Experiments and Simulations for Reactive Crystallization of Li₃PO₄ from Low Concentration Li-Rich Brine

Abstract

Li₃PO₄ is an ideal precursor for synthesizing high-performance LiFePO₄, as it simultaneously provides lithium and phosphorus sources. Extremely low solubility of Li₃PO₄ enables efficient lithium recovery from low-concentration Li-rich brine by reactive crystallization. A focused beam reflectance measurement (FBRM) system was employed to monitor the key optimization parameters for Li₃PO₄ crystallization, supersolubility, and metastable zone widths (MSZWs). The optimized process parameters were determined by systematically investigating the effects of operating conditions. Additionally, prediction of supersolubility and MSZWs was accomplished with theoretical models. Results demonstrate that both supersolubility and MSZWs exhibit a pronounced negative correlation with temperature. Supersolubility decreased sharply when LiCl concentration exceeded 5 mol·L⁻¹ or Na₃PO₄ concentration surpassed 0.8 mol·L⁻¹. Conversely, it increased exponentially with Na₃PO₄ feeding rate. The effect of impurity (NaCl/KCl) was non-monotonic, initially increasing and then decreasing supersolubility and MSZWs. Among these, Na₂B₄O₇ most significantly enhanced both parameters, followed by Na₂SO₄. The supersolubility data were well-fitted by an empirical equation (R² > 0.99). For MSZWs prediction, the self-consistent Nývlt-like model (R² > 0.9883) and the modified Sangwal’s model (R² > 0.994) achieved superior performance. Collectively, these findings establish a theoretical basis for optimizing lithium recovery via Li₃PO₄ crystallization, facilitating more efficient and sustainable production of high-purity lithium products.

1. Introduction

Driven by the global energy transition and carbon neutrality goals, the demand for lithium products in the battery industry has experienced explosive growth in recent years [1,2,3]. Lithium iron phosphate (LiFePO₄) is a preferred cathode material for electric vehicles [4,5], energy storage systems [6,7], and portable electronic devices [1] owing to its high energy density, long life cycle, and excellent safety performance. Furthermore, cathode materials constitute over 45% of the total battery cost [8]. Salt lake brine is a crucial lithium resource, accounting for approximately 66% of global lithium reserves [9]. In China, it represents a major lithium resource, accounting for 12.0% of the global total [10]. Dong-Taijinaier salt lake represents one of China’s largest and highest-quality brine-type lithium deposits [10]. Typically, the cost of producing lithium from salt lake brine is 30–50% lower than that from hard-rock sources [11]. Consequently, the extraction of lithium from salt lake brine has become a major trend in the industry.

Lithium phosphate (Li₃PO₄), a high-value-added lithium compound, has garnered significant interest for its applications in lithium-ion batteries [12,13,14,15,16], catalysts [17,18,19], phosphors [20], glass [21] and other advanced materials. Li₃PO₄ serves as an ideal precursor for synthesizing high-performance LiFePO₄ cathode materials because it simultaneously provides both lithium and phosphorus sources [12,13,22]. With its low solubility (Ksp = 2.37 × 10⁻¹¹ at 298.15 K), Li₃PO₄ can precipitate efficiently from solutions even at low lithium concentration, making it a promising candidate for enhancing lithium recovery rates [23,24,25,26]. Liu et al. [24] proposed a seed-induced precipitation method for recovering lithium from brine at low temperatures, utilizing facet-engineered Li₃PO₄ crystals as seeds and sodium phosphate dodecahydrate (Na₃PO₄·12H₂O) as the precipitant. Zhao et al. [26] investigated the recovery of Li₃PO₄ from a lithium-containing solution (2 g·L⁻¹) using solid sodium phosphate (Na₃PO₄) as the precipitant. Wang et al. [27] developed a tandem electro-leaching and solvent extraction method to selectively recover metals from spent LiNi_xCo_yMn_(1−x−y)O₂. In this process, over 98.0% of Ni, Co, and Mn were selectively separated from Li via solvent extraction. Subsequently, using trisodium phosphate as a precipitant, battery-grade Li₃PO₄ with a purity of up to 99.5% was obtained. Xiao et al. [28] investigated the thermodynamics of Li₃PO₄ recovery, performing a thermodynamic analysis of the Li⁺, Fe³⁺/PO₄³⁻-H₂O system at 298.15 K. Emmanuel et al. [29] studied a system in which lithium chloride and sodium phosphate are simultaneously injected into a microchannel, allowing the precipitation reaction to proceed under concentration gradients. The growth kinetics of individual precipitate particles were studied by monitoring particle growth both along and transverse to the flow direction. Collectively, these studies provide a foundation for enhancing lithium resource utilization efficiency and product quality to meet diverse market demands. Li₃PO₄ is typically produced by precipitation from lithium-containing solutions. Although the precipitation of Li₃PO₄ is a well-established method, there are no engineering and scientific approaches to the mechanism of Li₃PO₄ precipitation from Li-rich brine. This mechanism gap limits the efficiency of the Li₃PO₄ precipitation process and results in poor product quality. Therefore, a fundamental study on the reaction crystallization (precipitation) process of Li₃PO₄ is essential.

The supersolubility (c_sp) and metastable zone widths (MSZWs) are key parameters for optimizing the reactive crystallization of Li₃PO₄. Industrial crystallization processes are often operated within the metastable zone to obtain products with desirable characteristics, such as a uniform particle size distribution, high purity, and high yield. The supersolubility and MSZWs are influenced by several operational parameters, including temperature, impurities, reactant concentration, feeding rate, and stirring rate. Therefore, quantifying these parameters is essential for the rational design and operation of crystallization equipment. Wang et al. [30] investigated the supersolubility and solubility of Li₃PO₄ in Na₂CO₃ solution, providing fundamental physicochemical data and theoretical insights for the efficient separation and extraction of lithium from precipitation mother liquor. Several techniques, including conductivity [31], turbidity [32,33], laser method [34], and focused beam reflectance measurement (FBRM) [34,35,36,37], are employed to determine supersolubility and MSZWs. Among these, FBRM is a real-time analytical technique that rapidly provides crystal information based on chord length distribution (CLD), typically within seconds (e.g., a 2 s interval). The principles [38,39] and applications [34,35] of FBRM technology are detailed in the relevant literature. Furthermore, predictive models for supersolubility and MSZWs are important tools for generating necessary estimates. The Qinghai–Tibet Plateau in China contains hundreds of salt lakes rich in lithium resources. Currently, the production of Li₂CO₃ from the Dong-Taijinaier salt lake requires concentrating the lithium-containing brine to a Li⁺ concentration above 20 g·L⁻¹ prior to precipitation. However, this process is not only energy-intensive but also results in a relatively low lithium recovery rate (approximately 80%), owing to the higher solubility of Li₂CO₃ (13.3 g·L⁻¹ at 293.15 K) compared to that of Li₃PO₄. Therefore, developing a method for the direct recovery of lithium from low concentration Li-rich brine ([Li⁺] = 8.9 g·L⁻¹) in the Dong-Taijinaier salt lake via phosphate precipitation is of great significance. However, the presence of co-existing impurities (e.g., NaCl, KCl, Na₂SO_4, and Na₂B₄O₇) in Li-rich brine can significantly influence the crystallization behavior of Li₃PO₄, posing a major challenge for its efficient precipitation. Our previous work investigated the dissolution thermodynamics of Li₃PO₄ in mixed salt solutions (LiCl-NaCl-KCl-Na₂SO₄-Na₂B₄O₇) [40]. Building upon the foundation, this study employs FBRM to systematically investigate the supersolubility and MSZWs of Li₃PO₄ in both a synthetic mixed-salt system and an authentic Li-rich brine. The purpose of this study is to determine the optimal crystallization process parameters for the efficient recovery of Li₃PO₄ from Li-rich brine and to develop an accurate predictive model for the MSZWs of its reactive crystallization.

2. Theory

2.1. Metastable Zone Widths

The metastable zone (Figure 1), bounded by the solubility and supersolubility curves, represents a state of supersaturation where spontaneous nucleation does not occur within a sufficiently short time. Operating within the metastable zone enables precise control over crystal quality and particle size. Exceeding it leads to uncontrolled crystallization, characterized by burst nucleation and undesirable product qualities. The supersolubility is influenced by many factors, including reactant concentration, stirring rate, feeding rate, presence of impurities, and addition of crystal seeds. Consequently, the MSZWs are a key characteristic of any crystallization system. It is quantitatively defined as the difference between the supersolubility (c_sp) and solubility (c_s), expressed as Δc = c_sp − c_s.

2.2. Calculation of Supersolubility

Owing to the lack of activity coefficients for Li⁺ and PO₄³⁻ in the multi-component system, the supersaturation of Li₃PO₄ cannot be directly calculated using the rigorous definition in Equation (1). Consequently, a simplified approach, justified by the low solubility of Li₃PO₄, is employed in Equation (2) [41] for practical applications. The supersolubility can then be determined using Equations (3) and (4).

$$S={\left({\displaystyle \frac{{(a_{{\mathrm{Li}}^{+}})}^3(a_{{{\mathrm{PO}}_4}^{3-}})}{K_{\mathrm{sp}}}}\right)}^{1/4}$$

(1)

$$S={\left({\displaystyle \frac{{(c_{{\mathrm{Li}}^{+}})}^3(c_{{{\mathrm{PO}}_4}^{3-}})}{K_{\mathrm{sp}}}}\right)}^{1/4}$$

(2)

$$S_{\mathrm{sp}}={\displaystyle \frac{c_{\mathrm{sp}}}{c_{\mathrm{s}}}}$$

(3)

$$c_{\mathrm{sp}}=S_{\mathrm{sp}}\times c_{\mathrm{s}}$$

(4)

where K_sp denotes the solubility product constant of Li₃PO₄; S, c_sp, c_s, and Ssp represent the relative supersaturation, supersolubility (mol·L⁻¹), solubility (mol·L⁻¹), and the relative supersaturation at c_sp, respectively.

2.3. MSZWs Predictions Models

The MSZWs are influenced by process parameters and vary significantly across different chemical systems. Nývlt [42] proposed the first semiempirical model for MSZWs prediction. This model later evolved into Sangwal’s theory [43], both of which incorporate the nucleation order, system dimensions, and saturation temperature. In this work, the self-consistent Nývlt-like model, Sangwal’s theory model, and the modified Sangwal’s theory model were employed to predict the MSZWs for the reactive crystallization of Li₃PO₄.

Expressed in terms of concentration instead of temperature, the self-consistent Nývlt-like model, Sangwal’s theory model, and modified Sangwal’s theory model are given by Equations (5)–(7), respectively.

$$\ln {\displaystyle \frac{\mathsf{\Delta }c}{c_{\mathrm{s}}}}={\displaystyle \frac{1-m}{m}}\ln {\displaystyle \frac{\Delta H}{R{c}_{\mathrm{sp}}}}+{\displaystyle \frac{1}{m}}\ln {\displaystyle \frac{f}{K{c}_s}}+{\displaystyle \frac{1}{m}}\ln Uc$$

(5)

$${\left({\displaystyle \frac{c_{\mathrm{s}}}{\Delta c}}\right)}^2=F-F_1\ln Uc$$

(6)

$${\displaystyle \frac{{\left(\frac{c_{\mathrm{s}}}{\mathsf{\Delta }c}\right)}^2}{\Delta c-c_{\mathrm{s}}}}=M+N\ln \left({\displaystyle \frac{Uc}{c_{\mathrm{s}}(\mathsf{\Delta }c-c_{\mathrm{s}})}}\right)$$

(7)

where U is the feeding rate of Na₃PO₄ solution (mL·min⁻¹); c is the concentration of Na₃PO₄ solution (mol·L⁻¹); f is the proportionality constant; K is the nucleation rate constant, m represents the apparent nucleation order; F₁ represents the slope, and F represents the intercept. N represents the slope and M the intercept.

3. Experiment

3.1. Materials and Methods

All chemical reagents used in the experiments were of analytical grade, and their details are provided in

Table 1. Chemical reagents employed.

Chemical Name	Molar Mass/g·mol−1	Source	Initial Mass Fraction	Purification Method	Final Mass Fraction	Analysis Method
Na3PO4·12H2O	380.12	Shanghai Macklin Biochemical Co., Ltd., Shanghai, China.	≥98%	None	≥98%	ultraviolet-visible spectrophotometer
NaCl	58.44	Sinopharm Chemical Reagent Co., Ltd., Shanghai, China.	≥99.5%	Dried at 393 K	≥99.6%	Argentometric titration
KCl	74.55	Sinopharm Chemical Reagent Co., Ltd., Shanghai, China.	≥99.5%	Dried at 393 K	≥99.5%	Argentometric titration
LiCl	42.39	Tianjin Yongda Reagent Development Center, Tianjin, China.	≥99%	Dried at 393 K	≥99%	Argentometric titration
Na2SO4	142.04	Tianjin Guangfu Technology Development Co., Ltd., Tianjin, China.	≥99%	Dried at 393 K	≥99.1%	Barium chloride precipitation method
Na2B4O7·10H2O	381.37	Tianjin Hedong District Hongyan Reagent Factory, Tianjin, China.	≥99.5%	Dried at 393 K	≥99.6%	mannitol titration method
H2O	18.02	Chongqing Moore Co., Ltd., Chongqing, China.				Conductometry

. Ultrapure water (resistivity of 18.2 MΩ·cm at 298.15 K) was degassed by boiling prior to the preparation of all electrolyte solutions. The density and pH of the Li-rich brine were determined as described in previous studies [40]. In this study, “Li-rich brine” is defined as the unconcentrated, lithium-containing solution from the Dong-Taijinaier salt lake, used prior to Li₂CO₃ precipitation. Its detailed composition is listed in

Table 2. Major ion composition of the Li-rich brine [40].

Li+	Na+	K+	B2O3	Cl−	SO42−
mg·L−1
8912	9108	713	174	59,730	59,730

.

3.2. Experiment Setup

The supersolubility was determined using FBRM (G400, Mettler-Toledo, Columbus, OH, USA), as illustrated in Figure 1. To ensure data reliability, particle vision and measurement (PVM) (V19, Mettler-Toledo, Columbus, OH, USA) was employed concurrently for monitoring. The relative backscatter index (RBI) from PVM serves as a process trend indicator sensitive to particulate changes and was used to identify nucleation points [44]. The FBRM probe, with a measurement range of 0.5–2000 µm, was operated at a scan speed of 2 m/s with a 2 s measurement duration. Time averaging and chord length weighting were disabled to capture high-fidelity data during rapid nucleation events. PVM sampling was set to 1 frame per 2 s. All experiments were conducted in a 100 mL glass crystallizer within a fully automated synthesis reactor (Easymax 102, Mettler-Toledo, Columbus, OH, USA). The system was equipped with an internal overhead stirrer, and the temperature was controlled to within ±0.1 K throughout the experiments.

3.3. Supersolubility Measurements

To measure supersolubility, 50 mL of a 1.3 mol·L⁻¹ LiCl solution or Li-rich brine was added to the crystallizer. The overhead stirrer, temperature probe, FBRM, and PVM probes were then immersed in the solution adjacent to the stir rod. The system (Easymax102, iControl, version 6.0) was programmed to maintain a constant temperature and a stirring speed of 400 rpm. The experiment was initiated once the solution temperature had stabilized. An isothermal Na₃PO₄ solution (0.4 mol·L⁻¹) was then titrated into the LiCl solution or Li-rich brine using a liquid dosing system at a constant rate of 0.5 mL·min⁻¹. The initial baseline was characterized by a stable total chord count. Upon the onset of burst nucleation, indicated by a sharp increase in the total chord count (Figure 2), the dosing pump was immediately stopped. The time interval from the start of dosing to the nucleation point was recorded as the pumping time. The supersolubility was then calculated from the total amount of Na₃PO₄ added, which was the product of the dosing rate and the recorded pumping time. Each measurement was performed in at least triplicate to ensure reproducibility.

4. Results and Discussion

4.1. Solubility of Li₃PO₄

The direct determination of Li₃PO₄ solubility in pure water is challenging because the formation of a fine colloid impedes effective filtration [45]. Wu et al. [46] pointed to the ongoing hydrolysis of PO₄³⁻ and the dissolution-conversion of Li₃PO₄ with increasing time, which prevents the attainment of a stable equilibrium for accurate solubility measurement. The solubility value of Li₃PO₄ in pure water, widely used in important manuals at present, is 0.394 g/kg at 291.15 K. In our previous study [40], the solubilities of Li₃PO₄ in both pure water and Li-rich brine were measured via the isothermal dissolution equilibrium method [47,48] across a temperature range of 291.15 to 348.15 K; these data are summarized in

Table 3. The solubility of Li₃PO₄ at different temperatures.

Temperature (K)	Li3PO4-H2O a		Li3PO4-Li-Rich Brine (wLiCl = 5.21%) a
	S/(10−2 g/100 g H2O)	c1/(10−3 mol·L−1)	c2/(10−3 mol·L−1)
291.15	3.90	3.36	8.79
293.15	3.84	3.31	8.77
298.15	3.78	3.25	8.72
318.15	3.52	3.01	8.60
328.15	3.44	2.92	8.55
338.15	3.36	2.85	8.48
348.15	3.21	2.70	8.43

^a S, c₁ denote the solubility of Li₃PO₄ in pure water, while c₂ refers to its solubility in the Li-rich brine, respectively. The LiCl content in the Li-rich brine is expressed as the mass fraction, w_LiCl.

. The solubility values obtained in our previous work [40] were slightly higher than those reported by Wu et al. [46], a discrepancy likely attributable to differences in the dissolution time. The solubility data were fitted using the van’t Hoff equation, as shown in Figure 3. The high coefficient of determination (R² > 0.996) for both systems indicates an excellent fit to the model.

$$\ln c_s={\displaystyle \frac{\Delta H}{RT}}+{\displaystyle \frac{\Delta S}{R}}$$

(8)

where c_s is the equilibrium concentration of solute (molality mol·L⁻¹), and T is the temperature (K). ΔH is the dissolution enthalpy (kJ·mol⁻¹), ΔS is the dissolution entropy (J·mol⁻¹·K⁻¹), they are obtained from the slope and intercept of the fitted line, respectively. R is the gas constant 8.314 J·K⁻¹·mol⁻¹.

4.2. Supersolubility and MSZWs

4.2.1. Impact of Temperature

The supersolubility of Li₃PO₄ in LiCl solution and Li-rich brine was determined over the temperature range of 291.15 K to 363.15 K, as presented in

Table 4. Supersolubility c_sp of Li₃PO₄ in LiCl solution and Li-rich brine at temperatures T = (291.15–363.15) K, pressure p = 0.1 MPa. ^a

Temperature (K)	Supersolubility csp of Li3PO4 (10−3 mol·L−1)
	in LiCl Solution	in Li-Rich Brine
291.15	9.218	23.563
293.15	8.182	22.209
298.15	7.184	19.839
308.15	5.846	17.292
318.15	5.158	15.909
328.15	4.267	14.184
338.15	3.782	12.632
348.15	3.181	10.396
358.15	2.901	9.219

^a The standard uncertainties for u (T) = 0.1 K and u (p) = 5.0 kPa. The maximum absolute uncertainties of c_sp are ±2.0 × 10⁻⁶ mol·L⁻¹.

and Figure 4a. The supersolubility decreased markedly with increasing temperature, exhibiting a strong inverse dependence. The supersolubility results for Li₃PO₄ in LiCl solution are in satisfactory agreement with the literature values [30]. Specifically, the supersolubility values of Li₃PO₄ in Li-rich brine (19.837 × 10⁻³ mol·L⁻¹ at 298.15 K; 12.632 × 10⁻³ mol·L⁻¹ at 338.15 K) agrees well with the values reported in a 1.3Li-0.5K-1.3Na-0.1S-0.01B mixed salt solution (20.115 × 10⁻³ mol·L⁻¹ and 12.580 × 10⁻³ mol·L⁻¹ at the corresponding temperatures). This agreement suggests that the supersolubility in Li-rich brine is primarily governed by the impurities KCl, NaCl, Na₂SO_4, and Na₂B₄O₇, whereas the effects of other potential impurities are negligible. The metastable zone for Li₃PO₄ in LiCl solution and Li-rich brine is indicated by the yellow and green areas in Figure 4a, respectively. As shown in Figure 4b, the MSZWs gradually decrease with increasing temperature. Elevated temperatures reduce solution viscosity, which accelerates molecular motion, increases molecular collision frequency, and enhances mass transport. These factors collectively increase the likelihood of nucleation. Furthermore, higher temperatures lower the surface tension, thereby facilitating nucleus formation. Consequently, the MSZWs narrow with increasing temperature [49,50,51].

To maximize crystallization yield, a higher operating temperature is generally preferable. However, the crystallization process must be controlled within the metastable zone. If the MSZWs become too narrow for precise process control, various factors must be considered when determining the optimal operating conditions.

4.2.2. Impact of Operating Conditions

Stirring Speed: the supersolubility was determined across a stirring speed range of 100 to 900 rpm under the following conditions: a Na₃PO₄ solution feeding rate of 0.5 mL·min⁻¹, with LiCl and Na₃PO₄ concentrations of 1.3 and 0.4 mol·L⁻¹, respectively, as shown in Figure 5a. It can be seen that an increase in stirring speed leads to a slight decrease in supersolubility. Related studies suggest that stirring exerts two opposing effects: it accelerates mass transfer and facilitates nucleation, yet it may also disrupt crystal embryos and impede nucleus formation. The net effect on supersolubility is thus a synergy of these competing mechanisms. This dual mechanism may account for the slight increase in supersolubility observed at 400 rpm.

LiCl Concentration: the reactive crystallization of LiCl and Na₃PO₄ is an important industrial process. Therefore, determining an optimal LiCl concentration range is critical. The dependence of supersolubility on LiCl concentration (0–5.5 mol·L⁻¹) as investigated at a constant stirring speed of 400 rpm, with the Na₃PO₄ solution fed at 0.5 mL·min⁻¹ and a concentration of 0.4 mol·L⁻¹, as shown in Figure 5b. A steep decline in the supersolubility of Li₃PO₄ is observed when the LiCl concentration exceeds 5 mol·L⁻¹. This decline is attributed to the significantly accelerated reaction and nucleation rates at high LiCl concentrations, which markedly reduce the supersolubility. Supersaturation levels approaching or exceeding this limit promote burst nucleation, leading to the formation of fine crystals that are difficult to filter and wash, ultimately yielding a product with suboptimal purity and morphology. Moreover, concentrating dilute raw materials to such high levels is both energy- and time-intensive. Therefore, LiCl concentrations exceeding 5 mol·L⁻¹ should be avoided.

Na₃PO₄ Concentration: the dependence of supersolubility on Na₃PO₄ concentration (0.1–1 mol·L⁻¹) was determined under the following conditions: a Na₃PO₄ feeding rate of 0.5 mL·min⁻¹, a stirring speed of 400 rpm, and a LiCl concentration of 1.3 mol·L⁻¹, as shown in Figure 5c. It can be seen that the supersolubility initially increased with Na₃PO₄ concentrations, reaching a maximum before gradually decreasing at concentrations exceeding 0.8 mol·L⁻¹. This non-monotonic trend is governed by concentration-dependent nucleation kinetics. At lower Na₃PO₄ concentrations, Li₃PO₄ crystal nuclei may dissolve before growing to a detectable size for the FBRM probe. In contrast, higher concentrations significantly accelerate the reaction and nucleation rates, promoting rapid nucleus growth and detection, thereby reducing the supersolubility. Since a wider metastable zone width is generally desirable for process control, the Na₃PO₄ concentration should be maintained at or below 0.8 mol·L.

Feeding Rate of Na₃PO₄: the dependence of supersolubility on the feeding rate of Na₃PO₄ (0.03–3.2 mL·min⁻¹) was determined at a stirring speed of 400 rpm, with LiCl and Na₃PO₄ concentrations of 1.3 and 0.4 mol·L, respectively, as shown in Figure 5d. It can be seen that the supersolubility of Li₃PO₄ increases exponentially with the Na₃PO₄ feeding rate. This trend is attributed to the interplay between the continuous addition of reactant and the finite induction period required for nucleation. A higher feeding rate introduces more Na₃PO₄ into the LiCl solution before nucleation commences, thereby generating a greater degree of supersaturation and resulting in higher supersolubility. This behavior is analogous to that observed in cooling crystallization [52,53,54], where supersolubility increases with the cooling rate. In essence, both the reactant feeding rate and the cooling rate govern the rate of supersaturation generation. Thus, it can be concluded that for a given substance, a higher supersaturation generation rate leads to a higher supersolubility. Furthermore, the experimental data were well-fitted by an exponential equation (R² > 0.99), confirming the model’s effectiveness in predicting the supersolubility of Li₃PO₄.

4.2.3. Impact of Impurities

Single Salts System: the primary concomitant salts in the Li-rich brine of the Dong-Taijinaier salt lake are KCl, NaCl, Na₂SO_4, and Na₂B₄O₇; thus, these were selected as representative impurities for this study. The supersolubility c_sp of Li₃PO₄ in single salt solutions KCl (0–2 mol·L⁻¹), NaCl (0–2 mol·L⁻¹), Na₂SO₄ (0–1 mol·L⁻¹), and Na₂B₄O₇ (0–0.06 mol·L⁻¹) is determined at 298.15 K and 338.15 K (Figure 6a,b). A general trend of decreasing supersolubility with increasing temperature was observed across all systems.

In NaCl and KCl solutions, the supersolubility of Li₃PO₄ initially increased, reached a maximum, and then gradually decreased with increasing salt concentration. This non-monotonic behavior can be attributed to evolving ion-pair structures. As the concentration of NaCl or KCl increases, the solution microstructure transitions from containing free hydrated ions and solvent-separated ion pairs (SIPs) to an increasing proportion of contact ion pairs (CIPs) and more complex ion associates. The initial increase in supersolubility is likely promoted by free ions and SIPs, while the subsequent decline is dominated by the effects of CIPs. In contrast, the supersolubility in Na₂SO₄ and Na₂B₄O₇ solutions increased monotonically with concentration, with Na₂B₄O₇ exerting a more pronounced effect. This indicates a dominant “salting-in” effect. Furthermore, NaCl demonstrated a stronger salting-in influence on Li₃PO₄ supersolubility than KCl at equivalent concentrations. These observations suggest that the impurities alter the solubility product constant via a thermodynamic mechanism, consistent with reported trends for Li₂CO₃ in salt solutions [41,55], implying a potentially shared thermodynamic origin.

Li-Na-K-B-Cl-S Mixed-Salt System: the supersolubility c_sp of Li₃PO₄ was determined in four mixed-salt solutions (compositions detailed in

Table 5. Four solutions with different mixed-salt compositions.

Mixed-Salt Solutions	Salt Solution Concentration (mol·L−1)
	LiCl	KCl	NaCl	Na2SO4	Na2B4O7
1.3Li-0.5K-Na	1.3	0.5	0–2	-	-
1.3Li-0.5K-1.3Na-S	1.3	0.5	1.3	0–1	-
1.3Li-0.5K-1.3Na-B	1.3	0.5	1.3	-	0–0.06
1.3Li-0.5K-1.3Na-0.1S-B	1.3	0.5	1.3	0.1	0–0.06

) at 298.15 K and 338.15 K. The results are summarized in Figure 6c,d. A general decrease in supersolubility was observed with increasing temperature. In the 1.3Li-0.5K-Na system, the supersolubility exhibited a non-monotonic dependence on NaCl concentration, initially increasing to a maximum before gradually decreasing. This suggests that NaCl and KCl exert a salting-in effect on Li₃PO₄ at low concentrations and a salting-out effect at high concentrations. The supersolubility of Li₃PO₄ increased monotonically with the concentration of Na₂SO₄ or Na₂B₄O₇ in the 1.3Li-0.5K-1.3Na-S, 1.3Li-0.5K-1.3Na-B, and 1.3Li-0.5K-1.3Na-0.1S-B systems, indicating a dominant salting-in effect. Notably, Na₂B₄O₇ exerted a more pronounced effect on enhancing supersolubility than Na₂SO₄. These results demonstrate that the supersolubility of Li₃PO₄ during crystallization can be effectively modulated by controlling the concentrations of specific impurities (NaCl, KCl, Na₂SO_4, and Na₂B₄O₇) in the Li-rich brine.

MSZWs of Li₃PO₄ in Single and Li-Na-K-B-Cl-S Mixed-Salt Solutions: the MSZWs were calculated from previously reported solubility data for Li₃PO₄ in single-salt (NaCl, KCl, Na₂SO_4, and Na₂B₄O₇) and Li-Na-K-B-Cl-S mixed-salt solutions [40]. As shown in Figure 7a,b, the MSZWs of Li₃PO₄ decreased with increasing temperature across all systems. Notably, the variation in the MSZWs with solution composition closely paralleled the corresponding trends in supersolubility, indicating that the MSZWs are primarily governed by the supersolubility.

4.3. Supersolubility and MSZWs Predictions

4.3.1. Supersolubility Predictions

Based on experimental results identifying temperature and Na₃PO₄ feeding rate as the primary factors influencing supersolubility, the supersolubility of Li₃PO₄ in Li-rich brine was modeled as a function of these two variables. The experimental conditions were as follows: Li-rich brine with a LiCl concentration of 1.3 mol·L⁻¹, a temperature range of 291.15 K to 348.15 K, and Na₃PO₄ feeding rates of 0.1, 0.3, 0.5, 1.0, and 3.0 mL·min⁻¹. The supersolubility data were fitted to an empirical equation, as shown in Figure 8a,b. The high coefficient of determination (R² > 0.99) confirms that the equation effectively predicts supersolubility within the experimental range. Furthermore, the model demonstrates better accuracy at lower temperatures compared to higher temperatures.

4.3.2. MSZWs Predictions

The linear solubility of Li₃PO₄ in Li-rich brine indicates a constant temperature coefficient (dc_s/dT)_T, independent of the saturation temperature. This relationship justifies the application of both Nývlt’s and Sangwal’s theories to correlate the MSZWs data and estimate the nucleation kinetics [56]. Figure 9 presents the corresponding fitting results obtained from the self-consistent Nývlt-like model, Sangwal’s model, and the modified Sangwal’s model.

Self-consistent Nývlt-like Model: the MSZWs data were fitted using the self-consistent Nývlt-like model (Equation (5)), which establishes a linear relationship between ln(Δc/c_s) and lnUc. The fitting results at different temperatures are presented in Figure 9a and

Table 6. Estimated nucleation kinetics parameters and coefficient of determination from the self-consistent Nývlt-like model.

T (K)	Fitting Equation	1/m	103 f/K (K−1)	φ	R2
291.15	y = −4.3534 + 0.4652x	0.4652	8.5461	−4.3534	0.9954
293.15	y = −4.5794 + 0.4427x	0.4427	8.4912	−4.5794	0.9948
298.15	y = −4.5941 + 0.4806x	0.4806	7.4215	−4.5941	0.9951
318.15	y = −4.7309 + 0.5667x	0.5646	4.7139	−4.7309	0.9907
348.15	y = −5.0622 + 0.5646x	0.5667	4.0274	−5.0622	0.9883

. The slope of the linear regression, corresponding to 1/m, increased with temperature, indicating a corresponding decrease in the apparent nucleation order m. Concurrently, the intercept φ decreased from −4.3534 to −5.0622. The derived parameter f/K decreased from 0.008546 to 0.004027. The coefficients of determination (R²) for all fits were high, ranging from 0.9883 to 0.9954, confirming the model’s reliability. The overall strong agreement between the model and experimental data demonstrates that the Nývlt-like model is a reliable tool for predicting the MSZWs of Li₃PO₄ in Li-rich brine.

Sangwal’s Theory Model: according to Sangwal’s theory (Equation (6)), a linear correlation between (c_s/∆c)² and lnUc was evaluated, as shown in Figure 9b,c. However, the coefficients of determination (R²) for these fits were relatively low, ranging from 0.6844 to 0.7252 (

Table 7. Estimated nucleation kinetics parameters and coefficient of determination from Sangwal’s theory.

T (K)	F	F1	R2
291.15	−0.6098	0.2426	0.6844
293.15	−2.1734	0.7397	0.6942
298.15	−4.1496	1.3429	0.7025
318.15	−21.4589	6.3214	0.6990
348.15	−56.2418	15.0425	0.7252

), indicating a poor fit to the experimental data. Consequently, Sangwal’s theory is not suitable for predicting the MSZWs of Li₃PO₄ in Li-rich brine. This discrepancy is likely because Sangwal’s theory was developed for systems with fixed composition and is less applicable to reactive crystallization, where the solution composition changes over time [56]. This evolving composition affects the fundamental parameter F in the model, thereby compromising its predictive accuracy.

Modified Sangwal’s Theory Model: the modified Sangwal’s theory (Equation (7)) was applied, which posits a linear relationship between (c_s/∆c)²/(∆c − c_s) and lnUc/c_s(∆c − c_s), as illustrated in Figure 9d. The corresponding coefficients of determination (R²) were exceptionally high, ranging from 0.994 to 0.999 (

Table 8. Estimated nucleation kinetics parameters and coefficient determination from the modified Sangwal’s theory.

T (K)	M	N	R2
293.15	−7562.11	887.71	0.9940
298.15	−5102.59	580.60	0.9990

), confirming the model’s excellent fit and its suitability for predicting the MSZWs of Li₃PO₄ in Li-rich brine within the studied temperature range (293.15 K to 348.15 K). However, the modified theory has a significant limitation: the term (∆c − c_s) can attain negative values under certain conditions. When this occurs, the mathematical formulation of the model breaks down, preventing a physically meaningful prediction. This situation typically arises when the metastable zone width is too narrow for the model’s assumptions to hold, thereby limiting its practical application.

5. Conclusions

This study investigates the crystallization behavior of Li₃PO₄ from Li-rich brine by measuring its supersolubility and MSZWs using FBRM. The effects of key operating parameters, including temperature, stirring speed, LiCl concentration, Na₃PO₄ concentration, and feeding rate, and impurities were systematically examined. The results demonstrate that supersolubility and MSZWs exhibit a strong inverse dependence on temperature. An increase in stirring speed caused a slight decrease in supersolubility. A steep decline in supersolubility was observed at LiCl concentrations exceeding 5 mol·L⁻¹ and at Na₃PO₄ concentrations above 0.8 mol·L⁻¹. In contrast, supersolubility increased exponentially with the Na₃PO₄ feeding rate. The influence of NaCl and KCl impurities was non-monotonic, initially increasing and then decreasing supersolubility with concentration. Among the impurities, Na₂B₄O₇ had the most significant effect on enhancing supersolubility, followed by Na₂SO₄. Consequently, to optimize the reaction crystallization process, we recommend maintaining LiCl and Na₃PO₄ concentrations below 5 mol·L⁻¹ and 0.8 mol·L⁻¹, respectively, with a stirring speed of 400 rpm. This confines the crystallization to a controllable metastable zone, effectively suppressing burst nucleation and thereby facilitating precise control over product quality. Moreover, concentrating dilute raw materials to such high levels is both energy- and time-intensive. Furthermore, supersolubility can be tuned by controlling impurity levels. An empirical equation (R² > 0.99) effectively predicted supersolubility. For MSZWs prediction, the self-consistent Nývlt-like model (R² > 0.9883) and the modified Sangwal’s model (R² > 0.994) demonstrated superior accuracy. This study provides critical insights for the industrial implementation of an efficient crystallization process to recover Li₃PO₄ from Li-rich brine.