characterising the subdiffusion ($\alpha < 1$) or surdiffusion ($\alpha > 1$) process. $\Gamma(1+\alpha)$ is the gamma function, an extension of the factorial function for complex and real numbers. One note that in case of normal diffusion, $\alpha = 1$ and $\Gamma(1+\alpha) = 1$. The adjustable parameters in this model are $\alpha$, $D_{ion}$. The term $\langle \mathbf{r}_{ion}^2(t) \rangle^2$ accounts for the systematic drift of the charged particle under an external field, and must be considered to isolate the purely diffusive contribution to the mean square displacement. In such a case, the particle acquires a net average velocity, commonly referred to as the drift velocity, denoted by $\mathbf{v}_d$:

$$ \mathbf{v}_d = \frac{d}{dt} \langle \mathbf{r}_{ion}(t) \rangle = \frac{\beta \mathbf{v}}{\Gamma(1+\beta)} t^{\beta-1} \quad (4) $$

where $\mathbf{v} = (v_x, v_y, v_z)$ is a constant and $\beta$ a fractional coefficient that can be different from $\alpha$.

<figure><img src="image_4.png" /><figcaption>FIG. 3. Comparison of $|\langle z(t) \rangle|$ obtained with MD simulations with the predictions of Eq. 6 under elastic field with $q^M = 0.05e$. The ion concentration is 36g/L.</figcaption></figure>

In this context, the flux $\mathbf{J}$ of the $n$ ions, which quantifies the net transport of ions, can be expressed as:

$$ \mathbf{J} = n\mathbf{v}_d. \quad (5) $$
In our case, the electric field is applied only along z, thus $\mathbf{J} = (0, 0, J)$. By fitting the mean ion displacement from MD simulations to the anomalous drift expression

$$< \mathbf{r}_{ion}(t) > = \frac{\mathbf{v}t^{\beta}}{\Gamma(1 + \beta)} \qquad (6)$$

we extracted the anomalous drift exponent $\beta$ and the drift velocity $\mathbf{v} = (0, 0, v)$ (Fig. 3). Since the applied electric field is oriented along the z-axis, the mean displacement can be written as $< z(t) > = \frac{vt^{\beta}}{\Gamma(1+\beta)}$ where $z(t)$ represents the ion drift along the z-direction.

For each simulation, to confirm the relevance of a anomalous model, we compared Eq. 3 with fitted $\alpha$ and $D_{ion}$ (see black dashed lines Fig. 2 to Eq. 3 with $\alpha$ fixed at 1 (see purple lines on Fig. 2), corresponding to a normal diffusion model.

<figure><img src="image_5.png" /><figcaption>FIG. 4. Mean value of the fractional coefficient $\alpha$ from Eq. 3 as a function of the charge $q^M$ of metallic atoms composing the electrodes. The standard deviation is shown as a transparent overlay.</figcaption></figure>

We plot Fig. 3 an example of simulation without electric field for an ionic concentration of 36g/L. In the absence of an applied electric field (i.e., uncharged electrodes, $q^M = 0e$), our results show that a classical diffusion

model accurately describes the ion dynamics (see Fig. 3). In this regime, both $Na^+$ and $Cl^-$ ions exhibit quasi-Brownian motion, characterized by a diffusion exponent $\alpha \simeq 1$. Although their diffusion behaviors are qualitatively similar, the slopes of their mean squared displacements differ, reflecting different diffusion coefficients. We obtain $D_{Na^+}$ between 0.7 and $0.9 \times 10^{-9} \text{m}^2/\text{s}$ and $D_{Cl^-}$ between 0.8 and $1.1 \times 10^{-9} \text{m}^2/\text{s}$. These values are consistent with those reported in the literature that found around $1 \times 10^{-9} \text{m}^2/\text{s}$ for $D_{Na^+}$ and around $2 \times 10^{-9} \text{m}^2/\text{s}$ for $D_{Cl^-}$ [45–48]. Note that this diffusion coefficient is highly temperature-dependent. In this work, the temperature is fixed at 300K. A study at different temperatures will be carried out in future work.

We plot Fig. 2(b)-(c) some examples of simulations under electric field, respectively with $q^M = 0.025e$ and $q^M = 0.05e$, for a concentration of 9g/L and 36g/L. When the system is subjected to an electric field, the mean squared displacement (MSD) exhibits a nonlinear time dependence, indicating anomalous diffusion characterized by $\alpha \neq 1$ in Eq. 3. In this case, the transport of ions becomes biased, forcing negatively charged ions to migrate towards the positive electrode and vice versa. We can note that the behavior of the MSD over time follows the same trend regardless of the ions.

It is interesting to note that the type of anomalous diffusion depends on the electric field. Indeed, at low fields, the diffusion is slightly accelerated ($\alpha > 1$), whereas at high fields, the diffusion is slowed down ($\alpha < 1$). However, it is important to note that at high fields, the simulation times are very short (0.1 ns), and very quickly all ions become fixed on the electrodes (see Fig. 2(c)).

In order to obtain reliable statistics for the value of $\alpha$ as a function of ion type, concentration, and electrode charge $q^M$, we averaged the results from 50 simulations for each set of parameters. The results are shown Fig. 4. The standard deviation is displayed as a shaded region to illustrate the statistical uncertainty associated with the estimation of $\alpha$. We plot the variation of $\alpha$ as a function of the charge $q^M$, and thus the electric field, for the different concentrations tested.

The behavior of $\alpha$ is complex and highly dependent on concentration (Fig. 4). At low concentrations, a superdiffusion phenomenon is observed, which increases with the electric field ($\alpha > 1$). Regardless of the concentration studied, as previously shown we then find that, this effect reverses at high electric fields, where the ions quickly reach the electrode surfaces.

For electrode charges below $0.025e$, Fig. 4 shows that the degree of anomalous diffusion decreases as the ion concentration increases. Indeed, $\alpha$ approaches 1 for concentrations above 18 g/L, indicating a diffusion behavior increasingly close to the classical Brownian regime.

The same tendency can be anticipated from Fig. 5, which presents the evolution of the diffusion coefficient $D_{ion}$ as a function of the applied charge $q^M$ for the different concentrations studied. It can be hypothesized that, at low concentrations, diffusion might be boosted by the
<figure><img src="image_6.png" /><figcaption>FIG. 5. Mean value of the fractional coefficient $D_{ion}$ from Eq. 3 as a function of the charge $q^M$ of metallic atoms composing the electrodes.</figcaption></figure>

electric field due to enhanced ionic mobility. However, this effect could progressively vanish at higher concentrations, where ion-ion interactions and crowding effects are expected to play a dominant role. Above 18 g/L, the influence of $q^M$ on $D_{ion}$ may therefore become negligible.

At higher concentrations, it is possible that electrostatic interactions between oppositely charged ions lead to the formation of ionic pairs such as $Na^+/Cl^-$. These paired entities, being closer to electrically neutral, may respond less strongly to the applied electric field, since the net force acting on them could be reduced. As a result, their contribution to directed ion transport might be limited, which could appear as a decrease in the effective diffusion coefficient.

To gain a better understanding of this phenomenon, which appears to be linked to the increasing Coulombic interactions between ions at higher concentrations, we examined the average number of $Na^+/Cl^-$ ion pairs, denoted as $N_{pairs}$, as a function of ion concentration and charge $q^M$. A pair is considered to be formed when the distance between a $Na^+$ and a $Cl^-$ ion is less than $a_i = 2.5Å$, based on the first peak of the radial distribution function (RDF) obtained from our molecular dynamics simulations (Fig. 6(a)). Note that the value of $a_i$ found is in agreement with the experimental value of 2.35Å [49]. We then estimated an effective activity coefficient $\gamma$, being roughly the ratio of free (unpaired) ions to the total number of ions in solution:

<figure><img src="image_7.png" /><figcaption>FIG. 6. (a) $Na^+$-$Cl^-$ pair correlation function $g(r)$ as function of the distance from the center of the particle $r$ at $c=27$ g/L. (b) Comparison of the activity coefficient $\gamma$ deduced from our MD results compared to the Extended Debye-Hückel (EDH) model [49] (Eq. 8) and experiments of Ensor and Anderson [50].</figcaption></figure>

$$ \gamma \propto \frac{N_{ion} - N_{pairs}}{N_{ion}} \qquad (7) $$

We harshly adjust the coefficient of proportionality to 0.82 to fit the experimental results of Ensor and Anderson [50] obtained without applied electric field (Fig. 6(b)). Then we confront our results to the Extended Debye-Hückel (EDH) model [49]:

$$ - \log_{10} \gamma = \frac{A_m |q_{Na} + q_{Cl^-}| I_m^{1/2}}{1 + B_m a_i I_m^{1/2}} \qquad (8) $$
with A_m and B_m two coefficient and I_m the ionic strength:

$$A_m = \left( \frac{2\pi N_a}{1000} \right)^{1/2} \times \frac{e^3}{2.3026(k_B T \epsilon_r)^{3/2}} \quad (9)$$

$$B_m = \left( \frac{8\pi N_a}{1000} \right)^{1/2} \times \frac{e}{(k_B T \epsilon_r)^{1/2}} \quad (10)$$

and

$$I_m = \frac{1}{2} \frac{c}{m_{NaCl}} (v_{Na^+} + v_{Cl^-}) \quad (11)$$

where $m_{NaCl} = 58.44g/mol$ is the molar mass of NaCl, $q_{Na^+}$ and $q_{Cl^-}$ are respectively the charge of $Na^+$ and $Cl^-$, $v_{Na^+}$ and $v_{Cl^-}$ are respectively the valance of $Na^+$ and $Cl^-$, $N_a$ is the Avogadro constant, $k_B$ the Boltzmann constant and $\epsilon_r = 65$ is the dielectric permittivity constant [51].

We plot the results Fig. 6(b). As commonly observed, the EDH model (Eq. 8) tends to underestimate the activity coefficient $\gamma$ at high concentrations. This limitation arises because the model cannot fully capture the complexity of ion-ion interactions, particularly salt effects and short-range correlations [49, 51].

In agreement with experiments, we show that the activity coefficient $\gamma$ decreases significantly with increasing ion concentration, reaching a value of approximately 0.7 at 36 g/L (see Fig. 6(b)). This behavior can be attributed to the sharp increase in the number of ion pairs formed at higher concentrations: as the average distance between ions decreases, the likelihood of forming pairs between oppositely charged species ($Na^+$ and $Cl^-$) increases. When such ion pairs form, they can be considered quasi neutral over long distances. As a result, the electric field has a reduced effect on these neutral entities. This could explain why the deviation of $\alpha$ from 1 indicative of anomalous diffusion tends to diminish as the ion concentration increases.

Moreover, the results show that the stronger the applied electric field (i.e., for larger values of $q^M$), the higher the activity coefficient $\gamma$. This is because the electric field tends to counteract the attractive Coulomb interactions between oppositely charged ions. If the force induced by the electric field is strong enough, it can break apart the formed ion pairs, thereby increasing the number of free ions in solution and raising the value of $\gamma$.

In Fig. 7, we present the evolution of the fractional drift exponent $\beta$, extracted from Eq. 6 and used in the calculation of the ionic flux (Eq. 5). The results consistently show that $\beta < \alpha$, indicating that the collective ionic transport, the drift, is more hindered than individual diffusion. This difference suggests that while ions undergo subdiffusive motion, their macroscopic drift under an applied field exhibits an even stronger deviation from normal transport dynamics.

Furthermore, one note that as the applied charge $q^M$ increases, $\beta$ departs further from 1 and appears to converge toward a limiting value. This trend may reflect a

<figure><img src="image_8.png" /><figcaption>FIG. 7. Mean value of the fractional coefficient $\beta$ from Eq. 6 as a function of the charge $q^M$ of the metallic atoms composing the electrodes. The standard deviation is shown as a transparent overlay.</figcaption></figure>

transition from a weak-field regime, where ion motion remains partially Brownian, to a strong-field regime dominated by correlated and constrained dynamics. Under high electric fields, ion-ion interactions and local structuring effects (as illustrated in Fig. 6) become increasingly significant, imposing collective constraints that limit the system's ability to respond linearly to the external forcing.

Furthermore, the drift velocity $v$, shown in Fig. 8, increases markedly with the applied charge $q^M$. This trend reflects both an enhanced driving force under higher fields and a nonlinear response of the ionic system, suggesting that the electric field not only accelerates ion motion but also amplifies the underlying anomalous transport behavior.

It is interesting to note that we were able to decouple the contribution of migration (drift) from that of diffusion. Our results indicate that migration exhibits a more pronounced anomalous behavior than diffusion, suggesting
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>FIG. 8. Mean value of the drift velocity $v$ from Eq. 6 as a function of the charge $q^M$ of the metallic atoms composing the electrodes. The standard deviation is shown as a transparent overlay.</figcaption></figure>

ing that the collective transport of ions under an electric field is more strongly influenced by constraints and correlations than individual diffusive motion.

Bridging atomistic molecular dynamics with continuum or mesoscopic models remains an important but nontrivial task. From MD simulations, several microscopic quantities such as the diffusion coefficient, the anomalous exponents $\alpha$ (for diffusion) and $\beta$ (for drift), and the steady-state ionic flux (Eq. 5) can be extracted and used to parameterize or validate higher-scale transport models. These quantities provide direct physical input for generalized frameworks such as fractional Poisson–Nernst–Planck equations or memory-dependent Langevin formulations, where anomalous transport is described through non-local temporal kernels or distributed mobility [22–26]. However, establishing a rigorous correspondence between these atomistic observables and macroscopic transport coefficients requires system-

atic coarse-graining and averaging over time and space, which represents a significant research effort that we plan to undertake in future work.

## IV. CONCLUSION

In this work, we investigated the diffusion dynamics of ionic species in aqueous saline solution under the influence of an external electric field, using molecular dynamics simulations. By analyzing the mean squared displacement (MSD) of ions, we highlight that the transport process cannot always be accurately captured by a classical Brownian model. This is in agreement with impedance measurements and previous analyses [20, 21]. In particular, we found that the presence of an electric field induces anomalous diffusion, characterized by a non-linear time dependence of the MSD and quantified by a fractional exponent $\alpha \neq 1$.

Our analysis revealed that, in the absence of an electric field, ion dynamics closely follow normal diffusion behavior, characterized by a diffusion exponent $\alpha \approx 1$. Moreover, the diffusion coefficients extracted from our simulations are in good agreement with experimental values reported in the literature [45–48]. However, when an electric field is applied, the diffusion behavior becomes strongly dependent on both the strength field and ion concentration. At low concentrations and moderate electric field, we observed superdiffusive behavior ($\alpha > 1$), whereas at high field intensities, the dynamics became subdiffusive ($\alpha < 1$), likely due to rapid adsorption of ions onto the electrode surfaces.

Furthermore, we showed that ion concentration plays a critical role in modulating the diffusion regime. At high concentrations, ion–ion coulombic interactions promote the formation of quasi-neutral $Na^+/Cl^-$ pairs, which reduce the effectiveness of the electric field in driving ion transport. This results in a lower diffusion coefficient and a diffusion regime closer to normal behavior, with $\alpha$ approaching 1. Our results thus reveal a delicate interplay between Coulomb interactions, electric field-driven migration, and ion pairing, which governs the nature of ion transport across different regimes.

Importantly, we demonstrated that assuming normal diffusion in regimes where $\alpha \neq 1$ can lead to a significant overestimation of the diffusion coefficient, especially at high electric strengths. This finding highlights the necessity of incorporating anomalous diffusion models when analyzing ion transport in such systems.

Beyond diffusion, we also quantified the anomalous drift behavior of ions under an external electric field. By fitting the mean displacement to a fractional drift law, we extracted the drift exponent $\beta$ and the drift velocity $v$, which together determine the ionic flux $J = nv_d$. Interestingly, we found that $\beta$ farthest from 1 than $\alpha$, indicating that the collective migration of ions under an electric field is more hindered than their individual diffusion. This suggests that correlations and local struc-
turing effects play a dominant role in governing the over-
all transport dynamics, particularly under strong electric
fields.

This work opens the door to future investigations aim-
ing to bridge the gap between atomistic and macroscopic
descriptions. Comparing the results of our molecular dy-
namics simulations with continuum models such as the
Poisson–Nernst–Planck framework possibly extended to
include anomalous dynamics could provide a more com-
plete understanding of ion transport across scales. Such
a multiscale approach is essential for the accurate mod-
eling of electrochemical systems, energy storage devices,
and biological ion channels. Finally, it is challenging to
determine the exact origin of anomalous diffusion. This
behavior may arise from the system’s non-ergodicity, in-
duced by the non-periodic boundary conditions (surface)

in one direction, leading to a continuous system evolu-
tion as ions progressively attach to the electrodes. We
will aim to address these questions in our future work.

ACKNOWLEDGMENTS

We warmly thank Dr Malo Duportal for the very
fruitful discussions we had, which significantly helped
improve this work. The computational resources uti-
lized for this study were graciously provided by the
MCIA (Mésocentre de Calcul Intensif Aquitain) and the
NOETHER computing facilities at La Rochelle Univer-
sity, hosted at the LaSIE laboratory and administered by
Dr Antoine Falaize, to whom we are particularly grateful.

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