High-pressure solubility of methadone hydrochloride in supercritical CO2: Experimental investigation and correlations

Gholamhossein Sodeifian; Hamidreza Bagheri; Shima Golshani; Hamidreza Bakhshaei Ghorogh Aghaei; Adel Noubigh; Mohammadreza Rashidi-Nooshabadi

doi:10.25259/AJC_228_2024

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Original Article

2025

:18;

2282024

doi:

10.25259/AJC_228_2024

High-pressure solubility of methadone hydrochloride in supercritical CO₂: Experimental investigation and correlations

Gholamhossein Sodeifian^a,b,c,*, Hamidreza Bagheri^d, Shima Golshani^a,e, Hamidreza Bakhshaei Ghorogh Aghaei^a,b, Adel Noubigh^f, Mohammadreza Rashidi-Nooshabadi^g

aDepartment of Chemical Engineering, Faculty of Engineering, University of Kashan, 87317-53153 Kashan, Iran

bLaboratory of Supercritical Fluids and Nanotechnology, University of Kashan, 87317-53153 Kashan, Iran

cModeling and Simulation Centre, Faculty of Engineering, University of Kashan, 87317-53153 Kashan, Iran

dDepartment of Chemical Engineering, Faculty of Engineering, Shahid Bahonar University of Kerman, 76188-68366 Kerman, Iran

eDepartment of Chemical Engineering, Amirkabir University of Technology, 1591634311 Tehran, Iran

fCenter for Scientific Research and Entrepreneurship, Northern Border University, Arar 73213, Saudi Arabia

gDepartment of Pharmacology, School of Medicine, Kashan University of Medical Sciences, 87159-88141 Kashan, Iran

*Corresponding author: E-mail address: sodeifian@kashanu.ac.ir (G. Sodeifian)

Received: 2024-12-06, Accepted: 2025-07-06, Epub ahead of print: 2025-09-15, Published: 2025-10-15

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Abstract

Exact drug solubility determination and modeling in supercritical carbon dioxide (SC-CO₂) would address the optimal experimental conditions for processing substances with supercritical fluid (SCF). In this investigation, for the first time, the solubility of Methadone hydrochloride is measured and modeled using the perturbed chain-statistical associating fluid theory (PC-SAFT) equation of state (EoS) alongside a set of eight semi-empirical models. The 24 experimental data points were obtained at pressures ranging from 120 to 270 bar and temperatures between 308 and 338 K. The minimum and maximum mole fractions observed were 1.74 × 10^-5 and 4.67 × 10^-5, respectively, occurring within these ranges. The results indicate that both pressure and temperature significantly affect Methadone hydrochloride solubility, so that as both pressure and temperature rise, the solubility increases. Among the models, the Chrastil model and PC-SAFT EoS exhibited the lowest absolute average relative deviation (AARD) % of approximately 6.71% and 9.73%, respectively. Finally, the enthalpies of the methadone hydrochloride/CO₂ binary system were calculated as $Δ H_{t o t . C h r a s t i l}$ = 29.14 kJ/mol, $Δ H_{t o t . K J}$ = 23.04 kJ/mol, $Δ H_{v a p . a r t l e e t a l .}$ = 40.61 kJ/mol, $Δ H_{s o l . C h r a s t i l - B a r t l e e t a l .}$ = -11.47 kJ/mol, and $Δ H_{s o l . K J - B a r t l e e t a l .}$ = -17.57 kJ/mol.

Keywords

Equation of state

Methadone hydrochloride

PC-SAFT

Semi-empirical

Supercritical CO2

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1. Introduction

Green technology is changing from a choice to a necessity in modern industrial. Solvents are the foundation of the biotechnological, chemical, agrochemical, cosmetic, food, and pharmaceutical technologies [1]. In recent decades, deep eutectic [2], ionic liquids [3,4], and supercritical fluids (SCFs) [5,6] solvents have become the most considered as potential green solvents, particularly in the pharmaceutical industry. The utilization of compressed carbon dioxide as an environmentally friendly alternative to traditional solvents has experienced a significant surge in popularity, primarily attributable to its considerable availability, economical pricing, inherent non-toxic characteristics, the relative simplicity with which it can be removed from various systems, and the remarkable adaptability in its solvation power, which can be finely tuned to meet specific application requirements [7-12]. The solvating power of compressed carbon dioxide can be easily adjusted using variations of pressure, which propagate faster and more uniformly than temperature and composition variations in the conventional solvents [13-15]. This phenomenon has been employed to synthesize substances exhibiting distinct physico-chemical properties, including monodisperse particulates, polymorphic structures, and solids possessing regulated morphologies and porosity, which are frequently unachievable or challenging to procure through traditional liquid mediums [16-19]. The low solubility of most polar and ionic compounds in CO₂ significantly limits its use as a solvent medium. Alternative solutions include developing suitable surfactants, using ionic liquids and CO₂ simultaneously, or utilizing CO₂’s solvating properties when combined with an organic solvent [20-24].

The effectiveness of drug delivery is essential to attain therapeutic efficacy and reduce adverse effects. Administration of a drug alone is not sufficient to ensure that the drug reaches the target site in adequate concentration and duration [25,26]. Non-specific distribution, rapid degradation, poor bioavailability, and off-target interactions can all dramatically affect a drug’s performance and are often a barrier for the development of effective therapeutics; therefore, any advanced drug delivery format is necessary to convert promising drug molecules into viable therapeutic options [27-29]. The importance of controlled and targeted drug delivery is related to overcoming challenges associated with drug development and therapy: bioavailability, dosage and frequency, site specificity, and stability or protection of a sensitive therapeutic, and overcoming biological barriers [30,31]. SCFs have unique features that make them an appropriate solvent and medium for drug delivery uses. Development of the advanced drug delivery systems is vital to improve the efficiency and safety of drug treatments [32-35]. SCFs offer an environmentally friendly and flexible foundation to modify drug features, creating innovative drug formulations, and enabling targeted drug delivery. Also, these solvents propose a variety of advantages in drug delivery, like the formation of micro- and nanoparticles and drug encapsulation [36-38]. The technologies based on SCFs are an environmentally friendly method for pharmaceutical industries that utilize the unique properties of SCFs, including high density, low viscosity, higher diffusivity, and low interfacial tension, particularly carbon dioxide’s non-flammability and non-toxicity [39,40]. CO₂ offers low critical conditions, particularly in terms of critical temperature, making it advantageous for medical and pharmaceutical processing. It is gaseous in ambient conditions, allowing spontaneous separation upon depressurization at the end of the process [41,42]. Also, supercritical carbon dioxide (SC-CO₂) has biocidal features and medical applications due to its partition among the impregnated matrix and SCF phase, a key parameter in supercritical impregnation [43-45].

It is essential to assess the solubility of the pharmaceutical solute within the fluid media to proficiently regulate the supercritical impregnation procedure. Measurement solubility accuracy depends on operational conditions and device, with synthetic and analytical methods categorized based on equilibrium phase composition determination [46-49]. Synthetic methods involve indirect phase composition determination without sampling, but analytical procedures determine solubilities through analyses after reaching equilibrium. Analytical methods can be static or dynamic, with static methods introducing components in a closed vessel to reach thermodynamic equilibrium [50-53]. The non-toxicity and environmental friendliness of SC-CO₂ have made this type of solvent of interest for applying in pharmaceutical applications. However, predicting drug solubility in SC-CO₂ is an important step in the process based on SCFs [54]. Because of their simplicity, cubic EoS are widely used. To model drug solubility, mixing rules must be applied to calculate the cubic equation of state (CEoSs) of the mixture. Nevertheless, CEoSs show poor performance at high pressures [55-57]. To improve thermodynamic modeling of drug solubility, non-cubic EoSs such as statistical associating fluid theory (SAFT) family EoS and cubic plus association (CPA) EoS with the capability to consider complex molecular interactions were used. These types of EoSs provide more accuracy in comparison to CEoSs, especially at high pressures. Of course, non-cubic EoSs need more computational resources in comparison to CEoSs [58-64]. Activity models are another method for thermodynamic phase equilibria of drug/SC-CO₂. For this approach, SC-CO₂is considered a dense gas or a liquid-like solvent. Activity models are mostly valuable for systems with significant deviations from ideal behavior, where CEoS may fail [65-68]. Machine learning (ML) has been known as an authoritative alternative to conventional models to predict drug solubility in the SC-CO₂ [69]. ML method can efficiently investigate complex and non-linear relationships between drug and SCF that may be difficult to model with conventional thermodynamic methods. ML models need a great deal of experimental data to train, and the accuracy of the model is dependent on the quality of the training data [70-72]. Decision trees, support vector machines, artificial neural networks (ANNs), k-Nearest neighbors, and random forests are well-known ML algorithms that can learn difficult relationships among features and output variables [73-76].

Methadone hydrochloride, a synthetic opioid with analgesic activity, mimics the activities of endogenous peptides at central nervous system (CNS) opioid receptors, causing morphine-like influences such as physical dependence, bradycardia, miosis, depression, respiratory depression, sedation, euphoria, and analgesia [77]. Due to its prolonged half-life, methadone’s onset of withdrawal symptoms is slower, duration is longer, and withdrawal symptoms are less severe in comparison to supplementary morphine-like agents [78]. The solubility calculation of Methadone hydrochloride in supercritical carbon dioxide has not been studied to our knowledge. In this investigation, the Methadone hydrochloride drug solubility has been achieved, experimentally, under various circumstances. Additionally, the outcomes were modeled by eight density-based correlations in addition to PC-SAFT EoS. Some enthalpies were calculated for Methadone hydrochloride/CO₂.

2. Materials and Methods

2.1. Materials

Table 1 presents more information on materials, and no purification occurred. Furthermore, Figure 1 illustrates the methadone hydrochloride structure.

Table 1. The details about the used materials.

Substance	Formula	M_W	CAS number	Minimum purity^*	Supplier
Methadone hydrochloride	C₂₁H₂₈ClNO	345.91	1095-90-5	99.70%	Temad Pharmaceutical Co.
Deionized water	H₂O	18.01	7732-18-5	99.99%	Merck Group, Germany
Carbon dioxide	CO₂	44.01	124-38-9	99.99%	Barfak Co., Iran

* Determined by the supplier

The structure of the methadone hydrochloride drug.

Based on the schematic given in Figure 2, the solubility tests were performed at various temperatures and pressures using a laboratory united system and a UV-Vis spectrophotometer setup. The apparatus was composed of a flow meter, heaters, a sampler, a refrigerator, a micrometer valve, gas cylinders, a 6-way valve, and filters.

The schematic of the supercritical setup for measuring methadone hydrochloride drug solubility.

2.2. Method

All setup material, made of stainless steel (SS-316), containing connections, tubes, and valves, was workable at pressures up to 400 bar. The initial temperature of CO₂ was adjusted to 253 K using dispatching the filtered gas into the refrigerator. At 253 K, carbon dioxide was liquefied and ready for pumping. To bring the methadone hydrochloride drug and SCF into direct contact, a stainless steel extraction vessel was used. At first, 2 g of Methadone hydrochloride powder was moved into the extraction vessel. After bringing the CO₂ into the extraction vessel, the desired temperature and pressure according to the experimental circumstances of each experiment were attuned by a pressure gauge with an accuracy of 1 bar and using an oven with an accuracy of 0.1 K, respectively. To increase the rate of dissolution of methadone hydrochloride in SCF and therefore decrease the required time to reach the equilibrium, the mixture in the extraction vessel was mixed nonstop. For this aim, the extraction vessel was located on a magnetic stirring. Following equilibrium at the operating circumstances of any experiment, the saturated SCF known volume was discharged from the sampling tube into a graduated cylinder, including a suitable solvent. The Methadone hydrochloride was protected in the extraction vessel with sintered filters (1 μm) on both sides of the extraction vessel. Based on the study, methadone hydrochloride is soluble in H₂O (120 mg/mL at 298 K). Consequently, deionized water was utilized as the collection solvent. Finally, by adding an amount of deionized water, the drug particles that may be left in the sampling tube were moved into the graduated cylinder. The sampling method description has been provided in the previous investigations [79-90]. Methadone hydrochloride concentration in any experiments was obtained by attaining its λ_max = 295 nm (maximum absorbance wavelength) in the ultraviolet-visible spectrum. For preparing standard solutions for UV-Vis spectrophotometer analysis, a Methadone hydrochloride aqueous stock was provided with a 100 μg/mL concentration. The other solutions were attained consecutively. The mass concentration and mole fraction of Methadone hydrochloride drug in any test were calculated by using an absorbance calibration line that was recognized with standard solutions (R² = 0.994). Each drug solubility test was done in triplicate, and the mean value was applied for drug solubility measurements.

At the equilibrium state, the fugacity of Methadone hydrochloride in both solid and supercritical phases must be equaled ( Eqs 1-4) [91-93]:

(1)

{\hat{f}}_{1}^{S u p .} = f_{1}^{S o l i d}

where [92]

(2)

{\hat{f}}_{1}^{S u p .} = y_{1} P {\hat{φ}}_{1}^{S u p .}

(3)

\begin{matrix} f_{1}^{S o l i d} = φ_{1}^{s a t .} P_{1}^{S u b .} e x p (\int_{P_{1}^{S u b .}}^{P} (\frac{v_{1}^{S o l i d}}{R T}) d P) \\ = φ_{1}^{s a t .} P_{1}^{S u b .} e x p (\frac{v_{1}^{S o l i d} (P - P_{1}^{S u b .})}{R T}) \end{matrix}

Therefore, the solute solubility is as [94-96]:

(4)

y_{1} = \frac{φ_{1}^{s a t .} P_{1}^{S u b .} e x p [\frac{v_{1}^{S o l i d} (P - P_{1}^{S u b .})}{R T}]}{P {\hat{φ}}_{1}^{S u p .}}

To calculate the ${\hat{φ}}_{1}^{S u p .}$ of Methadone hydrochloride drug, we utilize the PC-SAFT EoS. Moreover, the capability of eight density-based models [97-104] was considered to model the Methadone hydrochloride drug solubility in the SC-CO₂. The details of all equations were presented in Appendix A [105-107].

3. Results and Discussion

3.1. Solubility data

The solubility of Methadone hydrochloride in supercritical CO₂ is obtained by a static solubility setup. Table 2 gives the equilibrium solubility and solubility data at six pressure values (120-270 bar) and four temperature levels (308-338 K). The tests were performed in triplicate.

The Methadone hydrochloride solubility and equilibrium solubility are calculated as [108-110]:

(5)

\begin{matrix} y_{1} = y_{d r u g} = \frac{C_{d r u g} \times V_{C} \times M_{w, C O_{2}}}{C_{d r u g} \times V_{C} \times M_{w, C O_{2}} + V_{S} \times ρ \times M_{w, d r u g}} \\ (\begin{matrix} V_{C} = c o l l e c t i o n s a m p l i n g l o o p v o l u m e \\ V_{S} = v i a l c o l l e c t i o n v o l u m e \end{matrix}) \end{matrix}

(6)

S = E q u i l i b r i u m s o l u b i l i t y p a r a m e t e r = \frac{ρ \times M_{w, d r u g} \times y_{1}}{M_{w, C O_{2}} \times (1 - y_{1})}

The measured values of mole fraction ranged from 1.74 × 10^-5 to 4.65 × 10^-5 for mole fractions and have been graphically illustrated in Figure 3. Also, equilibrium solubility is between 0.105 and 0.287 (kg/m³). The SC-CO₂ density was obtained from the National Institute of Standards and Technology (NIST) database [111].

Table 2. Solubility of Methadone hydrochloride in SC-CO₂ at various temperatures and pressures. The experimental standard deviation was obtained by

S D (y_{k}) = \sqrt{\frac{\sum_{j = 1}^{n} {(y_{j} - \bar{y})}^{2}}{n - 1}}

. Expanded uncertainty and the relative combined standard uncertainty are defined as (U) = k*u_combined and

u_{c o m b i n e d} / y = \sqrt{\sum_{i = 1}^{N} {(P_{i} u (x_{i}) / x_{i})}^{2}}

, respectively.

T (K)^a	P (bar)^a	ρ_CO2 (kg/m³) [111]	y×10⁵	SD (ȳ) × (10⁵)	S×10 (kg/m³)	Expanded uncertainty of mole fraction (10⁶U)
308	120	766.62	1.74	1.80	1.05	6.63
	150	814.73	2.20	2.11	1.41	6.23
	180	847.76	2.56	2.43	1.71	2.59
	210	873.43	2.64	2.75	1.81	3.59
	240	894.66	3.21	3.06	2.26	1.09
	270	912.88	3.27	3.37	2.35	1.08
318	120	657.07	2.11	2.18	1.09	4.60
	150	741.57	2.34	2.46	1.36	4.45
	180	788.93	2.98	2.79	1.85	5.52
	210	822.65	3.06	3.12	1.98	3.66
	240	849.16	3.28	3.45	2.19	2.46
	270	871.19	3.75	3.78	2.57	5.81
328	120	503.73	2.86	2.75	1.13	3.61
	150	653.02	2.94	2.89	1.51	2.45
	180	722.73	3.35	3.19	1.90	1.53
	210	767.60	3.77	3.53	2.30	3.06
	240	800.92	3.82	3.88	2.37	1.30
	270	827.62	4.45	4.23	2.89	4.14
338	120	382.44	3.20	3.30	0.97	2.08
	150	553.24	3.23	3.40	1.40	9.11
	180	649.67	3.84	3.65	1.96	1.34
	210	708.47	4.15	3.99	2.31	0.52
	240	750.13	4.43	4.35	2.61	1.61
	270	782.37	4.67	4.71	2.87	4.18

a : Standard uncertainty u are u (p) = ± 1 bar; u (T) = ± 0.1 K.

The solubility data of Methadone hydrochloride drug in SC-CO2.

As clarified in Figure 3, under isothermal circumstances, the solubility of Methadone hydrochloride exhibits an increase with pressure, which is consistent with the associated elevation in CO₂ density. Due to the direct relation between density and pressure and the effect of SCF density on the solvating power when temperature is constant, increasing the pressure leads to increased Methadone hydrochloride solubility in SCF. Similar results have been obtained in other research [112-117]. Also, Figure 3 makes it clear that there is no cross-over pressure behavior for Methadone hydrochloride. At isobaric circumstances, the variation of drug solubility in relation to temperature engenders two opposing effects: increasing the temperature causes a decrease in the density of CO₂, which in turn reduces the interactions among drug and solvent molecules, leading to a decrease in drug solubility. Conversely, when the temperature increases, the drug vapor pressure also increases, which in turn increases solubility. The drug solubility may be affected by two factors, which may have a decreasing or increasing influence in the temperature ranges. Obviously, at constant pressure, the solubility of Methadone hydrochloride increases with increasing temperature. More considerations have been described similarly [118-122].

3.2. Solubility modeling using PC-SAFT EoS

The CEoS requires fewer calculations than the SAFT family, but the accuracy of the SAFT family is higher, particularly at pressures above the critical point [123]. The parameters of the PC-SAFT EoS must be accessible to accurately depict the calculations relating to phase equilibrium. For some components like CO₂, the experimental data of vapor pressure and saturated liquid density are applied to obtain the parameters. In the case of PC-SAFT EoS, there are five parameters as follows:

$(\begin{matrix} m = S e g m e n t n u m b e r \\ \frac{ε}{κ} = I n t e r a c t i o n e n e r g y \\ σ = H a r d c o r e s e g m e n t r a d i u s \\ κ^{A B} = A s s o c i a t i o n c o - v o l u m e \\ ε^{A B} = A s s o c i a t i o n e n e r g y \end{matrix})$

In the present investigation, the carbon dioxide molecules have been modeled as a non-associative component. Similar considerations are presented in the literature [124,125]. Nonetheless, in the case of Methadone hydrochloride drug, similar to a method that was reported in the literature [126,127], the PC-SAFT parameters were estimated using a curve-fitting method with solubility data, such that $k_{i j} = 0$ . The parameters of the PC-SAFT EoS have been tabulated in Table 3.

Table 3. The PC-SAFT EoS parameters.

Substance	m	σ (Å)	ε/k_B (K)	ε^AB/k_B (K)	κ^AB	Ref.
Methadone hydrochloride	6.1849	3.0541	314.5436	6053.2678	0.0614	This study
CO₂	2.8246	2.8147	160.1094			125, 127

To obtain the PC-SAFT parameters, the following objective function was used:

(7)

\begin{matrix} Objective Function = \frac{1}{N} \sum i = 1 (\frac{y_{i}^{E x p .} - y_{i}^{C a l c .}}{y_{i}^{E x p .}}) \times 100 \\ (\begin{matrix} y_{i}^{E x p .} = e x p e r i m e n t a l s o l u b i l i t y \\ y_{i}^{C a l c .} = c a l c u l a t e d s o l u b i l i t y \end{matrix}) \end{matrix}

Figure 4 reveals thermodynamic results based on the PC-SAFT EoS. Solubility data suggests that an increase in pressure will result in more Methadone hydrochloride solubility in the solvent. The PC-SAFT EoS follows the experimental data trend with satisfactory accuracy. The PC-SAFT EoS reveals an S-shaped trend, which suggests a good predictive capability of this EoS at different temperatures. Binary interaction parameter (k_ij) was applied to improve PC-SAFT EoS accuracy. Table 4 tabulated k_ij parameter average absolute relative deviation percent (AARD%) values of PC-SAFT EoS.

The thermodynamic modeling of Methadone hydrochloride drug solubility based on PC-SAFT EoS.

Table 4. The density-based correlations and AARD% values of the PC-SAFT EoS.

Correlation	a	b	c	d	e	f	AARD%^*
KJ [97]	-3.1364	0.0018	-2770.4176				7.21
GM [98]	-2.4706	-2839.3055	1.0766				8.54
Chrstil [102]	4.7784	-22.6144	-3504.3767				6.71
Bartle et al., [101]	9.7007	0.0049	-4884.1540				10.39
Sung-Shim [99]	0.5365	182.5569	-3704.1576	-6.1814			8.38
Bian et al., [100]	15.3475	-4437.0069	1.8338	-2.4680	1.3814×10^-5		7.48
Sodeifian et al., (Model I) [103]	-25.1697	-0.0064	1.6749	4.0558×10^-4	0.0037	-807.4517	13.19
Sodeifian et al., (Model I) [104]	-14.0829	-14375.2365	-305675.0647	-377.0156			20.95
EoS	k_ij
PC-SAFT EoS
T = 308 K	0.105						7.36
T = 318 K	0.110						9.22
T = 328 K	0.116						10.56
T = 338 K	0.123						11.78
Total							9.73

* $A A R D % = \frac{1}{N} \sum i = 1 (\frac{y_{i}^{E x p .} - y_{i}^{C a l c .}}{y_{i}^{E x p .}}) \times 100 (\begin{matrix} y_{i}^{E x p .} = e x p e r i m e n t a l s o l u b i l i t y \\ y_{i}^{C a l c .} = c a l c u l a t e d s o l u b i l i t y \end{matrix})$

Figure 4 includes the results of experimental data and modeling by PC-SAFT EoS. Table 4 reveals that AARD values attained by the PC-SAFT EoS varied from 7.36% at 308 K to 11.78% at 338 K. It is paramount significance to acknowledge that, under conditions of elevated operational pressure, the intricate interactions that transpire between molecules become exceedingly critical to the overall behavior of the system, and it is therefore imperative to employ a suitably formulated equation of state (EoS) that possesses the capability to precisely represent and characterize these complex interactions in order to ensure accurate predictions and analyses. According to Figure 4, PC-SAFT EoS indicated the capability to foresee the methadone hydrochloride solubility, particularly at high-pressure conditions. The PC-SAFT EoS has been used in some research to predict the solubility of solids in the SCF. The satisfactory outcomes have been attained [63,124,126,128-132].

The importance of optimization in the engineering field is demonstrated, particularly to solve complex problems and improve performance in the engineering field. Synthesis, control, and material selection (like adsorbents, solvents, and catalysts) are some applications of optimization in engineering [133-136]. Each mentioned application needs decisions based on available information. Creating physiochemical models for engineering processes can be costly and time-consuming from an efficiency perspective. Though optimization methods present an engineering-appropriate alternative [137-141]. They learn complex processes using learning from data rather than necessarily programming explicit models. Genetic Algorithms (GAs) and ML, containing their subsets, i.e., deep learning and ANNs, are transforming optimization, providing a data-driven solution to complex and high-dimensional problems [142-147]. These methods can be a potential substitute for mathematics models that are expensive to derive or cannot derive specific behaviors with conventional approaches. The evolution of ML methods such as deep ANNs has developed the potential for optimization in engineering fields. This is because the capability of the mentioned approaches to address high-dimensional non-linear problems is often encountered in engineering applications. For instance, ANNs have been widely applied in engineering as a way for modeling or optimizing a process [148-154]. The incorporation of ML, deep learning, and ANNs in optimization indicates a paradigm shift for application-based fields like engineering, pharmaceutical, and science. The usage of the mentioned approaches lets an efficient decision-making process according to data-driven approaches while considering the limitations of conventional modeling methodologies [155,156]. GAs propose a powerful and useful method to solve optimization problems in various science and engineering fields. GAs are strong optimization and search methods based on genetics and natural selection. They are particularly valuable for approximating complex non-linear multi-modal optimization problems in many fields of science and engineering. In contradiction to other derivative-based techniques, GAs do not depend on derivatives, and they are better fit for solutions characterized using discontinuity and noise [157-162]. In the present communication, among all the optimization approaches mentioned, we employed the GAs to 1) determine the parameters of the PC-SAFT EoS and 2) identify the coefficients of the semi-empirical correlation. We are confident that the best coefficients have been achieved through optimization using the GAs.

3.3. Semi-empirical correlations

This investigation used eight semi-empirical correlations to model the experimental data for Methadone hydrochloride. Acceptable arrangements between the solubility data and the outcomes of semi-correlation are considered a confirmation of reliable experimental data. The semi-empirical correlations of the present investigation correlate with various adjustable parameters, are tabulated in Table 4. The optimum parameters were achieved by minimizing the AARD% values, which shows an indicator of the correctness of the used correlations. The arrangement of the drug solubility with the eight semi-empirical correlations yields the AARD% that fluctuates between 6.71% and 20.95% (see Table 4), suggesting a commendable regression of the findings and consequently facilitating a potential forecast of Methadone hydrochloride solubility under the examined temperature and pressure conditions, utilizing the correlation parameters derived from this investigation. The obtained results show that among eight correlations, the Chrastil correlation reveals the most accurate outcomes with AARD%, 6.71%. Moreover, Figure 5 illustrates the comparison of experimental data and model data.

Correlated Methadone hydrochloride solubility using semi-empirical models. The obtained results are based on (a) KJ, (b) GM, (c) Chrastil, (d) Bartle et al., (e) Sung-Shim, (f) Bian et al., (g) Sodeifian et al., I, and (h) Sodeifian et al., II models.

Similar results from the use of semi-empirical correlations for modeling various drug solubility in SC-CO₂ have been presented in the literature [163-172].

To understand the drug molecule dissolution, drug molecule vaporization from its solid state, and its consequent dissolution into SCF should be studied. So, total and vaporization enthalpies are associated with the dissolution mechanism. The abovementioned were calculated as [173-178]:

(8)

Δ H_{t o t .} (t o t a l e n t h a l p y) = - c_{C h r a s t i l} \times R = 29.14 k J . m o l^{- 1}

(9)

Δ H_{t o t .} (t o t a l e n t h a l p y) = - c_{K J} \times R = 23.04 k J . m o l^{- 1}

(10)

Δ H_{v a p .} (v a p o r i z a t i o n e n t h a l p y) = - c_{B a r t l e e t a l .} \times R = 40.61 k J . m o l^{- 1}

And solvation enthalpy can be obtained based on the following relationship [174, 175]:

(11)

\begin{matrix} Δ H_{s o l .} (s o l v a t i o n e n t h a l p y) = Δ H_{t o t . C h r a s t i l} - Δ H_{v a p . B a r t l e e t a l .} \\ = - 11.47 k J . m o l^{- 1} \end{matrix}

(12)

\begin{matrix} Δ H_{s o l .} (s o l v a t i o n e n t h a l p y) = Δ H_{t o t . K J} - Δ H_{v a p . B a r t l e e t a l .} \\ = - 17.57 k J . m o l^{- 1} \end{matrix}

Other research in the field of drug solubility in SC-CO₂ calculated the mentioned enthalpies based on the coefficients of semi-empirical correlations [179-183].

Finally, regarding to the parameters optimization in EoSs and semi-empirical models, one point must be mentioned that generally, adjustable parameters in EoSs or other models may be determined by different methods such as nonlinear regression methods [184-185].

4. Conclusions

In this communication, Methadone hydrochloride solubility in supercritical CO₂ is considered at various pressures (120 to 270 bar) and isotherms (308-338 K). The outcomes showed the drug solubility in the range of 1.74 × 10^-5 and 4.67 × 10^-5, in mole fraction, and in an equilibrium solubility in the range of 0.97 to 2.89 kg/m³. The experimental data were modeled by eight semi-empirical models (Sung-Shim, Sodeifian et al. (model I and II), Chrastil, GM, KJ, Bian et al. and Bartle et al.) and PC-SAFT EoS. Among the used semi-empirical models, Chrastil was acceptably correlated to Methadone hydrochloride solubility data with the lowest AARD value (AARD = 6.71%), and PC-SAFT EoS AARD value was 9.73%. Thermodynamic features of Methadone hydrochloride were measured ( $Δ H_{t o t .}$ of 23.04 kJ/mol, $Δ H_{v a p .}$ of 40.61 kJ/mol and $Δ H_{s o l .}$ of -11.47 kJ/mol).

Acknowledgment

The authors sincerely would like to thank the deputy of research, University of Kashan regarding the fund (Pajoohaneh # 1403-14) for supporting this valuable and fruitful project. The authors extend their appreciation to Northern Border University, Saudi Arabia, for supporting this work through project number (NBU-CRP-2025-1497).

CRediT authorship contribution statement

Gholamhossein Sodeifian: Conceptualization, Funding acquisition, Investigation, Methodology, Resources, Supervision, Validation, Writing–review & editing. Hamidreza Bagheri: Investigation, Methodology, Software, Writing-original draft. Shima Golshani: Experimental measurement, Resources. Hamidreza Bakhshaei Ghorogh Aghaei: Data curation, experimental measurement. Adel Noubigh: Investigation, Software, Writing-original draft. Mohammadreza Rashidi-Nooshabadi: Resources.

Declaration of competing interest:

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability:

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declaration of generative AI and AI-assisted technologies in the writing process

The authors confirm that there was no use of artificial intelligence (AI)-assisted technology for assisting in the writing or editing of the manuscript and no images were manipulated using AI.

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1. Introduction

2. Materials and Methods

3. Results and Discussion

4. Conclusions

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