Methods of separating rare earth elements by capillary electrophoresis and related systems

Abstract

A method for separating rare earth elements by capillary electrophoresis that comprises introducing a ligand solution and a rare earth element solution into a separation chamber, the ligand solution comprising one or more ligands comprising propylamine, butylamine, diethylamine, dipropylamine, dibutylamine, triethylamine, tripropylamine, 15-crown-5, 18-crown-6, iminodiacetic acid (IDA), lactic acid (LA), formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), citric acid (CA), propionic acid (PA), butyric acid, and hydroxyisobutyric acid (α-HIBA), and the rare earth element solution comprising two or more rare earth elements; positioning terminal ends of the separation chamber in respective buffer solutions; applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber to separate one or more of the rare earth elements from the rare earth element solution; and recovering the one or more of the rare earth elements. Additional methods and related systems are also disclosed.

Description

[0001]METHODS OF SEPARATING ELEMENTS BY CAPILLARY RELATED SYSTEMS PRIORITY CLAIM This application claims the benefit of the filing date of United States Provisional Patent Application Serial No. 63 / 514,313, filed July 18, 2023, and of United States Provisional Patent Application Serial No. 63 / 568,959, filed March 22, 2024, for “A CAPILLARY ELECTROPHORETIC SYSTEM AND RELATED METHODS OF SEPARATING RARE EARTH ELEMENTS,” the disclosure of each of which is hereby incorporated herein in its entirety by this reference. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT This invention was made with government support under Contract No. DE‑AC07‑05- ID14517 awarded by the United States Department of Energy. The government has certain rights in the invention. TECHNICAL FIELD This disclosure relates generally to the separation of analytes by capillary electrophoresis. More particularly, embodiments of the disclosure relate to a system for the ligand-assisted separation of rare earth elements by capillary electrophoresis and methods for the same. BACKGROUND The rare earth elements (REEs) include lanthanide series elements plus scandium and yttrium. REEs are used in a wide variety of applications such as in electronics, electric vehicle batteries, permanent magnets, satellites, environmental probes and sensors, and defense weapons systems. REEs fall under the category of critical materials, as they are crucial to both the advancement of clean energy and at high supply risk. Since the early 2000s, the United States has been reliant on foreign imports of REEs from China and other countries. In 2002, the Mountain Pass Mine in California closed its doors to REE mining due to the implementation of more strict environmental regulation policies, effectively ceasing U.S. REE mining and production. The 2021 Annual Critical Materials Report from the Department of Energy showed that the United States has only a 15% share in production, and a 1% stake in reserves. Sustainable methods for the extraction and separation of REEs into their counterparts must be achieved to build a stable domestic supply of these materials. Perhaps the most difficult challenge lies in separating lanthanide elements from one another, as they each possess similar properties such as a stable +3 oxidation state. Across the series of the lanthanide elements, the lanthanide elements differ chemically only in terms of the filling of their f-orbitals, which are highly shielded and do not participate in valence activity. The radial wavefunctions of 4f lanthanide orbitals predict high, Gaussian-like probability distributions at distances close to the nucleus that shift to even smaller distances as the number of f electrons increases, causing what is known as the lanthanide contraction. This behavior is responsible for the observed decrease in ionic radius and corresponding increase in Lewis acidity across the series. Conventional large-scale separation approaches such as solvent extraction and ion-exchange chromatography are used to exploit the subtle differences present in these two properties to achieve separations. However, these separations are inefficient and use large volumes of volatile organics and acids and, therefore, are not sustainable. DISCLOSURE A method of separating rare earth elements by capillary electrophoresis is disclosed and comprises introducing a ligand solution and a rare earth element solution into a separation chamber of a capillary electrophoresis system, the ligand solution comprising one or more ligands comprising propylamine, butylamine, diethylamine, dipropylamine, dibutylamine, triethylamine, tripropylamine, 15-crown-5, 18-crown-6, iminodiacetic acid (IDA), lactic acid (LA), formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), citric acid (CA), propionic acid (PA), butyric acid, and hydroxyisobutyric acid (α- HIBA), and the rare earth element solution comprising two or more rare earth elements. The method further comprises positioning terminal ends of the separation chamber in respective buffer solutions of the capillary electrophoresis system, applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber to separate one or more of the rare earth elements from the rare earth element solution, and recovering the one or more of the rare earth elements. A method of separating rare earth elements by capillary electrophoresis is disclosed and comprises introducing a ligand solution into a separation chamber of a capillary electrophoresis system, the ligand hydroxyisobutyric acid and propionic acid or hydroxyisobutyric acid and iminodiacetic acid. The method further comprises introducing a rare earth element solution into the separation chamber, the rare earth element solution comprising two or more of scandium, yttrium, lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium. The method further comprises positioning terminal ends of the separation chamber in respective buffer solutions of the capillary electrophoresis system, the buffer solutions comprising an anion, applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber to separate one or more of the rare earth elements from the rare earth element solution, and recovering the one or more of the rare earth elements. A system for separating rare earth elements is disclosed and comprises a separation chamber extending between reservoirs, an anode in one of the reservoirs and a cathode in another of the reservoirs and terminal ends of the separation chamber configured to be positioned in the reservoirs. The system also comprises a voltage source coupled to the separation chamber and configured to apply an electrical field between the anode and the cathode, and a detection system coupled to the separation chamber. The separation chamber is configured to contain a sample solution comprising two or more of scandium, yttrium, lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium, and a ligand solution comprising one or more ligands comprising propylamine, butylamine, diethylamine, dipropylamine, dibutylamine, triethylamine, tripropylamine, 15-crown-5, 18- crown-6, iminodiacetic acid (IDA), lactic acid (LA), formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), citric acid (CA), propionic acid (PA), butyric acid, and hydroxyisobutyric acid (α-HIBA). BRIEF DESCRIPTION OF THE DRAWINGS For a detailed understanding of the disclosure, reference should be made to the following detailed description, taken in conjunction with the accompanying drawings, wherein: FIG. 1 is a schematic of a system for separating rare earth elements by capillary electrophoresis, according to embodiments of the disclosure; FIG.2 is a flow diagram of a separating rare earth elements by capillary electrophoresis, according to embodiments of the disclosure; FIG. 3 is an electropherogram taken in NaNO3 background electrolyte (BGE), with an inset showing two distinct Ce peaks, according to embodiments of the disclosure; FIG. 4 is a graph of measured diffusion coefficients plotted against ionic radii for each lanthanide, according to embodiments of the disclosure; FIG. 5 is a graph of the effective mobilities of four different lanthanides in each investigated ligand system, according to embodiments of the disclosure; FIG. 6 is a raw electropherogram showing enhanced separation between four lanthanides in increasing iminodiacetic acid (IDA) concentration, according to embodiments of the disclosure; FIG. 7 is a graph of the mobilities of four lanthanides in increasing IDA concentration, according to embodiments of the disclosure; FIG.8 is an electropherogram of lanthanides in nitrate and chloride BGE using IDA as a ligand for separation, according to embodiments of the disclosure; FIG. 9 is a graph of the diffusion coefficients for lanthanides in nitrate and chloride BGE, according to embodiments of the disclosure; FIG. 10 is a graph of the mobilities of lanthanides in increasing IDA concentration, according to embodiments of the disclosure; FIG. 11 is a graph of the change in diffusion coefficients for lanthanides using IDA as a ligand, according to embodiments of the disclosure; FIG. 12 is a graph of the species fraction of various ligands according to pH, according to embodiments of the disclosure; FIG. 13 is a visualization of the separation of various lanthanides using IDA as a ligand, according to embodiments of the disclosure; FIG. 14 is a graph of the diffusion coefficients of four lanthanides, according to embodiments of the disclosure; FIG.15 is a graph of the stability constants of four lanthanides using different ligands, according to embodiments of the disclosure; FIG. 16 is a graph of the separation factor between lanthanides at increasing HIBA concentration, according to embodiments of the disclosure; FIG.17 is a graph of the effective mobilities of four lanthanides in HIBA, according to embodiments of the disclosure; FIG.18 is a graph of the number plates for each lanthanide, according to embodiments of the disclosure; FIG.19 is a graph of the stepwise stability constants determined for each lanthanide, according to embodiments of the disclosure; FIG. 20 is a contour map of the separation factors between three lanthanide pairs in increasing α-HIBA- concentration, according to embodiments of the disclosure; FIG.21 is a graph of the electrophoretic mobilities and stability constants for several ligands in a separation of a lanthanide pair, according to embodiments of the disclosure; FIGS. 22A-22D are graphs of the effects of α-hydroxyisobutyrate on the separation of lanthanides, according to embodiments of the disclosure; FIG. 23 is a graph of separation factors for three lanthanide pairs in different combinations of ligands, according to embodiments of the disclosure; FIGS. 24A-24C are graphs of the effects of several parameters on separation of a lanthanide pair, according to embodiments of the disclosure; FIGS. 25A-25C are graphs of the effects of several parameters on separation of a lanthanide pair, according to embodiments of the disclosure; FIGS.26A and 26B are graphs of separation factors and number of theoretical plates for three lanthanide pairs, according to embodiments of the disclosure; FIGS. 27A-27C are graphs of complied data of a separation of a solution of 14 lanthanides, according to embodiments of the disclosure; FIGS. 28A-28F are graphs of complied data of carboxylate ligand solutions used to separate a solution of four lanthanides, according to embodiments of the disclosure; FIGS. 29A-29F are graphs of compiled data of amine ligand solutions used to separate a solution of four lanthanides, according to embodiments of the disclosure; FIGS. 30A-30F are graphs of compiled data of sulfate ligand solutions used to separate a solution of four lanthanides, according to embodiments of the disclosure; FIGS. 31A-30F are graphs of compiled data of phosphate ligand solutions used to separate a solution of four lanthanides, according to embodiments of the disclosure; FIGS. 32A-32D are graphs of normalized electropherograms and corresponding La mobility graphs for crown ethers, according to embodiments of the disclosure; and FIG. 33 is a graph of mobility of the electroosmotic flow for a set of ligands, according to embodiments of the disclosure. MODE(S) FOR OUT THE INVENTION As used herein, the singular forms following “a,” “an,” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. As used herein, the term “may” with respect to a material, structure, feature, or method act indicates that such is contemplated for use in implementation of an embodiment of the disclosure, and such term is used in preference to the more restrictive term “is” so as to avoid any implication that other compatible materials, structures, features, and methods usable in combination therewith should or must be excluded. As used herein, any relational term, such as “first,” “second,” “top,” “bottom,” “upper,” “lower,” “above,” “beneath,” “side,” “upward,” “downward,” etc., is used for clarity and convenience in understanding the disclosure and accompanying drawings and does not connote or depend on any specific preference or order, except where the context clearly indicates otherwise. As used herein, the term “substantially” in reference to a given parameter, property, or condition means and includes to a degree that one skilled in the art would understand that the given parameter, property, or condition is met with a small degree of variance, such as within acceptable manufacturing tolerances. By way of example, depending on the particular parameter, property, or condition that is substantially met, the parameter, property, or condition may be at least 90.0% met, at least 95.0% met, at least 99.0% met, or even at least 99.9% met. As used herein, the term “about” used in reference to a given parameter is inclusive of the stated value and has the meaning dictated by the context (e.g., it includes the degree of error associated with measurement of the given parameter, as well as variations resulting from manufacturing tolerances, etc.). According to embodiments described herein, a system for the separation of one or more analytes from a sample solution by capillary electrophoresis (CE) may include a separation chamber, the sample solution, a buffer reservoir containing a buffer solution, a ligand solution, a voltage source configured to apply an electrical field across the separation chamber, and a detection system. The analyte(s) in the sample solution include one or more rare earth elements (REEs). The analyte(s) include one or more lanthanide series elements, scandium, yttrium, or a combination thereof. One or more ligands (e.g., one or more complexing agents) in the ligand solution react with one or more analytes of interest in the sample solution to form one or more complexes (e.g., chemical complexes, REE-ligand complexes, lanthanide-ligand formation of the chemical complexes between the analytes of the sample solution and the ligand of the ligand solution alters electrophoretic properties (e.g., electrophoretic mobilities) and separation factors of the analytes through the separation chamber, allowing for effective separation of the analytes. By appropriately selecting the ligand, changes in the charge / diffusion coefficient of the analyte-ligand complex may occur, effectively increasing mobility differences during CE, which enhances separation. The electrophoretic properties of the analyte-ligand complex may depend on charge, size, and / or denticity of the ligand. An analyte-ligand complex of interest may be separated from another analyte-ligand complex if there is a sufficiently large difference in stability constants for the analyte-ligand complexes. Different metals may form analyte-ligand complexes exhibiting a large difference in stability constant, even when the same ligand is used in both analyte-ligand complexes. Electrolyte concentration, pH, or separation voltage may also be tailored to increase separation between different analyte- ligand complexes. The systems and methods according to embodiments of the disclosure may enable environmentally neutral separations to be conducted since aqueous solutions are able to provide efficient separations with a high selectivity. The systems and methods according to embodiments of the disclosure may also enable REEs adjacent to one another on the Periodic Table to be separated. Without being bound to any theory, the ligand is believed to partially complex with free metal ions (e.g., free REE ions, free lanthanide ions) in solution, effectively altering the chemical environment of the metal. The resulting differences in the transport properties of the REE-ligand complexes enable selective separations to be achieved in a short amount of time. Differences in the transport properties of the REE-ligand complexes may enable selective separations of the REE-ligand complexes and, thus, of the REEs, in a short amount of time. The system according to embodiments of the disclosure may, for example, be used to separate REEs that are present in ore or electronic waste from one another. The REEs include lanthanide series elements plus scandium and yttrium. The ore or electronic waste may be in solution, such as in the sample solution (e.g., the REE solution). The sample solution may be an aqueous solution. The ore or electronic waste may include multiple REEs. One or more REEs of interest may be separated from the other REEs in the ore or electronic waste based on electrophoretic properties rather than on size of the REEs. The REE of interest may be separated by a transport-based technique. In contrast, separating REEs using conventional systems and processes may based on chemical differences alone. The conventional systems, such as chromatographic systems, also use large volumes of solvents and other chemical compounds, which increase environmental and safety considerations. A system 100 according to embodiments of the disclosure is depicted in FIG. 1 and includes a separation chamber 102, a detection system 104, a ligand solution 106, a sample solution 108, a reservoir 110 (e.g., a buffer reservoir), a buffer solution 112, a voltage source 114, an anode 116, and a cathode 118. The separation chamber 102 may be a capillary tube. The separation chamber 102 may be a fused silica capillary tube or another type of capillary tube, such as a quartz capillary tube, a plastic capillary tube, or a glass capillary tube. In some embodiments, the system 100 may include multiple separation chambers 102. The separation chamber 102 may have a length of from about 25 cm to about 100 cm, such as from about 25 cm to about 50 cm or from about 50 cm to about 100 cm. The separation chamber 102 may have an internal diameter of from about 25 µm to about 100 µm, such as from about 25 µm to about 50 µm or from about 50 µm to about 100 µm. In some embodiments, the internal diameter of the capillary tube is about 50 µm. In other embodiments, the internal diameter of the capillary tube is about 75 µm. However, other dimensions (e.g., length, internal diameter) of the separation chamber 102 may be used depending on the scale of the separation process and the REE analyte(s) of interest. The detection system 104 may be coupled to the separation chamber 102. The detection system 104 may, for example, be a capacitively-coupled contactless conductivity detector (C4D), a UV-Vis detector, or another type of detector. Upon passing through the detection system 104, the groups of analyte-ligand complexes may be detected, such as a peak on an electropherogram. The temperature at which the separation chamber 102 is maintained during the separation process may be from about 10°C to about 30°C, such as from about 10°C to about 20°C or from about 20°C to about 30°C. To minimize energy consumption of the system, the temperature of the separation chamber 102 may be about room temperature. The ligand solution 106 may include one or more ligands having one or more binding sites formulated to react with an ion of the analyte (e.g., an ion of the metal) of the sample solution 108. The ligand may be an ionizable compound that includes one or more heteroatoms, such as one or more nitrogen-, sulphur-, phosphorous-, or oxygen-containing groups. The ligand may be a carboxylate, an amine, a sulfate, a phosphate, a thiol, or a crown ether. The ligand may include at least donating atom, such as an oxygen atom. If multiple electron donating atoms are present in the ligand, the electron donating atoms in addition to the oxygen atom may provide additional binding capability between the ligand and the free metal ions. By way of example only, the ligand may be an organic compound that includes at least one carboxylate group. By way of example only, the organic compound may include one carboxylate group, two carboxylate groups, or three carboxylate groups. The organic compound may include from one carbon atom to about eight carbon atoms. The ligand may include, but is not limited to, one or more of propylamine, butylamine, diethylamine, dipropylamine, dibutylamine, triethylamine, tripropylamine, 15-crown-5, 18- crown-6, iminodiacetic acid (IDA), lactic acid (LA), formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), citric acid (CA), propionic acid (PA), butyric acid, and hydroxyisobutyric acid (e.g., α-HIBA), or another molecule. If nitrogen atoms are present in the ligand, the ligand may be a primary, a secondary, or a tertiary amine. In some embodiments, the ligand is propionic acid. In other embodiments, the ligand is citric acid. In yet other embodiments, the ligand is α-HIBA. The ligand solution 106 may include two or more ligands, such as a primary ligand and an auxiliary ligand. The primary ligand and auxiliary ligand may be selected to exhibit a large difference in stability constant relative to one another. By exhibiting the large difference in stability constant, a secondary equilibrium may be introduced. The primary ligand may be present in the ligand solution 106 at a relatively higher amount (e.g., relatively higher concentration) than the auxiliary ligand. If two or more ligands are present in the ligand solution 106, the primary ligand may be present at a concentration of from about 0.1 mM to about 40 mM and the auxiliary ligand may be present at a concentration of from about 0.1 mM to 20 mM. In some embodiments, the auxiliary ligand includes propionic acid. In other embodiments, the auxiliary ligand includes citric acid. In some embodiments where the ligand solution 106 includes two or more ligands, the ligand includes citric acid and α- HIBA. In other embodiments, the ligand includes propionic acid and α-HIBA. In yet other embodiments, the ligand includes iminodiacetic acid and α-HIBA. The ligand solution 106 may be an aqueous solution. The pH of the ligand solution 106 may be from about 1.0 to about 5.5, such as from about 1.0 to about 3.0 or from about 3.0 to about 5.5. The pH of the ligand solution 106 may be adjusted so that a dominant species present is in the singly deprotonated L- form (e.g., a pH of about 4.5 for FA, about 5.5 for AA and BA, about 3.0 for OA, about 5.0 for IDA, and about 4.0 for CA). In other words, the singly deprotonated L- form of the may account for a majority of the ligands in the ligand solution. The ligand may be present at a concentration of from about 0.1 mM to about 40 mM, such as from about 0.1 mM to about 5 mM; from about 5 mM to about 10 mM; from about 10 mM to about 15 mM; from about 15 mM to about 20 mM; from about 20 mM to about 25 mM; from about 25 mM to about 30 mM; from about 30 mM to about 35 mM; or from about 35 mM to about 40 mM. The sample solution 108 (e.g., the REE solution) may include two or more REEs. The REEs may include, for example, one or more of scandium (Sc), yttrium (Y), lanthanum (La), cerium (Ce), praseodymium (Pr), neodymium (Nd), promethium (Pm), samarium (Sm), europium (Eu), gadolinium (Gd), terbium (Tb), dysprosium (Dy), holmium (Ho), erbium (Er), thulium (Tm), ytterbium (Yb), and lutetium (Lu). The REEs in sample solution 108 may be obtained from, for example, an ore or electronic waste material. The ore or electronic waste material may be processed (e.g., preprocessed) by conventional techniques so that non- rare earth metals and general impurities are removed from the sample solution 108, and so that the sample solution 108 includes the REEs in an aqueous solution. For instance, the REEs may be leached from the ore or electronic waste material to form the aqueous sample solution 108. The sample solution 108 may include two or more REEs at varying concentrations, such as five REEs, ten REEs, fifteen REEs, or twenty REEs. The sample solution 108 may have a total metal concentration of from about 0.008 mM to about 3.2 mM, such as from about 0.008 mM to about 0.4 mM; from about 0.4 mM to about 1.0 mM; from about 1.0 mM to about 2.0 mM; or from about 2.0 mM to about 3.2 mM. The sample solution 108 may include the two or more REEs at various ratios. In some embodiments, the two or more REEs are present at equimolar concentrations. A ratio of the ligand concentration in the ligand solution 106 to the total metal concentration in the sample solution 108 may be from about 5 to about 100, such as from about 5 to about 10, from about 10 to about 20, from about 20 to about 30, from about 30 to about 40, from about 40 to about 50, from about 50 to about 60, from about 60 to about 70, from about 70 to about 80, from about 80 to about 90, or from about 90 to about 100. The sample solution 108 may have a pH of from about 1.0 to about 5.5, such as from about 1.0 to about 3.0 or from about 3.0 to about 5.5. The reservoir 110 may include a buffer solution 112. The buffer solution 112 may include a background electrolyte (BGE). The background electrolyte may include an anion, for example, such as one or more of chloride, nitrate, triflate, sulfate, phosphate, acetate, carbonate, bicarbonate, perchlorate, oxalate, citrate, or another anion. The background electrolyte may be present at a concentration of from about 0.1 mM to about 40 mM, such as from about 0.1 mM to about 5 mM; from about 5 mM to about 10 mM; from about 10 mM to about 15 mM; from about 15 mM to about 20 mM; from about 20 mM to about 25 mM; from about 25 mM to about 30 mM; from about 30 mM to about 35 mM; or from about 35 mM to about 40 mM. The buffer solution may have a pH of from about 1.0 to about 5.5, such as from about 1.0 to about 3.0 or from about 3.0 to about 5.5. The background electrolyte may contribute to the conductivity of the system 100 and the establishment of an electroosmotic flow through the separation chamber 102. FIG. 2 is a flow diagram showing a process 200, according to embodiments of the disclosure, for separating rare earth elements by capillary electrophoresis that includes act 202 of introducing a ligand solution and a sample solution comprising rare earth elements into a separation chamber; act 204 of positioning terminal ends of the separation chamber in respective buffer solutions; act 206 of applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber to separate one of the rare earth elements from the sample solution; and act 208 of recovering the one of the rare earth elements. By way of example only, the system 100 may be used to separate the rare earth elements. Act 202, of introducing a ligand solution and a sample solution comprising rare earth elements into a separation chamber, may include introducing the ligand solution into the separation chamber before introducing the sample solution to achieve on-capillary complexation, or combining the ligand solution and sample solution and incubating before introducing them into the separation chamber simultaneously to achieve pre-capillary complexation. To achieve on-capillary complexation, the ligand solution may be introduced (e.g., loaded) into the separation chamber by, for example, electrokinetic injection, vacuum injection, hydrodynamic injection, micropipette injection, or another method. The ligand solution may fill (e.g., substantially fill) the separation chamber. The sample solution may then be introduced to the separation chamber by electrokinetic injection, vacuum injection, hydrodynamic injection, micropipette injection, or another method. On-capillary complexation may allow for partial complexation to occur during the separation process, which may facilitate the maintenance of a dynamic equilibrium between the ligands of the ligand solution and the REEs of the sample solution. Alternatively, to achieve pre-capillary complexation, the ligand solution may with the sample solution and incubated before being introduced into the separation chamber. The ligand solution and sample solution may be incubated for from about 10 minutes to about 10 hours, such as for from about 10 minutes to about 30 minutes, from about 30 minutes to about 1 hour, from about 1 hour to about 5 hours, or from about 5 hours to about 10 hours. Pre-capillary complexation may allow for full complexation of the ligands of the ligand solution and the REEs of the sample solution. Upon introduction of the sample solution to the ligand solution, ligands of the ligand solution may displace H2O molecules from an inner hydration sphere of the analytes of the sample solution, forming complexes between the analytes and the ligands. The complexes may exhibit a stoichiometry of ML2+, ML2+, or ML3. On-capillary complexation may result in the displacement of fewer H2O molecules from the inner hydration sphere of the analytes than pre-capillary complexation as the analytes and ligands have less time to react than during pre-capillary complexation. Different analytes may have different numbers of H2O molecules displaced by the same ligand. For example, IDA may displace three H2O molecules from the inner hydration sphere of Sm, while only displacing two H2O molecules from the inner hydration sphere of Dy. By way of example only, in the case of Sm, IDA may coordinate with the nitrogen atom, the deprotonated oxygen atom on one carboxyl group, and the protonated oxygen atom from the opposite carboxyl group. In contrast, in the case of Dy, IDA may coordinate in a bidentate fashion through both oxygen atoms on the deprotonated carboxyl group. Act 204, of positioning terminal ends of the separation chamber in respective buffer solutions, may include submerging the terminal ends of the separation chamber in buffer reservoirs containing the buffer solution. An anode may be positioned in the buffer solution containing the terminus of the separation chamber where the sample solution is introduced. A cathode may be positioned in the buffer reservoir containing the opposite terminus of the separation chamber. Act 206, of applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber to separate one of the rare earth elements from the sample solution, may include applying a voltage between the anode and cathode to create an electric field across the entirety of the separation chamber. The electric field may extend from the anode to the cathode. The voltage may be from about 0.001kV to about 30kV. The electric field may have a strength of from about 0.1 kVcm-1to about 3.0 kVcm-1. Larger electrophoresis systems may use a lower voltage. Upon application of the voltage across the length of the separation chamber, ions in the background electrolyte migrate towards the cathode. The migration of ions towards the cathode may establish an electroosmotic flow that causes the charged analyte-ligand complexes to migrate through the separation chamber, e.g., from the anode towards the cathode. The analyte-ligand complexes may migrate at different speeds based on their effective charge, size, or denticity, enabling separation of the analyte(s). For example, an analyte-ligand complex with an effective charge of +2 may migrate through the separation chamber at a faster rate than an analyte-ligand complex with an effective charge of +1. Neutrally charged analyte-ligand complexes may not migrate through the separation chamber. By appropriately selecting the ligand and REE combination, the electrophoretic mobility of the analyte-ligand complexes may be adjusted, enabling separation of the REEs in the sample solution. This may result in separation of analyte-ligand complexes into distinct groups based on their effective charge. Act 208, of recovering the one of the rare earth elements, may include recovering the separated REEs at single REE purities of from at least about 70% by weight to at least about 98% by weight, such as at least about 70%, at least about 80%, at least about 85%, at least about 90%, at least about 95%, or at least about 98%. The methods according to embodiments of the disclosure may be used to separate REEs from the sample solution, where the REEs are adjacent to one another on the Periodic Table of Elements. For instance, adjacent REEs having a large difference in stability constant may be separated from one another using the methods according to embodiments of the disclosure. Therefore, pairs of REEs that are adjacent to each other on the Periodic Table of Elements may be separated. By way of example only, 20 mM α-hydroxyisobutyrate (α- HIBA) and 15 mM propionic acid (PA) adjusted to pH 5.0 and using a 20 kV separation voltage with 10 mM sodium nitrate electrolyte, has been shown to provide baseline resolution between 10 / 13 adjacent lanthanide pairs in less than 8 minutes through capillary electrophoresis separations experiments, as well as enhance the separation factor between samarium (Sm) and dysprosium (Dy) by a factor of 2.4 compared to using α-HIBA alone. The methods according to embodiments of the disclosure may be implemented in a large-scale system such as a continuous free-flow capillary electrophoresis (CFFCE) system. CFFCE allows for the efficient and scalable separation of REEs in a continuous flow setup, significantly enhancing throughput compared to conventional CE. In a CFFCE system, the sample solution containing REEs is introduced into a thin, flat separation chamber. The separation chamber may be made of materials such as poly(methyl methacrylate) (PMMA), polydimethylsiloxane (PDMS), polycarbonate (PC), or glass. The separation chamber may be designed to maintain a laminar flow, ensuring that the sample solution flows smoothly without turbulence. A buffer reservoir may provide a steady supply of buffer solution, maintaining consistent ionic strength and pH throughout the separation process. The ligand solution may be mixed with the sample solution to form analyte-ligand complexes. A voltage source may apply an electric field perpendicularly across the flow direction of the sample solution within the separation chamber. The charged analyte-ligand complexes in the sample solution deviate laterally from their initial flow paths based on their electrophoretic mobilities. The CFFCE system may process industrial quantities of REEs, making it suitable for high-throughput separations. For example, flow rates of from about 10 mL / h to about 1000 mL / h or more may be achieved, allowing for environmentally sustainable, continuous, and efficient separation processes suitable for industrial-scale operations. Compared to conventional systems of separation, the CE system and process according to embodiments of the disclosure offers improved separation of REEs by reducing the need for large volumes of volatile organics and acids. The CE system and processes according to embodiments of the disclosure are also faster and more efficient. For example, an ion chromatographic method may take about 30 minutes to separate REEs with a theorical plate (i.e., separation efficiency) of 1000, while the CE system and processes according to embodiments of the disclosure may be conducted in under about 5 minutes with a theorical plate between 100,000 and 500,000. The methods according to embodiments of the disclosure may be used to separate one or more of the REEs from other of the REEs. In some embodiments, Sm is separated and recovered. In other embodiments, Dy is separated and recovered. In yet other embodiments, Sm and Dy are recovered. The recovered REE may be used in various applications including, but not limited to, electronics, electric vehicle batteries, permanent magnets, catalysts, satellites, probes and sensors (e.g., environmental probes and sensors), and defense weapons systems. The separation by capillary electrophoresis and recovery may be achieved in a shorter amount of time than conventional extraction processes. By way of example only, the REEs may be separated and recovered in less than about 30 minutes, such as less than about 20 minutes or less than about 10 minutes. EXAMPLE 1 Lanthanides were purchased in their nitrate, chloride, and triflate forms through SIGMA ALDRICH® and used as-is. Formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), iminodiacetic acid (IDA), and citric acid (CA) were purchased through SIGMA ALDRICH® and used as-is. A pH of lanthanide solutions and ligand solutions was adjusted through the addition of HNO3 / NaOH for nitrates, HCl / NaOH for chlorides, and NaOTf / NaOH for triflates. For diffusion studies, NaCl, NaNO3, and NaOTf were used as the BGE, each at 5 mM concentrations and pH of 3.75, and lanthanides were measured separately each at 0.4 mM concentration. The ionic strength was kept constant at 5 mM for each solution through the addition of the respective counterion. For studies involving ligands, NaNO3was used as the counterion solution in the BGE, and lanthanide solutions were prepared to have a total metal concentration of 0.4 mM (La, Sm, Dy, and Lu each at 0.1 mM). The ligand solutions were adjusted to reach a pH value that reflected nearly 100% of the singly deprotonated ligand form (4.5 for FA, 5.5 for AA and BA, 3.0 for OA, 5.0 for IDA, and 4.0 for CA) through the addition of HNO3 / NaOH. Separate lanthanide solutions were prepared to have a pH value that matched that of its corresponding ligand solution, adjusted through HNO3 / NaOH addition. The ionic strength of each solution was maintained at 5 mM through the addition of NaNO3. Nanopure water (MILLIPORE®, 18 MΩ) was used for solution preparation and all final solutions were filtered through a 0.2 µm nylon syringe filter from AcrodiscTM. An AGILENT® 7100 CE (AGILENT TECHNOLOGIES®) was used for all experimental work and was equipped with a conductivity detector (TraceDec® Contactless Conductivity). Fused silica capillaries with a length of 56 cm and internal diameter of 50 µm were used (AGILENT®). All capillaries were conditioned at the beginning and end of each day by applying a 940-mbar flush of HCl for 10 minutes, H2O for 10 minutes, NaOH for 10 minutes, and, finally, the running BGE for 10 minutes. Samples were injected electrokinetically at 5 kV for 12 seconds. A BGE flush was applied before each experimental run at 940 mbar for 5 minutes. Experiments were conducted at 25°C with a 20 kV separation voltage and 12 seconds voltage ramp time. The detector settings were as follows: frequency = 19.79 Hz, filter = 1 / 3, cutoff = 0.5, voltage = -6 dB, and gain = 100%. CE was used to measure the transport properties of individual trivalent lanthanides. Initial diffusion studies were performed with three different anions (chloride, nitrate, and triflate) present in the BGE. FIG.3 electropherograms taken of each trivalent lanthanide in nitrate media, with an inset showing two distinct cerium peaks when cerium was measured in triflate media. Without being bound by any theory, this dual-peak behavior was likely due to the two most probable oxidation states of cerium, Ce3+and Ce4+. In the presence of nitrate media alone, very little separation was observed between the lanthanides. FIG.4 shows the diffusion coefficients measured for each trivalent lanthanide species with the different anions present. The diffusion coefficients were calculated using the Nernst- Einstein equation for the diffusion of an individual species in solution: where is the effective mobility, constant, is the temperature, and is elementary unit of charge ne, where n is the integer of charge and e the electronic charge. Each lanthanide was measured individually, and all solutions were adjusted to pH 3.75 to prevent hydrolysis of the lanthanides and formation of hydroxyl groups. The S- shape of the curve in FIG. 4 is ascribed to the presence of the ‘gadolinium break,’ where many studies have shown an abrupt decrease in inner coordination number of aquo-ions from lighter (~9) to heavier (~8) lanthanides in the middle of the series. Of the three anions investigated, triflate appeared to provide the largest interaction with lanthanides, evident from the relatively large difference in Dibetween the light (e.g., La to Gd) and heavy (e.g., Tb to Lu) lanthanides. This is likely due to its bulky properties, which are believed to slow diffusion overall. The chloride ion is the smallest and most electronegative of the three anions and is believed to essentially act as a ligand itself and provide enhanced separation between lanthanides. Larger differences in Dibetween adjacent lanthanides were more apparent with the lighter lanthanides compared to the heavier lanthanides for the cases of chloride and triflate. The nitrate ion has been evidenced to be present in the first inner hydration sphere of a lanthanide complex, whereas chloride likely does not inhabit this region. The intermediate nitrate ion showed the least difference between Difor adjacent lanthanides. Table 1 shows the experimental diffusion coefficients for three trivalent lanthanides. The values showed minimal variation between lanthanides, but a trend of decreasing diffusivity across the series was observed. TABLE 1 Lanthanide Species D / x10-6cm2s-1Nd3+3.708±0.005 Eu3+ Dy3+3.588±0.005 To investigate the transport of lanthanide-carboxylate complexes, formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), iminodiacetic acid (IDA), and citric acid (CA) were used as ligands. The pH of each ligand was adjusted such that the dominant species was in the singly deprotonated L- form for consistency. FIG.5 depicts the electropherograms of four lanthanides (La, Sm, Dy, and Lu) taken in different concentrations of each ligand and their mobilities. The mobilities were calculated using the following equations: where is the observed mobility and is the mobility of the electroosmotic flow where is the total length length of the capillary from the point to the detector, is the applied potential, is the migration time of the species, and is the time during which the potential linearly from zero to its Of the ligands investigated, CA provided the most immediate and rapid decrease in mobility for each lanthanide, attributed to its bulkier size and available binding sites. OA decreased mobility in a similar manner, though it had fewer available binding sites and is much smaller compared to CA, suggesting complexation happens with multiple ligands. For both ligands, µeff = 0 cm2V-1s-1at higher concentrations, attributed to the formation of neutrally charged species that migrated at the same rate as the EOF. In the cases of AA and BA, resolution was not achieved between La-Sm and Dy-Lu, however mobility was still decreased at higher concentrations. At an AA concentration of 5 mM, peaks for Sm and Dy just began to resolve. Using FA allowed for resolution between each lanthanide. However, mobility decreased gradually, and separation was not improved at high concentrations. FIG.6 depicts the separation between lanthanides achieved in varying concentrations of IDA. The separation between with the concentration of IDA. FIG.7 depicts the decrease in lanthanide mobility achieved as the concentration of IDA increased from 0 mM to 5 mM. FIG.8 shows the separation of various lanthanides (La, Sm, Dy, Lu, Ce, Eu, Ho, Yb, Pr, Gd, Er, La, Nd, Tb, and Tm) in nitrate and chloride BGE using IDA as the ligand. FIG.9 shows the diffusion coefficients for various lanthanides (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu) in nitrate and chloride BGE. FIG. 10 shows mobilities of various lanthanides (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu) in increasing IDA concentration ranging from 0 mM to 10 mM. FIG. 11 shows the change in diffusion coefficients for various lanthanides (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu) using IDA as the ligand. FIG.12 shows the speciation of IDA at various pH levels. At a pH of from about 3 to about 8, the singly deprotonated form of IDA is the dominant species. FIG.13 shows the speciation of various lanthanides (Ce, Eu, Ho, and Yb) using IDA as a ligand. As shown in FIG. 8 to FIG. 13, using IDA as the ligand provided effective separation of the REEs. The use of IDA in the BGE was demonstrated to have a direct influence on the electromobility of each lanthanide studied, as well as on the separation factor between each lanthanide. Without being bound by any theory, it is believed that IDA was able to provide partial complexation and speciation at the pH used. The counterion used had minimal effect on the separation. FIG.14 shows the diffusion coefficients of various lanthanides (La, Sm, Dy, and Lu), with an inset showing an electropherogram with no ligand present. FIG.15 is a graph of the stability constants of various lanthanides (La, Sm, Dy, and Lu) using HIBA and IDA as ligands. β₁, β₂, and β₃ refer to the first, second, and third stability constants, respectively. These stability constants quantify the affinity between a ligand and a metal ion in successive binding steps. Specifically, β₁ represents the equilibrium constant for the formation of the first complex, β₂ for the formation of the second complex, and β₃ for the formation of the third complex. Higher values of these constants indicate stronger binding and greater stability of the respective complexes. Two lanthanide-ligand complexes with a large difference in stability constants may separate more effectively than two lanthanide-ligand complexes with a small difference in stability constants. FIG.16 is a graph of the separation factor between various lanthanides (La-Sm, Sm-Dy, and Dy-Lu), with an inset showing an electropherogram taken at 7.5 mM α-HIBA-. FIG.17 is a graph of the effective mobilities of various lanthanides (La, Sm, Dy, and Lu) in increasing concentrations of HIBA-. FIG.18 is a graph of the number of theoretical plates for various (La, Sm, Dy, and Lu) at increasing concentrations of HIBA-. FIG. 19 is a graph of the stepwise stability constants for various lanthanides (La, Sm, Dy, and Lu). FIG. 20 is a contour map of separation factors between three lanthanide pairs (La-Sm, Sm-Dy, and Dy-Lu) in increasing concentrations of HIBA-. The electropherograms above the contour map reflect the ligand concentrations at which separation is largest for the respective pair: 2.5 mM IDA- and 0.0 mM α-HIBA- for La‑Sm, 0.08 mM IDA- and 3.75 mM α-HIBA- for Sm-Dy, and 5.0 mM IDA- and 1.88 mM α-HIBA- for Dy-Lu. As shown in FIG. 14 to FIG. 19, using HIBA as the ligand provided effective separation of the REEs. As shown in FIG. 20, there were distinct combinations of IDA and HIBA that enhanced separations between each metal pair. FIG.20 also shows that the separation factors were reduced in certain regions. Without being bound by theory, it is believed that this may be due to the electrostatic repulsion between negatively charged carboxyl groups. EXAMPLE 2 Ln(NO3)3 (Ln = La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, ≥ 99.9%), sodium nitrate (≥ 99.0%), nitric acid (1 M, Titripur), sodium hydroxide (1 M, Titripur), formic acid (≥ 96.0%), acetic acid (99.7%), propionic acid (≥ 99.5%), butyric acid (≥ 99.0%), lactic acid (≥ 85.0%), α-hydroxyisobutyric acid (99.0%), and iminodiacetic acid (98.0%) were purchased through Sigma Aldrich® and used as-is. For initial screening studies, 5 mM NaNO3solutions were prepared and adjusted to pH = 5.0 using HNO3 / NaOH, and ligand solutions were prepared to 40 mM in nanopure water (Millipore®, 18 MΩ), adjusted to pH = 5.0 using HNO3 / NaOH, and diluted using the previously prepared 5 mM NaNO3solution. For all experiments, lanthanide sample solutions were prepared to have a total metal concentration of 0.4 mM. All final solutions were filtered through a 0.2 μm nylon syringe filter (Acrodisk®). An Agilent® 7100 CE (Agilent Technologies®) equipped with a capacitively- coupled conductivity detector (C4D, TraceDec® Contactless Conductivity) was used for all electrophoretic studies. The capillaries (fused silica, 50 μm internal diameter x 56 cm length, Agilent) were conditioned with a flush (~950 mbar) of 1 M HCl, nanopure water, 1 M NaOH, and 5 mM NaNO3each for 10 min. Fresh ligand solution was used to flush and fill the capillary for 10 min before each run. All samples were injected electrokinetically at 5 kV for 12 s, and experiments conducted at 25°C. For initial screening experiments, the separation voltage was kept at 20 The detector settings were as follows: frequency = 34.13 Hz, filter = 1 / 3, cutoff = 0.5, voltage = -6 dB, and gain = 75%. The ligands investigated included lactate (LA-), α-hydroxyisobutyrate (HIBA-), iminodiacetate (IDA-), formate (FA-), acetate (AA-), propionate (PA-), and butyrate (BA-). The electrophoretic mobilities (μeff) and first stability constants (β1) for each ligand were determined by conventional techniques and are shown in FIG.21 for Sm and Dy. FIGS.22A-22D show an in-depth study of HIBA-. FIG.22A shows separation factors between lanthanides for 5 mM - 25 mM HIBA-, with an inset showing an electropherogram taken at 7.5 mM HIBA-. FIG. 22B shows effective mobilities for four lanthanides at increasing ligand concentration. The dotted lines represent the fitting equations used to determine stability constants in FIG.22D. FIG. 22C shows the number of theoretical plates for each lanthanide at increasing ligand concentration. FIG. 22D shows stepwise stability constants determined for each lanthanide. According to the stability constants determined, the largest differences occurred between IDA- / HIBA- and PA- / BA-. IDA- performed well at selectively separating light and heavy lanthanides from one another due to a preferential difference in geometric configuration with lanthanides before and after the gadolinium break. HIBA- has been shown to enhance the separation of middle-series lanthanides both on its own and alongside an auxiliary ligand but has only been investigated with a limited number of competing ligands. At high concentrations, LA- performed well at separating light and heavy lanthanides but separating those in the middle of the series was reduced. FIG.23 shows separation factors of each Ln pair (La-Sm, Sm-Dy, and Dy-Lu) with the primary ligand, L1, on the left and the auxiliary ligand, L2, specified on top. Stars represent ligand combinations where separation was enhanced compared to that obtained at the same concentration of L1 alone. To encompass a wide range of possible metal-ligand complexes, three specific ligand combinations were chosen, represented as A, B, and C on the bottom of the figure: 1.25 mM L1and 20 mM L2(A), 20 mM L1and 1.25 mM L2(B), and 5 mM of both L1and L2(C), where L1 = LA-, HIBA-, and IDA-, and L2 = FA-, AA-, PA-, and BA-. Solutions were prepared for each possible ligand combination, and their pH values adjusted to 5.0. In each case, the sample included a 0.4 mM Ln(NO3)3 solution that contained 0.1 mM each of La(NO3)3, Sm(NO3)3, Dy(NO3)3, and Lu(NO3)3. Ligand performance was based on the calculated separation factors, γ, between Sm and Dy due to the difficulty that lies in separating middle- series lanthanides. The separation factors were calculated as: where t1 and W1 are the width for the earlier migrating peak, and t2and W2the migration for the later migrating peak. FIG.23 shows a visual representation of the separation factors obtained for each Ln pair (La-Sm, Sm-Dy, and Dy-Lu) at combinations A, B, and C. The separation factors were calculated for La-Sm, Sm-Dy, and Dy-Lu, respectively. In a majority of cases, separations were improved compared to when only a primary ligand was present. LA- is most susceptible to competition as nearly every combination that included a secondary ligand yielded improved separation factors compared to independent LA- measurements. The competitive ligands had a more pronounced effect on the separation of Sm-Dy compared to La-Sm and Dy-Lu in the case of IDA-. The best separation between Sm and Dy occurred when 20 mM of HIBA- and 1.25 mM PA- were used. FIG.22A shows that when HIBA- was used independently, the highest Sm-Dy separation factor also occurred at 20 mM. The separation factor between Sm-Dy was observed as increasing concentrations of PA- were included with 20 mM HIBA- as a complexing electrolyte to separate a lanthanide mixture electrophoretically. FIG. 24A shows the raw electropherograms at different concentrations where the shading of the electropherogram corresponds to γSm-Dy. FIG.24B shows effective electrophoretic mobilities of Sm and Dy recorded from the electropherograms in FIG. 24A. FIG. 24C shows separation factors between Sm and Dy at increasing [PA-]. The effective mobilities of Sm and Dy show very little change as [PA-] increased except for at high concentrations where the two metals began to coalesce as precipitates. Separation factors, however, increased up to 20 mM PA. At 15 mM PA-, the separation factor reached its highest value (6.163), which was markedly improved compared to when no PA- was present. To further enhance the separation of Sm and Dy, several parameters were investigated. FIG. 25A shows the effect of pH, with an inset showing separation factors at each value. FIG. 25B shows electropherograms of lanthanide separation using different concentrations of stock NaNO3 where Ci = 5 mM, 10 mM, 20 mM, and 30 mM (Cf = 0.625 mM, 1.25 mM, 2.5 mM, and 3.75 mM NaNO3, respectively), and an inset showing average electrophoretic mobilities of La, Sm, Dy, and Lu with error. FIG. 25C shows separation between Sm and Dy at different separation voltages. First, the pH of both the optimized ligand and sample were adjusted between 4.0 - 5.0 and electropherograms recorded at each value in increments of 0.2 pH units. separation occurred at pH = 5.0 (γSm-Dy= 7.520), and the worst separation occurred at pH = 4.2 (γSm-Dy= 3.769). Changing the pH of both the background and sample solution had an effect on the electroosmotic flow (EOF), as well. Higher pH values resulted in a larger degree of deprotonation of silanol groups on the wall surface of the capillary, which caused faster EOF. The pKa values for PA and HIBA are 4.88 and 3.95, respectively. Thus, each ligand should exist as > 50% in its deprotonated species at pH = 5.0. Though both pH = 4.6 and 5.0 showed similar separation factors, 5.0 was chosen as the optimal pH as it provided a faster analysis time due to increased EOF. The effect of [NaNO3] was investigated in a similar manner, where 5 mM, 10 mM, 20 mM, and 30 mM solutions of NaNO3were used to prepare lanthanide sample solutions and to dilute stock ligand solutions, shown in FIG. 25B. The relationship between electrolyte concentration and current was calculated using the following equation: where V is the separation R is the resistance, r is the inner radius of the capillary, L is the length of the capillary, λ is the molar conductivity of the electrolyte, and c is the concentration of the electrolyte. Increasing the concentration of electrolyte in the experiments directly influenced the amount of current generated in the system, which led to excessive Joule heating that affected both the profile of analyte peaks, as well as the electropherogram baseline. The electropherograms in FIG.25B began to suffer the effects of Joule heating starting at [NaNO3] = 20 mM where the peak for Lu became interrupted. At [NaNO3] 30 mM, the baseline became unstable, and peaks were unable to be reliably identified. The electrolyte concentration did not seem to have a notable effect on the mobility of lanthanide species, illustrated in the inset. FIG.26A shows separation factors of lanthanide pairs at increasing [NaNO3], and FIG.26B shows the number of theoretical places for each lanthanide at increasing [NaNO3]. While separation factors and number of theoretical plates were overall increased at higher NaNO3 concentrations, the best electrolyte concentration was taken as 10 mM as this best preserved peak clarity. The final experimental condition investigated was the separation voltage. A large separation voltage should result in fast analysis time, though may be subject to excessive Joule heating in a similar manner to electrolyte concentration according to Eq. 2. The separation voltage was investigated at 10 kV, 15 kV, 20 kV, and 25 kV. The separation factors for Sm-Dy were similar in value, suggesting that the separation voltage had little effect on the mobility of species and rather on the analysis time. Increasing the separation voltage led to faster analysis times (31 10 kV, 20 min for 15 kV, and 17 min for 20 kV), however, the capillary broke due to Joule heating at 25 kV. Therefore, 20 kV was considered the best separation voltage. Combining the results from each study, the conditions for the separation of Sm-Dy including a buffer composition of 20 mM HIBA- and 15 mM PA- at pH = 5.0, prepared with 10 mM NaNO3, and a separation voltage of 20 kV, allowed for the achievement of a separation factor of γSm-Dy= 9.457, which was notably enhanced compared to separations conducted with only HIBA- at pH = 5.0, [NaNO3], and 20 kV separation voltage, where gave a separation factor of γSm-Dy= 3.921. Combining all of the experimental parameters and ligand solution that were investigated, a sample solution was examined that contained 14 lanthanide nitrates (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu) to see how well adjacent lanthanide pairs could be resolved. Resolution was used as a performance metric instead of separation factor as it takes into consideration the full width at half maximum (FWHM) of peaks rather than just the baseline peak width, which was not a concern when examining the four- lanthanide sample solution due to the large mobility differences. The resolution was calculated as: FIG.27A shows a of these separations, which were repeated in triplicate. Peaks were fit to an asymmetric sigmoidal function to extract accurate FWHM values for each peak. The resolution between each adjacent Ln pair is also shown in FIG.27A on the bottom, where pairs that are baseline separated (R ≥ 1.5) being distinguished from those that are not by the region below the dotted line. The most resolved pair is Nd-Sm ( = 4.597), likely due to this pair having the largest difference in ionic radius compared to the rest. Resolution of heavier pairs (Er-Tm, Tm-Yb, and Yb-Lu) was not able to be accurately determined due to precipitation of Yb and Lu, which migrated at the same rate as the EOF and thus had zero electrophoretic mobility. Excluding Tm, Yb, and Lu from a calibration curve in FIG. 27B, the correlation coefficient was 0.993, showing good agreement between the relationship of ionic radius and effective electrophoretic mobility. The electrophoretic mobilities of lanthanides without ligand present are represented in FIG.27B. The addition of ligand into the background electrolyte increased electrophoretic mobility differences between all 14 lanthanides through partial complexation with the ligands present. Further, the separation between and Gd was investigated under the same experimental conditions, as these two are directly in the middle of the series and occur immediately prior to the gadolinium break before a change in the inner-sphere water coordination environment, making separation difficult compared to other pairs. Pr-Nd is another challenging pair to separate because of their similar ionic radii, however they were not investigated separately apart from the 14-Ln mixture in this study. A sample solution that contained 0.4 mM Ln(NO3)3 (Ln = Eu and Gd, 0.2 mM each) was run in triplicate. FIG.27C shows the measured resolution between Eu and Gd for each run, with electropherograms shown in the inset. From these experiments, = 1.195 ± 0.039 and from the study with 14 lanthanides, = 1.664 ± being bound by any theory, the discrepancy between is most likely due to an increased individual concentration of Eu and Gd in the isolated experiments, which caused peak broadening. EXAMPLE 3 Formic acid (≥ 96.0%), acetic acid (99.7%), propionic acid (≥ 99.5%), butyric acid (≥ 99.0%), propylamine (PAm, 99.0%), butylamine (BAm, 99.5%), diethylamine (DEA, ≥ 99.5%), dipropylamine (DPA, 99.0%), dibutylamine (DBA, 99.5%), triethylamine (TEA, 99.5%), tripropylamine (TPA, ≥ 98.0%), sulfuric acid (SA, 98.0%), methanesulfonic acid (MSA, ≥ 99.0%), ethanesulfonic acid (ESA, 95.0%), ethanethiol (ET, 97.0%), methanethiol (MT, ≥ 90.0%), propanethiol (PT, 99.0%), trimethylphosphate (TMP, ≥ 99.0%), triethylphosphate (TEP, ≥ 99.8%), tripropylphosphate (TPP, 99.0%), 15-crown-5 ether (15c5, 98.0%), 18-crown-6 ether (18c6, ≥ 99.0%), La(NO3)3(≥ 99.9%), Sm(NO3)3(≥ 99.9%), Dy(NO3)3 (≥ 99.9%), Lu(NO3)3 (≥ 99.9%), NaNO3 (≥ 99.0%), HNO3 (1 M, Titripur), and NaOH (1 M, Titripur) were purchased through Sigma-Aldrich and used as-is. A solution of 0.4 mM Ln(NO3)3 (Ln = La, Sm, Dy, and Lu) was prepared and its pH adjusted to 5.0 through the addition of HNO3 / NaOH. All ligand stock solutions were prepared at 5 mM in nanopure water (Millipore, 18 MΩ) and diluted with 5 mM NaNO3. The ligand solutions and NaNO3 were adjusted to pH = 5.0 through the addition of HNO3 / NaOH. Fresh ligand solutions were prepared before each experiment, and all final solutions were filtered through a 0.2 μm nylon syringe filter (Acrodisk®) before use. An Agilent 7100 CE instrument (Agilent Technologies) was used for all experimental work and equipped with a capacitively coupled contactless conductivity detector (TraceDec Contactless Conductivity). The fused silica capillaries used were 56 cm in length and 50 μm in inner diameter (Agilent). All conditioned before running a new ligand solution by applying a 940 mbar flush of HCl for 10 min, H2O for 10 min, NaOH for 10 min, and finally running the ligand solution for 10 min. Between ligands and before preconditioning, the capillary was flushed with acetone for 20 min to remove any organic adsorbates. Lanthanide samples were injected electrokinetically at 5 kV for 12 s. A ligand solution flush was applied before each experimental run at 940 mbar for 3 min. Experiments were conducted at 25°C with a 20 kV separation voltage and 12 s voltage ramp time. The detector settings were as follows: frequency = 19.79 Hz, filter = 1 / 3, cutoff = 0.5, voltage = 0 dB, and gain = 100%. Four different carboxylates ligands (FA, AA, PA, and BA) were investigated as ligand additives in a solution of NaNO3 for the separation of La, Sm, Dy, and Lu. Electropherograms were recorded at ligand concentrations ranging from 0 mM - 5.0 mM to observe the mobility, separation factor, and separation efficiency at each ligand concentration. FIGS. 28A-28F show compiled data of carboxylate ligand solutions used to separate a solution of four lanthanides. FIG. 28A shows normalized electropherograms for 0.8 mM of each ligand. FIG. 28B shows normalized electropherograms for 0.63 mM of each ligand. FIG.28C shows normalized electropherograms for 2.5 mM of each ligand. The acid dissociation constants for FA, AA, PA, and BA are 3.751, 4.756, 4.874, and 4.871, respectively. At a working pH of 5.0, it would be expected that > 50% of each ligand exist as base, with that value increasing for ligands with acid dissociation constants less than 5.0. While FA may have more base species present and available for complexation, separation between the four lanthanides was not observed even at high concentrations, and the same was true for AA. The acid dissociation constants for PA and BA were nearly identical, so it can be assumed that they are present in their base forms at nearly equal amounts in solution. FIG.28D shows mobility values of lanthanum for each ligand at increasing concentration. PA appeared to interact more strongly with the lanthanides, illustrated by its effect on the mobility curve of La in FIG.28D. Because partial complexation occurs in dynamic equilibrium, Di is not a constant value throughout these experiments. The amount of time La spends complexed with a given ligand is governed by the species formation constant as well as the chemical environment, or “accessibility” of ligands to complex the metal. When complexed with La, BA would be expected to yield a lower Di compared to PA, decreasing mobility as it contains an additional methyl group compared to PA. The stability constant for a LaPA2+complex was, however, larger than that of BA and appeared to of an impact on the electrophoretic mobility rather than the size of the ligand. Without being bound by any theory, the additional methyl group in BA may have also influenced the ligand’s accessibility to La, and its hydrophobic tail may have hindered penetration through the hydration spheres of the lanthanide. FIG.28E shows separation factors between La-Lu for each ligand at increasing concentration. FIG.28F shows the number of theoretical plates for La at each concentration. Of all ligand groups investigated, carboxylates provided the highest separation factors between La-Lu and the second highest number of theoretical plates. While no trend was detected with separation factor, there seemed to be a steady increase in plate number as concentration increased along with a relationship among ligand size. FIGS. 29A-29F show compiled data of amine ligand solutions used to separate a solution of four lanthanides. The amines evaluated included primary, secondary, and tertiary amines. Due to their high (> 10.0) acid dissociation constants, at a working pH of 5.0 each amine was predominantly protonated, having a single positive charge. A positively charged ligand should have very little interaction with the trivalent lanthanides in solution, therefore what little interaction can be expected should mainly be governed by diffusion. FIG.29A shows normalized electropherograms for 0.8 mM of each ligand. FIG.29B shows normalized electropherograms for 0.63 mM of each ligand. FIG. 29C shows normalized electropherograms for 2.5 mM of each ligand. Peaks were positive for primary and secondary amines, but then shifted to negative for tertiary amines as the lanthanides became less conductive than the background electrolyte using these ligands. FIG. 29D shows mobility values of lanthanum for each ligand at increasing concentration. The mobility graph in FIG.29D shows that TPA lowered the mobility of La the most, followed by TEA, which was expected due to these ligands’ bulky structures. It would be expected for DBA to follow this, but instead DPA took its place. FIG. 29E shows separation factors between La-Lu shown for each ligand at increasing concentration. The separation factors presented in FIG.29E show relatively large values for DPA at most concentrations, while values were negligible for all other ligands (with the exception of DEA at 3.75 mM). It should also be noted that peaks began to broaden amongst all ligands as concentration increased, leading to a decrease in number of theoretical plates. Because most of the ligand species in these experiments spent a majority of their time uncomplexed with lanthanides, this led to the observation of ligand peaks in the electropherograms. As the charge is known for these species, the diffusion coefficient of each ligand was calculated, assuming Stokes-Einstein diffusion where the geometry of ligands as spherical. FIG. 29F shows the relationship between the calculated diffusion coefficients of the amines and their estimated ionic radii. A linear (R2= 0.96) trend was observed, which confirmed the relationship between ligand size and diffusive properties. The sulfates investigated in this study were SA, MSA, and ESA. At a pH of 5.0, the majority of MSA and ESA species exist with a single negative charge, and SA will exist with a double negative charge. FIGS. 30A-30F show compiled data of sulfate ligand solutions used to separate a solution of four lanthanides. FIG. 30A shows normalized electropherograms for 0.8 mM of ligand. FIG. 30B shows normalized electropherograms for 0.63 mM ligand. FIG. 30C shows normalized electropherograms for 2.5 mM ligand. FIG.30D shows mobility values of lanthanum for each ligand at increasing concentration. FIG.30E shows separation factors between La-Lu shown for each ligand at increasing concentration. FIG.30F shows the number of theoretical plates for La at each concentration. The mobilities of La using ESA and MSA compared to SA showed little change as ligand concentration increased, suggesting minimal interaction. However, SA provided a profound mobility decrease, likely due to its -2 charge and small ionic radius. It is difficult to compare the three sulfates on the basis of size due to the difference in charge, however, there did not appear to be a size trend amongst ESA and MSA. It is also difficult to assess the contribution of the sulfur atom, as the surrounding oxygen atoms of the sulfate functional group likely contributed to partial complexation as well. Though SA complexed with La fairly strongly (β1= 3.597 ± 0.105), separation between La-Lu was poor as a result of severe peak broadening. Even though SA was capable of binding lanthanides efficiently, it was not suitable for distinguishing lanthanide complexes from one another electrophoretically as the formation constants of complexes were too similar between lanthanides. Similar to the sulfates, phosphorus is not the only atom contribution to electron donation in this group of ligands as oxygen atoms are present in the functional group. The phosphates studied included TMP, TEP, and TPP, all neutral in charge and differing only by the addition of a methyl group onto each branch. Phosphate-containing ligands are widely used in solvent extraction methods for their soft electron donating ability but are usually incorporated with various other electron donating atoms. FIGS. 31A-31F show compiled data of phosphate ligand solutions used to separate a solution of four lanthanides. FIG.31A shows normalized electropherograms for 0.8 mM of ligand. FIG. 31B shows normalized electropherograms for 0.63 mM ligand. FIG. 31C shows normalized electropherograms for 2.5 mM ligand. FIG. 31D shows values of lanthanum for each ligand at increasing concentration. FIG.31E shows separation factors between La-Lu shown for each ligand at increasing concentration. FIG.31F shows the number of theoretical plates for La at each concentration. The electropherograms in FIGS.31A-31C show a higher degree of peak broadening for each of the three ligands at increasing concentration, and peaks for TPP remain relatively broad even at low concentrations. At the high ligand concentrations in FIG.31C, considerable band broadening was observed that increased with ligand size. This demonstrated the increase in overall diffusion coefficient of the complexed lanthanide species as additional methyl groups are added to the ligand. The mobility of La with each ligand became irregular throughout the course of increasing concentration, which, without being bound by any theory, may be due to ligand adsorption to the capillary wall, causing inconsistency in ligand concentration in the bulk solution within the capillary. Two crown ethers, 15-crown-5 (15c5) and 18-crown-6 (18c6), were investigated. Crown ethers contain a cavity suitable for complexing metals through electron donation via the surrounding oxygen atoms. FIGS.32A-32D show the normalized electropherograms and corresponding La mobility graphs for each crown ether. FIG. 32A shows the normalized electropherograms for 15c5 and FIG.32B shows the normalized electropherograms for 18c6 ligands used to separate La, Sm, Dy, and Lu. FIG. 32C shows La mobility data using 15c5 and FIG. 32D shows La mobility data using 18c6. For 15c5, only one lanthanide peak was observed throughout all ligand concentrations, and there was no discernible trend amongst the peak shape / size. For 18c6, multiple lanthanide peaks were observed which seemed to disappear at 0.63 mM and gradually return at higher concentrations. The effective electrophoretic mobilities of La overall increased with ligand concentration for both 15c5 and 18c6. The crown ethers tend to preferentially bind metal cations with an ionic diameter that closely matches their cavity diameter.15c5 shows a clear preference for binding Na+, which was the co-ion used. Without being bound by any theory, this may help to explain why one lanthanide peak was present in the electropherograms using 15c5. Rather than working to complex with the lanthanides, the 15c5 bound sodium ions in solution. This presents two possibilities: 1) 15c5 bound with free sodium ions in the bulk solution, effectively removing all potential for lanthanide complexation, and / or 2) 15c5 was drawn to sodium ions among the rigid portion of the electric double layer at the capillary wall. If these sodium ions are removed from the double layer, electroosmotic flow (EOF) mobility will decrease as a result of decreased zeta potential. Both of possibilities may have occurred in these experiments due to the solo lanthanide peak present, as well as a decrease in EOF mobility that was not seen with 18c6 as shown in FIG.33, which shows the mobility of the EOF for each set of ligands. 18c6 should show more of a preference for lanthanides which was initially seen in the electropherograms, but it is likely 18c6 bound sodium ions to some degree as well. The peaks that reappeared for 18c6 may also be ‘ghost’ peaks, which appear due to adsorbates freeing from the capillary wall. Non-limiting, example embodiments may include the following, alone or in combination: Embodiment 1: A method of separating rare earth elements by capillary electrophoresis, the method comprising introducing a ligand solution and a rare earth element solution into a separation chamber of a capillary electrophoresis system, the ligand solution comprising one or more ligands comprising propylamine, butylamine, diethylamine, dipropylamine, dibutylamine, triethylamine, tripropylamine, 15-crown-5, 18-crown-6, iminodiacetic acid (IDA), lactic acid (LA), formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), citric acid (CA), propionic acid (PA), butyric acid, and hydroxyisobutyric acid (α-HIBA), and the rare earth element solution comprising two or more rare earth elements; positioning terminal ends of the separation chamber in respective buffer solutions of the capillary electrophoresis system; applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber to separate one or more of the rare earth elements from the rare earth element solution; and recovering the one or more of the rare earth elements. Embodiment 2: The method of Embodiment 1, wherein introducing a ligand solution and a rare earth element solution into a separation chamber comprises sequentially introducing the ligand solution and the rare earth element solution into the separation chamber. Embodiment 3: The method of Embodiment 1, wherein introducing a ligand solution and a rare earth element solution into a separation chamber comprises introducing the rare earth element solution comprising one or more of scandium, yttrium, lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium into the separation chamber. Embodiment 4: The method of Embodiment 1, wherein introducing a ligand solution and a rare earth element solution into a separation chamber comprises introducing the ligand solution comprising one or more ligands to react with an ion of the two or more rare earth elements into the separation chamber. Embodiment 5: The method of any one of Embodiments 1 to 4, wherein introducing a ligand solution and a rare earth element solution into a separation chamber comprises introducing the ligand solution comprising one or more of formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), iminodiacetic acid (IDA), and citric acid (CA) into the separation chamber. Embodiment 6: The method of any one of Embodiments 1 to 4, wherein introducing a ligand solution and a rare earth element solution into a separation chamber comprises introducing the ligand solution comprising propionic acid, citric acid, IDA, or α-HIBA into the separation chamber. Embodiment 7: The method of any one of Embodiments 1 to 4, wherein introducing a ligand solution and a rare earth element solution into a separation chamber comprises introducing the ligand solution exhibiting a pH of from about 1.0 to about 5.5 into the separation chamber. Embodiment 8: The method of any one of Embodiments 1 to 4, wherein applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber to separate one or more of the rare earth elements from the rare earth element solution comprises reacting the one or more ligands with the two or more rare earth elements to produce one or more rare earth element-ligand complexes. Embodiment 9: The method of Embodiment 8, wherein reacting the one or more ligands with the two or more rare earth elements to produce one or more rare earth element- ligand complexes comprises reacting the one or more ligands with the two or more rare earth elements at a temperature of from about 10°C to about 30°C. Embodiment 10: The method of any one of Embodiments 1 to 4, wherein recovering the one or more of the rare earth elements comprises recovering the one or more rare earth elements comprising one or more of lanthanum, samarium, dysprosium, and lutetium. Embodiment 11: The method of any one of Embodiments 1 to 4, wherein recovering the one or more of the rare earth elements comprises recovering the one or more rare earth elements comprising lanthanum and samarium, samarium and dysprosium, or dysprosium and lutetium. Embodiment 12: The method of any one of Embodiments 1 to 4, further comprising recovering another of the rare earth elements. Embodiment 13: A method rare earth elements by capillary electrophoresis, the method comprising introducing a ligand solution into a separation chamber of a capillary electrophoresis system, the ligand solution comprising hydroxyisobutyric acid and propionic acid or hydroxyisobutyric acid and iminodiacetic acid; introducing a rare earth element solution into the separation chamber, the rare earth element solution comprising two or more of scandium, yttrium, lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium; positioning terminal ends of the separation chamber in respective buffer solutions of the capillary electrophoresis system, the buffer solutions comprising an anion; applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber to separate one or more of the rare earth elements from the rare earth element solution; and recovering the one or more of the rare earth elements. Embodiment 14: The method of Embodiment 13, wherein introducing a ligand solution into a separation chamber comprises introducing the hydroxyisobutyric acid at a concentration of from about 0.1 mM to about 40 mM and introducing the propionic acid at a concentration of from about 0.1 mM to 20 mM. Embodiment 15: The method of Embodiment 13 or Embodiment 14, wherein applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber comprises applying a voltage of from about 0.001 kV to about 30 kV across electrodes positioned in the respective buffer solutions to create an electric field across the separation chamber of from about 0.1 kV / cm to about 3.0 kV / cm. Embodiment 16: The method of Embodiment 13 or Embodiment 14, wherein applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber comprises applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through a capillary tube. Embodiment 17: A system for separating rare earth elements, the system comprising a separation chamber extending between reservoirs, an anode in one of the reservoirs and a cathode in another of the reservoirs and terminal ends of the separation chamber configured to be positioned in the reservoirs; a voltage source coupled to the separation chamber and configured to apply an electrical field between the anode and the cathode; and a detection system coupled to the separation separation chamber configured to contain: a sample solution comprising two or more of scandium, yttrium, lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium; and a ligand solution comprising one or more ligands comprising propylamine, butylamine, diethylamine, dipropylamine, dibutylamine, triethylamine, tripropylamine, 15-crown-5, 18-crown-6, iminodiacetic acid (IDA), lactic acid (LA), formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), citric acid (CA), propionic acid (PA), butyric acid, and hydroxyisobutyric acid (α-HIBA). Embodiment 18: The system of Embodiment 17, wherein the separation chamber comprises a capillary tube. Embodiment 19: The system of Embodiment 17 or Embodiment 18, wherein the voltage source is configured to apply a voltage of from about 0.001 kV to about 30 kV across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber. Embodiment 20: The system of Embodiment 17 or Embodiment 18, wherein the detection system comprises a capacitively-coupled contactless conductivity detector or a UV-Vis detector. The embodiments of the disclosure described above and illustrated in the accompanying drawings do not limit the scope of the disclosure, which is encompassed by the scope of the appended claims and their legal equivalents. Any equivalent embodiments are within the scope of this disclosure. Indeed, various modifications of the disclosure, in addition to those shown and described herein, such as alternate useful combinations of the elements described, will become apparent to those skilled in the art from the description. Such modifications and embodiments also fall within the scope of the appended claims and equivalents.

Claims

What is claimed is:

1. A method of separating rare earth elements by capillary electrophoresis, the method comprising: introducing a ligand solution and a rare earth element solution into a separation chamber of a capillary electrophoresis system, the ligand solution comprising one or more ligands comprising propylamine, butylamine, diethylamine, dipropylamine, dibutylamine, triethylamine, tripropylamine, 15-crown-5, 18-crown-6, iminodiacetic acid (IDA), lactic acid (LA), formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), citric acid (CA), propionic acid (PA), butyric acid, and hydroxyisobutyric acid (α-HIBA), and the rare earth element solution comprising two or more rare earth elements; positioning terminal ends of the separation chamber in respective buffer solutions of the capillary electrophoresis system; applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber to separate one or more of the rare earth elements from the rare earth element solution; and recovering the one or more of the rare earth elements.

2. The method of claim 1, wherein introducing a ligand solution and a rare earth element solution into a separation chamber comprises sequentially introducing the ligand solution and the rare earth element solution into the separation chamber.

3. The method of claim 1, wherein introducing a ligand solution and a rare earth element solution into a separation chamber comprises introducing the rare earth element solution comprising one or more of scandium, yttrium, lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium into the separation chamber.

4. The method of claim 1, introducing a ligand solution and a rare earth element solution into a separation chamber comprises introducing the ligand solution comprising one or more ligands formulated to react with an ion of the two or more rare earth elements into the separation chamber.

5. The method of any one of claims 1 to 4, wherein introducing a ligand solution and a rare earth element solution into a separation chamber comprises introducing the ligand solution comprising one or more of formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), iminodiacetic acid (IDA), and citric acid (CA) into the separation chamber.

6. The method of any one of claims 1 to 4, wherein introducing a ligand solution and a rare earth element solution into a separation chamber comprises introducing the ligand solution comprising propionic acid, citric acid, IDA, or α-HIBA into the separation chamber.

7. The method of any one of claims 1 to 4, wherein introducing a ligand solution and a rare earth element solution into a separation chamber comprises introducing the ligand solution exhibiting a pH of from about 1.0 to about 5.5 into the separation chamber.

8. The method of any one of claims 1 to 4, wherein applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber to separate one or more of the rare earth elements from the rare earth element solution comprises reacting the one or more ligands with the two or more rare earth elements to produce one or more rare earth element-ligand complexes.

9. The method of claim 8, wherein reacting the one or more ligands with the two or more rare earth elements to produce one or more rare earth element-ligand complexes comprises reacting the one or more ligands with the two or more rare earth elements at a temperature of from about 10°C to about 30°C.

10. The method of any one of claims 1 to 4, wherein recovering the one or more of the rare earth elements comprises recovering the one or more rare earth elements comprising one or more of lanthanum, samarium, dysprosium, and lutetium.

11. The method of any one of claims 1 to 4, wherein recovering the one or more of the rare earth elements comprises recovering the one or more rare earth elements comprising lanthanum and samarium, samarium and dysprosium, or dysprosium and lutetium.

12. The method of any one of claims 1 to 4, further comprising recovering another of the rare earth elements.

13. A method of separating rare earth elements by capillary electrophoresis, the method comprising: introducing a ligand solution into a separation chamber of a capillary electrophoresis system, the ligand solution comprising hydroxyisobutyric acid and propionic acid or hydroxyisobutyric acid and iminodiacetic acid; introducing a rare earth element solution into the separation chamber, the rare earth element solution comprising two or more of scandium, yttrium, lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium; positioning terminal ends of the separation chamber in respective buffer solutions of the capillary electrophoresis system, the buffer solutions comprising an anion; applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber to separate one or more of the rare earth elements from the rare earth element solution; and recovering the one or more of the rare earth elements.

14. The method of claim 13, wherein introducing a ligand solution into a separation chamber comprises introducing the hydroxyisobutyric acid at a concentration of from about 0.1 mM to about 40 mM and introducing the propionic acid at a concentration of from about 0.1 mM to 20 mM.

15. The method of claim 13 14, wherein applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber comprises applying a voltage of from about 0.001 kV to about 30 kV across electrodes positioned in the respective buffer solutions to create an electric field across the separation chamber of from about 0.1 kV / cm to about 3.0 kV / cm.

16. The method of claim 13 or claim 14, wherein applying a voltage across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber comprises applying a voltage across the electrodes positioned in the respective buffer solutions to facilitate ionic movement through a capillary tube.

17. A system for separating rare earth elements, the system comprising: a separation chamber extending between reservoirs, an anode in one of the reservoirs and a cathode in another of the reservoirs and terminal ends of the separation chamber configured to be positioned in the reservoirs; a voltage source coupled to the separation chamber and configured to apply an electrical field between the anode and the cathode; and a detection system coupled to the separation chamber, the separation chamber configured to contain: a sample solution comprising two or more of scandium, yttrium, lanthanum, cerium, praseodymium, neodymium, promethium, samarium, europium, gadolinium, terbium, dysprosium, holmium, erbium, thulium, ytterbium, and lutetium; and a ligand solution comprising one or more ligands comprising propylamine, butylamine, diethylamine, dipropylamine, dibutylamine, triethylamine, tripropylamine, 15-crown-5, 18-crown-6, iminodiacetic acid (IDA), lactic acid (LA), formic acid (FA), acetic acid (AA), oxalic acid (OA), benzoic acid (BA), citric acid (CA), propionic acid (PA), butyric acid, and hydroxyisobutyric acid (α-HIBA).

18. The system of claim 17, wherein the separation chamber comprises a capillary tube.

19. The system of claim 17 or 18, wherein the voltage source is configured to apply a voltage of from about 0.001 kV to about 30 kV across electrodes positioned in the respective buffer solutions to facilitate ionic movement through the separation chamber.

20. The system of claim 17 or claim 18, wherein the detection system comprises a capacitively-coupled contactless conductivity detector or a UV-Vis detector.

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