† Corresponding author. E-mail:
Project supported by the Science Foundation of Civil Aviation Flight University of China (Grant Nos. J2019-059 and JG2019-19).
Both ionic solutions under an external applied static electric field E and glassy-forming liquids under undercooled state are in non-equilibrium state. In this work, molecular dynamics (MD) simulations with three aqueous alkali ion chloride (NaCl, KCl, and RbCl) ionic solutions are performed to exploit whether the glass-forming liquid analogous fractional variant of the Stokes–Einstein relation also exists in ionic solutions under E. Our results indicate that the diffusion constant decouples from the structural relaxation time under E, and a fractional variant of the Stokes–Einstein relation is observed as well as a crossover analogous to the glass-forming liquids under cooling. The fractional variant of the Stokes–Einstein relation is attributed to the E introduced deviations from Gaussian and the nonlinear effect.
Ionic solutions play a significant role in many areas including chemical engineering[1–3] and biological systems.[4,5] The microscopic structure and transport properties of ions are two classical topics in chemical physics. The Debye–Hückel theory[6] describes ions as fully decomposed and solvated in solutions. However, recent studies suggested that ions tend to form clusters even in diluted solution.[7–10] The microscopic structures of ions are essential for many processes, including nucleation,[11,12] battery performance,[13–15] and biological systems.[16–18] Moreover, ion transport properties usually dominantly determine the performance of ionic solutions working in a non-equilibrium processes, e.g., driven by an external electric field E.
The transport properties, including diffusion and viscosity, are significantly influenced by the microscopic structure of solvation shells. Many works have been carried out to explore transport properties of ion under E. Lee and Rasaiah[19] systematically studied the extremely dilute solutions by molecular dynamics simulations. The relation between solvation structure and viscosity was also explored in extremely dilute solutions with real or virtual particles.[20] The frictional coefficients of ion were investigated under the ambient condition.[21] It was found that the charge transport and ion-ion correlation are different for different ionic solutions.[22] In addition, the concentrated NaCl solution under E was studied by several groups.[23,24] Kerisit et al. exploited the dynamic and transport properties of alkali ions in dimethyl sulfoxide solutions under E.[25]
The diffusion and viscosity are closely connected by the Stokes–Einstein relation D∼ T/η in the linear response region. However, the Stokes–Einstein relation is invalid when external applied force is large, such as under a strong E.[24,25] Ionic solution is in a non-equilibrium steady state for the steady flux caused by E. The glass-forming liquids under undercooled state are also out of equilibrium for the slow dynamics. The Stokes–Einstein relation is observed to break down in glass-forming liquids and liquid metal.[13,14,26–32] Because of the difficulty to accurately determine shear viscosity η via simulation, the structural relaxation time τ is usually adopted as a substitute to evaluate the shear viscosity,[27,33–36] and the corresponding variant of the Stokes–Einstein relation is D ∼ τ−1.[34,35] As glass-forming liquids undergo supercooling, D ∼ τ−1 is breakdown and follows a fractional form as D ∼ τξ with ξ ≠ −1. Since both glassy-forming liquids under undercooled state and ionic solution under strong E are out of equilibrium, but different forms of Stokes–Einstein relation are adopted in the two cases, an interesting question arises: is there an analogous fractional variant of Stokes–Einstein relation D ∼ τξ or dynamic heterogeneity existing in ionic solution under strong E?
In this work, we take three aqueous alkali chloride solutions (NaCl, KCl, and RbCl) to explore the variant of the Stokes–Einstein relation in ionic solution under strong E. The paper is organized as follows: Section
To explore the variant of the Stokes–Einstein relation in ionic solution under E, we choose three aqueous alkali chloride (NaCl, KCl and RbCl) solutions. Each system contains 128 ion pairs and 3072 water molecules, the concentration is ∼ 2.2 mol/L. Fourteen different E’s uniformly distributed among 0–1.3 V/nm are applied in the X direction. The diffusion and viscosity of alkali ions have been studied by Lee et al.[20] In this work, we choose the same model for alkali ions and SPC/E water model. The related force field data are listed in Table
![]() | Table 1. The force field parameters for the Na+, K+, Rb+ and SPC/E water model. . |
Ions exists as solvated ions in aqueous ionic solution. To characterize the influence of E on solvation structure, we adopt the radial distribution function (RDF).[19] The RDF is defined as g(r) = 〈 δ (r–r1) 〉, where r1 is the position of the central ion. The RDFs for Na+–Cl−, K+-Cl−, and Rb+–Cl− solution show almost the same changes under E. We only plot the RDFs for Na+–Cl− solution to illustrate the influence of E and the related results are shown in Fig.
![]() | Fig. 1. Radial distribution functions (RDFs) of solvation shells under different E: (a) Na+–Cl− shell, (b) Na+–H2O shell, (c) Cl−–H2O shell. |
Figure
The variant of the Stokes–Einstein relation D ∼ τ−1 is based on the exponential relaxation of the self-intermediate scattering function Fs(k,t) ∼ e−t/τ in simple liquids, which can be described by Fs(k, t) = exp (−k2Dt) if the displacements of particle follow Gaussian distribution.[28] To evaluate the variants D ∼ τ−1, the diffusion constant of ions is calculated via the mean square displacement as
![](cpb_29_3_036101/cpb_29_3_036101_ieqn1.gif)
![]() | Fig. 3. The drift velocities v for Na+ and its corresponding Cl− versus E. The symbols are the simulated data. The solid lines are fitted by v ∼ E and the dotted lines are fitted by v ∼ E + a ⋅ E2. |
The mean square displacements (MSDs) of Na+, K+, Rb+, and their corresponding Cl− are calculated. To illustrate the MSDs under E, the results of NaCl solution is plotted in Fig.
![]() | Fig. 4. The logarithm of mean square displacements for Na+ and Cl− as a function of the logarithm of time t under different E’s: (a) Na+, (b) Cl−. |
The structural relaxation of ion is described by the self-intermediate scattering function, namely,
![]() | Fig. 6. The structure factor S(k) for each ion under E = 0. S(k) for Na+, K+, and Rb+ are plotted in (a) and their corresponding Cl− are plotted in (b). |
![]() | Fig. 7. The self-intermediate scattering function Fs(k, t) for Na+ and Cl− as a function of time t under different E’s: (a) Na+, (b) Cl−. |
The simulation can be well fitted by log (D),log(τ−1) ∼ a + bE + cE2. The fitted a, b, and c are listed in Table
Figures
To examine the existence of the fractional variant of the Stokes–Einstein relation in ionic solution under E, the variant D ∼ τ−1 is evaluated by D ∼ τξ. It will be valid if ξ = −1, otherwise it will follow a fractional form. The logarithms of τ−1 and D for Na+, K+, Rb+ and their corresponding Cl− are plotted in Fig.
![]() | Table 2. Fitting data of the logarithm of the diffusion constant D and structural relaxation time τ by log(D), log(τ−1) ∼ a + bE + cE2. . |
The variant of the Stokes–Einstein relation D ∼ τ−1 will be a strict result if the displacements of particle follow Gaussian distribution. It is proposed that the fractional form is resulted from dynamic heterogeneity in glass-forming liquids.[47] The dynamic heterogeneity can be characterized by the deviations from Gaussian distribution in particle displacement. We calculate the self van Hove function Gs (r, t) and its first-order approximation G0 (r, t) to describe the deviation.[48] Gs (r, t) is the Fourier transformation of Fs(k,t) and G0(r, t) takes the Gaussian form. If the distribution of ion displacement takes a Gaussian form, G0 (r, t) and Gs (r, t) are the same, otherwise Gs (r, t) deviates from G0 (r, t). The calculated results for Na+ and Cl− in NaCl solution at t* = 10 ps are shown in Fig.
To explore the reasons for the crossover and the fractional form observed in D ∼ τ−ξ, as data plotted in Fig.
The atomistic MD simulations for aqueous NaCl, KCl and RbCl solutions have been carried out to exploit the fractional variant of the Stokes–Einstein relation in ionic solutions under strong E. Alkali ion and Cl− get more spatially strongly correlated under a stronger E but the hydration shell is weaker. Ion gets a larger velocity with a larger E that leads the system further away from the equilibrium state. E accelerates the diffusion and structural relaxation of ion, and ion gets a greater diffusion constant and smaller relaxation time under a stronger E. The variant of the Stokes–Einstein relation is found to be breakdown under E, a fractional form and a crossover are observed as E increases. The fractional form and crossover are resulted from nonlinear response introduced by E. The fractional form and crossover are caused by the dynamic heterogeneity of ion and the linear–nonlinear response under E. Our results indicate the similarities between ionic solutions under E and glass-forming liquids, which may help us to improve our understanding and applications of ionic solutions. Moreover, the fractional variant of the Stokes–Einstein relation is caused by E in our work. Actually, some evidences have shown that both the dynamics and structures of glass-forming liquids are heterogeneous. Thus our results may make some hints on how to explain the fractional variant and crossover in glass-forming liquids under cooling, such as the heterogeneity cause molecular mean field.
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