1 Introduction

Rare-earth elements (REEs) are indispensable for numerous high-tech products, such as motors for electric cars and permanent magnets for generators [1]. As energy production shifts to a higher proportion of renewable sources, the demand for REEs is foreseen to increase greatly over the next few decades [2]. However, the beneficiation of REEs involves the environmentally and economically costly Solvent eXtraction (SX) process. Ever since its introduction in the 1960s, this continues to be a low-efficiency procedure on the industrial scale, limited by the small separation factor and high consumption of non-recyclable chemicals [3]. Furthermore, there is increased interest in a stabler REE supply chain with an additional focus on innovative separation methods that are more efficient in terms of both REE beneficiation and secondary resources, e.g., recycling from end-of-life REEs [4]. In recent years, reports on the working principle of enriched REE ions close to their magnetic source [5,6,7,8] have raised the prospect of green REE separation technology, which should be fully explored. The stand-alone technology can also be combined with conventional SX to produce an innovative magnet-assisted solvent extraction process. By addressing the differences in REEs’ magnetic susceptibility, an intrinsic property of the respective metals [8], the selective enhancement of individual REEs’ extraction kinetics is expected. An enhanced separation factor is expected to be achieved by sampling prior to equilibrium, as is done nowadays in SX plants. However, to elucidate the mechanism for selectively enhancing the extraction kinetics in magnet-assisted solvent extraction, a novel approach for investigating the kinetic mechanism is crucial.

The mixer-settler is the main device within which solvent extraction takes place in a separation plant. Thus, hundreds of mixer-settler stages linked in a countercurrent circuit might be needed to achieve sufficient separation quality for individual REE products [3]. The phosphorous acids di-2-ethylhexylphosphoric acid (D2EHPA) and 2-ethylhexylphosphonic acid mono-2-ethylhexyl ester (PC88A) are two widely used cation exchanger extractants [9]. They tend to form dimers, i.e., \(H_{2} A_{2,O}\), in a non-polar organic phase. The extraction reaction equation for Dy reads

$$ Dy_{A}^{3 + } + 3H_{2} A_{2,O} \leftrightarrow Dy\left( {{\text{HA}}_{2} } \right)_{3,O} + 3H_{A}^{ + } . $$
(10.1)

Knowledge of the distribution coefficient

$$ D_{RE} = \frac{{c_{RE\left( o \right)} }}{{c_{RE\left( a \right)} }} $$
(10.2)

is defined by the concentration ratio of REEs, i.e., Dy in Eq. (10.1), in the organic phase and that in the aqueous phase at equilibrium; and the separation factor

$$ \beta_{RE1/RE2} = \frac{{D_{RE1} }}{{D_{RE2} }} $$
(10.3)

which is the ratio between the distribution coefficient, respectively, for two RE species, namely RE1 and RE2 in subscript. The \(c_{RE,o}\) and \(c_{RE,a}\) are REE concentrations in the organic and aqueous phases at equilibrium, respectively, and are critical in designing the process. This information could further feed into equilibrium-based modeling to improve and optimize the process [10]. Furthermore, the kinetic mechanism offers critical information about the timescale and hence the size of the plant for the separation process. Nevertheless, few tools have been developed to investigate chemical changes occurring at the liquid–liquid interface. The exploration of the kinetics is mainly based on indirect experimental investigations. Several sets of apparatus have been used to study the initial boundary mass flux where the concentration is known and the partial rate order for individual species and the reaction rate constant can be individually studied. Since only the species at the initial time step have defined concentration values, the forward rate equation normally takes a semi-empirical form

$$ R\left[ {\frac{mol}{{m^{2} \cdot s}}} \right] = k_{f} c_{RE\left( a \right)}^{a} c_{{H_{2} A_{2} \left( o \right)}}^{b} c_{H + }^{c} . $$
(10.4)

The Lewis cell and its variants represent a conventional experimental approach to quantifying the solvent extraction kinetics of this heterogeneous extraction process [11]. The aqueous phase and organic phase with their respectively defined volumes, V, are stirred separately/spontaneously so that respective bulk phases are well mixed while the water–oil interface is still stratified. The average extracted rare-earth \(\overline{c}_{RE\left( a \right)}\) is quantified by characterizing the sampled species at different time intervals during the reaction. Consequently, the reaction rate R of RE(III) is readily computed with

$$ R\left[ {\frac{mol}{{m^{2} \cdot s}}} \right] = \frac{V}{A} \cdot \frac{{d\overline{c}_{RE\left( a \right)} }}{dt} $$
(10.5)

and the initial rate is then extrapolated at the zero time stamp. There are some obvious disadvantages to Lewis-type cells. Tracking the reaction at different time intervals requires independent, repetitive experiments. It further entails the need to independently study the influence of the REE concentration, extractant concentration and pH. The overall sample consumption is enormous (100–1000 mL per experiment). Plus, at an increased stirrer speed, the hydrodynamic situation is ill-defined at the interface. Kelvin–Helmholtz instability and entrainment can occur at interfaces with an increased shear rate [12]. Consequently, a complicated wave pattern, turbulence and emulsification occur and the initially well-defined interface area A changes. Using a Lewis-type cell with a membrane, Chitra et al. [13] found a first-order rate law for Nd(III) and a third-order law for a dimeric PC88A concentration for the forward reaction, while the backward reaction is of the first order for the REE complex and of the third order for H+. The reaction order is, surprisingly, the same as the stoichiometric coefficient of the reactants in Eq. (10.1). Mishra found a quasi-first order rate for La(III) and D2EHPA and 0.08 order for H+ in the presence of lactic acid [14] for the forward reaction. Alternatively, using highly stirred apparatus AKUFVE [15], continuous sampling at different time intervals is achieved by sacrificing the quantified interface area. Wang et al. [16] found a reaction order of 1 for Sm(III), Eu(III), Gd(III) and PC88A, and a reaction order of −1 for H+ at a stir speed of 420 rpm. At the same time, there was one attempt [17] to minimize the organic phase volume in a single drop experiment where the organic phase rises freely due to buoyancy in a REE-laden aqueous column. The extraction time is adjusted by changing the column height, leading the setup to deliver more flexible measurements, with the advantage of sampling at different time intervals during the reaction in a single run. Mostaedi et al. [17] found a forward reaction law of the first order for Sm(III) and D2EHPA and one of −0.8 for H+, while Huang [18] found a quasi-first order rate for La(III), Gd(III), Ho(III) and Lu(III) concentrations and a PC88A monomeric concentration and quasi-negative first order for H+. An even smaller amount of sample is needed when employing the hollow-fiber membrane extractor technique, using which Nakashio et al. [19] reported very different rate laws compared with other studies. The increased effective diffusivity might cause an over-estimated mass flux due to Taylor diffusion [20] as a result of shear flow in the porous medium.

One major reason for the current disagreements over the rate law is un-decoupled fluid dynamics in the investigation apparatus. One obvious dilemma lies in the need to stir the cell to achieve active mixing in the accurate sampling that enters \(\overline{c}_{RE\left( a \right)}\) in Eq. (10.5) and the clearly defined reaction area when the interface is free from convection. In this paper, we propose and deliver preliminary validation for using a Mach–Zehnder interferometer to quantify the rate law with minimum reagent consumption, in the order of 1 mL per experiment. The heterogeneous extraction process, similar to a Lewis cell without active stirring, takes place at the water–oil interface, with both bulk phases in a state of stagnancy. Therefore, a space- and time-resolved Dy(III) boundary layer is monitored in the vicinity of the interface, where cation exchange takes place with PC88A as an extractant. A diffusion layer of Dy(III) is formed gradually with a positive concentration gradient in the direction of gravity. Further integrated spatially, a Dy(III) boundary flux is quantified from which the forward reaction rate order and rate constant are studied by parametrically varying the initial Dy(III), H2A2 concentration.

2 Experimental Aspects

This experiment addresses solvent extraction in an immiscible water–oil system. For that purpose, DyCl3 up to 1 M (purity 99.9%, Abcr GmbH) with the pH adjusted to 1 by HCl in an aqueous phase is brought into contact with an organic phase, a low-viscosity paraffin (Sigma-Aldrich 76,235), dissolving the extractant PC88A up to 1 M. Hence, a Dy(III) cation exchange [3] following Eq. (10.1) is triggered at the water–oil interface. The reaction is conducted using a quartz-glass cell (Hellma 404-1) with inner dimensions of 36.5 × 18.5 × 1 mm3. The narrow gap width of the cell allows us to study the diffusion-extraction kinetics process in a quasi-2D Hele-Shaw configuration (Fig. 10.1a), a widely used fluid mechanics model setup for flow visualization and qualification, incorporating advective transport [21, 22]. The cell is then placed in the measurement arm of a 632.8 nm monochromatic laser-based Mach–Zehnder interferometer [5,6,7], with the cell’s depth in the direction of the laser beam. The refractive index of the aqueous phase, which is proportional to the RE(III) concentration change, is monitored by the phase shift of the interference fringes at a frame rate of 5 fps with a resolution of 2160 × 2160 pixels (Jai Go-5100 M-USB). The interferogram, see Fig. 10.1b, is processed [5,6,7] into a space- and time-resolved RE(III) concentration field as shown in Fig. 10.2 in the following chapter.

Fig. 10.1
An experimental setup with a transparent container holding two layers of fluids labeled organic and aqueous with chemicals including D y H A 2 3, H 2 A 2, H plus D u 3 plus. A coordinate plane has vertical stripes observed at 2 millimeters with labels for interface and laser.

Illustration of a experimental configuration. The coordinate system is selected so that gravity is parallel to the z direction with the water–oil interface at z = 0 mm. b interferogram at aqueous phase with water–oil interface as upper boundary

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Fig. 10.2
A distribution graph of z versus t has a vertical colored scale for delta c ranging from negative 2 to 2. Most distribution lies in the middle ranges.

Dy(III) concentration distribution along z axis against different time intervals after the reaction layer with an initial concentration of DyCl3 1 M and PC88A 0.25 M

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To prepare the cell, the aqueous phase is first poured into the bottom until a fixed height of approximately half the cell is reached, where we set the interface coordinate at z = 0 mm, see Fig. 10.1b. A fixed camera view is maintained so that the measurement region at any parametric variation observes the same respective area in the aqueous phase, with the water–oil interface acting as a boundary. Then, after the aqueous phase stabilizes, an equal amount of organic phase is injected simultaneously from both ports at the top of the cell. Using this method, the two fronts of organic phase meet in the middle of the cell and the interface remains relatively flat. Menisci near the cell edge can be observed when the two phases are in contact (see Fig. 10.1a). As a result of RE(III) consumption, the density gradient is parallel to gravity. Therefore, the hydrodynamically stable system, together with a flat interface in the middle of the cell, further simplifies the model into a quasi-1-D diffusion reaction transport [5,6,7].

3 Results

Figure 10.2 shows an experimentally resolved Dy(III) mass boundary layer in the aqueous phase over time, obtained by interferometry, see Fig. 10.1b. The diffusion front scales with a t1/2 law, which indicates a diffusive transport process of Dy(III) concentration stratifications. The quasi-1D reaction–diffusion scenario makes it possible to de-noise the experimental illustrations by averaging the concentration profile along the x axis, i.e., in the direction of the interface (see Fig. 10.1b), for every z-axis value, i.e., in the direction of gravity. The interface position is located at z = 0 and the Dy(III) cation exchange starts at time t = 0 s. The experimental conditions of Fig. 10.2b in this case are a DyCl3 and PC88A with a molarity of, respectively, 0.25 M and 1 M in their respective phases. The concentration profile is shown as the reduction in concentration (Δc) versus the distance (z axis) from the interface at selected times t.

To this end, we revisit Eqs. (10.4) and (10.5) to explicitly correlate the rate law and the results gained from the interferometer. Under the condition that the reverse reaction is negligible,

$$ \begin{gathered} \log_{10} \frac{V}{A} \cdot \frac{{d\overline{c}_{RE\left( a \right)} }}{dt} \hfill \\ = \log_{10} k_{f} + a\log_{10} c_{RE\left( a \right)} + b\log_{10} c_{{H_{2} A_{2} \left( o \right)}} + \log_{10} c_{H + } \hfill \\ \end{gathered} $$
(10.6)

the interferometer results (left) can be correlated with the rate law (right) by applying the log operator on both sides. In this way, varying the individual species, i.e., Dy(III) and PC88A concentration or pH value, the rate order can be individually extracted.

3.1 Dy(III) Concentration Influence

The Dy(III) concentration is varied in an aqueous phase with a salinity of 0.25, 0.5, 0.75 and 1 M while the extractant concentration remains the same, i.e., 0.25 M PC88A in the organic phase. The result, shown in Fig. 10.2, is then integrated along the z axis so that the value \(V/A \cdot\overline{c}_{RE\left( a \right)}\) is obtained at different time intervals during the reaction, see Fig. 10.3a. A linear fit (dashed line in Fig. 10.3a) is then applied to individual parametric variations with its gradient feeding to the left-hand side of Eq. (10.6). Summarizing the \(V/A \cdot{\text{d}}\overline{c}_{RE\left( a \right)} /{\text{d}}t\) value against different Dy(III) concentrations in a double logarithmic diagram (Fig. 10.3b), a quasi-first order dependence of Dy(III) is found for experimental system.

Fig. 10.3
A scatterplot of V over A versus t has decreasing plots for C P C 88 A = 0.5, 0.75, and 1 M and almost constant plots for C P C 88 A = 0.25 M. A scatterplot of log 10 of V over A versus log 10 of C P C 88 A over 2 has plots along an increasing regression line with slope equivalent to 2.3.

The interface-averaged spatial integral of Dy(III) concentration, \(V/A \cdot\overline{c}_{RE\left( a \right)}\), with PC88A 0.25 M, i.e., dimeric concentration 0.125 M, at different initial Dy(III) concentrations (a). The dashed lines are linear fits for different initial Dy(III) concentrations; their slope is summarized in b. b initial forward reaction rate at different initial Dy(III) concentrations in a double logarithmic diagram

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3.2 Influence of Dimeric PC88A Concentration

The extractant PC88A(III) concentration is varied in organic phases of 0.25, 0.5, 0.75 and 1 M while the Dy(III) concentration remains the same, i.e., 0.25 M DyCl3 in the aqueous phase. The resulting dimeric PC88A concentration is half of the respective values. The result, shown in Fig. 10.2, is then integrated along the z axis so that the value \(V/A \cdot\overline{c}_{RE\left( a \right)}\) is obtained at different time intervals during the reaction, see Fig. 10.4a. A linear fit (dashed line in Fig. 10.4a) is then applied to individual parametric variations with its time derivative feeding to the left-hand side of Eq. (10.6). Summarizing the \(V/A \cdot{\text{d}}\overline{c}_{RE\left( a \right)} /{\text{d}}t\) value against different PC88A concentrations in a log10-log10 diagram, Fig. 10.4a, b quasi-second order dependence of Dy(III) is found.

Fig. 10.4
A scatterplot of V over A versus t has decreasing plots for C P C 88 A = 0.5, 0.75, and 1 M and almost constant plots for C P C 88 A = 0.25 M. A scatterplot of log 10 of V over A versus log 10 of C P C 88 A over 2 has plots along an increasing regression line with slope equivalent to 2.3.

The interface-averaged spatial integral of the Dy(III) concentration, \(V/A \cdot\overline{c}_{RE\left( a \right)}\), with Dy(III) 0.25 M at different initial PC88A concentrations of 0.25 M, 0.5 M and 1 M, i.e., dimeric concentrations of 0.125 M 0.125 M, 0.25 M and 0.5 M, respectively (a). The dashed lines are linear fits for different initial PC88A concentrations, with their slopes summarized in b. b the initial forward reaction rate at different initial dimeric PC88A concentrations in a double logarithmic diagram

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Consequently, the rate law is readily computed by combining the intercept and slope of the linear fit in Figs. 10.3b and 10.4b, respectively. To this end, the forward reaction rate takes a fixed pH value, hence \(k_{f}{\prime} = k_{f} c_{H + }^{c} = 7.77 \times 10^{ - 4}\)

$$ R\left[ {\frac{mol}{{m^{2} \cdot s}}} \right] = 7.77 \times 10^{ - 4} \left[ {\frac{m}{{M^{ - 2.4} \cdot s}}} \right]c_{RE\left( a \right)}^{1.1} c_{{H_{2} A_{2} \left( o \right)}}^{2.3} $$

4 Conclusion

We report a novel approach for measuring the forward rate law of a rare earth (RE) solvent extraction system. A cation exchange system of Dy(III)-PC88A-HCl is used for validation. A Mach–Zehnder interferometer monitors the space- and time-resolved Dy(III) concentration boundary layer in a Hele-Shaw cell. The dependence of both the Dy(III) and the dimeric extractant PC88A concentration are studied independently and a rate law is found following a quasi-first order and quasi-second order, respectively. Our method is superior to conventional methods as it requires 2–3 orders of magnitude less material, and Dy(III) can be traced at different time intervals after the reaction in one experiment without the probe disturbance that is an issue with conventional methods. In addition, the approach can genuinely be applied to all kinetic systems, provided the medium is transparent or translucent. The potentially interesting aspects to be followed up include the extraction kinetic law for all trivalent RE ions, not limited to the model system of the Dy(III)-PC88A-HCl solvent extraction system we validated here. While an active follow-up investigation is underway extending the parametric range and systems to refine the accuracy and expand the rate law to encompass a broader spectrum of reaction constant determination, this work focuses on establishing and validating the general method.