A Computational Meta-Analysis of UCl₃ Cyclic Voltammograms in LiCl-KCl Electrolyte

CV data was extracted from several published papers using the Engauge plot digitizer tool. Several criteria were used in deciding which CVs to use:

• High reference electrode concentrations to maintain stability
• Good image quality and good image scaling
• High data resolution
• Small electrodes

The CVs being used are summarized in Table I. They were obtained over several temperatures, hydrodynamic conditions, and solute concentrations. The U(III) reduction peak occurring at about −1.4 vs Ag/AgCl was modeled for scans on electrodes traveling from the anodic to the cathodic direction. Because the software cannot model prepeaks, they were not included in the data extracted from the CVs.

ERAD is a 1-dimensional electrochemical kinetics and transport analysis code based on an earlier model, REFIN.^11,15 ERAD simulates each reaction at each electrode using three equations. The equilibrium state is evaluated with the Nernst equation, which defines the equilibrium potential as

Where E₀ is the standard potential, R is the gas constant, T is temperature, n is the number of electrons transferred in the reaction, F is Faraday's constant, γ is the activity coefficient, and X is the mole fraction of the dissolved species. The activity coefficient was assumed to be concentration independent, so in this paper, the activity coefficient is combined with the standard potential to produce the apparent standard potential.

The Bulter-Volmer equation addresses reaction rate as

In the above equation, i is the current density, i_ex is the exchange current density, α is the transfer coefficient, E is the electrode potential, X_Ox is the mole fraction of the oxidized species. A super script * indicates bulk solution value, and a superscript % indicates the surface of the electrode. i_ex is evaluated as

on solid electrodes where i₀ is the standard exchange current density.

The diffusion equation with an added migration term evaluates mass transfer near each electrode.

where C is concentration, x is location, D is diffusion coefficient. The derivation for the previous equation can be found in.¹¹

Equation 4 was solved for concentration, using the LSODE solver package's backwards differentiation method. Equations 1 and 2 were used to enforce interface boundary conditions at electrode surfaces. When electrode potential is set, like the working electrode in a CV, equation 2 is applied directly. However, if the electrode current density is set, the inverse of equation 2 is applied by using a root finding function.^11,15

It is important to note that ERAD focuses on describing rate-limiting steps in mass transport in electrorefining operations thus does not model adsorption, pseudocapacitance, ohmic drop, or dendrite formation. Additionally ERAD does not simulate electrolyte fluid flow, or current flow in more than one dimension.

ERAD results were evaluated with a least squares function in order to create an objective function, which is

Its resulting value is the sum of squares of the difference between experimental CV data (i_{CV, j}) and simulated CV data (i_{ERAD, j}). This is a function of apparent standard potential (E₀), exchange current density (i₀), diffusion coefficient in molten salt (D), and diffusion layer thickness at the working electrode (δ). The objective function was then minimized using the MATLAB function fminseach, which employs a Nelder-Mead minimum search to yield the four unknowns.

The transfer coefficient was not fitted because the U(III)->U reaction is mass transfer limited,^16,17 and most of the observable kinetic effects will lie in an approximately linear portion of the i-E curve. Because that portion of the curve is linear, it can only yield two parameters. The first parameter used is the intercept of the i-E curve, which yields standard potential. The second parameter can consequently yield the transfer coefficient or the exchange current density, but not both. Because the transfer coefficient was assumed to be 0.5, the second parameter fitted is exchange current density. This assumption is valid because the curve will rarely go far enough from equilibrium for the transfer coefficient assumption to seriously affect the result.

Standard potentials were obtained for uranium and plutonium vs. the Ag/AgCl reference electrodes used in the various experiments. In order to provide a uniform scale for the data, the results were then converted to potentials vs. a Cl₂/Cl⁻ reference electrode using

which was obtained by applying the Nernst equation. The standard potential of the silver chloride electrode is obtained from¹⁸ and¹⁹ as:

An example fit can be seen in Figure 1. The computer simulations converged to CV data in a similar manner. As can be noted, the prepeak that often occurs on uranium CV data is not modeled. Diffusion layer thickness was found to range from 0.004 to 0.019 cm.

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The resulting apparent standard potentials of uranium are shown in Figure 2. The potential for the reduction of UCl₃ to metal lies within the range of existing data, and follows the expected trend. There is some noise in the data, likely due to changes in experimental conditions.

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The diffusion coefficients for uranium fall within the range of measured data (Figure 3). It also follows the expected increasing trend with temperature with the exception of the data from.²⁰ It is not known why this CV varies from the others for the diffusion coefficient. Regardless, the outlier was removed from the data, and it was fitted to the Arrhenius equation, yielding

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The exchange current density for uranium was also resolved from CV data (Figure 4). It appears to have an increasing trend with respect to temperature. It should be noted however, that it likely varies depending on electrode surface preparation. This is exemplified in the case of the data from¹² which had a value of over 5 A/cm2 as opposed to the more typical values. The outliers were removed from the data, and the exchange current density was fitted to the Arrhenius equation, yielding

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This meta-analysis of CV data revisits a broad cross-section of CV data for the UCl₃/U reduction peak, and extends knowledge of several important parameters—especially in obtaining estimates of exchange current density, which is important in dendrite formation in molten salt systems, as well as modeling non-uniform electrodes.

Due to the match between experimentally measured data and data obtained through CV fitting for the UCl₃ reduction reaction, it can be said that the proposed approach, i.e., the computer model-based derivation of the values of key parameters using a least square fit of the CV data works well—or at least to the level of variation in the data. For the UCl₃ reduction reactions it can be seen that there is some variation, which should be considered in any predictive modeling work.

This work was supported by Korea Research Foundation under the contact number 2011-0031839. The authors would also like to acknowledge the support from Prof. Il Soon Hwang of Seoul National University who provided the REFIN code.