X-ray scattering reveals ion clustering of dilute chromium species in molten chloride medium

Enhancing the solar energy storage and power delivery afforded by emerging molten salt-based technologies requires a fundamental understanding of the complex interplay between structure and dynamics of the ions in the high-temperature media. Here we report results from a comprehensive study integrating synchrotron X-ray scattering experiments, ab initio molecular dynamics simulations and rate theory concepts to investigate the behavior of dilute Cr3+ metal ions in a molten KCl–MgCl2 salt. Our analysis of experimental results assisted by a hybrid transition state-Marcus theory model reveals unexpected clustering of chromium species leading to the formation of persistent octahedral Cr–Cr dimers in the high-temperature low Cr3+ concentration melt. Furthermore, our integrated approach shows that dynamical processes in the molten salt system are primarily governed by the charge density of the constituent ions, with Cr3+ exhibiting the slowest short-time dynamics. These findings challenge several assumptions regarding specific ionic interactions and transport in molten salts, where aggregation of dilute species is not statistically expected, particularly at high temperature.

Raman spectra for the KCl-MgCl2 (50:50 mol%) molten salt (black dashed line) and CrCl3 (5 mol%) in the KCl-MgCl2 (red dashed line) at 1073 K. Note that the addition of dilute Cr 3+ ions slightly modifies the KCl-MgCl2 spectrum in the ~260-350 cm -1 region, however, it is impossible to clearly assign frequencies related to Cr 3+ species in the molten salt, revealing the limitations of this form of spectroscopy for investigation of dilute metal ion species. The most intense band at ~232 cm -1 corresponds to the Mg-Cl symmetric stretching vibrations based on the previous reports. 1,2 Figure ESI-2 Furnace and molten salt samples used for X-ray scattering studies at NSLS-II. a) Experimental setup at the PDF beamline, NSLS-II (the inset shows inside features of the furnace for clarity): 1, molten salt sample in quartz capillary; 2, sample port; 3, X-ray entrance port; 4, heating wire; 5, thermocouple port. b) Molten KCl-MgCl2 (50:50 mol%) (left) and CrCl3 (5 mol%) in the KCl-MgCl2 (right) salts at ~1073 K. , where is one for i=j and zero for i≠j) calculated at Q=0 for the ionic pair correlations in the KCl-MgCl2 system. As can be seen, most of the scattering comes from the pairs containing chlorides and thus the X-ray diffraction patterns in Fig. 1 (main text) are primarily dominated by Cl-Cl, K-Cl, and Mg-Cl correlations.

Figure ESI-4
Optical absorption spectra of Cr 3+ (5 mol%) in KCl-MgCl2 (50:50 mol%), as a function of temperature. It is a well-defined spectrum with two bands in the visible region, assigned as follows: to 4 A2→ 4 T2 ( 4 F) for peak at 12,000 cm -1 , and to 4 A2→ 4 T1( 4 F) for peak at 18,000 cm -1 , and a charge transfer band corresponding to 4 A2→ 4 T1( 4 P) transition. There are no prominent features in the near-IR range. Based on the spectra, the symmetry of Cr 3+ can be attributed to the octahedral CrCl6 3ions. The spectral features and hence the geometry of the complex remain consistent over the temperature range investigated. The results obtained are in agreement with the Cr 3+ spectrum reported by Harrington and Sundheim, 3 and by Gruen and McBeth,4 in the LiCl-KCl eutectic melt.

Figure ESI-5
Comparison of radial distribution functions, g(r), obtained from RMC modeling (circles) and AIMD simulations (solid lines) for the CrCl3-KCl-MgCl2 system.        Density measurements. Given the bright purple color of the KCl-MgCl2 molten salt after addition of 5 mol% CrCl3 (Fig. ESI-2b), the density of the resulting CrCl3-KCl-MgCl2 mixture can be accurately determined by measuring the volume of the melt at 800 0 C. A similar technique to measure the density of molten NaCl-CrCl3 salt was applied by Li et al. 5 To determine the volume, the salt mixture was melted under an inert atmosphere in a calibrated quartz tube. The molten salt visibly colored the quartz and the colored part of the tube was used to deduce the height, from which the volume can be calculated for the molten salt sample of known mass. The measurements were repeated three times with different amounts of the CrCl3-KCl-MgCl2 salts, giving the density of 1.65 g/cm 3 at 800 0 C.
X-ray structure function and pair distribution function. The X-ray structure function, S(Q), is defined as the following: where Icoh is coherent scattering intensity, xi and fi(Q) are the molar fraction and Q-dependent Xray ionic form factor of species i, respectively, and Q denotes the magnitude of the scattering vector (Q = 4πsin(θ)/λ), where 2θ is the scattering angle, and λ is the incident X-ray wavelength.
Notice that as defined here, S(Q) goes to zero at large Q. In other words, our S(Q) is F X (Q) as defined in the review article by Keen; 6 we highlight this to avoid confusion with different terminologies and definitions commonly used in total scattering.
Computationally we define the X-ray weighted total scattering structure function, S(Q), as Where 0 is the ionic number density of the system, and are molar fractions of ionic species and , ( ) and ( ) are X-ray atomic form factors for species and and ( ) is the radial pair distribution function for species and .
Real space pair distribution functions (PDF), ( ) and ( ), are obtained from ( ) via the expressions One can also define the partial subcomponents of ( ) and ( ) via the Fourier transformation of the partial subcomponents of ( ). These provide information as to which pair interactions contribute to the PDF at specific distances. The differential PDF, d ( ), is accordingly defined as in Eq. 3, but with ( ) replaced by the difference between two ( )s for the CrCl3-loaded KCl-MgCl2 and the pristine KCl-MgCl2 molten salt mixture. Note that a direct subtraction of ( ) for the KCl-MgCl2 mixture from ( ) for the CrCl3-KCl-MgCl2 leads to the identical d ( ) as follows from the above definitions.
Coordination number. If is the distance between the i th Clion out of a total number of Clions ( Cl ) and a cation, and † is the location of the boundary of the first chloride solvation shell determined from the first minimum of the cation-Clradial distribution function (RDF), the coordination number of the cation is defined in terms of a smooth function, fi, (0 <fi <1): Eq. 7 allows smooth transitions of Clacross the boundary of the first chloride solvation shell.
Number of shared Cl -. in Eq. 1 represents the contribution of the i th Clto the coordination structure of a cation. Thus, if and are the contributions of the i th Clrespectively to the coordination structures of the j th and k th cations, the number of Clshared between these two cations is given by: (8) Notice that, Shared is maximum for Clions that are located between these two cations, contributing the most to their shared overlapping coordination shells. Eq. 8 can also be used to determine the number of cations coordinating with two different cations through chloride ions, but in this case, f is determined using the cation-cation RDF and corresponding † .
Free energy calculations. The 1D-free energy profile for a reaction coordinate is calculated using its probability distribution function (Ω (x)) computed from the AIMD trajectory: , where x is the reaction coordinate such as coordination number (CN) or distance (r). The 2D-free energy surfaces were computed using the joint probability distribution function (Ω (r,y)): ( , ) = − B [Ω( , )], where y is a coordination number (CN), or number of shared chloride ion between two cations, or number of cations simultaneously coordinating with two different cations via chloride ions ( Shared ), or electric field (E). Note that the averaging for the free energies was done considering all possible ion pairs and the production length of the AIMD trajectories.
TS-Marcus approach. We describe rate processes in a Coulombic system of molten salts in terms of the reaction coordinate, E, which is the electric field exerted by the solvent ions on the solute ion projected along a specific direction (û), and the ionic solvent bath coordinate, B. Following the work of Darve and Pohorille, 7 the Hamiltonian for this system can be determined through a coordinate transformation from a set of conventional Cartesian coordinates X of 3N components (N is the total number of ions) to the set of (E, B), where B has 3N−1 components. The associated conjugate momenta in this new set are pE and PB with 3N−1 components-these are also transformed from the Cartesian momenta PX with 3N components. Thus, the Hamiltonian can be expressed as: The first two terms in the above equation represent respectively the kinetic energy associated with the motion of the electric field and bath coordinates (notice that the cross-terms between pE and PB have been ignored for simplicity) and the last one is the potential energy. 1/ is a masslike quantity moving with the momentum pE in the electric filed space and can be obtained as: Here, xi is a component of the Cartesian coordinate X and mi is the associated mass. If the i th solvent ion with charge Qi (considering formal charges +3, +2, +1, and -1 respectively for Cr 3+ , Mg 2+ , K + , and Cl -) is located at a distance ri, the electric field on a solute ion exerted by all the solvent ions projected along û is obtained as =̂. ∑ where ′ = and ′ = , and M and are the mass of the solute ion and the solute ion-i th solvent ion reduced mass, respectively. ̂ and ̂ are the unit vectors pointing respectively from the i th and j th solvent ions to the solute ion. Now, the rate of transition between an initial (Ei) and final (Ef) electric field states via the transition state, E=E † , can be obtained as: 8 Here, Θ is the Heaviside step function ensuring a positive flux ( ≥ 0) through the transition state. =1/kBT is the inverse thermal energy at temperature T and kB is the Boltzmann constant.
W(E) is the potential of mean force (PMF) in the electric field space. Note that, TST assumes that after arrival at the transition state (barrier-top) from the reactant minimum on the PMF, a trajectory immediately moves to the product minimum. However, numerous studies on solvation and ion/charge transport indicated that solvent bath-induced barrier-recrossing is inevitable and must be accounted for to determine correct transition rates. The methods of reactive flux by Chandler, 9 Kramer's theory, 10 Grote-Hynes theory, 11 and the semiclassical Landau-Zener 12 approach are typically employed to examine such non-equilibrium solvent effects, wherein transmission coefficient is determined as a measure of the fraction of the flux of the trajectories through the transition state that finally arrives at the product minimum. The product of the TST rate and the transmission coefficient provides the correct rate of transition.
Recently, Marcus theory of electron transfer has been extended by Roy et al. to investigate ion pairing, solvent exchange, and ion exchange processes in condensed phase systems. [13][14][15][16][17] Following this theory, we can express the reactant (R) and product (P) free energy states using parabolic functions of electric fields, R ( ) and P ( ), respectively: Here, R and P are the minima of the reactant product parabolas located at = R and = P , and R and P are corresponding curvatures. These parabolas are diabatic states and the reactant-to-product transition occurs through their crossing locations. These diabats can be extracted as slices from a 2D-free energy surface (W(r,E)) spanned by interionic distance (r) and electric field. The first slice is for the equilibrium close-contact distance in the reactant state ( = R ) and the second one is for the equilibrium solvent-separated distance in the product state ( = P ). These slices, which cross at = † , are modeled with parabolic functions as presented in Eq. 12. An exact Marcus theory suggests that solvent reorganization in the form of electric field change drives the equilibrium reactant state, e.g., the state of close-contact ion pair, to an activated transition state (the crossing point of the two parabolas; = † ) where the barrier along the distance (Δ † ) are expected to vanish or reduce significantly, resulting in rapid dissociation of the ion pair, i.e., rapid transition to the equilibrium product state. However, in practice, slower transition rates are anticipated due to a large value of Δ † and recrossing events of the crossing point. Then depending on specific cases, in addition to accounting for this additional barrier, the recrossing events are treated with either the adiabatic or nonadiabatic prescription of Marcus theory.
When the reactant and product parabolas start to couple (with the coupling strength C) as they start to cross, they are modified by each other resulting in a lower ( − ) and a higher ( + ) adiabatic free energy surfaces: ± = R ( )+ P ( )