Macroscopic conductivity of aqueous electrolyte solutions scales with ultrafast microscopic ion motions

Despite the widespread use of aqueous electrolytes as conductors, the molecular mechanism of ionic conductivity at moderate to high electrolyte concentrations remains largely unresolved. Using a combination of dielectric spectroscopy and molecular dynamics simulations, we show that the absorption of electrolytes at ~0.3 THz sensitively reports on the local environment of ions. The magnitude of these high-frequency ionic motions scales linearly with conductivity for a wide range of ions and concentrations. This scaling is rationalized within a harmonic oscillator model based on the potential of mean force extracted from simulations. Our results thus suggest that long-ranged ionic transport is intimately related to the local energy landscape and to the friction for short-ranged ion dynamics: a high macroscopic electrolyte conductivity is thereby shown to be related to large-amplitude motions at a molecular scale.

T he electrical conductivity of an electrolyte solution is arguably its most critical property, as it limits the electrolyte performance in, e.g. batteries 1 , fuel cells 2 , or supercapacitors 3 . Yet, the molecular-level understanding of factors improving or suppressing charge mobility is still at its infancy 4 . Consequently, macroscopic measures, such as the product of electrolyte viscosity and single ion conductance (i.e. the so-called Walden product), are common-yet controversialmeasures for the electrolyte performance 5 . Even for relatively simple electrolyte solutions, like aqueous salt solutions at moderate concentrations, which are both technologically 6,7 and biologically 8 relevant, it has been challenging to understand longranged charge transport on a molecular scale. Our lack of understanding of the microscopic, molecular-level mechanisms determining macroscopic conductivity of electrolyte solutions precludes a rational design of, and search for, new electrolytes.
The challenge in understanding ionic conductance can be traced back to marked structuring 9 and correlated motions 9,10 in electrolytes. An ion cannot diffuse independently without displacement of the surrounding ions 11 . The same restriction applies to ionic solvation shell dynamics: 10 An ion has to strip at least part of its solvation shell to be transported, and exchange of solvent molecules in ionic solvation shells is obviously involved in ion conduction 12 . Thus, molecular-scale motion on a picosecond scale impacts macroscopic charge transport on long timescales (>nanoseconds) 13 . These slow collective dynamics have been theoretically linked to the molecular-level fast (picosecond scale) dynamics for dilute salt solutions 14 . For technologically relevant concentrated electrolytes 1 , the fast dynamics of both the ions [15][16][17] and their solvation shells 10,[18][19][20] , have been elucidated using spectroscopic techniques. Yet, the relevance of such fast molecular motion to macroscopic transport has remained elusive 21 .
Both, the dynamics of water in the solvation shell of ions and the motion of ions itself, go along with a change of the macroscopic dipole moment of the sample and can thus be probed using spectroscopy experiments 10,[22][23][24] . Here microwave and Terahertz spectroscopies have been extensively used. At field frequencies ranging from 100 MHz to~100 GHz, hydration of ions has been intensively studied by detecting the dynamics of water and also the rotational dynamics of long-lived ionic aggregates (i.e. ion-pairs) have been elucidated in detail [24][25][26] . At frequencies ranging from~2 THz up to 20 THz, at which hydrogen-bonding vibrations, librations, and also ion dynamics contribute, the cooperativity and spatial extent of ion hydration, as well as ion-pairing, have been investigated in great detail 10,23,27,28 . At intermediate frequencies (100 GHz-2 THz) computational studies predict the very weak contribution of ionic currents to peak 22 . Experimentally, this intermediate frequency range is, however, challenging to study and experiments on electrolytes are scarce 29 . As such, the potential of using this spectral information to understand ion dynamics has so far not been exploited 22,29 .
Here, by combining GHz to THz dielectric relaxation spectroscopy (DS) 30 and molecular dynamics (MD) simulation, we find that the contribution of ions to these spectra, which peaks at 0.3 THz, arises from the microscopic "cage" motion of ions in their potential energy minimum. This cage is imposed by (counter)ions and water molecules in their immediate surrounding. Surprisingly, the amplitude of this fast cage motion shows universal scaling with the macroscopic conductivity, independent of the nature of the ions. Using a harmonic potential model, we illustrate that the amplitude of motion in this "cage" is related to the ionic conductivity.

Results
Experimental polarization dynamics of aqueous KI solutions. We probe the response of the electrolytes to an externally applied electrical field with frequency ν, using broadband DS at frequencies spanning from a few hundred of MHz to~1.5 THz. DS probes the electrical polarization of a sample in an external field and is thus sensitive to any dynamics that go along with the displacement of charges from their equilibrium positions. Such displacement reflects either the rotation of molecules with an electrical dipole moment or the translation of ions. Typically, the response is expressed as the complex dielectric permittivity, where the response out of phase with respect to the external field is the dielectric loss and representative of absorption of the electric field. The frequency-dependent permittivity is a measure for the polarization in phase with the external field (for details on the experimental determination of permittivity spectra see "Methods" section). For neat water at ambient temperature, DS spectra ( Fig. 1a) are dominated by the collective re-orientational motion of the dipolar water molecules at~20 GHz 31 , as evident from the dispersion in the dielectric permittivity and a peak in the dielectric loss. The effect of various electrolytes on the reorientational motion of water has previsously been studied both experimentally 24,26,32 and computationally 22,33,34 in great detail. Adjacent to the dominant relaxation at~20 GHz, at least one low amplitude mode is present at 0.3-1 THz in neat water (light red shaded area in Fig. 1a) 29,35,36 . The molecular origin of this 0.3 THz dynamics is still under debate 37 and has been ascribed to the relaxation of weakly hydrogen-bonded water molecules 29 , to small angular motions of water molecules due to rapid fluctuations 37 , or to the dynamics of the low-density liquid phase of water 38 .
With increasing concentration of salt (here KI, c KI ), we find the spectral contributions at~20 GHz to be reduced. This reduced polarization has been reported for various salts and has been attributed to the reduced correlation between the rotational motion of water molecules in ionic hydration shells 22,34,39 and also-to a lesser extent-to kinetic depolarization 33,40-42 . Conversely, the spectral contributions to the 0.3-1 THz response increase with increasing c KI . The inset of Fig. 1a shows an increasing dispersion in the dielectric permittivity and a slight increase in the dielectric loss, as c KI increases: With increasing salt concentration, molecular or ionic motion at these frequencies goes along with an enhanced displacement of charges. To quantify the changes to the spectra, we model the experimental spectra: to describe the main relaxation at~20 GHz, we use a Cole-Cole mode, in line with previous studies 32 . To capture the faster dynamics at~0.3 THz, we use a Debye-type mode 32 (see "Methods" section for details). We note that both, motion of ions and motion of water molecules, contribute to the spectra at these frequencies 22 . As such, we use the Debye mode as a means to quantify the contribution of all fast dynamics. The variation of the parameters of the Debye mode with concentration thus reflects the salt-induced changes to these dynamics. Using this approach, we find the relaxation amplitudes for the fast dynamics (relaxation time 200-500 fs) to be very sensitive to the presence of the salt (Fig. 1b): Starting from an initial value of~2 (neat water) the amplitude increases linearly with increasing concentration of KI, in line with what has been found for solutions of NaCl 29 and several alkali-halide salts at >2 THz 28 .
Dissecting ions' contributions using MD simulations. To pinpoint the origin of the response at 0.3-1 THz, we perform forcefield MD simulations (for details, see "Methods" section, Supplementary Figs. 1-9, and Supplementary Notes 1 and 2). The dielectric response can be readily obtained from the timecorrelation function of the macroscopic dipole moment (the electrical dipole moment due to both water molecules and ions). By computing the time-correlation function due to only water molecules, one can isolate the contribution of water to the spectra (I WW in Fig. 1c). The dominant dielectric loss for water peaks at 20 GHz. The lower peak amplitude in the simulations as compared to the experiments arises from the used TIP4P/2005 water force-field, which is a shortcoming of this force-field 43 . Nevertheless, the TIP4P/2005 model is known to reproduce both structure and (fast) dynamics 44 , which are relevant to the present work, very well: the experimentally observed peak shoulder at 0.3 THz is also apparent in the simulated loss spectra.
Upon increasing c KI , the polarization dynamics at 20 GHz is reduced, again consistent with the experimental data. In contrast, water's contribution at 0.3 THz is hardly affected by the presence of KI. Hence, the simulations suggest that the experimentally observed salt-induced response at~0.3 THz does not originate from salt-induced changes to the water dynamics, as opposed to previous interpretations of experimental data; 29 rather, the ions' contribution (I II in Fig. 1c), which is polarization due to the displacement of both anions and cations out of their equilibrium position, increases with increasing salt concentrations. The correlation between the dynamics of water and the ions (I WI in Fig. 1c) has a negative sign and 'counters' the ionic polarization, similar to what has been found at higher frequencies by Marx and coworkers 10 . Nevertheless, the net-polarization due to the presence of salts (I II + I WI ) is positive at ≳1 THz (for a more detailed discussion on I WI , see Supplementary Note 2 and Supplementary Fig. 9). Hence, based on the simulation results, the ion-induced changes to the polarization dynamics at THz frequencies can be primarily ascribed to the (hindered translational) motion of ions.
Harmonic oscillator to model fast ion motions. To rationalize the molecular level origins of the observed ion dynamics, we constructed a harmonic potential model. First, we compute the cation-anion radial distribution function (RDF), g(r), for KI. g(r) shows a distinct peak at 3.3-3.8 Å (Fig. 2b), which corresponds to the first 'shell' of counterions around a given ion. Through the Boltzmann relation F(r) = −k B T In(g(r)), we obtain the potential of mean force F(r) for anion-cation pairs (Fig. 2c), where k B and T are the Boltzmann constant and temperature, respectively 45 . F(r) thus contains excluded volume effects of both water and ions as well as attractive intermolecular interactions. Approximating the potential to be harmonic (for details see "Methods" section), we obtain a force constant of k = 8.5 kg s −2 for 0.5 mol L −1 KI. Together with the reduced mass μ = 30 g mol −1 for KI, the frequency of these ion motions is predicted to be ω 0 /2π ≈ (k/μ) 1/2 /2π = 2 THz (see also Supplementary Note 3). By further assuming the damping coefficient of the ionic motion to be given by the macroscopic hydrodynamic drag coefficient γ = 10 −12 kg mol −1 s −1 for dilute KI solutions 46 , the thus obtained damped harmonic oscillator reproduces the spectral shape and amplitude of the ionic dynamics in the frequency domain very well (purple line in Fig. 2a). The predicted maximum of the spectral response of the harmonic oscillator model is shifted by a factor of~4 to higher frequencies as compared to the simulated I II response (Fig. 2a), which is attributable to the anharmonicity of the potential and the presence of ions beyond the first coordination shell, neglected in our harmonic approximation (Fig. 2c, see also Supplementary Note 4 and Supplementary  Fig. 10). Also an underestimation of the reduced mass, as hydration of ions and/or electrostatic interaction with other ions may effectively result in a higher reduced mass, could contribute to this difference.
Note that while the harmonic oscillator model does not capture all details of the simulated I II spectra, it serves to illustrate the underlying molecular-level dynamics. Within this model, the peak maximum (or similar, the peak integral) of the oscillator inversely scales with damping, as higher friction attenuates the ions' motion and thus reduces the ionic polarization according to ε HO 00 ω 0 ð Þ $ c salt =ω 0 γ (see Supplementary Eq. (13)). The oscillator amplitude also scales with the inverse fluctuation frequency, ω 0 , since steeper (shallower) potentials and/or heavier (lighter) ions narrow (widen) the spatial extent of thermally accessible excursions of the ions out from their equilibrium. We conclude that the I II spectral contribution sensitively reports on the ionic distribution and dissipative ion effects (as quantified by the friction coefficient) in solution.
Effect of nature of the ions on fast motion. The sensitivity of the ions' response to the ionic distribution can be directly demonstrated using MD simulations. To this end, we reduce (or increase) the charge of the ions. Altered ionic charges lead to altered interaction with water, leading to a modification of the friction experienced by the ion and a marked change in the width of the first ionic coordination peak in the RDFs (Fig. 2b). As such, the potential of mean force of the ions becomes more shallow (steep), and based on the oscillator model, the magnitude of the fluctuations is expected to be accordingly enhanced (reduced). Quantitatively, we find the height of the simulated I II peaks (Fig. 2a) to vary by +50% (−25%), upon decreasing (increasing) the charge by 30%, while the diffusivities of the ions vary by +30% (−30%) (see Supplementary Table 1). In light of the oscillator model described above, the higher sensitivity of I II to the ionic charge, as compared to the diffusivity, shows that both the altered friction and the altered ionic distribution give rise to the changes in I II upon charge scaling. Thus, the ionic contributions to the spectra sensitively report on the interaction and distribution of ions in electrolytes.
To demonstrate the sensitivity of the~0.3 THz absorption to the ionic species experimentally, we measure the DS spectra for different ions at various concentrations (for spectra and fits see Supplementary Figs. [11][12][13][14][15][16][17]. As the nature of the ions and their concentration markedly affects their pair distribution, their dynamics are expected to be ion-specific, and the magnitude of the ionic polarization will depend on both concentration of electrolyte and the nature of the salts (e.g., ionic radii, van der Waals repulsion, "soft or hard" nature of the ions, etc.). In contrast to the simulations, which can disentangle the spectral contribution of ions from the contribution of water, we experimentally monitor all fast dynamics due to water and ions by studying the amplitude of the fitted Debye mode. As the motions of both water and ions contribute to the spectral intensity at~0.3 THz (Fig. 1c), only the variation of the intensity with salt concentration allows for drawing conclusions on ioninduced dynamics, while the absolute values of S fast (and also τ fast ) also contain information on the dynamics of water. As can be seen in Fig. 3a, the values for S fast vary widely for different monoand bivalent salts. However, salt concentration does not exclusively determine the magnitude of S fast when comparing all studied salts (as would follow from asymptotic electrolyte theories that treat ions as point-like charges): The data in Fig. 3a are scattered, and we find a Pearson's correlation coefficient r = 0.64 for S fast (c salt ).
Conversely, the oscillator model suggests that the amplitude of the ionic contributions scales with the inverse damping and the ions concentration. Thus, given that the macroscopic friction for long-range transport and the friction for the short-scale THz motion are correlated, the spectral contributions of the ions to the fast dynamics are expected to scale with the macroscopic conductivity (see "Methods" section), rather than with concentration (see also Supplementary Note 5 and Supplementary  Fig. 18). Moreover, the curvature of the potential for the ions, which also affects the spectral contributions, may also be related to the energetic barrier to escape the potential minimum (for our simulations on solutions of KI the height of the maximum at Δr ≈ 1 Å in Fig. 2c is related to the curvature at the minimum, see Supplementary Note 6 and Supplementary Fig. 19). This suggests, that the ionic dynamics at~0.3 THz are related to the energy barrier for the ion to 'escape' the solvation cage: The shallower the cage potential, the easier the ion can escape its solvation cage. Translating the ion out of the solvation cage is again the ionic conductivity.
To experimentally testify the relation between local ion dynamics and macroscopic conductivity, we plot in Fig. 3c  value of~2 for the fast dynamics at~0 S m −1 (neat water), the amplitude increases to~3.5 for electrolytes with a d.c. conductivity of~25 S m −1 . We note that even for the strong acid HCl the correlation holds at low concentrations, while it breaks down at higher concentrations (see Supplementary Fig. 20), which can be related to the very different charge transport mechanism for the proton (Grothuss-type transport with charge transport being decoupled from mass transport 47 ). The experimentally determined Debye relaxation times (~inverse center frequency) of the fast relaxation lie in the range 100-700 fs (Fig. 3b).
Spatial correlation of ions' motion. Together, the experimental and simulation results show that short-ranged motions of ions and long-ranged diffusive transport of ions are correlated (for quantitative comparison with the harmonic oscillator model, see Supplementary Note 7 and Supplementary Fig. 21), even at relatively high salt concentrations. This correlation stems in part from similar trends in friction. For concentrated electrolytes, friction governing conductivity contains not only hydrodynamic (Stokes friction) contributions, but also ionic friction (ion cloud relaxation and electrophoretic drag, see also Supplementary  Fig. 14a) 11 . Thus, ionic motions are highly correlated, and one question that remains is the degree of collectivity of the ionic motion at~0.3 THz discussed here. The simple harmonic oscillator model based on the RDF takes only ion pairs into account. The dynamics in the first coordination shell are not independent of the motion of ions in the other coordination shells, an effect that is not contained in the RDF. To determine the degree of collectivity of the ionic dynamics, we performed MD simulations with water molecules beyond the first, second, and third hydration shell fixed in their coordinates. These simulations for a 0.5 mol L −1 KI solution (Fig. 4) indicate that the ionic dynamics approach the bulk dynamics only if molecules within the third coordination shell are mobile. This means that the observed ion dynamics at THz frequencies for the 0.5 mol L −1 KI solution (Fig. 4) are governed by the correlated motions of ions and water, with the correlation extending up to three coordination shells (~0.9 nm). This correlation length is comparable to the~0.4 nm Debye length for a 0.5 mol L −1 salt solution. Hence, despite classical mean-field theories like Debye-Hückel being unable to predict the observed ion dynamics as they cannot explicitly account for the interaction between ions and water, the Debye screening length still provides a satisfactorily accurate estimate for the spatial extent of ionic correlations.

Discussion
In summary, we show that the fast (~0.3 THz) dynamics in the dielectric relaxation spectra of electrolytes reflects the dynamics of ions in their solvation cages. These dynamics are ion-specific, but the magnitude of the motion scales universally with conductivity. This means that the microscopic motion of the ions in a given solvation cage and the macroscopic transport of the ions are intimately connected. This correlation can be understood by noting that microscopic and macroscopic frictions are related and that the potential energy landscape of the ions in the cage determines the ions' thermally accessible excursions, which are also related to the ion escaping its cage to allow for ion transport. This includes the frequently inferred concept of ion-pairing 23,48 : electroneutral ion-pairs do not contribute to conductivity, and the ions reside in very steep potentials that restrict the amplitude of the ionic motions. Although a simple harmonic oscillator model for the ions' motion captures large parts of the observed polarization dynamics, correlated motion involving coordination shells beyond the classical Debye length has to be included to fully capture the observed polarization dynamics. Despite the Debye length being a reasonable estimate for the spatial extent, the ionic dynamics, in fact, probe ionic distributions that go beyond a classical Debye-Hückel charge distribution, as Debye-Hückel does not take the molecular nature of the solvent (cage potentials) into account. As such, the dynamics of ions at 0.1-1 THz are a sensitive experimental measure for the ion distribution in concentrated electrolyte solutions that goes beyond mean-field theories and provide a means to test more advanced electrolyte theories that can explicitly account for molecular solvents. The scaling reported here provides a rationale for understanding, and possibly engineering, the macroscopic conductivity in electrolytes, e.g., ionic liquids or battery electrolytes. Dielectric relaxation spectroscopy. Complex permittivity spectra,ε ν ð Þ ¼ ε 0 ν ð ÞÀ iε 00 ðνÞ, were recorded at 0:96 ≤ v=GHz ≤ 125 using a frequency domain reflectometer based on an Anritsu Vector Star MS4647A 40,49 . Terahertz frequencies 0:3 ≤ v=THz ≤ 1:5 ð Þ were covered with a Terahertz time domain spectrometer 49 with the samples contained in a fused silica cuvette with a path length of 100 μm 50 . All experiments were performed at 296 ± 2 K.

Methods
In line with literature reports 29,31,35,40,51 , we fit a relaxation model based on a Cole-Cole equation and a Debye mode to the spectra: The first (Cole-Cole) term of Eq. (1) represents the main relaxation at~20 GHz with relaxation time, τ water , the relaxation strength, S water , and α CC the Cole-Cole parameter 52 . The second (Debye) term models the fast mode at~0.3 THz with relaxation time, τ fast , and relaxation strength, S fast . The limiting permittivity ε ∞ subsumes all polarization components at higher frequencies. The last term of Eq.
(1) accounts for Ohmic loss contributions due to the electrolyte conductivity, with ε 0 the permittivity of free space. We assume the conductivity, κ, to be real and independent of frequency. Thus, any frequency-dependent conductivity will be modeled by the Cole-Cole and the Debye term. Equation (1) was used to model the spectra for solutions of HCl, KCl, KI, LiCl, CsCl, NaCl, MgCl 2 , and GdmCl. Obtained parameters are shown in Fig. 3 [15][16][17]. Spectra were fit reducing the deviations of both ε′(ν) and ε″(ν) on a logarithmic scale (for fits using linear deviations see Supplementary Note 9 and Supplementary Figs. 22 and 23).
Molecular dynamics simulations. The MD simulations were performed using the CP2k package 55 . We used the TIP4P/2005 model 43 for water. The potentials for K + and I − ions are taken from ref. 56 . The long-range part of the electrostatic interactions was computed using the particle mesh Ewald scheme. The 0 mol L −1 system contained 1600 water molecules. The 0.5 and 1.0 mol L −1 KI solutions consisted of 14 and 29 KI ion pairs together with 1572 and 1542 water molecules, respectively. The neat water, 0.5 mol L −1 KI, and 1.0 mol L −1 KI systems were contained in cubic periodic cells with lengths of 36.343, 36.266, and 36.246 Å, respectively. To vary ionic distributions, we scaled ions' charges from ±0.7e to ±1.3e. All other parameters were kept identical to the 0.5 mol L −1 KI system. All systems were equilibrated for 180 ps in the canonical (NVT) ensemble using the CSVR thermostat set to 300 K 57 . After equilibration, 40 initial conditions were sampled from the canonical simulations at time intervals of 200 ps to initialize 2 ns long microcanonical (NVE) simulations. We also performed microcanonical simulations with fixed molecules (fixed-NVE) at 0.5 mol L −1 KI. The initial configurations of the fixed-NVE simulations were identical to the NVE simulations, whereas the position of ions and water molecules beyond the cut-off radii (r c K þ and r c I À ) were constrained in the Cartesian space. The cut-off radii were determined from the first, second, and third minimum of the ion-oxygen RDFs (r c K þ , r c I À )/Å = Harmonic oscillator model. We approximate the contributions of the ions to the dielectric response by a harmonic oscillator: where c salt is the salt concentration, q the ions charge, ω = 2πν the angular frequency, and μ the reduced mass. The force constant, k, was obtained from fitting a harmonic potential (F(Δr) = 1/2kΔr 2 ) to the potential of mean force (Fig. 2c). The drag coefficient γ (=k B T/D) was approximated based on the experimental diffusivity D = 0.4 Å 2 ps −146 . Expressing γ in terms of the conductivity γ = c salt q 2 /κ, yields the maximum in the dielectric loss at ω 0 ¼ τ À1 % ffiffiffiffiffiffiffi ffi k=μ p to scale with conductivity: For further details, see Supplementary Note 3.

Data availability
Data supporting the findings of this study are available within the article and its Supplementary Information, or from the corresponding authors upon reasonable request.