Antisolvent Effects of C₁–C₄ Primary Alcohols on Solid-Liquid Equilibria of Potassium Dihydrogen Phosphate in Aqueous Solutions

Abstract

The focus of this study was to examine antisolvent effects, which hold significance in particulate processes, such as crystallization and precipitation. In the first section, an experimental investigation revealed that C₁–C₄ primary alcohols significantly reduced the solubility of potassium dihydrogen phosphate (KDP) in water. The solid–liquid equilibria of KDP solutions were determined using an innovative polythermal method, demonstrating time and labor efficiency compared to the traditional isothermal method while maintaining solubility determination quality. This achievement established an efficient tool for high-throughput solvent screening, a crucial aspect of particulate process development. In addition to the experimental approach, in the second part, the influence of these alcohols on KDP solubility was analyzed using the eNRTL thermodynamics model. The model’s estimated parameters confirmed that the addition of these alcohols induced strong non-ideal behavior in the solutions, altered interactions between solute species and solvent components, and reduced KDP solubility. Under the effects of these alcohols, KDP solubility generally increased with the length of the alkyl chain in the added alcohols, although methanol deviated from this observation. Furthermore, the present work also discussed the limitation of the well-known Bromley’s equation, particularly when applied for KDP in alcohol–water mixed solvents. Consequently, binary and ternary systems consisting of KDP, water, and C₁–C₄ primary alcohols were successfully modeled using eNRTL. Furthermore, it was determined that the obtained model was insufficient for quaternary systems with a higher alcohol content, particularly when high-order interactions were neglected as in the cases of binary and ternary systems. In short, these investigated alcohols have potential for future applications in the design of particulate processes, with a particular emphasis on antisolvent crystallization.

1. Introduction

Many studies have reported that the presence of special co-solvents (so-called antisolvents) can reduce the solubility of substances [1,2]. Some cases of antisolvents are introduced below. For instance, acetonitrile can decrease the solubility of anhydrous sodium acetate in methanol by 75% when adding this antisolvent to the main solvent with a ratio of 50/50 mass percentage [3]. Deionized water also acts as a potent antisolvent for lutein fatty acid ester solutions in which tetrahydrofuran serves as the primary solvent [4]. Generally, the comprehension and utilization of antisolvent effects play important roles in various scientific and industrial fields. Firstly, antisolvents find extensive application in the preparation of solid products in processes such as crystallization and precipitation [5,6,7]. In these processes, the induction of supersaturation is typically achieved through cooling or solvent evaporation [8,9,10]. However, these heat-related processes at industrial scales often necessitate the use of fossil fuels, posing both economic and environmental challenges due to potential air pollution and contributions to global warming [11,12]. Alternatively, harnessing antisolvent effects provides a means to control solubility without the need for heat consumption. The addition of precise quantities of an antisolvent to a system increases supersaturation, thereby facilitating particle formation processes. Secondly, comprehending the roles of antisolvents could help to efficiently advance some applications, particularly in drug formulation [13,14]. In pharmaceutical production, unwanted antisolvents can lead to adverse effects, such as converting soluble pharmaceutical active compounds into insoluble forms within the human body or living organisms. Thus, studying antisolvent effects is an indispensable task, but it is often considered as time-consuming and intricate work. Herein, the antisolvent effects of several primary alcohols for potassium dihydrogen phosphate (KDP) in aqueous solutions are investigated.

Potassium dihydrogen phosphate (KDP) serves as a versatile compound and multi-nutrient fertilizer, supplying essential macronutrients simultaneously, namely, potassium and phosphorus [15,16]. High-quality KDP crystals are interesting materials for nonlinear optical (NLO) applications due to their high conversion efficiencies of the second, third, and fourth harmonics generation. These crystals also exhibit excellent transparency across a wide spectral range [17,18]. Furthermore, Deng et al. discovered that high-purity KDP crystals possess the ability to effectively separate dual laser beams, a property of significance in various laser applications, such as frequency control and optical switching [19]. High-quality KDP or doped crystals are commonly synthesized via crystallization, a process in which a solid is formed by cooling or evaporating solutions under suitable controlled temperature programs to achieve supersaturation [20,21]. Many parameters are needed to design these processes, encompassing kinetic and thermodynamic data, such as nucleation, crystal growth, agglomeration, and equilibrium state [8,9,10]. Among these factors, equilibrium is of principal importance, as it serves as the foundation for process design by providing solubility data. Essentially, solubility data characterize the thermodynamics of the solid–liquid equilibrium (SLE) [22,23], enabling a quantitative assessment of component interactions once equilibrium is established. Beyond its relevance to particulate processes, SLE also plays a pivotal role in many other fields. For instance, based on SLE data, it is possible to determine the optimal/maximal quantities of soluble compounds that are required in beverages or foods to form desired flavors, aromas, or colors [24]. In industrial settings, engineers must operate certain units with a profound understanding of SLE to prevent the deposition of substances, such as sedimentation in boilers and pipes [25]. Nevertheless, SLE is vital for drug formulation since the accepted quantities (again, related to solubility) of these compounds in living species, injections, or oral solutions must be ensured [26]. SLE data are also a prerequisite for coating processes [27]. Thus, developing new or improved methods for the rapid and accurate determination of SLE is an urgent necessity.

Although the determination of SLE is a classic problem, it is frequently underappreciated, which could lead to many types of errors. Essentially, there are two general methods used to detect SLE as described by Nývlt [28]. In the first method, known as conventional isothermal determination, the temperature remains constant as the system is driven towards equilibrium. Subsequently, the concentration of the resulting saturated solution can be determined through different techniques, particularly gravimetric, optical, and chromatographic methods [29,30,31]. In the second approach, known as the polythermal method, the temperature is gradually increased at a slow rate to completely dissolve a suspension with a known composition. During this process, the state of the suspension undergoes a gradual transition from turbidity (cloud point) to transparency (clear point). The region where this transition occurs represents the equilibrium state of the system, and the moment of this transition corresponds to the saturated temperature of the known composition system [32,33]

In this work, the antisolvents effects of several primary alcohols on KDP solutions were studied considering the roles of different alkyl chain lengths using the two above SLE determination methods. Besides the experimental approach, the thermodynamics of SLE establishment was also parametrically investigated for a quantitative analysis. Specifically, the eNRTL method was applied to evaluate and compare the interactions between the solute and solvents (or mixed solvents) for binary, ternary, and quaternary systems considering model compatibility with different alcohol types, and a wide range of alcohol concentrations was examined.

2. Materials and Methods

2.1. Materials

The following chemicals were used without further purification: KH₂PO₄ (KDP) from Merck Co., Germany, purity 98%; methanol (MeOH) purity 99.9%; and butan-1-ol (ButOH) purity 99.5% from Sigma Aldrich (USA); ethanol (EtOH) purity 99% from Fisher Chemical (Loughborough, UK); and propan-1-ol (PrOH) purity 99.5% from Merck (Darmstadt, Germany). Millipore MiliQ (Temecula, CA, USA) water was used as a solvent. Apparatuses were used in solubility measurement, including an electronic balance with a precision of 0.0001-g (Mettler Toledo ME203, Greifensee, Switzerland), a magnetic stirrer (IKA CMAG HS7, Königswinter, Germany), a waterbath (Ika C–Mag HS 7, Selangor, Malaysia), a sieve apparatus (Thermo Fisher Scientific, Waltham, MA, USA), a thermostat (Stuart SRC5, London, UK), and an oven (Memmert UN110, Schwabach, Germany). Furthermore, a special system was self-built to serve the polythermal measurements.

2.2. Polythermal Solubility Determination

A self-engineered SLE determination setup was constructed as depicted in Figure 1 which based on the polythermal concept. The functionality is briefly described below.

The solid was pre-treated in a fine grinding step (using an agate mortar and pestle), and a sample (passed through a 45 µm sieve) was collected for SLE measurement. Determined amounts of this solid and solvents (or mixed solvents) were placed in a 5 mL glass vial to prepare a given composition suspension. As seen in Figure 1, this turbid system was placed in a vial (1) and stirred with a 5 mm magnetic stir bar (6), and the vial was placed in a Teflon-made heating chamber (2). The magnetic stirrer (7) was set to the appropriate speed (150–200 rpm) to create a homogeneous suspension and to avoid air bubble formation and sedimentation. The temperature was raised at a heating rate of 5 K/min with a digital temperature control system (3). The suspension was exposed to a laser beam (4), which had a wavelength of 680 nm and a power of 5 mW. Attenuation data were collected using a BH1750 sensor (10). The intensity of the laser focused on the sensor module was determined using a microprocessor (Arduino Uno, EXP GmbH, Germany) and sent to a computer (9). Specifically, the internal temperature of the suspension was recorded in real time with an Omron temperature sensor (8).

2.3. The Isothermal Method

Solid–liquid suspensions were placed in 5 mL glass vials and stored at fixed temperatures under well-agitated conditions for a sufficiently long period, which ensured that the systems reached equilibrium states. These vials were immersed in temperature-controlled media placed in a double-layer glass flask, which was connected to a thermostat. Specifically, these suspensions were heated in advance to dissolve even the smallest particles and then stored (well agitated) at 5 °C higher than the investigated temperatures until recrystallization occurred. Then, these suspensions were magnetically stirred at 300 rpm for 2 days before performing solid–liquid phase separation using a syringe mounted with a 0.22 µm PTFE filter. The obtained saturated solutions were weighed before and after drying at 80 °C for 2 days. The concentration of the solute was calculated according to the mole fraction (moles of solute/total moles of solution). Each measurement was repeated at least three times to obtain mean values.

2.4. SLE Thermodynamic Description for Electrolyte KDP in Single and Mixed Solvents

Solid–liquid equilibrium occurs when the chemical potentials of each component in the solid and liquid phases are equal [34]. For an electrolyte solution, the equilibrium condition can be described using Equation (1), where μ, υ, γ, and m- are the chemical potential, stoichiometry, activity coefficient, and molality, respectively. Indices s, l, c, and a refer to the solid, liquid, cation, and anion, respectively. The standard state is referred to as the hypothetical ideal diluted solution at unit concentration in a solvent at the system temperature and pressure.

$${\mu }_s={\mu }_l^0+RT{\upsilon }_{c}ln\left(m_c{\gamma }_c\right)+RT{\upsilon }_{a}ln\left(m_a{\gamma }_a\right)$$

(1)

Hereby, the mean ionic activity coefficient γ_± of a compound is introduced and defined from activity coefficients of its constituent’s cation and anion, as seen in Equation (2).

$${\gamma }_{\pm }=\frac{1}{2}\sqrt[\left({\upsilon }_c+{\upsilon }_a\right)]{{{\gamma }_c^{{\upsilon }_c}\times \gamma }_a^{{\upsilon }_a}}$$

(2)

The mean ionic activity coefficient γ_± is expressed as the sum of two contributions including long-range (LR) and short-range (SR) interactions, as described in Equation (3).

$$ln{\gamma }_{\pm }={ln{\gamma }_{\pm }}^{LR}+{ln{\gamma }_{\pm }}^{SR}$$

(3)

On the one hand, the long-range interaction is calculated from the extended Debye–Hückel equation (Equation (4)), which was first proposed by Pitzer.

$$ln{\gamma }_{\pm }^{LR}=-{\left(\frac{1000}{M_s}\right)}^{1/2}{A}_{\varphi }\left[\frac{2Z_i^2}{\rho }ln\left(1+{\rho I}_x^{1/2}\right)+\frac{Z_i^2{I}_x^{1/2}-2I_x^{3/2}}{1+{\rho I}_x^{1/2}}\right]$$

(4)

In this equation, Z_i and M_s are the ionic charge and the solvent’s molar mass, respectively. ρ is the closed approach parameter (an adjustable parameter) and was suggested at a value of 14.9 by Pitzer [35]. I_x is ionic strength (mole fraction-based) and is calculated from Equation (5). The Debye–Hückel constant, A_ϕ, is defined as Equation (6).

$$I_x={\sum }_{i=1}^n\left(c_i{Z}_i^2\right)$$

(5)

$$A_{\varphi }=\frac{1}{3}{\left(\frac{2\pi N_A{d}_0}{1000}\right)}^{1/2}{\left(\frac{e^2}{D{k}_{B}T}\right)}^{3/2}$$

(6)

Herein, π, N_A, k_B are pi, Avogadro, and Boltzmann constants, while e and d₀ are electric charge and solvent density, respectively. As a function of temperature, the dielectric constant $D\left(i\right)$ of the single solvents (i) is adopted from the literature data [36,37,38,39] and presented in Figure 2 via a calculation from Equation (7); the relevant coefficients are summarized in Table S1 (Supplementary Information). The dielectric constant of mixed solvents (D) is calculated based on the volume fraction $v\left(i\right)$ of constituent solvents according to Equation (8) [40].

$$D(i)=B_0+B_1\frac{1}{T}+B_{2}T+B_3{T}^2+B_4{T}^3$$

(7)

$$D={\sum }_{i=1}^{n}D\left(i\right)·v\left(i\right)$$

(8)

On the other hand, the short-range interaction needs to be quantified using appropriate methods; e.g., the electrolyte non-random two-fluid model (eNRTL) is one of the frequently utilized models [41]. This concept was first described by Renon based on the theory of local composition, and it is applicable to both partially and fully mixed systems. By applying the eNRTL equation for the Gibbs residual energy, the mean activity coefficient can be calculated as shown in Equation (9). (Ignoring second-order and higher-order interactions.)

$$ln{\gamma }_{\pm }={\displaystyle \sum _m^{}\frac{x_m{G}_{ca,m}}{{\displaystyle \sum _j^{}{G}_{j,m}{x}_j}}}\left({\tau }_{ca,m}-\frac{{\displaystyle \sum _j^{}{x}_j{\tau }_{j,m}{G}_{j,m}}}{{\displaystyle \sum _{l=1}^m{G}_{j,m}{x}_j}}\right)+\frac{{\displaystyle \sum _{}^{}{x}_j}{G}_{j,ca}{\tau }_{j,ca}}{{\displaystyle \sum _{l=1}^m{G}_{j,ca}{x}_j}}\left(1-\frac{x_{ca}}{{\displaystyle \sum _{}^{}{x}_j{G}_{j,ca}}}\right)-G_{ca,w}{\tau }_{ca,w}-{\tau }_{w,ca}$$

(9)

In these equations, three terms τ_ij, τ_ji, and G_ij are calculated from the non-randomness factor (α_ij) and interaction parameters (g_ij, g_ji) using Equations (10) and (11).

$${\tau }_{ji}=\frac{g_{ji}-g_{ii}}{RT}$$

(10)

$$G_{ji}=exp\left(-{\alpha }_{ji}{\tau }_{ji}\right)$$

(11)

Thus, these three parameters are characteristics of the state of the electrolyte solution and need to be identified. The estimation procedure is described as follows: First, the eNRTL parameters are estimated for a solution containing only KDP and water. Then, these parameters are assumed to remain unchanged when upgrading the system from a single solvent to binary mixed solvents with the presence of other primary alcohols (varied from C₁ to C₄). Thus, in these ternary systems, besides the known parameters for the KDP–water interactions from the previous step, the parameters for the interactions between the alcohols and water are also required. In fact, these parameters have already been published in the literature [42,43,44,45] and are utilized in this work. These binary interaction parameters are listed in Table S2 (Supplementary Information). This assumption helps to reduce the mathematical complexity of the ternary systems (9 parameters) to the problem of solving interactions only between KDP and the co-solvents, which involves 3 unknown parameters.

The mean activity coefficient in Equation (9) is mole fraction basis, and it will be converted to mean molality activity coefficient using Equation (12), where m is molality.

$$ln{\gamma }_{\pm ,molal}=ln{\gamma }_{\pm }-ln\left[1+0.001M_s\left({\upsilon }_c+{\upsilon }_a\right)m\right]$$

(12)

Solubility (x) as a function of temperature is expressed by the modified Apelblat’s equation (Equation (13)) [46,47]. Herein, three parameters ($a, b,\mathrm{a}\mathrm{n}\mathrm{d} c$) were estimated from experimental solubility measurements. As a consequence, the apparent partial molar enthalpy of the solution of the solute (ΔH) is calculated as Equation (14) [48]. R is the ideal gas constant.

$$lnx=-\frac{a}{R}\frac{1}{T}+\frac{b}{R}lnT+c$$

(13)

$$∆H=a+bT$$

(14)

The experimental activity coefficient is calculated according to Bromley’s equation (Equation (15)) [49]. Here, the pre-parameters (${pa}_{Brom.}^{\left(i\right)}$) were adopted from the literature [50]. Indeed, Barata et al. reported the activity coefficients for KDP in some solvents, such as water, mixtures of water and ethanol, propan-1-ol, and propan-2-ol. First, we used Barata’s data to calculate two parameters in Bromley’s equation. Then, the obtained parameters were applied to calculate ${\gamma }_{\pm }^{exp}$ by using the solubility data measured in the current work. Finally, ${pa}_{Brom.}^{\left(1\right)}$ and ${pa}_{Brom.}^{\left(2\right)}$ were fitted for each of the solvent mixtures using the solubility data in this work.

$$log{\gamma }_{\pm }=-A\frac{\sqrt{I}}{1+\sqrt{I}}+{pa}_{Brom.}^{\left(1\right)}\frac{I}{{\left(1+1.5I\right)}^2}+{pa}_{Brom.}^{\left(2\right)}I+B·I^2$$

(15)

In this equation, A is the Debye–Hückel parameter, and it is a function of temperature and the dielectric constant of solvent/mixed solvents (calculated as Equation (16)) [51]. The role of B is herein assumed to be negligible and is set to zero.

$$A=\frac{1.82455\times {10}^6}{D\times T}$$

(16)

2.5. Parameter Estimation

In this work, parameters were estimated by minimizing the objective functions, which described the differences between the experimental data and the model values presented in Equation (17). The Levenberg–Marquardt algorithm with multi-start was applied. Indeed, the function lsqnonlin in the MATLAB 7.0.4 environment was used. The standard deviation with 95% confidence intervals was calculated using Equation (18). Herein, the calculated values were denoted by “cal.” and a variable subscripted by “exp.” assigned to the experimental data. “Val” is a variable representing the mean activity coefficient in eNRTL and Bromley’s equations, or mole fraction in Apelblat’s equations, respectively.

$$OF={\sum }_{i=1}^n{\left({Val}_{\left(i\right)}^{exp.}-{Val}_{\left(i\right)}^{cal.}\right)}^2$$

(17)

$$\sigma ={\left(\frac{OF}{n-p}\right)}^{1/2}{t}_{0.95}$$

(18)

3. Results

3.1. Solid Phase Analysis

First, it is important to verify the solid state of KDP in the presence of the studied solvents. In other studies relating to KDP, this information is rarely mentioned. As shown in Figure 3, the solid-phase structure remains unaltered after subjecting it to isothermal solubility measurements with a variety of solvents, comparing the reference XRD pattern (Reference—purchased solid from Merck) with cases of KDP in water and other mixed water–alcohol solvents. The characteristic peaks are typically found at the 2theta positions of 5.7°, 23.9°, 28.4°, 29.1°, 33.7°, etc., which correspond to the faces (101), (112), (201), (121), (212), etc., respectively. Furthermore, additional XRD tests of small amounts of collected solid prior to equilibrium in the polythermal measurements also showed the same results as the isothermal measurements. This confirms that there is no polymorphism affecting our SLE determinations. The application of polythermal measurements is highly advantageous compared to the conventional isothermal procedure, primarily due to the consistent solid state of KDP across all SLE determinations. Specifically, the isothermal measurement requires a long period of up to few days for one single measurement and could be influenced by various types of errors due to the multi-step procedure, such as solid–liquid equilibration, phase separation, and absolute evaporation.

3.2. SLE Determination

In Figure 4a, SLE determined in the cases of water and water–ethanol mixtures shows that the isothermal (closed symbols) and polythermal (open symbols) methods achieved a good agreement over a wide temperature range for all studied EtOH/water ratios. The maximal error in the studied conditions reached a value of 1.1%, which proves the high applicability of the self-engineered setup for polythermal determination. Since the solid phase at equilibrium was identical for all studied solvents, the polythermal determination was considered quicker and more favorable for the SLE determination of KDP solutions. This conclusion is likely the case for the L-lactide system, another case that was studied using the same apparatus [32]. In addition, the results of this study are comparable to those of the literature (Supplementary information, Tables S3–S10) [50]. A comparison between the current work and the literature revealed a maximal error of less than 4.7% for both water and the mixed solvents. In the following sections, SLE was mainly determined via polythermal measurements due to its convenience. By utilizing the acquired SLE data, the parameters for Apelblat’s equation were estimated and are presented in

Table 1. Estimated parameters for the modified Apelblat’s and Bromley’s equations. Mixed solvents are composed of x% alcohols and (100 − x)% water (on a weight basis). See Table S11.

SLE Set	Solvent	Apelblat’s Parameter			Bromley’s Parameter
		$\mathit{a}$	$\mathit{b}$	$∆{\mathit{H}}_{298.15}$, [J/mol]	${\mathit{p}\mathit{a}}_{\mathit{B}\mathit{r}\mathit{o}\mathit{m}.}^{\left(1\right)}$	${\mathit{p}\mathit{a}}_{\mathit{B}\mathit{r}\mathit{o}\mathit{m}.}^{\left(2\right)}$
1st	Water	−6862.38	23.44	127.89	0.67 (48)	−0.05 (31)
2nd	5%EtOH	−6937.20	33.25	2978.07	0.21 (98)	−0.05 (18)
3rd	10%EtOH	−6980.43	39.07	4670.01	0.19 (88)	−0.06 (12)
4th	15%EtOH	−6986.25	39.40	4763.34	0.18 (14)	−0.08 (82)
5th	20%EtOH	−6980.43	39.07	4670.01	1.01 (52)	−0.02 (61)
6th	5%MeOH	−6900.62	28.35	1552.15	0.21 (97)	−0.05 (18)
7th	10%MeOH	−6979.60	39.07	4670.84	0.22 (89)	−0.06 (12)
8th	15%MeOH	−6945.52	33.92	3168.06	0.21 (17)	−0.08 (83)
9th	5%PrOH	−6869.03	24.11	319.54	0.22 (01)	−0.05 (18)
10th	10%PrOH	−6947.18	34.66	3389.49	0.21 (94)	−0.05 (18)
11th	15%PrOH	−6936.37	32.75	2830.17	0.21 (94)	−0.05 (17)
12th	5%ButOH	−6861.54	23.94	277.45	0.21 (84)	−0.05 (17)
13th	10%ButOH	−6934.71	33.08	2930.99	0.22 (07)	−0.05 (18)
14th	15%ButOH	−6965.47	36.99	4065.27	0.22 (07)	−0.05 (18)
15th	2.5%:2.5% MeOH + EtOH	−6936.37	32.75	2830.17	0.21 (97)	−0.05 (18)
16th	5.0%:5.0% MeOH + EtOH	−6992.07	40.82	5178.92	0.21 (98)	−0.05 (18)
17th	7.5%:7.5% MeOH + EtOH	−6886.49	26.35	971.37	0.21 (97)	−0.05 (18)

(standard deviations are enclosed in parentheses).

Bromley’s parameters were synthesized using the data adopted from the literature [50]. First, published data relating to the mean activity coefficients from Barata et al. were used to deduce two parameters in Bromley’s equation for the cases of propan-1-ol, propan-2-ol, and ethanol at the mean values of ${pa}_{Brom.}^{\left(1\right)}$ and ${pa}_{Brom.}^{\left(2\right)}$, about 0.2 and −0.05, respectively. Indeed, the values of ${pa}_{Brom.}^{\left(1\right)}$ and ${pa}_{Brom.}^{\left(2\right)}$ were almost unchanged for the above three alcohols. These parameters do not seem to be affected by carbon chain length in such low-molecular-weight alcohols. Hence, it is expected that these results could be extended to other similar alcohols, such as MeOH and ButOH. Second, these parameters were used to calculate${\gamma }_{\pm }^{exp}$ with solubility in this work. The empirical model (Equation (15)) was exploited. Pairs ${pa}_{Brom.}^{\left(1\right)}$ and ${pa}_{Brom.}^{\left(2\right)}$ were estimated for each solvent system, and the results are shown in Table 1.

The limitations of Bromley’s equation for the KDP solution were herein analyzed with the representative cases of EtOH–water mixed solvents. Figure 4b shows that Bromley’s equation described relatively well the mean activity coefficient of KDP in the mixed solvents of up to 15% (w/w) EtOH in water with a maximal deviation lower than 1.1%. However, an elevated content of EtOH, e.g., 20% (w/w), led to strong deviations between ${\gamma }_{\pm }^{exp}$ and ${\gamma }_{\pm }^{cal}$ of up to 8.3% (see Figure 4b). The limitation of Bromley’s equation for concentrated solutions was also pointed out in the literature [49]. Therefore, in the following sections, three levels of alcohol contents were considered for model correlation, i.e., 5%, 10%, and 15% (w/w) of the four C₁–C₄ alcohols used as co-solvents.

3.3. Antisolvent Effects of C₁–C₄ Primary Alcohols for KDP Solution

As depicted in Figure 4a, adding EtOH resulted in proportionate reductions in KDP solubility. For instance, adding 5% (w/w) EtOH reduced solubility by up to 21.1%, while adding 20% (w/w) EtOH resulted in a reduction of 60.7% (a comparison at 333.15 K). This trend was also apparent for the other alcohols, as shown in Figure 5a when other C₁–C₄ primary alcohols were introduced into the KDP aqueous solution. Only the cases of 5% (w/w) of these alcohols were plotted in this figure for comparison purposes; details of the other contents for each alcohol are separately presented in Figure 4a and Figure 5b–d for the cases of EtOH, MeOH, PrOH, and ButOH to enhance readability. These findings indicate that these alcohols exhibited antisolvent properties in the studied system. This observation could be explained since these alcohols change the properties of the mixed solvents, such as the dielectric constant, density, and polarity. For instance, Figure 1 clearly shows that alcohol significantly lowers the dielectric constant compared to water. Thus, the decrease in the total dielectric constant of the KDP aqueous solution upon the addition of alcohols is one of the factors contributing to the reduction in solubility. A comprehensive quantitative analysis is necessary to gain deeper insights into these alterations and their associated effects. As indicated in Table 1, the values of the apparent molar enthalpy of the solution for the solute corroborated this observation. For example, when comparing the data at 5% w/w alcohol content, the ΔH_298.15 values for EtOH, MeOH, PrOH, ButOH, and water were determined to be 2978.07, 1552.15, 319.54, 277.45, and 127.89 (J/mol), respectively. These enthalpy values are positive, indicating that the dissolution processes were endothermic. Obviously, a decrease in the energy required for the dissolution process corresponds to an increase in solubility.

Based on the aforementioned observations, the antisolvent properties of these primary alcohols in the KDP solutions have been conclusively demonstrated. Notably, effects similar to those reported by Barata et al. [50] were also observed in the cases of ethanol, propan-1-ol, and propan-2-ol. In fact, these phenomena occur quite often in many other systems, such as when alcohols are added to amino acids [52]. Herein, we were interested in whether the strength of the antisolvent effects depended on the alcohol type. As depicted in Figure 5a, the solubility of KDP decreased in the following order: EtOH > PrOH > ButOH. It appears that a shorter alkyl chain length corresponds to a more pronounced decrease in solubility. This trend is likely attributed to the interactions between the hydrophobic groups of alcohols and water molecules within the solution. However, when the alkyl group size was further reduced to C₁, corresponding to methanol (MeOH), the solubility curve increased once more and surpassed that of ethanol (EtOH). This phenomenon may be linked to the special ability of methanol (MeOH) to form hydrogen bonds [53]. Consequently, the addition of alcohol induces solvent restructuring, which, in turn, affects solubility.

Figure 4a and Figure 5b–d show a consistent tendency that increasing the alcohol content leads to a decrease in solubility for all the studied co-solvents. However, for a comprehensive understanding of the competitive interactions among solute–solvent, solute–antisolvent, and solvent–antisolvent components, it is essential to incorporate the results obtained from the thermodynamic eNRTL modeling.

3.4. eNRTL Model Characterization

Parameter estimation was conducted for the KDP–water system, and the resulting mean activity coefficients obtained through both experimental and modeling methods are plotted in Figure 6a. First, even though α_ij has a strong influence on the eNRTL model, it is commonly fixed at 0.2 in many previous studies [41,50]. Herein, all parameters were simultaneously determined via the nonlinear estimation method, which resulted in a relative different value of α_ij at 0.36. Based on the obtained α_ij value, the solution of KDP in water is highly deviated from the ideal case where α_ij = 0 for a complete randomness state. Second, the other two parameters were obtained at reasonable values: τ_ij = −1705.7 and τ_ji = 10,973. Normally, according to Chen et al., τ_ij receives negative values and τ_ji is positive [41]. Third, regarding the long-range interactions (relating to the ion-ion long-range electrostatic interactions), the magnified figure in Figure 6a shows that γ_±^LR is rapidly decreased when increasing temperatures up to 318 K, then began to plateau at about 325 K. However, as seen in Figure 6a, the major contribution to γ_± comes from the short-range interaction (accounting for ion–molecule and molecule–molecule interactions). Long-range interactions possess extremely small quantities compared to short-range interactions.

For comparison purposes, the γ_± values are plotted in Figure 6b for all antisolvents at the same ratio of alcohol/water at 5% (w/w). Notable trends emerged from these observations: Firstly, γ_± decreased with an increase in temperature. Secondly, the sequence of change showed the following order: ButOH < PrOH < MeOH < EtOH. Furthermore, as shown in Figs 6c–f (which share the same y-axis scale for ease of comparison), γ_± can be considered an increasing function of alcohol content. These figures show that the eNRTL model describes relatively well the mean activity coefficients for ternary systems involving KDP–water–alcohol. The discrepancy between the experimental data and the model correlations ranged between 0.3% and 2.8%.

Table 2. eNRTL parameters for KDP aqueous solutions under antisolvent effects.

SLE	Solvent	eNRTL			τij	τji	$\left|{\mathit{\tau }}_{\mathit{i}\mathit{j}}-{\mathit{\tau }}_{\mathit{j}\mathit{i}}\right|$
		αij	gij	gji
1st	Water	0.36 (2)	−1705. (7)	1097 (3)	−0.68	4.42	5.1
2nd	5%EtOH	0.12 (6)	−5236. (3)	3636 (8)	−2.11	14.65	16.76
3rd	10%EtOH	0.12 (3)	−4211. (7)	2325 (7)	−1.69	9.38	11.07
4th	15%EtOH	0.10 (1)	−4911. (6)	2448 (6)	−1.98	9.88	11.86
6th	5%MeOH	0.16 (2)	−1397. (9)	3041 (7)	−5.64	12.27	17.91
7th	10%MeOH	0.19 (3)	−2049. (5)	2340 (6)	−8.27	9.44	17.71
8th	15%MeOH	0.20 (5)	−2055. (2)	2344 (3)	−8.29	9.46	17.75
9th	5%PrOH	0.06 (9)	−2481. (2)	5869 (9)	−1.00	23.69	24.69
10th	10%PrOH	0.08 (9)	−7715. (3)	5565 (4)	−3.11	22.46	25.57
11th	15%PrOH	0.05 (1)	−2101. (3)	3995 (6)	−8.48	16.12	24.6
12th	5%ButOH	0.03 (1)	−7497. (8)	5751 (0)	−3.02	23.21	26.23
13th	10%ButOH	0.06 (9)	−2481. (4)	5869 (9)	−1.00	23.69	24.69
14th	15%ButOH	0.29 (8)	5743. (2)	7838 (5)	2.31	31.63	29.32
15th	2.5%:2.5% MeOH + EtOH	0.02 (9)	−89,526. (5)	−504,241. (2)	−36.23	−203.73	167.50
16th	5.0%:5.0% MeOH + EtOH	0.01 (2)	4025. (8)	2413 (3)	1.63	9.74	8.22
17th	7.5%:7.5% MeOH + EtOH	0.01 (1)	4006. (7)	1394 (5)	1.62	5.63	4.01

shows that the eNRTL model parameters were estimated within suitable ranges.

On the one hand, as discussed in the previous section (Figure 4b), high alcohol contents revealed a disagreement with Bromley’s model due to the strong deviation between the experimental and model correlation γ_± values. On the other hand, the applicability of the obtained eNRTL model was re-evaluated in this context. The results presented in Figure 6b demonstrate the successful modeling of γ_± values for all C₁–C₄ alcohols, with the maximum error between the experimental and model correlations being 6.3%. At lower alcohol contents, differences in alkyl groups did not significantly impact the eNRTL model’s correlation. However, when increasing the content of alcohols, the interactions underwent significant changes, and it was necessary to re-evaluate these effects. An example was selected to analyze this influence; the case of EtOH was chosen, as shown in Figure 6c. At higher contents, such as 20% (w/w) EtOH, the eNRTL model exhibited a relatively weak correlation of the mean activity coefficient, with the maximum error reaching 19.6%. This conclusion aligns with the findings regarding Bromley’s equation in the previous section, emphasizing the limitations of these models at high antisolvent contents. Furthermore, Table 2 highlights the weak dependency of these parameters on the solvent composition.

As a matter of fact, the dissolution of electrolytes mainly depends on the amount of free water available in the solution. These dipole molecules surround dissolved KDP species to form cation and anion hydration, which stabilizes the dissolved species and promotes dissolution. Consequently, the more pronounced these processes become, the higher the solubility of KDP. Any factor that disrupts this pathway will inevitably lead to a reduction in solubility. Besides the variation in the dielectric constant, hydrogen bond formation is also one of the factors that could reduce the number of free water molecules. Thus, the observed antisolvent effects of the C₁–C₄ primary alcohols are intricately related to the formation of hydrogen bonds. The hydroxyl groups in these alcohols forming (H)-bonding with water (reducing the amount of free water) leads to a decrease in solubility. Therefore, these alcohols function as antisolvents in KDP solutions.

The data in Table 2 show that the values of the estimated non-randomness parameter α_ij for all cases were found in a wide range up to 0.36, which reveals the states of real electrolyte solutions. Normally, the larger the α_ij factor, the stronger the non-randomness behavior (referred to as α_ij = 0 for the complete randomness state). This behavior can be explained by the formation of strong hydrogen bonds between water and the hydroxyl group present in these alcohols. Such selective stabilization leads to higher α_ij values. In the presence of alcohols as co-solvents, the following order of α_ij was observed: ButOH < PrOH < MeOH < EtOH (Table 2). This trend aligns well with the formation of (H)-bonding between these alcohols and water. Firstly, the intermolecular force decreases in the order of ButOH < PrOH < EtOH due to the difference in the hydrophobic groups. Secondly, the competitive (H)-bonding formation between EtOH/MeOH and water must be taken into consideration [53]. In MeOH, there are three alpha hydrogen atoms that form stronger hyperconjugation with the lone pairs of oxygen than in the case of EtOH (only two alpha hydrogen atoms) [54]. This results in less availability of these lone pairs of oxygen to form hydrogen bonds with the hydrogen atoms of adjacent molecules. Thus, intermolecular hydrogen bonding with water is stronger in the case of EtOH than in the case of MeOH. Hence, less free water is available in the case of EtOH, which, in turn, results in a lower solubility than in the case of MeOH.

Concerning the interaction parameters, the obtained τ_ij and τ_ij in Table 2 were found in suitable ranges and evaluated as weakly dependent on composition. Besides the main interaction of KDP and water, there exists another factor that contributes to the establishment of SLE, i.e., KDP–alcohol interactions. The absolute difference in binary interaction parameters (τ_ij − τ_ij) provides insight into the strength of interactions between the solute and co-solvents. As indicated in Table 2, the absolute (τ_ij − τ_ij) values with alcohols as co-solvents showed the following order: EtOH < MeOH < PrOH < ButOH. Corresponding to this order, an increase in KDP solubility was observed, as evident in Figure 4a and Figure 5. This phenomenon can be attributed to the stabilization of the dissolved species by alcohols, which serves as a secondary contributing factor alongside the primary interaction between the dissolved species and water.

3.5. Validation of the Obtained eNRTL for Quaternary System

Due to the distinctive behavior of MeOH, it was herein selected together with EtOH as an antisolvent pair to assess the validity of the obtained eNRTL model for quaternary systems. The solubility of KDP in mixed solvents composed of water-MeOH-EtOH was measured at three total alcohol contents of 5%, 10%, and 15% (w/w). These mixtures were prepared with equal amounts of MeOH and EtOH (see the 15th–17th SLE sets in Table 1). An attempt was made to describe the activity coefficients in these quaternary systems, which inherently involve six interaction pairs, as depicted in Figure 7a, necessitating the estimation of 18 parameters. Herein, it was assumed that the binary interactions determined in the previous sections remained unaffected by the presence of other compounds in the system. Consequently, the 18-parameter estimation problem was significantly reduced to just three variables, specifically related to the interaction between MeOH and EtOH.

Correlation was successful in the case of 5% (w/w) MeOH-EtOH mixed in water, as shown in Figure 7b. The γ_± values exhibited a strong agreement between the experiment and model correlation (the estimated parameters are listed in Table 2). The large absolute difference (τ_ij − τ_ji) revealed an important clue that the extremely strong interaction between these two alcohols was due to the intermolecular force and hydrogen bond formation between them. Further investigations were carried out for quaternary systems with higher mixed alcohol contents. Unfortunately, these attempts failed to accurately describe other quaternary systems at higher alcohol contents, as shown in Figure 7b. The experimental and simulated results exhibited substantial deviations, reaching up to 20.8% for the case of the 15% (w/w) mixed alcohol content (in comparison at 304.65 K). This discrepancy implies that the binary interactions previously determined for KDP–water, water–alcohols, and alcohols–water are insufficient to account for the properties of quaternary systems with such high alcohol contents. Consequently, it suggests that higher-order interactions may play a significant role in establishing equilibrium in quaternary solid–liquid KDP solutions.

4. Conclusions

The developed polythermal solubility measurement apparatus, based on a self-engineered setup, serves as a high-throughput tool supporting solvent screening studies. The maximal error between this development and the conventional method is less than 1.1%, while the time requirement is significantly reduced (from a few days to several hours for one single measurement). This achievement holds considerable significance for solid–liquid equilibrium (SLE) determination, which typically serves as a crucial starting point for the development of chemical and biochemical processes. By applying this technique to KDP solutions, the antisolvent effects of C₁–C₄ primary alcohols were elucidated. It was observed that the presence of these alcohols led to a significant reduction in the solubility of KDP, following the order of EtOH > MeOH > PrOH > ButOH. This trend is influenced not only by the length of the alkyl chain but also by the complexity of the hydrogen bond formation in each mixed solvent. Consequently, one of these alcohols could be selected to perform antisolvent crystallization. Furthermore, the non-randomness and interaction properties in these solutions were successfully characterized via the eNRTL model considering antisolvent roles. This modeling approach yielded good agreement with the changing trends in solubility observed in the experimental section. The binary interaction parameters remained applicable as the systems expanded from two components (KDP–water) to ternary systems (KDP–water–alcohol). However, it should be noted that the obtained eNRTL model was only able to describe the quaternary systems at relatively low alcohol contents. When the content of mixed MeOH-EtOH reached up to 15% (w/w), the error in determining the mean activity coefficient increased to 20.8%. Therefore, the model improvement is suggested in consideration of second- or higher-order interactions in these complex systems, taking into account the challenges of solving the multi-variable estimation problem.