Santanu
Roy
*^{a},
Fei
Wu
^{b},
Haimeng
Wang
^{c},
Alexander S.
Ivanov
*^{a},
Shobha
Sharma
^{b},
Phillip
Halstenberg
^{ad},
Simerjeet K
Gill
^{e},
A. M.
Milinda Abeykoon
^{f},
Gihan
Kwon
^{f},
Mehmet
Topsakal
^{e},
Bobby
Layne
^{g},
Kotaro
Sasaki
^{g},
Yong
Zhang
^{c},
Shannon M.
Mahurin
^{a},
Sheng
Dai
^{ad},
Claudio J.
Margulis
*^{b},
Edward J.
Maginn
*^{c} and
Vyacheslav S.
Bryantsev
*^{a}
^{a}Chemical Sciences Division, Oak Ridge National Laboratory, 1 Bethel Valley Rd., Oak Ridge, TN 37830, USA. E-mail: roys@ornl.gov; bryantsevv@ornl.gov; ivanova@ornl.gov
^{b}Department of Chemistry, The University of Iowa, USA. E-mail: claudio-margulis@uiowa.edu
^{c}Department of Chemical and Biomolecular Engineering, University of Notre Dame, USA. E-mail: ed@nd.edu
^{d}Department of Chemistry, University of Tennessee, Knoxville, USA
^{e}Nuclear Science and Technology Department, Brookhaven National Lab, USA
^{f}National Synchrotron Light Source II (NSLS-II), Brookhaven National Lab, USA
^{g}Chemistry Division, Brookhaven National Lab, USA
First published on 14th August 2020
Molten salts are of great interest as alternative solvents, electrolytes, and heat transfer fluids in many emerging technologies. The macroscopic properties of molten salts are ultimately controlled by their structure and ion dynamics at the microscopic level and it is therefore vital to develop an understanding of these at the atomistic scale. Herein, we present high-energy X-ray scattering experiments combined with classical and ab initio molecular dynamics simulations to elucidate structural and dynamical correlations across the family of alkali-chlorides. Computed structure functions and transport properties are in reasonably good agreement with experiments providing confidence in our analysis of microscopic properties based on simulations. For these systems, we also survey different rate theory models of anion exchange dynamics in order to gain a more sophisticated understanding of the short-time correlations that are likely to influence transport properties such as conductivity. The anion exchange process occurs on the picoseconds time scale at 1100 K and the rate increases in the order KCl < NaCl < LiCl, which is in stark contrast to the ion pair dissociation trend in aqueous solutions. Consistent with the trend we observe for conductivity, the cationic size/mass, as well as other factors specific to each type of rate theory, appear to play important roles in the anion exchange rate trend.
This article focuses on the alkali-chloride family of molten salts for which we make predictions on what the important driving forces are for transport. The accuracy of these predictions strongly depends on the quality of our simulations. For example, the 2D free energies that will be introduced in our rate theory study are directly related to the structural distributions of ions in different coordination shells. A comparison between computed and measured X-ray/neutron scattering data including structure functions and concomitant pair distribution functions (PDFs)^{11} can help validate the accuracy of models and simulations in determining molten salt coordination environments and associated free energies.^{12–17} Early experimental data already exist for monovalent chloride salts obtained through pioneering neutron diffraction experiments employing the isotopic substitution technique, albeit with limited resolution and momentum transfer (q) range.^{18–23} Modern X-ray diffraction studies on the simple chloride melts are scarce, and old data were typically obtained from standard X-ray sources with a narrow q-range,^{24,25} causing uncertainties in the positions and heights of peaks in the corresponding PDF due to truncation effects.^{11} In our current study, we overcome these obstacles by utilizing high-energy X-rays (λ = 0.1667 Å, 74.4 keV) to measure structure functions, S(q), with an extended q-range and low statistical noise, which by Fourier transformation provide more accurate PDFs.
One of the critical objectives of molten salt-based technologies in electrochemical applications is to attain high ionic conductivity.^{26,27} Trends in diffusivities and conductivities will be contrasted with available experimental data in the literature allowing us not only to test models but also to establish possible connections with rate theories^{28–54} of ion exchange. Several computational studies on molten alkali halide salts and their mixtures at different concentrations and temperatures already exist in the literature,^{9,55–70} but questions still remain about transport and ion exchange dynamics on a microscopic scale. For example, it has been established that a small cation (such as Li^{+}) in neat molten salts has high internal mobility, which is reduced significantly when mixed into a salt of a larger cation (such as K^{+}) at a low mole fraction of the smaller cation.^{63,68,71} Although this finding about the cationic dynamics is remarkable, rates and mechanisms describing the dynamics of anion-exchange around cations remain unknown even in the case of the neat salts. We will attempt to obtain such mechanistic information and reveal the effect of barriers, mass, volume, and the coupling to solvent by applying and in cases extending the formalism of Marcus theory^{28–36} and transition state theory.^{37–39,42–45}
The containment used for the X-ray scattering experiments was a cylindrical quartz capillary of 1 mm internal diameter and 0.01 mm wall thickness. Each capillary was fitted with a quartz tube of 4 mm internal diameter. The addition of the extra length of quartz tubing allowed for the attachment of a compression fitting so that vacuum could be pulled to flame seal the capillaries, once they were loaded with samples (Fig. S1 in the ESI†). After the extra tubing was fitted to each capillary, they were taken into the glove box. Each salt was crushed into a fine powder using a mortar and pestle and then the additional tubing added to each capillary was used as a funnel, and the powder was added until the capillary was full. After the capillary was packed with the appropriate amount of salt, it was taken out of the glove box with a compression fitting and flame sealed under a 1 × 10^{−3} Torr vacuum.
(1) |
In this study, the reduced pair-distribution function (D(r)) in the range q_{min} = 0.6 Å^{−1} to q_{max} = 16 Å^{−1} is defined^{72} as
(2) |
(3) |
Final snapshots from the RIM simulations were taken as initial points for simulations using the PIM.^{80,81} Each system was first subject to a 200 ps simulated annealing equilibration at constant pressure (at 1 bar) that increased their temperature to 1640 K and cooled back to target temperatures. Further equilibration and production runs in the NPT ensemble (at 1 bar and target temperature) as well as the calculation of S(q) and D(r) followed the same scheme as described for the RIM simulations. PIM simulations were performed using the CP2K package^{83} with a time step of 1 fs. Both the RIM and PIM simulations used the Nosé–Hoover thermostat and barostat^{84–86} with a time constant of 1 ps to control the temperature and pressure. The non-bonded cutoff was set to 15 Å.
Additionally, following a similar protocol to the one described above, all system were equilibrated to a target temperature of 1100 K. The last 2 ns of 3 ns runs in the NPT ensemble were used to compute average densities. Volumes in the last snapshot of these NPT runs were rescaled to match these average densities and used as starting points for constant volume–temperature (NVT) runs. In the case of the RIM (PIM), we further equilibrated the system for 1 ns (0.5 ns) in the NVT ensemble and collected data from a production run of 1 ns in duration for computing 2D-free energy surfaces. We verified convergence by noting that any 100 ps segment could reproduce identical 2D-free energy surfaces.
AIMD simulations used the Quickstep module of the CP2K software,^{75} where trajectories were run in the NVT ensemble using the Nosé–Hoover thermostat with a 1 ps time constant.^{84,85,88} Valence electrons were treated explicitly at the DFT level employing the revPBE functional^{89} and the DZVP-MOLOPT basis set^{90} with density from a cutoff of 400 Ry. The core electrons on all atoms were represented by revPBE pseudopotentials, and we used the Grimme's D3 long-range dispersion correction to the DFT functional.^{91} A time step of 0.5 fs was used to generate a 120 ps trajectory and the last 100 ps of that trajectory were used for computing S(q) and D(r).
(4) |
(5) |
To compute the ionic conductivity of the RIM and PIM, 30 independent trajectories were simulated using the same NVT simulation protocol and data were saved every 10 fs for analysis. These trajectories were 1 ns in duration in the case of the RIM and 200 ps in the case of the PIM. Data at 100 fs intervals from these 30 independent trajectories were also used to evaluate standard deviation errors in the calculated diffusion coefficients.
Ionic conductivity was computed using the Green–Kubo formula:
(6) |
(7) |
(8) |
For our 2-D reaction coordinate, we compute the joint probability distribution (P(r,n)) of r and n from which a 2D-free energy landscape can be obtained through the expression W(r,n) = −k_{B}TlnP(r,n). We also define the quantity dN = 4πr^{2}ρ_{A}P(r,n)drdn as the number of cases for which the solute X^{+}–Cl^{−} distance is between r and r + dr and the X^{+}–solvent Cl^{−} coordination number is between n and n + dn (ρ_{A} is the anionic number density). As required in our rate theory formalism, we define n in terms of a smooth function (continuously differentiable), f(r_{i});^{34,35}
(9) |
According to Marcus theory, an anion exchange process can be described by considering the reactant (R) and product (P) states as 1D parabolic functions of n, which are extracted from W(r,n) as follows:
(10) |
The free energy barrier (ΔW^{†}) for the reactant-to-product transition takes a simple expression (eqn (11)) when K_{R} = K_{P}. In eqn (11), ΔW^{†} depends on ΔW and the solvent reorganization energy (λ = W_{R}(n_{P}) − W_{R}(n_{R})) which is the energy cost required for changing the equilibrium reactant coordination state to the equilibrium product coordination state.
(11) |
(12) |
Using the free energy barrier derived from Marcus theory, eqn (13)^{34,94} provides a simple approximation to the transition rate between reactant and product states based on Wigner's transition state theory (TST).^{38,39,94}
(13) |
(14) |
(15) |
(16) |
If Marcus theory was exact, upon arrival at the crossing point of the reactant and product parabolas from the reactant equilibrium coordination state, an anion spontaneously dissociates from the paired state with the cation leading to the product coordination state. However, it is possible that there is a non-vanishing barrier along the ion pair distance at the crossing point , making the dissociation event nonspontaneous. can be determined by extracting a slice from W(r,n) at n = n^{†}; the first barrier on that slice provides . If , it must be accounted for as an additional barrier for the reactant-to-product transition. Furthermore, rapid fluctuation of the surrounding solvent coupled to the motion along the coordination number can cause recrossing at the crossing point, resulting in effectively slower transition rates. Such nonequilibrium solvent effect can be treated by utilizing the semiclassical approach of Landau^{30,95} and Zener,^{31} which corrects Marcus rates through the determination of the transmission coefficient (κ_{LZ}). κ_{LZ} is defined in terms of the probability (P) of reactive transitions through the crossing region and the location of the crossing region:^{32,34}
(17) |
Since for all the molten alkali chloride studied here the Marcus parabolas cross on opposite sides, only the “normal region” expression in eqn (17) is relevant to our study. P depends on the coupling (C) between the reactant and product parabolas and the positive traversal velocity (v_{n}) in coordination number space at the crossing point:
(18) |
(19) |
A complementary analysis to the one discussed above in the case of n as a reaction coordinate can be carried out instead utilizing TST with the distance r as the reaction coordinate. This approach has been extensively used to study ion-pairing and solvent exchange kinetics,^{46,48–50,96–108} where nonequilibrium solvent effects are treated by computing the transmission coefficient with a variety of methods such as that of Krammer,^{46} the Grote–Hynes's stable state picture,^{48} or the reactive flux (RF) method^{49,50} by Chandler and Bennett. We see from eqn (20), corresponding to Roy's approach,^{54} that the expression is also influenced by a mass-weighted configuration space “reactant volume”, , in addition to the barrier (W(r^{†})) and the transmission coefficient (κ_{RF}).
(20) |
(21) |
(22) |
Fig. 2 For KCl at 1173 K, experimental S(q) (top) and reduced pair distribution function D(r) (bottom) from this work compared to that of Takagi and coworkers from ref. 24.‡ Takagi's real-space data were generated by digitizing the pair distribution function, G(r), in Fig. 1 from ref. 24 and converting it to D(r) using the expression: D(r) = 4π*ρ_{0}*r*G(r),^{72,74} where ρ_{0} = 0.02345 atoms per Å^{3}.^{87} (G(r) = g*(r) − 1 in Takagi's notation). Takag's reciprocal-space reduced intensity data were digitized from Fig. 2 in ref. 24 from which I_{coh} was derived based on eqn (8) in the same article. I_{coh} was then used to compute S(q) based on eqn (1) in our article. |
To examine whether our MD simulations can accurately describe the structure of the molten chloride salts, we computed X-ray structure functions, S(q), at different levels of theory to compare with those from the synchrotron X-ray scattering experiments. Fig. 3a–c show that across systems, simulations using each of three models (PIM, RIM, and ab initio) capture features of the experimental S(q) fairly well, but the PIM model appears to be overall slightly more accurate when comparing peak positions and intensities. This is in part because the chloride ion is significantly polarizable, because AIMD results may be sensitive to the chosen DFT flavor, and also because DFT simulations boxes are necessarily smaller than those used for the PIM and RIM simulations.
Fig. 3 Comparison between our NSLS-measured and our simulated X-ray structure functions for LiCl at 958 K (a), NaCl at 1148 K (b), and KCl at 1173 K (c) and corresponding decompositions of the total structure function into cation–cation, anion–anion, and cation–anion subcomponents (partial structure functions) as defined in the ESI of ref. 69 and 70 obtained from PIM simulations for LiCl (d), NaCl (e), and KCl (f). |
The interpretation of peaks in S(q) can be achieved by decomposing the function into contributions from the different ion-pair subcomponents (the partial structure functions, S_{αβ}(q)). Fig. 3d–f show how cation–cation, anion–anion, and cation–anion correlations contribute to the total S(q) determined from the PIM simulations. The inherent structural characteristic of molten salts is charge alternation, and its signature is a positive-going peak arising from the contributions of same-charge ions at about the same q value as a negative-going peak (also known as an antipeak^{109–116}) resulting from spatial correlations between opposite-charge ions. For LiCl, these charge-alternation peaks and the concomitant antipeak are present around ∼2 Å^{−1}; peaks and antipeaks corresponding to this feature move to lower q values (∼1.75 Å^{−1} for NaCl and ∼1.6 Å^{−1} for KCl) with larger cation size. This is because these peaks and the corresponding antipeak are linked to the distance between ions of the same charge alternated by ions of opposite charge which becomes larger as the cation size increases. Summing the three contributions (cation–cation, anion–anion and cation–anion) results in a prominent peak in the total S(q) for LiCl but the peak diminishes in intensity for NaCl and is completely absent for KCl. This is purely related to lack of contrast in X-ray experiments; the X-ray form factor for Li is small but that for Cl is large and the sum of intensities of the two peaks (Li–Li and Cl–Cl) are not cancelled by that of the Li–Cl antipeak at the same q value. Instead, there is essentially complete cancellation of peaks and antipeaks for KCl resulting in a missing charge alternation feature in the overall S(q).^{69} As is obvious from the two large (and nearly identical) peaks and the antipeak in Fig. 3f, this does not mean that there is no charge alternation in KCl but instead, that there is a poor contrast in the technique. For example, the result would be different if one was to use neutron weighting factors instead of those for X-ray. The peak at a larger q value than charge alternation is what we commonly call the “adjacency peak”.^{109–116} In general, this peak is associated with short-range structural interactions between nearest neighbors. In the case of salts, these are commonly opposite-charge ion interactions. This can be clearly gleaned from the similarity between the total S(q) and the cation–anion partial S(q) above ∼2 Å^{−1} in Fig. 3e and f. The case of LiCl is particularly interesting since there is a major weight difference in the X-ray form factors between the cation and the anion. This results in no apparent Li–Cl adjacency peak; in Fig. 3d, the black line does not look like the blue line between 2.5 and 3 Å^{−1}.
Real space D(r) functions obtained by the Fourier transform of the total structure functions (eqn (2)) are depicted in Fig. 4a–c, and compositionally resolved partial D_{αβ}(r) from the PIM simulations (as defined in eqn (3)) in Fig. 4d–f. Corresponding pair distribution functions g_{αβ}(r) are provided in Fig. S3 of the ESI.† These figures show that the shortest contact ion pair distance (dominating the first peak in D(r)) is achieved for LiCl, followed by NaCl and KCl as is obviously expected from the trend in cationic sizes. It is worth noting that the reduced pair distribution function, D(r), emphasizes the large-r portion of the data and this helps highlight ionic correlations beyond short-range interactions. Partial D_{αβ}(r) from our MD simulations facilitate the interpretation of real-space data by showing how particular ion–ion correlations are manifested in the total D(r). As the number of electrons in the cation increases, so does the cation–cation contribution to D(r). Analogous to the case of the partial subcomponents of S(q) discussed in the previous paragraph, this is simply an issue of contrast in the X-ray technique. Overall, our results appear to indicate that simulations, particularly the less expensive classical ones, can be used for the interpretation of experimental results.
Fig. 4 Comparison between our simulated and experimental D(r) functions for LiCl at 958 K, NaCl at 1148 K, and KCl at 1173 K (top row) and interpretation of different peaks in terms of the sub-components of the total D(r) obtained from the PIM simulations (bottom row) (see also pair distribution functions g_{αβ}(r) in Fig. S3 of the ESI†). |
Fig. 5 Variation in the diffusion coefficient, molar conductivity, and conductivity with increasing ionic radius of the cation obtained from experiments^{119,120} as well as the PIM and RIM for LiCl, NaCl, and KCl at 1100 K. |
The ionic conductivity, is made of contributions from the collective dynamics of same- and different-charge species that goes beyond the self-diffusion dynamics of cations or anions. This can be gleaned from eqn (6) which can be expanded^{110} as
(23) |
Fig. 6 shows all correlation functions (C_{v}) as well as the weighted contributions of the different diffusion coefficients to the conductivity. As can be seen from Fig. 6, for the most part, all C_{v} functions decay to zero within a fraction of a picosecond (although small oscillations still persist beyond this time that contribute to their time integrals) confirming the idea that what happens at a very short time is extremely important to the overall value of the conductivity in these systems. A few things can be learned from the bottom panel in Fig. 6. First, as expected from AB-type ionic systems where there is no third neutral solvent component to act as a “momentum buffer”,^{118} the cation–anion contribution to the conductivity is positive. This is in contrast to what happens to ions in solution where ion-pairing reduces conductivity. Second, the behavior of LiCl across the different weighted D_{i} functions is very different from that of NaCl and KCl which are much more similar. This is not to say that the C_{v} functions are similar for NaCl and KCl, but in most cases, the long-time weighted integrals are when compared to those for LiCl. This is particularly clear in the case of the cationic self contribution, but is also apparent for the anionic self and distinct contributions. In the case of LiCl the positive contribution of the anionic self term is roughly cancelled by that of its distinct contribution; since cation–anion and cation distinct contributions across the three salts are not that different, one can fairly state that it is the self diffusion component for Li^{+} that makes the conductivity for LiCl distinctly different from that of NaCl and KCl. Notice that distinct cation and anion contributions in Fig. 6i and j across the three salts appear to be in reverse order and in the case of NaCl and KCl these differences roughly cancel each other resulting in overall conductivities for NaCl larger than for KCl that follow the trends in Fig. 6f–h. It is now easier to see why the drop in conductivity is much larger when going from LiCl to NaCl than from NaCl to KCl. The dominating effect is simply the high self diffusivity of Li^{+} and the similarity in trends between Fig. 5a and b confirms this.
Fig. 6 Various types of self (s) and distinct (d) velocity correlation functions for LiCl, NaCl, and KCl at 1100 K (a–e). The time-dependent weighted contribution of the different diffusion coefficients to the conductivity (f–j); positive contributions from the cation–anion, self cation–cation, and self anion–anion increase the conductivity, whereas cation–cation and anion–anion distinct contributions decrease conductivity. See Fig. S4 (ESI†) highlighting the short-time behavior of the weighted diffusion coefficients. |
The first two methods require knowledge of the free energies, W(r), W(r,n), and W(n), which are presented in Fig. 7a–g for the PIM simulations at 1100 K. Fig. S5 in the ESI† provides a comparison between the PIM and RIM results, indicating a good agreement between them. The reader is reminded that n in the y-axis of W(r,n) is defined as the number of counter-anions to a central alkali metal ion not counting the special anion defined to be its pair (the progression from reactant to transition and finally product states as well as the definition of n are depicted in Fig. 1). This theoretical construct is used so that n is different (smaller) in the reactant state than in the product state even though the actual number of Cl^{−} ions surrounding the cation is the same.
Fig. 7 One-dimensional free energy along the solute cation–anion distances (W(r)) (a) and their extensions to two-dimensions (W(r,n)) by including the coordination number of the solute cation (n) with the solvent anions (evenly spaced (1 kcal mol^{−1}) contours between 0 and 10 kcal mol^{−1}) (b, d and f). White arrows on W(r,n) display the Marcus pathways of ion-pair dissociation. Slices through W(r,n) at r = r_{R} and at r = r_{P} (dotted lines) and their parabolic fits (solid lines) represent reactant and product states in coordination number space used in Marcus theory (c, e and g). Using Marcus theory, the ion pair dissociation and anion exchange mechanism around a cation in molten salts can be described as the adiabatic traversal of the crossing point of the Marcus parabolas (dashed red and blue, respectively) that couple to generate lower and higher free energy surfaces (solid red and blue, respectively) (h, i and j). These results are from PIM simulations at 1100 K for LiCl, NaCl, and KCl, whereas other cases are discussed in the ESI.† |
For the different salts, W(r) clearly distinguishes the equilibrium distances corresponding to the contact ion pair (r_{R}) in the reactant state (R) and the solvent-separated ion pair (r_{P}) in the product state (P). These equilibrium distances systematically increase going from LiCl to NaCl to KCl. However, we notice that barriers along the solute cation–anion distance are very similar (∼4.6–4.8 kcal mol^{−1}) for all the salts and are significantly higher than the thermal energy at 1100 K (2.18 kcal mol^{−1}). Overcoming this barrier makes ion-pair dissociation an activated process. Of course, a 1D-free energy hides the fact that barriers can be circumvented if pathways are considered in multi-dimensions such as those shown by the white arrows on the 2D-free energy landscapes in Fig. 7b–f. Physically, the white arrows highlight three mechanistic steps: first, solvent-rearrangement induces activation in coordination number (n) leading to anionic overcrowding (notice that while at the transition state n < n_{P}, the actual number of anions surrounding the cation including the one considered as the ion pair is larger than in either of the wells); second, the overcrowding reduces the barrier along the solute cation–anion distance (r), and third the dissociation of the solute ion-pair along r takes place where the solvent anion takes the place of the solute anion as the product state. Cuts on the 2D-free energy surface at r = r_{R} and r = r_{P} (the diabats) in Fig. 7c–g have minima that are well separated and cross in the “normal region” (opposite sides) in the Marcus theory sense. Lower and higher adiabatic free energy curves can be derived from these as discussed in the methods section and are shown in Fig. 7h–j.
Our first method to study rates of chloride exchange across the family of alkali chlorides, is the hybrid Marcus-TST approach associated with eqn (19). All essential parameters entering this equation in the case of the PIM and RIM are provided in Table 1 (see Table S2 in the ESI† for the remaining parameters including those defining Marcus parabolas and their couplings and traversal velocities at the crossing point). τ values in Fig. 8 (labeled Marcus), which are the reciprocal of the exchange rate, show a clear trend of increasing exchange times with an increase in cation size. There are several factors contributing to this; a large contributor is what we defined in the methods section as the mass-weighted “reactant volume” in coordination number space, . The reader should notice that for a given system, Z_{n} = Λ/μ is essentially a conserved quantity (its value as a function of time is shown in Fig. S6 of the ESI†), which has complex contributions of the solvent anions incorporated in Λ (see eqn (15)). However, Λ is almost the same for all three salts (considering their error bars) as shown in Table 1. Thus, the trend in Z_{n} is mostly dominated by μ, therefore by the mass of the cation (as all three salts have the same anion). On the other hand, V^{n}_{R} does not show any specific trend (see Table 1), which leads to the conclusion that is predominantly governed by the mass of the cation. For the PIM, a little more than doubles (the ratio is ∼2.19) in going from LiCl (∼0.36) to KCl (∼0.79), and since its role is multiplicative (see eqn (19)) so does the value of τ. The second most important factor in the trend for τ is the Landau–Zener transmission coefficient which is larger for LiCl (κ_{LZ} ∼ 0.97–0.98) than for NaCl (κ_{LZ} ∼ 0.91–0.94) and KCl (κ_{LZ} ∼ 0.64–0.80). For the PIM, in going from LiCl to KCl, its effect is to increase τ by about 52%. The large values of κ_{LZ}, particularly for LiCl and NaCl, are indicative of a weak coupling to the solvent bath and small nonequilibrium solvent effects for n. Yet, differences in κ_{LZ} are significant in distinguishing LiCl from KCl when it comes to the anion exchange rate. The last factor that contributes to differences in the anion exchange rate is the barrier , which is slightly larger for KCl (4.8–4.9 kcal mol^{−1}) than for NaCl and LiCl (4.4–4.6 kcal mol^{−1}). The effect from the barrier (see the term, in Table 1) in differenciating the cations is small, on the order of 9% in going from LiCl to KCl for the PIM.
The reader is reminded that the conductivity across the family of alkali chlorides in Fig. 5c shares the decreasing trend shown by the exchange rate, which is inverse to the increasing trend for τ shown in Fig. 8. Furthermore, we note that as the cationic mass (size) and concomitantly increase, so does the macroscopic volume across the family of molten salts at the same temperature and pressure. In going from LiCl to KCl, this increase in the overall macroscopic volume has an explicit effect on their conductivity. This can be clearly gleaned from the prefactor in eqn (6) and the fact that conductivity and molar conductivity have different trends. The latter, which loses this prefactor, flattens out significantly as a function of cationic size, particularly when going for NaCl to KCl. It is therefore notable that an increase in cationic mass/size consistent with an increase in the macroscopic sample volume goes along with a decrease in conductivity as well as in the anion exchange rate.
The second TST approach that we used to study anion exchange rates and derive τ (see Fig. 8 labeled TST), also follows the formulation by Roy et al.^{54} but in this case for r as the reaction coordinate. The expression for the anion exchange rate is given in eqn (21) for which the relevant quantities are W(r) (shown in Fig. 7a), the reactive flux transmission coefficients (shown in Fig. 9 for the PIM simulation and in Fig. S7 in the ESI† for the RIM simulation), the quantity ṽ_{r} linked to the thermal average relative velocity between the cation and the anion, the volume of the sample, and the average coordination number, n_{ave}, for the different cations (all provided in Table 2).
Fig. 9 For the three molten salts at 1100 K, time-dependent transmission coefficients (κ_{RF}(t)) determined from the PIM simulations at 1100 K using the reactive flux method showing strong interionic distance-solvent coupling. The values of κ_{RF} used in the expression for τ are derived from the average of κ_{RF}(t) in the region between 1 and 2 ps (same protocol for the RIM in Fig. S7, ESI†). |
LiCl | NaCl | KCl | |
---|---|---|---|
μ (g mol^{−1}) | 5.81 | 13.95 | 18.59 |
v _{r} (nm ps^{−1}) | 1.26 | 0.81 | 0.70 |
PIM | |||
κ _{RF} | 0.20 | 0.22 | 0.22 |
V (nm^{3}) | 42.54 | 53.89 | 71.46 |
n _{ave} | 4.34 | 5.16 | 5.18 |
r ^{†} (nm) | 0.36 | 0.41 | 0.45 |
W(r^{†}) (kcal mol^{−1}) | 4.83 | 4.65 | 4.75 |
e^{−βW(r†)} | 0.11 | 0.12 | 0.11 |
RIM | |||
κ _{RF} | 0.21 | 0.21 | 0.22 |
V (nm^{3}) | 45.87 | 57.63 | 78.82 |
n _{ave} | 3.90 | 4.82 | 4.89 |
r ^{†} (nm) | 0.36 | 0.41 | 0.45 |
W(r^{†}) (kcal mol^{−1}) | 4.60 | 4.61 | 4.58 |
e^{−βW(r†)} | 0.12 | 0.12 | 0.12 |
In this case, the analysis appears to be much simpler. For all three liquids, κ_{RF} is essentially the same; the structural component would make the rate of anion exchange for KCl > NaCl > LiCl, but both the anionic number density, N_{an}/V, and ṽ_{r} contribute to the opposite trend making the anion exchange significantly faster for systems with a smaller and lighter cation. Since N_{an} is the same for all systems, the trend observed in anion exchange rates appear to be simply determined by μ (through ṽ_{r}), and the volumetric differences across samples. This correlates well with findings in Fig. 5 where Li has the faster diffusivity, conductivity, and molar conductivity, whereas for NaCl and KCl the larger conductivity for the former is majorly influenced by the overall sample volume.
Notice that interpretation on how the similar values across theories for τ in Fig. 8 are arrived at can be quite different when the analysis is based on r vs. n as reaction coordinates. For example, nonequilibrium solvent effects depends on how a reaction coordinate couples with the solvent bath, and when n is the reaction coordinate, those are reflected in κ_{LZ}. Values for κ_{LZ} hinted at a relatively weak coupling to the solvent, but the numerical value differences across salts made a very significant difference to the observed τ values as we went from LiCl to KCl. Previous studies on aqueous electrolytes^{98–108} indicate that the distance coordinate strongly couples with the solvent resulting in very small transmission coefficients (≪1). Herein, we also find the interionic distance-solvent bath coupling to be strong and affecting equally all three molten salts, since the reactive flux transmission coefficients are small and similar for all of them (∼0.20–0.22). Thus, while nonequilibrium solvent effects on the distance coordinate equally influence the anion exchange rates for all salts, the same effects on n as a reaction coordinate can slow down the rate a small amount for NaCl but a significant amount for KCl when compared to LiCl. It is therefore clear that the very nature of the nonequilibrium solvent effects strongly depends on our choice of reaction coordinate and the specific rate theory method used, and therefore, it is not an unambiguous factor. The reader is reminded that, for the TST approach where r is the reaction coordinate, the barrier for anion exchange is almost identical (W(r^{†}) ∼ 4.6–4.8 kcal mol^{−1}) across the family of alkali halide salts. The barrier enters the structural factor through the value of the pair distribution function at r^{†}, and the overall contribution of this factor is opposite to the actual trend of anion exchange. In the Marcus-TST approach where n was the reaction coordinate, the total barrier was also quite similar (∼4.4–4.9 kcal mol^{−1}) across systems. Thus, consistent across the two methods so far discussed, barrier effects do not determine the cationic size effect on the rates of anion exchange. This leads us to the overall conclusion that μ (and therefore the mass of the cation) is an important factor in both rate theories, while solvent effects are important to the Marcus-TST approach based on n, and differences in molar volume of the melt are important in the TST approach based on r.
As a final and more direct way to interrogate the process of anion exchange across our systems, we utilized the method proposed by Impey, Madden, and McDonald (IMM) to compute the residence time of a solvent molecule (in our case the chlorides) in the first solvation shell of the alkali metal cations.^{121} The IMM method determines the survival probability (P(t,t + δt,t*)) of a anion in the first solvation shell of a cation where P is assigned a value of 1 when an anion is found in the first solvation shell at both times t and t + δt and during this time interval the anion does not leave the shell for any continuous period of time greater than t*; otherwise, P is assigned a value of 0. Numerically, in the condensed phase, particles at an artificial boundary are constantly recrossing it, but this does not mean that they have left the first solvation shell. This is why a tolerance value t* is defined to treat unsuccessful recrossing, i.e., when anions cross the boundary of the first solvation shell but rapidly return to the shell. Obviously the choice of t* will change P(t,t + δt,t*), but sensible values should keep trends unmodified. The residence time of the ith anion is obtained from the normalized time-correlation function of the survival probability
C(t,t*) = 〈P_{i}(t,t + δt,t*)〉_{i,t}/〈P(t,t,t*)〉_{i,t}. | (24) |
Fig. 10 (a and c) Survival probability correlation functions for t* = 0.5 ps at 1100 K; circles are simulation results and solid lines are bi-exponential fits. (b and d) Anionic escape times derived from the fits as a function of t*. (b and d) Show that independent of the model and the choice of t*, faster chloride escape is observed for LiCl than for NaCl and KCl. This is the same trend observed in Fig. 8 for τ. |
The experimental and computational conductivity of these systems decreases with increasing cationic size. Two factors contribute to this decrease, the first one is the very high self contribution of Li^{+} compared to other ions and the second is the change in molar volume across the salts. If one corrects for the change in volume by focusing instead on the molar conductivity, there is still a significant drop in going from LiCl to NaCl, but the experimental change in going from NaCl to KCl is small and computationally there is essentially no change in molar conductivity between these two. The same type of trend is observed when considering the diffusion coefficients instead of the molar conductivity.
We used three different approaches to understand the time scale and mechanisms for chloride exchange (or escape in the case of the IMM technique) across the family of cations. In all cases, we found that consistent with the trends in conductivity, the reciprocal rate of anion exchange, τ, is larger for the larger cations. Different reaction coordinates and methods provide unique perspectives to the process; for example, the solvent ions couple strongly to the inter-ionic distance but less so to the coordination number. Whereas anion exchange is definitively an activated process, barriers to anion exchange do not appear to determine trends of exchange. The mass of the cation and solvent effects are salient factors in the Marcus-TST approach based on n whereas the mass of the cation and the overall sample volume are important in the TST approach based on r. While none of the methods are superior to any other-they all generate reasonably consistent τ values bolstering confidence in our analysis, the TST approach based on r appears to provide the most intuitive link to simple factors (the cation size/mass and the sample volume) that directly correlate with trends for the conductivity of the different melts.
In the future, it may be interesting to study mixtures of these salts, particularly the low-melting eutectics which are very relevant to technology applications. We suspect that at these lower temperatures there could be a crossover between the high-temperature dynamics (where free energy barriers are not fundamentally important in defining anion exchange) and the lower-temperature regime in which free-energetics may be all that matters. We have mentioned earlier that whereas lithium has a very high mobility in molten LiCl, when mixed in a salt with a larger cation such as KCl this is reduced significantly. This phenomenon is also common when a small cation such as Li^{+} is mixed in an room temperature ionic liquid; in this case it is understood that the hard Li^{+} cation becomes an integral part of the ionic liquid charge network stiffening it and causing an increase in viscosity. How this happens with high temperature molten salts is something we would like to explore.
Footnotes |
† Electronic supplementary information (ESI) available: Experimental details and comparison between different simulation models. See DOI: 10.1039/d0cp03672b |
‡ We believe a typo has been introduced in expression of eqn (7) and (10) from ref. 24 where a factor of N appears to be missing. Eqn (7) also appears to be missing a factor of r. To the best of our understanding, to convert the reduced intensity data, si(q), in Fig. 2 of the aforementioned article to our notation as shown in Fig. 2 (top) we must use . In the case of Fig. 2 (bottom) we use the definition of g*(r) provided in eqn (9) of ref. 24. |
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