Binary and ternary approach of solubility of Rivaroxaban for preparation of developed nano drug using supercritical fluid

Mahshid Askarizadeh; Nadia Esfandiari; Bizhan Honarvar; Seyed Ali Sajadian; Amin Azdarpour

doi:10.1016/j.arabjc.2024.105707

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Original article

04 2024

:17;

105707

doi:

10.1016/j.arabjc.2024.105707

Binary and ternary approach of solubility of Rivaroxaban for preparation of developed nano drug using supercritical fluid

Mahshid Askarizadeh^{^a}, Nadia Esfandiari^{^a,⁎}, Bizhan Honarvar^{^a}, Seyed Ali Sajadian^{^b,^c}, Amin Azdarpour^{^d}

aDepartment of Chemical Engineering, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran

bDepartment of Chemical Engineering, Faculty of Engineering, University of Kashan, Postal Code, 87317-53153, Kashan, Iran

cSouth Zagros Oil and Gas Production, National Iranian Oil Company, Postal Code, 7135717991, Shiraz, Iran

dDepartment of Petroleum Engineering, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran

⁎Corresponding author. esfandiari_n@miau.ac.ir (Nadia Esfandiari)

Received: 2023-10-22, Accepted: 2024-3-3,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Abstract

This study addressed the solubility of Rivaroxaban in supercritical carbon dioxide at a temperature range of 308–338 K and a pressure range of 12–30 MPa with and without a Co-solvent in binary and ternary systems. The impact of ethanol Co-solvent was also examined. Furthermore, the examined systems were modeled using semi-empirical approaches once the tentative solubility data were determined. Rivaroxaban solubility in the binary and ternary systems ranged based on mole fraction from $1.0 \times 10^{- 6}$ to $2.57 \times 10^{- 5}$ and $1.9 \times 10^{- 5}$ to $2.02 \times 10^{- 4}$ , respectively. Based on the results, the use of a Co-solvent can greatly boost the solubility of Rivaroxaban. The highest Co-solvent effect on the Rivaroxaban-Ethanol-CO₂ mixture was observed at 18.73 (338 K and 12 MPa). Furthermore, empirical and semi-empirical models can effectively fit the solubility values of the analyzed materials by AARD% and R_adj for binary and ternary approaches. The Jouyban et al. (AARD%=7.40 and R_adj = 0.993) model for the binary system and the Garlapati-Madras (AARD%=6.16 and R_adj = 0.991) and Sodeifian-Sajadian (AARD%=6.13 and R_adj = 0.979) and Soltani-Mazloumi (AARD%=6.89 and R_adj = 0.987) models for the ternary system are the most accurate models.

Keywords

Rivaroxaban

Solubility

Supercritical carbon dioxide

Semi-empirical modeling

Co-solvent

Nomenclature

a0-a6: Adjustable parameters for density-based models
AARD%: Average absolute relative deviation
C_s: Solute concentration in the collection vial $(g / L)$
e: Co-solvent enhancement effects
M_CO2: CO₂ molecular weight (g/mol)
Ms: Solute molecular weight (g/mol)
M_W: Molecular weight (g/mol)
N: The number of experimental data, dimensionless
n_CO2: Mole of CO₂
n_solute: Moles of solute (RXN)
P: Pressure (MPa)
Pc: Critical pressure (MPa)
P_ref: Reference pressure (0.1 MPa)
Q: The number of self-determining parameters
R²: Correlation coefficient
R_adj: Adjusted correlation coefficient
S: Equilibrium solubility (g/L)
SS_E: Sum square error
SS_T: Total sum of squares
T: Temperature (K)
Tc: Critical temperature (K)
T_m: Melting temperature (K)
y₂: Equilibrium mole fraction
$y_{2}^{'}$: Mole fraction in ternary system
y₃: Mole fraction of Co-solvent
V_s: Volume of the collection vial (L)
V_L: Volume of the sampling loop (L)
Z: Number of adjustable parameters

Cal: Calculated
Exp: Experimental
i, j: Component

2: Solute
i, j: Component

ASES: Aerosol solvent extraction system
BCS: Biopharmaceutics Classification System
cEoS: Cubic equations of state
DMSO: Dimethyl sulfoxide
EC number: European Community number
EoS: Equations of State
GAS: Supercritical gas antisolvent
GRAS: Generally Recognized as Safe
GUM: Guide of uncertainty measurement
HBA: Hydrogen-bond acceptor
HBD: Hydrogen-bond donor
HSP: Hansen solubility parameter
KJ: Kumar and Johnston model
KT: Kamlet-Taft solvent parameters
LFHB: Lattice Fluid Hydrogen Bonding
PGSS: Particles from the gas saturated solution
PC-SAFT: Perturbed-chain SAFT
PCP-SAFT: Perturbed-chain polar SAFT
PR: Peng-Robinson
MST: Méndez-Santiago and Teja model
RESS: Rapid expansion of the supercritical solution
RESOLV: Rapid expansion of a supercritical solution into a liquid solvent
RESSAS: Rapid expansion of supercritical solution into aqueous solutions
RV: Retrograde vaporization
RXN: Rivaroxaban
SA: Simulated annealing
SAS: Supercritical antisolvent
SCF: Supercritical fluid
SC-CO₂: Supercritical CO₂
SEDS: Solution-enhanced dispersion by supercritical fluid
SRK: Soave-Redlich-Kowng

α: H-bond donor
β: H-bond acceptor
π*: Kamlet-Taft dipolarity/polarizability
δ: Hildebrand solubility parameter
λ_max: Maxim wave length (nm)
$ρ_{1}$: Density of SC-CO₂ (kg m⁻³)
$ρ_{ref}$: Reference density (700 kg m⁻³)

1 Introduction

Rivaroxaban (RXN) is the first authorized oral direct factor Xa inhibitor (xabans) and a direct oral anticoagulant. Inhibiting Factor Xa diminishes the activation of coagulation and platelets. RXN can be used to minimize the risk of coronary heart disease and embolism individuals with nonvalvular atrial fibrillation, to prevent and/or treat venous thromboembolism, and to treat bioprosthetic mitral valves. It has emerged as an acceptable alternative to vitamin K antagonists, which are more susceptible to drug-drug interactions and more complicated to administer. However, RXN has an inherent risk of bleeding and can increase the risk of hemorrhage when used with other hemostasis-weakening medications. It is not recommended in pregnant or lactating women, children, or those with severe hepatic (ChildPugh C), renal, antiphospholipid syndrome, or artificial heart valves (Kubitza et al., 2010, Patel et al., 2011, Samama et al., 2013, Thomas et al., 2013, Costa et al., 2020, Duarte et al., 2020, Evans et al., 2020, Fernandez et al., 2021, Galiuto and Patrono, 2021 ). Capell et al. discovered that RXN improved the incidence of thrombotic events, hospitalizations, and deaths among symptomatic outpatients with COVID-19 (Capell et al., 2021).

RXN is categorized as a high-permeability and low-solubility substance by the Biopharmaceutical Classification System (BCS) (Class II) (Mueck et al., 2014, Kushwah et al., 2021). It exhibits low pH-independent solubility in aqueous solution. Xarelto is the commercial brand of RXN, and 685–132-2 is the and EC number (European Community) of RXN, respectively (Seshamamba and Sekaran, 2017, Kushwah et al., 2021).

The bioavailability of drugs is limited by their solubility in aqueous media, which is governed by their dissolution time. Reducing the particle size of drugs that are normally water-insoluble is a typical strategy for enhancing their solubility and dissolution rate (Esfandiari, 2015, Esfandiari and Ghoreishi, 2015a,b, Sodeifian et al., 2020a, Esfandiari and Sajadian, 2022a ). Supercritical carbon dioxide (SC-CO₂)-based particle production technology is a cutting-edge method for creating nano-sized pharmaceuticals. The rapid mass transfer rate and superior dissolving capability of supercritical fluids can be assigned attributed to their viscosities that are more similar to those of gases rather than liquids. Additionally, SC-CO₂ is harmless, colorless, odorless, and leaves no residue in the finished product, further promoting its extensive application in the paramedical industry (Cheng et al., 2018, Ardestani et al., 2020, MacEachern et al., 2020, Pishnamazi et al., 2020a,b, Sodeifian et al., 2020f, Zabihi et al., 2020a, Pishnamazi et al., 2021a ). A supercritical fluid (SCF) is frequently used as a dense solvent or anti-solvent to manufacture therapeutic nanoparticles. The solubility of the medicine in the solvent is one of the prerequisites for employing supercritical technology. In general, SCF can be utilized in particle production processes through three different approaches: (i) SCF as a solvent, such as RESS, RESSAS, and RESOLV; (ii) SCF as an anti-solvent, such as GAS, SAS, SEDS, and ASES; and (iii) SCF as a Co-solvent, such as PGSS and PGSS-drying (Esfandiari and Ghoreishi, 2013, Esfandiari and Ghoreishi, 2014, Esfandiari, 2015, Esfandiari and Ghoreishi, 2015a, Cheng et al., 2018, Sodeifian et al., 2019d, MacEachern et al., 2020, Pishnamazi et al., 2020c,b, Najafi et al., 2021, Pishnamazi et al., 2021a, Esfandiari and Sajadian, 2022a ).

Throughout the last few decades, the estimation of the solubility of medications in SCFs has become one of the main subjects. So far, a few studies have addressed the reliability and correlation of the solubility documentation of different sorts of medications in SCFs. Table 1 sorts the solubility (crossover and mole fraction points of these medicinal compounds in SC-CO₂) of certain medications examined in the years between 2017 and the present. The solubility of solid components in SCFs offers fundamental facts on the development of small-scale medicinal particles with the ideal size dispersion, to achieve better dissolution rates (Ardestani et al., 2020, Saadati Ardestani et al., 2020, Askarizadeh et al., 2023 ). Although numerous experimental techniques can evaluate the solubility of a substance, correlations and mathematical models are frequently implemented to estimate the solubility of the substances in SC-CO₂ due to the high cost of experimental measurements. In general, semi-empirical models and equation of state (EoS)-based models can be used to correlate the solubility data. EoS-based models require complex computational techniques and data for a variety of physical variables (cubic equations of state (Peng-Robinson (PR) (Peng and Robinson, 1976) and Soave-Redlich-Kowng (SRK) (Soave, 1972)) or perturbation equations (perturbed-chain polar SAFT (PCP-SAFT) (Gross, 2005), PC-SAFT (Gross and Sadowski, 2001))). Semi-empirical equations, like density-based models, only require easily accessible control items i.e., temperature, pressure, and the density of CO₂ with no need for thermophysical properties such as molar volume, acentric factor, and critical point, which cannot be estimated (Sodeifian et al., 2019e, Ardestani et al., 2020, Zhan et al., 2020, Zabihi et al., 2021a ). Numerous semi-empirical models have been geared towards connecting the solubility data of solids in SC-CO₂; among which, Kumar and Johnston (KJ) (Kumar and Johnston, 1988), Bartle et al. (Bartle et al., 1991), Khansary et al. (Khansary et al., 2015), Jouyban et al. (Jouyban et al., 2002b), Chrastil (Chrastil, 1982), Adachi-Lu (Adachi and Lu, 1983), Garlapati − Madras (Garlapati and Madras, 2010), González et al. (González et al., 2001), Mendez − Santiago − Teja (MST) (Sauceau et al., 2003), Li et al. (Li et al., 2003), Soltani − Mazloumi (Soltani and Mazloumi, 2017), Reddy − Madras (Reddy and Madras, 2011), Keshmiri et al. (Keshmiri et al., 2014), Bian et al. (Bian et al., 2011), Sodeifian et al. (Sodeifian et al., 2019), Sparks et al. (Sparks et al., 2008), Del Valle and Aguilera (Del Valle and Aguilera, 1988), Tan (Yeoh et al., 2013), Gordillo (Gordillo et al., 1999), Yu (Yu et al., 1994), Sung and Shim (Sung and Shim, 1999) can be mentioned.

Table 1 Review of some published works on the crossover and mole fraction points of various pharmaceutical compound in SC-CO_2.

Compound	Pressure range (MPa)	Temperature range (K)	Cross over (MPa)	Mole fraction (y)	M_W (g/mol)	Ref
Esomeprazole (C₁₇H₁₉N₃O₃S)	12–27	308.2–338.2	22	1.11 × 10^-5 to 9.10 × 10^-4	345.42	(Sodeifian et al., 2019b)
Amiodarone hydrochloride (C₂₅H₂₉I₂NO₃. HCl)	12–30	313.2–343.2	19	2.510 × 10⁻⁵ to 1.012 × 10⁻³	681.77	(Sodeifian et al., 2017b)
Ketotifen fumarate (C₂₃H₂₃NO₅S)	12–30	308.2–338.2	20	2.11 × 10⁻⁵ to 1.07 × 10⁻³	425.5	(Sodeifian et al., 2018a)
Aprepitant (C₂₃H₂₁F₇N₄O₃)	12–33	308.15–338.15	15–18	4.50 × 10⁻⁶ to 7.67 × 10⁻⁵	534.4	(Sodeifian et al., 2017a)
Imatinib mesylate (C₃₀H₃₅N₇O₄S)	12–27	308.2–338.2	18–21	1.0 × 10⁻⁷ to 4.4 × 10⁻⁶	589.71	(Sodeifian et al., 2019e)
Loratadine (C₂₂H₂₃N₂O₂Cl)	12–27	308.15–338.15	18–21	4.50 × 10⁻⁶ to 1.30 × 10⁻³	382.88	(Sodeifian et al., 2018b)
Loxoprofen (C₁₅H₁₈O₃)	12–40	308–338	20	1.04 × 10⁻⁵ to 1.28 × 10⁻³	246.10	(Zabihi et al., 2020a)
Quetiapine hemifumarate (C₂₁H₂₅N₃O₂S.0.5C₄H₄O₄)	12–27	308–338	13–14	0.30 × 10⁻⁶ to 9.03 × 10⁻⁶	441.54	(Sodeifian et al., 2021a)
2,4,7-Triamino-6-phenylpteridine (Triamterene) (C₁₃H₁₃N₇)	12–27	308–338	19.2–19.5	0.03 × 10⁻⁵ to 2.89 × 10⁻⁵	253.26	(Sodeifian et al., 2020a)
Tolmetin (C₁₅H₁₅NO₃)	12–40	308–338	16	5.00 × 10⁻⁵ to 2.59 × 10⁻³	257.29	(Pishnamazi et al., 2020c)
Amlodipine besylate (C₂₆H₃₁ClN₂O₈S)	12–27	308–338	NO	4.15 × 10⁻⁶ to 23 × 10⁻⁶	567.05	(Sodeifian et al., 2021c)
Busulfan (C₆H₁₄O₆S₂)	12–40	308–338	16	3.27 × 10⁻⁵ to 8.65 × 10⁻⁴	246.30	(Pishnamazi et al., 2020b)
Sunitinib malate (C₂₆H₃₃FN₄O₇)	12–27	308–338	NO	0.5 × 10⁻⁵ to 8.56 × 10⁻⁵	532.56	(Sodeifian et al., 2020c)
Fenoprofen (C₁₅H₁₄O₃)	12–40	308–338	16	2.01 × 10⁻⁵ to 4.20 × 10⁻³	242.3	(Zabihi et al., 2020b)
Azathioprine (C₉H₇N₇O₂S)	12–27	308–338	12–15	0.27 × 10⁻⁵ to 1.83 × 10⁻⁵	277.26	(Sodeifian et al., 2020b)
Sorafenib tosylate (C₂₈H₂₄ClF₃N₄O₆S)	12–27	308–338	NO	0.68 × 10⁻⁶ to 12.57 × 10⁻⁶	637.03	(Sodeifian et al., 2020d)
Capecitabine (C₁₅H₂₂FN₃O₆)	10–––35	308.15–––348.15	19	3.18 × 10⁻⁵ to 120.29 × 10⁻⁵	359.35	(Ardestani et al., 2020)
Aspirin (C₉H₈O₄)	10–––30	308.15–––328.15	13–14	0.33 × 10⁻⁴ to 3.45 × 10⁻⁴	180.15	(Ardestani et al., 2020)
Ibuprofen (C₁₃H₁₈O₂)	10–––30	308.15–––333.15	10	0.72 × 10⁻³ to 3.8 × 10⁻³	206.28	(Ardestani et al., 2020)
Repaglinide (C₂₇H₃₆N₂O₄)	12–27	308–338	16–18	2.89 × 10⁻⁶ to 9.53 × 10⁻⁵	452.29	(Sodeifian et al., 2019d)
Sodium Valproate (C₈H₁₅NaO₂)	12–27	308.15–––338.15	22–24	0.05 × 10⁻⁵ to 3.71 × 10⁻⁵	166.19	(Sodeifian et al., 2020f)
Chloroquine (C₁₈H₂₆ClN₃)	12–40	308–338	16–20	1.64 × 10⁻⁵ to 8.92 × 10⁻⁴	319.87	(Pishnamazi et al., 2021a)
Decitabine (C₈H₁₂N₄O₄)	12–40	308–338	16	2.84 × 10^–5 to 1.07 × 10^–3	228.41	(Pishnamazi et al., 2021b)
Oxcarbazepine (C1₅H₁₂N₂O₂)	12–27	308–338	17–19	1.10 × 10^-7 to 2.675 × 10^-5	252.27	(Sodeifian et al., 2019c)
Sulfabenzamide (C₁₃H₁₂N₂O₃S)	12–27	308–338	NO	1.53 × 10^-6 to 22.35 × 10^-6	276.3	(Sodeifian et al., 2021d)
Galantamine (C₁₇H₂₁NO₃)	12–27	308–338	17–19	0.006 × 10⁻⁴ to 0.233 × 10⁻⁴	287.35	(Sodeifian et al., 2021e)
Gliclazide (C₁₅H₂₁N₃O₃S)	10–18.6	308.2–328.2	15–17	1.26 × 10⁻⁷ to 5.01 × 10⁻⁶	323.41	(Wang et al., 2021)
Captopril (C₉H₁₅NO₃S)	10–18.6	308.2–328.2	14–16	3.59 × 10⁻⁶ to 9.32 × 10⁻⁵	217.28	(Wang et al., 2021)
Salsalate (C₁₄H₁₀O₅)	12–40	308–338	16	3.77 × 10⁻⁵ to 3.88 × 10⁻³	258.23	(Zabihi et al., 2021a)
Lansoprazole (C₁₆H₁₄F₃N₃O₂S)	12–27	308.2–338.2	21	1.15 × 10⁻⁵ to 7.36 × 10⁻⁴	369.36	(Sodeifian et al., 2020g)
(Letrozole) (C₁₇H₁₁N₅)	12–36	318.2–348.2	16–18	1.6 × 10⁻⁶ to 8.51 × 10⁻⁵	263.33	(Sodeifian and Sajadian, 2018)
Rivaroxaban (C₁₉H₁₈ClN₃O₅S)	12–27	308–338	22.5	0.0104 × 10⁻⁴ to 0.2062 × 10⁻²	435.90	(Sodeifian et al., 2023e)
Tamsulosin (C₂₀H₂₈N₂O₅S)	12–27	308–338	21	0.18 × 10⁻⁶ to 1.013 × 10⁻⁵	408.05	(Hazaveie et al., 2020)
Gambogic acid (C₃₈H₄₄O₈)	10–30	308.15–328.15	20	1.63 × 10⁻⁶ to 22.62 × 10⁻⁶	628.76	(Xiang et al., 2019)
Tamoxifen (C₂₆H₂₉NO)	12–40	308–338	20	1.88 × 10⁻⁵ to 9.89 × 10⁻⁴	371.51	(Pishnamazi et al., 2020a)
Losartan potassium, Cozaar (C₂₂H₂₂ClN₆O)	12–27	308–338	19	2.03 × 10⁻⁶ to 1.88 × 10⁻⁵	461	(Sodeifian et al., 2021f)
Gatifloxacin (C₃₈H₅₀F₂N₆O₁₁)	12–36	313–333	14	0.106 × 10⁻⁶ to 1.605 × 10⁻⁶	375.4	(Shi et al., 2017, Padrela et al., 2018)
Enrofloxacin (C₁₉H₂₂FN₃O₃₎	17–36	313–333	17	0.022 × 10⁻⁶ to 5.605 × 10⁻⁶	359.4	(Shi et al., 2017, Padrela et al., 2018)
Ciprofloxacin (C₁₇H₁₉ClFN₃O₃)	24–36	313–333	NO	0.0265 × 10⁻⁶ to 0.1887 × 10⁻⁶	331.34	(Shi et al., 2017, Padrela et al., 2018)
Penicillin G (Benzylpenicillin) (C₁₆H₁₈N₂O₄S)	10–35	313.15–333.35	10	0.420 × 10⁻⁵ to 6.330 × 10⁻⁵	334.4	(Gordillo et al., 1999, Padrela et al., 2018)
Lenalidomide (C₁₃H₁₃N₃O₃)	12–30	308–338	18	0.02 × 10⁻⁴ to 1.08 × 10⁻⁴	259.25	(Sajadian et al., 2022a)
Glibenclamide (C₂₃H₂₈ClN₃O₅S)	12–30	308–338	21	0.8 × 10⁻⁶ to 8.03 × 10⁻⁵	494	Esfandiari and Sajadian, 2022b
Montelukast (C₃₅H₃₆ClNO₃S)	12–30	308–338	15	0.4 × 10^-6 to 6.12 × 10^-5	586.18	(Sajadian et al., 2022c)
Minoxidil (C₉H₁₅N₅O)	12–27	308–338	19	0.24 × 10⁻⁶ to 3.39 × 10⁻⁶	209.25	(Sodeifian et al., 2020e)
Ketoconazole (C₂₆H₂₈C_l2N₄O₄)	12–30	308–338	13–15	0.20 × 10^–6 to 8.02 × 10^–4	531	(Sodeifian et al., 2021g)
Sertraline. HCl (C₁₇H₁₇C_l2N. HCl)	12–30	308–338	17–19	0.61 × 10⁻⁴ to 0.89 × 10⁻³	342.69	(Sodeifian et al., 2019)
Favipiravir (C5H₄FN₃O₂)	12–30	308–338	18	3.0 × 10^-6 to 9.05 × 10^-4	157.1	(Sajadian et al., 2022b)
Dasatinib Monohydrate (C₂₂H₂₈ClN₇O₃S)	12–27	308–338	NO	0.45 × 10⁻⁶ to 9.08 × 10⁻⁶	505.16	(Sodeifian et al., 2022h)
Clemastine Fumarate (C₂₁H₂₆ClNO·C₄H₄O₄)	12–27	308–338	NO	1.61 × 10^–6 to 9.41 × 10^–6	460	(Sodeifian et al., 2021b)
Teriflunomide (C₁₂H₉F₃N₂O₂)	12–27	308–338	19.5	8.84 × 10^–5 to 5.43 × 10^–4	270.21	(Sodeifian et al., 2022g)
Metoclopramide hydrochloride (C₁₄H₂₃C_l2N₃O₂)	12–27	308–338	22	0.15 × 10^–5 to 5.56 × 10^–5	336.26	(Sodeifian et al., 2022f)
Pholcodine (C₂₃H₃₀N₂O₄)	12–27	308–338	16–16.5	2.06 × 10^–4 to 5.93 × 10⁻⁴	398.55	(Sodeifian et al., 2022a)
Lacosamide (C₁₃H₁₈N₂O₃)	12–30	308–338	12–18	1 × 10⁻⁶ to 2.29 × 10⁻⁴	250.3	(Esfandiari and Ali Sajadian, 2022)
Febuxostat (C₁₆H₁₆N₂O₃S)	12–27	308–338	21	0.05 × 10^-4 to 7.42 × 10^-4	316.37	Abourehab et al., 2022b; Zabihi et al., 2021b
Paracetamol (C₈H₉NO₂)	9.5–26.5	311–358	11	0.305 × 10^-6 to 16.358 × 10^-6	151.16	(Bagheri et al., 2022)
Methylparaben (C₈H₈O₃)	12–35.5	308–348	15.2	1.13 × 10^-5 to 1.213 × 10^-3	152.16	(Mahesh and Garlapati, 2022)
Ethylparaben (C₉H₁₀O₃)	8–21	308–328	8	1.64 × 10^-6 to 1.755 × 10^-5	166.17	(Mahesh and Garlapati, 2022)
Propylparaben (C₁₀H₁₂O₃)	9.41–22.02	308.15––328.15	14	4.4 × 10^-6 to 6.12 × 10^-5	180.2	(Mahesh and Garlapati, 2022)
Empagliflozin (C₂₃H₂₇ClO₇)	12–27	308–338	16.5	5.14 × 10^–6 to 25.9 × 10^–6	450.91	(Sodeifian et al., 2022c)
Pantoprazole sodium sesquihydrate (C₁₆H₁₄F₂N₃NaO₄S × 1.5 H₂O)	12–27	308–338	16	0.0301 × 10^–4 to 0.463 × 10^–4	432.4	(Sodeifian et al., 2022d)
Prazosin hydrochloride (C₁₉H₂₂ClN₅O₄)	12–27	308–338	NO	1.59 × 10⁻⁵ to 7.2 × 10⁻⁵	419.9	(Sodeifian et al., 2022i)
Temozolomide (C₆H₆N₆O₂)	12–40	308–338	20	4.30 × 10⁻⁴ to 5.28 × 10⁻³	194.1	(Zabihi et al., 2021b)
Cefuroxime axetil (C₂₀H₂₂N₄O₁₀S)	8–25	308–328	Higher than 25 MPa	2.2 × 10⁻⁷ to 11.24 × 10^-6	510.47	(Ongkasin et al., 2019)
Ethosuximide (C₇H₁₁NO₂)	9–15	313.15–328.15	NO	3.45 × 10⁻³ to 8.71 × 10⁻³	141.168	(Zha et al., 2019)
(Octatrimethylsiloxy) Polyhedral oligomeric silsesquioxanes (POSS) C₂₄H₇₂OO₂₀Si₁₆	1–30	308–328	10.6	0.0083 to 2 × 10^-3	1146.18	(Demirtas and Dilek, 2019)
Chlorothiazide (C₇H₆ClN₃O₄S₂)	13–29	308–338	17	0.417 × 10⁻⁵ to 1.012 × 10⁻⁵	295.73	(Majrashi et al., 2023)
Pazopanib hydrochloride (C₂₁H₂₄ClN₇O₂S)	12–27	308–338	NO	1.87 × 10⁻⁶ to 14.25 × 10⁻⁶	474	(Sodeifian et al., 2022b)
Crizotinib (C₂₁H₂₂Cl₂FN₅O)	12–27	308–338	14.5	0.156 × 10⁻⁵ to 1.219 × 10⁻⁵	450.3	(Sodeifian et al., 2022e)
Alendronate (C₄H₁₃NO₇P₂)	12–30	308–338	18	0.01 × 10⁻⁴ to 1.5 × 10⁻⁴	271.08	Abourehab et al., 2022a
Sildenafil citrate (C₂₂H₃₀N₆O₄S)	12–30	308–338	15–18	2.40 × 10⁻⁷ to 6.48 × 10⁻⁶	474.6	(Honarvar et al., 2023)
Riluzole (C₈H₅F₃N₂OS)	12–27	308–338	22	4.95 × 10⁻⁵ to 1.49 × 10⁻⁴	234.2	(Abadian et al., 2022)
Fludrocortisone acetate (C₂₃H₃₁FO₆)	12–30	308–338	18–21	0.211 × 10^-6 to 0.653 × 10^-5	422.5	(Amani et al., 2022)
Metformin (C₄H₁₁N₅)	14–29	308–328	NO	0.39 × 10^-6 to 1.23 × 10^-6	129.16	(Venkatesan et al., 2022)
Haloperidol (C₂₁H₂₃ClFNO₂)	12–22	313.2–323.2	17–19	3.4 × 10^-7 to 1.4 × 10^-5	375.9	(Khudaida et al., 2023a)
Retinol Vitamin A (C₂₀H_3O)	9–23.3	303–323	11	2.18 × 10^-5 to 1.964 × 10^-4	286.45	(Naikoo et al., 2021)
Famotidine (FAM) (C₈H₁₅N₇O₂S₃)	12–30	308–338	18	1.4 × 10^-6 to 1.11 × 10^-4	337.43	(Saadati Ardestani et al., 2023)
Erlotinib hydrochloride (C₂₂H₂₄N₃O₄Cl)	12–30	308–338	19–22	1.2 × 10^-6 to 2.12 × 10^-5	429.9	(Bazaei et al., 2023)
Phemytoin (C₁₅H₁₂N₂O₂)	9.5–25	313–345	11	0.68 × 10^-6 to 15.7 × 10^-6	252.268	(Notej et al., 2023)
Raloxifene (C₂₈H₂₇NO₄S)	9.5–25	313–345	12	0.79 × 10^-5 to 8.09 × 10^-5	473.59	(Notej et al., 2023)
Clonazepam (C₁₅H₁₀ClN₃O₃)	12–30	308–338	20	3.9 × 10^-6 to 7.26 × 10^-5	315.71	(Alwi et al., 2023)
Curcumin (C21H20O6)	8–20	308.15–328.15	13	1.82 × 10^-8 to 1.97 × 10^-6	368.38	(Zhan et al., 2017)
Dibutylbutyl phosphonate (C₁₂H₂₇O₃P)	10–25	313–333	11	0.087 to 0.117	250.31	(Pitchaiah et al., 2017)
Diamylamyl phosphonate (C₁₅H₃₃O₃P)	10–25	313–333	12	0.065 to 0.09	292.4	(Pitchaiah et al., 2017)
Ipriflavone (C₁₈H₁₆O₃)	10–20	308.2–328.2	15	1.4 × 10⁻⁴ to 2.2 × 10⁻⁴	280.3	(Wang and Su, 2020)
Tolbutamide (C₁₂H₁₈N₂O₃S)	10–30	313.15–353.15	17–20	1.66 × 10^–5 to 40.5	270.35	(Manna and Banchero, 2018)
Chlorpropamide (C₁₀H₁₃N₂O₃S)	10–30	313.15–353.15	17–20	2.29 × 10^–6 to 72.2 × 10^–6	276.74	(Manna and Banchero, 2018)
1-aminoanthraquinone (C₁₄H₉NO₂)	12.5–25	323.15–383.15	17	5.5 × 10^–7 to 351 × 10^–7	223.23	(Tamura et al., 2017)
1-nitroanthraquinone (C₁₄H₇NO)	12.5–25	323.15–383.15	18–20	9.8 × 10^–7 to 252.3 × 10^–7	253.21	(Tamura et al., 2017)
Phthalocyanines green (Pc-G)	10–35	308.15–338.15	22	0.01 × 10^-5 to 12.12 × 10^-5	1127.154	(Sodeifian et al., 2019f)
Fampridine (pyridin-4-amine, 4-aminopyridine) (C₅H₆N₂)	10–22	308.2–328.2	10.5–12.5	2 × 10^-5 to 2 × 10^-4	94.11	(Chen et al., 2017)
Vitamin E acetate (α-tocopheryl acetate) (VEA) (C₃₁H₅₂O₃)	8–15	308.15–328.15	NO	2.76 × 10⁻⁴ to 7.26 × 10⁻⁴	472.76	(Han et al., 2017)
Anthraquinone violet 3RN (AV3RN) (C₂₈H₂₀N₂Na₂O₈S₂)	10–34	308–338	10	0.047 × 10⁻⁵ to 0.546 × 10⁻⁵	622.58	(Saadati Ardestani et al., 2020)
Phosphatidylcholine (PC) (C₄₂H₈₀NO₈P)	12.4–17.2	313–353	NO	5.082 × 10⁻⁶ to 11.758 × 10⁻⁶	758.1	(Jash et al., 2020)
Coumarin-7 (C₂₀H₁₉N₃O₂)	9–33	308–338	13–16	0.415 × 10⁻⁵ to 1.009 × 10⁻⁵	333.38	(Sodeifian et al., 2019a)
Vanillin (C₈H₈O₃)	8–28	313–353	16	0.14 × 10⁻³ to 13 × 10⁻³	152.15	(Maqbool et al., 2017)
Phenol (C₆H₆O)	10–35	333–363	28	1.14 × 10⁻³ to 9.064 × 10⁻²	94.11	(Maqbool et al., 2017)
Flufenamic acid (FFA (C1₄H₁₀F₃NO₂)	8–21	313.2–333.2	14	0.8 × 10⁻⁶ to 2.13 × 10⁻⁴	281.23	(Tsai et al., 2017)
Nystatin (C₄₇H₇₅NO₁₇)	12–30	308–338	22	0.40 × 10⁻⁶ to 1.20 × 10⁻⁵	926.1	(Sajadian et al., 2023)
Aripiprazole (C₂₃H₂₇CL₂N₃O₂)	12–30	308–338	18	1.83 × 10⁻⁶ to 1.036 × 10⁻⁵	448.39	(Ansari et al., 2023)
Nifedipine (C₁₇H₁₈N₂O₆)	12.5–27.5	333.15–353.15	18	7.9 × 10⁻⁶ to 53.6 × 10⁻⁶	346.3	(Li et al., 2017)
Quinine (C₂₀H₂₄N₂O₂)	12.5–27.5	323.15–343.15	15–19	12 × 10⁻⁶ to 50.4 × 10⁻⁶	324.4	(Li et al., 2017)
Nilotinib hydrochloride monohydrate (C₂₈H₂₅ClF₃N₇O₂)	12–27	308–338	12–15	0.1 × 10^–5 to 0.59 × 10^–5	584	(Nateghi et al., 2023)
Palbociclib (C₂₄H₂₉N₇O₂)	12–27	308–338	12–15	0.081 × 10^–5 to 2.027 × 10^–5	447.533	(Sodeifian et al., 2023c)
Oxaprozin (C₁₈H₁₅NO₃)	12–40	308–338	NO MEN	3.31 × 10^–5 to 1.24 × 10^–3	293.317	(Alshehri et al., 2022)
lutein (β,ε-carotene-3,3′-diol) (C₄₀H₅₆O₂)	18.7–33.55	313–333	NO	0.82 × 10^–6 to 2.45 × 10^–6	568.89	(Araus et al., 2019)
Metoprolol (C₁₅H₂₅NO₃)	12–30	308–338	18	0.02 × 10⁻⁵to 8.11 × 10⁻⁵	267.36	(Alshahrani et al., 2023)
Chlorpromazine (C₁₇H₁₉ClN₂S)	17–41	308–348	20	3.21 × 10^-5to 5.25 × 10^-5	318.9	(Alharby et al., 2023)
Hyoscine (C₁₇H₂₁NO₄)	17–40	308–348	20	0.79 × 10^-4to 2.83 × 10^-4	303.3	(Hani et al., 2023)
Verapamil (C₂₇H₃₈N₂O₄)	12–30	308–338	12–15	3.6 × 10^–6to 7.14 × 10^–5		(Esfandiari et al., 2023)
Buprenorphine hydrochloride (C₂₉H₄₂ClNO₄)	12–27	308–338	15–18	0.131 × 10^-4to 4.752 × 10^-4	504.1	(Sodeifian et al., 2023a)
Hydroxychloroquine sulfate (C₁₈H₂₈ClN₃O₅S)	12–27	308–338	NO	0.0304 × 10^–5 to 0.5515 × 10^–5	434	(Sodeifian et al., 2023b)
Probenecid (C₁₃H₁₉NO₄S)	15–21	313.2–353.2	15–19	0.13 × 10⁻⁵to 1.45 × 10⁻⁵	285.36	(Khudaida et al., 2023b)
Warfarin (C₁₉H₁₆O₄)	10–18	308.2–328.2	18	1.48 × 10⁻⁶ to 4.32 × 10⁻⁶	434	(Ciou et al., 2018)
Ibrutinib (C₂₅H₂₄N₆O₂)	12–27	308–338	17	3.90 × 10^–6 to 1.30 × 10^–5	440.51	(Sodeifian et al., 2023d)
Sitagliptin phosphate (C₁₆H₁₈F₆N₅O₅P)	12–30	308–338	15–16.5	3.02 × 10^–5 to 6.98 × 10⁻⁵	407.31	(Ardestani et al., 2023)

NO: In the pressure range, the crossover is not exit.

NO MEN: In the article, the crossover is not calculated.

One of the biggest challenges in the development of the SCF process is the limited solubility of polar solutes in SC-CO₂. As the majority of pharmaceuticals are polar molecules, the interaction of carbon dioxide (a nonpolar structure) with medications is limited. Therefore, supercritical CO₂ is employed in combination with other solvents, known as “Co-solvent”, to enhance the solubility. Co-solvents can alter the polarity of the solvent. Consequently, the use of Co-solvents can enhance the solubility in SCFs (polar or non-polar). Moreover, the incorporation of small amounts (less than 10 %) of polar solvents, such as acetone, dimethyl sulfoxide (DMSO), ethanol, menthol, and methanol can significantly increase the solute solubility in SC-CO₂ (Hosseini et al., 2018, Bitencourt et al., 2019, Ardestani et al., 2020, Saadati Ardestani et al., 2020, Sodeifian et al., 2021g ). These Co-solvents can participate in hydrogen bonding with solute molecules and increase the solvation power of a specific supercritical fluid in solvents with lower solvation ability like water. Additionally, the impact of the Co-solvent is related to an improvement in solubility by a rise in solvent density or by intermolecular cooperation between the Co-solvent and the solute. Furthermore, an increase in the specific intermolecular interactions between the Co-solvent and one or more components of the mixed components can enhance the separation selectivity (Knez et al., 2017, Bitencourt et al., 2019, Saadati Ardestani et al., 2020, Zhan et al., 2020 ).

The Co-solvent effect is primarily influenced by heightened intermolecular interactions and solvent density. In systems with multiple components, solubility can be significantly increased, but selectivity remains unaffected if the increase is solely due to higher solvent mixture density. The density input to the Co-solvent impact is affected by pressure, temperature, and the addition of a Co-solvent, which can lead to the formation of clusters of SCF molecules around it, boosting overall density. The greatest density increase is observed near the critical point of the solvent mixture. An increase in pressure decreases clustering, increases SCF density, and decreases density differences between the SCF and SCF mixture, ultimately crossing density isotherms. The addition of a Co-solvent can strengthen the SCF solvent while decreasing the molar density of the solvent. Key factors influencing the Co-solvent effect include various physical interactions like dipole-induced dipole, dipole–dipole, and induced dipole-induced dipole (dispersion), as well as specific interactions like charge transfer and H-bonding complexes. A comprehensive understanding of the Co-solvent effect requires a thorough comprehension of intermolecular interactions between solvents and solutes (Prausnitz et al., 1999, Güçlü-Üstündağ and Temelli, 2005, Cui et al., 2018, Li et al., 2018, Pitchaiah et al., 2018, Peyrovedin and Shariati, 2020, Matin et al., 2022, Sajadian et al., 2023 ).

The Hildebrand solubility parameter (δ) and the Hansen solubility parameter (HSP) are commonly used to assess suitable solvents for specific applications based on similar solubility parameters. HSP categorizes molecular interactions into dispersion, hydrogen-bond, and polar contributions, making it applicable to both polar and non-polar mixtures. Kamlet-Taft solvent parameters (KT) like α (H-bond donor) (HBD), β (H-bond acceptor) (HBA), and solvent dipolarity/polarizability (π^*) help evaluate total solvent polarity. The entertainer effect enhances selectivity and solubility through specific intermolecular interactions like H-bonding between Co-solvent and solutes. When selecting a binary mixed-solvent, the one with higher KT-acidity is the HBD solvent, and the local composition of the HBD-HBA pair influences KT-parameters of complex molecules. In SC-CO₂, non-aqueous and aqueous HBD-HBA solvent pairs act as Co-solvents, interacting with polar solutes and CO₂. The HBD-HBA complex molecule impacts selectivity by specific interactions with solutes and CO₂ affinity (CO₂ philicity). Adjusting the HBD-HBA Co-solvent composition can enhance basicity, CO₂ philicity, and specific interactions for solute dissolution in the SC-CO₂ phase (Güçlü-Üstündağ and Temelli, 2005, Cui et al., 2018, Duereh and Smith, 2018, Li et al., 2018, Pitchaiah et al., 2018 ).

This research is aimed at understanding the solubility of RXN in SC-CO₂ with or without ethanol as a Co-solvent. The static equilibrium test conditions involve the pressure range of 12, 15, 17, 21, 24, 27 and 30 $M P a$ and temperatures of 308, 318, 328, and 338 K. The solubility of RXN in SC-CO₂ was experimentally studied to investigate the impacts of the operational factors such as temperature, pressure, and the presence of a Co-solvent. The density models of Jouyban et al., Soltani-Mazloumi, Méndez-Santiago-Teja (MST), Sodeifian-Sajadian, González et al., Garlapati–Madras, Chrastil, Kumar and Johnston (KJ), Bian et al. and Bartle et al. were used to correlate the solubility data of RXN in binary and ternary procedures. The model parameters were established, and the average absolute relative deviation (AARD (%)) was also utilized to evaluate the prediction effectiveness of the method.

2 Experiments

2.1

2.1 Materials

Rivaroxaban was acquired from Tofigh Darou drug company (Tehran, Iran) with a purity of 99 %. Additionally, further information regarding other components including carbon dioxide and ethanol can be found in Table 2. The structure of Rivaroxaban is provided in Table 3.

Table 2 The sources and purity of the materials used in this work.

Material	Source	Initial mass fraction (Purity)	Final mass fraction Purity	Analysis method
Rivaroxaban	Tofigh Daru Research & Engineering Co.	0.99	0.99	HPLC^a
Ethanol	Merck Co.	0.999	0.999	GC^b
Carbon dioxide	Novin Oxygen Co.	0.9999	0.9999	GC

aHigh-performance liquid chromatography.

bGas chromatography.

2.2

2.2 Experimental apparatus

Based on Fig. 1, the experimental pilot plant was equipped with a spectrophotometer and included a CO₂ tank, an air compressor (Finac, China), a high-pressure pump (Haskel pump, Burbank CA 91502, USA), a refrigeration machine, a magnetic stirrer with 100 rpm, a filter, flow control valves like a needle valve, a back-pressure valve, a metering valve, an equilibrium cell, and an oven (Memert, Germany). All the components of this high-pressure unit, including the pipeline and fittings, have a diameter of 1/8 in. and are made of 316 stainless steel. The impurities in the CO₂ flow from the tank were eliminated by passing through a molecular filter with a pore size of 1 µm. The flow then reached the cooling unit where the CO₂ flow liquefied due to the low interior temperature of the refrigerator (∼-15 °C). From the pressure of the CO₂ tank, the liquid CO₂ enters the high-pressure pump at a pressure of about 6 MPa. A manometer and transmitter were used to evaluate the pressure with an accuracy of 0.1 MPa.

Schematic diagram of experimental apparatus used for measuring solubility.

The drug was homogenized in SC-CO₂ using a magnetic stirrer in a 300-mL cell to achieve a lab balance cell (binary system). In the case of a ternary system or a Co-solvent approach, 3000 mg of the RXN was added to the cell with a certain amount (3 mol %) of ethanol as a Co-solvent. The temperature control was achieved using an oven. A sintered filter (1 $μ m$ ) was placed to keep the RXN in place on either side of the cell. Before being fed to the cell, CO₂ was compressed to an appropriate pressure. Based on the preliminary test, the static time was 120 min. Saturated SC-CO₂ (600 $μ L$ ± 0.6 % $μ m$ ) was inserted into the injection loop using a three-valve two-position device (shown by V₁-V₃) after 120 min. Upon rerouting the injection valve, the loop can be depressurized into the collecting vial to keep a particular amount of ethanol (solvent). This mechanism is summarized in Fig. 1 in four cases. After the static time, the loop should be filled. So, the valve (V₁) has been opened. After filling the loop, V₁ is closed. Next, V₂ is opened. The loop goes to the collection vial, then V₃ is opened and all the lines and the loop are rinsed with ethanol (1 mL). Also, in the picture, the green and black colors show the open and closed positions of the valves, respectively. This process was repeated three times for every data point and each system.

The solution was collected in a container with a final volume of 5 mL (±0.2 %). Each test was repeated multiple times. The absorbance was measured spectrophotometrically using a Jenway UV-V equipped with a quartz cell, at a maximum wavelength ( $λ_{m a x}$ ) of 249 nm, to monitor the solubility. By utilizing the alignment bend (with a correlation coefficient of 0.989) and the UV absorbance, the solubility was calculated based on the solute concentration. The calibration curve and the linear relationship of the regression data over a wide concentration range confirmed the suitability of the method. Additionally, the melting point was estimated using the DSC test, as shown in Fig. 2. The calculation method for pressure fluctuation during sampling was also presented in the supplementary information, Table (S1).

The equilibrium mole fraction y₂ and solubility, $S (g / L$ of RXN in SC-CO₂ were calculated at various pressure and temperature levels as follows (Sodeifian et al., 2023e):

(1)

y_{2} = \frac{n_{solute}}{n_{solute} + n_{{CO}_{2}}}

(2)

n_{solute} = \frac{C_{s} (\frac{g}{L}) \times V_{s} (L)}{M_{s} (\frac{g}{mol})}

(3)

n_{{CO}_{2}} = \frac{V_{l} (L) \times ρ (\frac{g}{L})}{M_{{CO}_{2}} (\frac{g}{mol})}

where C_s interprets the RXN concentration

(g / L)

in the collecting flask as determined by the standardized curve. n_solute and n_CO2 also depict the moles of solute (RXN) and CO₂ in the sampling loop, respectively. The volumes of collecting vial and sampling loop were V_s

(L)

and V_l

(L)

, respectively. The solute molecular weight is also shown by M_s

(g / m o l)

, while M_CO2 stands for the molecular weight of carbon dioxide Eq. (4) was used to determine S

(g / L)

, which is the concentration of RXN in the SC-CO₂.

(4)

S (\frac{g}{L}) = \frac{C_{s} (\frac{g}{L}) \times V_{s} (L)}{V_{l} (L)}

As a result, Eq. (5) is produced by combining Eq. (2) and (3) with Eq. (1):

(5)

y_{2} = \frac{C_{s} (\frac{g}{L}) \times V_{s} (L) \times M_{{CO}_{2}} (\frac{g}{mol})}{C_{s} (\frac{g}{L}) \times V_{s} (L) \times M_{{CO}_{2}} (\frac{g}{mol}) + V_{l} (L) \times ρ (\frac{g}{L}) \times M_{s} (\frac{g}{mol})}

2.3

2.3 Empirical and semi-empirical density-based models

Several empirical and semi empirical models can be used to determine the solubility of solids (drugs) in SC-CO₂. For the binary system in this study, the density-based models of Chrastil, Bian et al., Jouyban et al., Bartle et al., Mendez − Santiago − Teja (MST), and Kumar-Johnston (KJ) were adopted. Concerning the ternary approach with ethanol (Co-solvent), the Garlapati–Madras, Sodeifian-Sajadian, MST, González et al., Soltani-Mazloumi, and Jouyban et al. models were used. Empirical and semi-empirical models were employed to establish the connection between RXN solubility. Tables 4.a and 4.b provide a summary of the equations applied in binary and ternary approaches respectively and Table 5 outlines the parameters employed in the models.

Table 4 A summary of the binary semi-empirical and empirical models used in this work Table 4. b Summary of the Ternary semi-empirical and empirical models used in this work.

Model	Formula/ explain	constant	ref
Chrastil Semi-empirical	$\ln y_{2} = a_{0} + a_{1} l n ρ_{1} + \frac{a_{2}}{T}$	3	(Chrastil, 1982)
Chrastil Semi-empirical	An equation describes the formation of a solvate complex AB_k in a system where one unit of solute A combines with k units of solvent B. It highlights a correlation between solubility and density in a supercritical fluid, as well as a relationship between solubility and temperature. However, Chrastil's equation has limitations, such as being unsuitable for solubility levels above 100–200 kg m⁻³ and lacking validity across a wide range of temperatures. This model is designed for pure fluids and can be applied in mixtures with consistent Co-solvent mole fractions, assuming these mixtures behave like pure fluids at constant concentrations. Overall, the model provides a macroscopic view of the molecular environment in the fluid phase without requiring knowledge of the solute's properties.	3	(Sauceau et al., 2003, Hojjati et al., 2007, Sparks et al., 2008, Kostrzewa et al., 2019 )
Bartle et al. Semi-empirical	$l n (\frac{y_{2} P}{P_{ref}}) = a_{0} + \frac{a_{1}}{T} + a_{2} (ρ_{1} - ρ_{ref})$	3	(Bartle et al., 1991)
Bartle et al. Semi-empirical	This model illustrates the relationship between solubility and solvent density. The correlation is expressed in a linear manner using the enhancement factor of the solute with respect to the density of the solvent. By fitting the correlation to experimental data, the coefficients a₀, a₁, and a₂ can be determined. The parameter a₂ is particularly useful in estimating the heat of vaporization of the solute, H_vap (H_vap = − a₂R). By utilizing the values of H_total and H_vap, the heat of solvation can be estimated for each solute-CO₂ system. The Bartle model, which includes an individual pressure term, is expected to provide more reliable correlated results for solubility data at different pressures.	3	(Hojjati et al., 2007, Sparks et al., 2008)
Mendez − Santiago − Teja (MST) Semi-empirical	$T l n (y_{2} P) = a_{0} + a_{1} ρ_{1} + a_{2} T$	3	(Sodeifian et al., 2018a)
Mendez − Santiago − Teja (MST) Semi-empirical	This model utilizes the principles of dilute solutions and employs the algorithm of the Henry constant of solute in a supercritical fluid. Within this theoretical framework, the enhancement factor is ascertained by the solvent's density, leading to straightforward equations for a range of thermodynamic properties of dilute near-critical binary mixtures. Additionally, this model allows for the normalization of data across varying temperatures.	3	(Sauceau et al., 2003, Hojjati et al., 2007, Sparks et al., 2008 )
Jouyban et al. Empirical	$\ln y_{2} = a_{0} + a_{1} P + a_{2} P^{2} + a_{3} P T + \frac{a_{4} T}{P} + a_{5} \ln (ρ_{1})$	6	(Jouyban et al., 2002a, 2002b)
Jouyban et al. Empirical	The solubilities of organic solids in SC-CO₂ can be accurately predicted using the empirical model developed by Jouyban et al. This model takes into account the interplay between solute mole fraction, linear pressure, and temperature, allowing for the estimation of solubility data that has not been measured. Additionally, it can be used to identify any outliers in experimental solubility data, providing valuable insights into the behavior of these systems.	6	(Jouyban et al., 2002a, Sridar et al., 2013)
Bian et al. Empirical	$y_{2} = {ρ_{1}}^{a_{0} + a_{1} ρ} e x p (\frac{a_{2} + a_{3} ρ_{1}}{T} + a_{4})$	5	(Bian et al., 2011)
Bian et al. Empirical	The density-based empirical model proposed by Bian et al. offers a comprehensive understanding of the solubility of compounds in SC-CO₂. It accounts for the intricate interplay between solubility and density of the supercritical fluid at varying temperatures and pressures. Additionally, it considers the correlation between solubility and temperature under isopycnic conditions, as well as the impact of temperature and pressure on the association number. This model is derived from Chrastil's equation.	5	(Sridar et al., 2013)
Kumar and Johnston (KJ) Semi-empirical	$\ln y_{2} = a_{0} + a_{1} ρ_{1} + \frac{a_{2}}{T}$	3	(Kumar and Johnston, 1988)
Kumar and Johnston (KJ) Semi-empirical	In 1988, a thermodynamic formalism was introduced to explain the connection between the solubility of a nonvolatile solute in a SCF and the density of the fluid phase. This model suggests that the logarithm of the solute's mole fraction in the fluid phase shows a nearly linear relationship with either the logarithm or the density of the SCF phase in the vicinity of the critical point, depending on the specific system. The slope of this linear correlation is determined by both the partial molar volume of the solute in the SCF phase and the isothermal compressibility of the fluid. Through the analysis of solubility data from existing literature, scientists have been able to calculate partial molar volumes using this framework, and these calculated values are in good agreement with independently measured data.	3	(Kumar and Johnston, 1988, Yan et al., 2022)

Model	Formula	constant	Ref
Mendez−Santiago−Teja (MST) semi-empirical	$T l n (\frac{y_{2}^{'} P}{P_{ref}}) = a_{0} + a_{1} ρ_{1} + a_{2} T + a_{3} y_{3}$	4	(Méndez-Santiago and Teja, 1999)
Mendez−Santiago−Teja (MST) semi-empirical	A correlation with four adjustable parameters was derived by combining the Mendez-Santiago and Teja equation with a Clausius-Clapeyron-type equation and including sublimation pressure. This correlation is used to assess the impact of density, temperature, and Co-solvent composition on the solubility of the ternary system.	4	(Sauceau et al., 2003)
Sodeifian-Sajadian semi-empirical	$l n (y_{2}^{'}) = {(a}_{0} + \frac{a_{1} ρ_{1}}{T}) l n (ρ_{1}) + a_{2} ρ_{1} + a_{3} l n (y_{3} P)$	4	(Sodeifian et al., 2019c)
Sodeifian-Sajadian semi-empirical	Four experimental data points were selected as the minimum requirement from the collected data sets to train the proposed model for determining the solubilities of organic solids in SC-CO₂ when a Co-solvent is present. The development of this model is based on the works of González et al. and Chrastil models.	4	(Rojas et al., 2023)
González et al. semi-empirical	$l n (y_{2}^{'}) = a_{0} l n (ρ_{1}) + a_{1} l n (y_{3}) + \frac{a_{2}}{T} + a_{3}$	4	(González et al., 2001)
González et al. semi-empirical	González and colleagues introduced a thermodynamic model based on the Chrastil model, utilizing the mass-action law to predict solute solubility in non-entrained supercritical fluids. This model has shown effectiveness, especially in systems where the presence of an entrainer boosts solute solubility significantly, particularly in cases with strong solute-entrainer interaction. The model incorporates the logarithmic dependence of solubility on fluid density along with an exponential relationship between solubility and Co-solvent concentration. It is built on the assumption of cluster or solvate complex formation involving the solute, entrainer, and solvent, which is consistent with the observed decrease in solute solubility with temperature. Therefore, the model may not accurately forecast solubility in systems where the Co-solvent only serves as a Co-solvent for CO₂, lacking the entrainer effect that enhances both solubility and extraction selectivity.	4	(González et al., 2001)
Soltani-Mazloumi Empirical	$l n (y_{2}^{'}) = a_{0} + \frac{a_{1}}{T} + \frac{a_{2}}{T} ρ_{1} - a_{3} l n (P) + a_{4} \ln (y_{3} ρ_{1} T)$	5	(Soltani and Mazloumi, 2017)
Soltani-Mazloumi Empirical	Soltani Mazloumi is an innovative experimental framework that incorporates five parameters to forecast solid solubility in supercritical carbon dioxide with the presence of a Co-solvent. This model considers various input data, including temperature, pressure, and density correlations. It is important to highlight that this model is derived from Hozhabr et al.'s model, showcasing a linear relationship between ln $y_{2}^{'}$ and ln P, a nonlinear association between ln $y_{2}^{'}$ ' and temperature as well as density, a linear correlation between ln $y_{2}^{'}$ and a linear correlation between ln $y_{2}^{'}$ ' and ln $y_{3}$ (Co-solvent mole fraction).	5	(Soltani and Mazloumi, 2017)
Garlapati–Madras semi-empirical	$l n (y_{2}^{'}) = a_{0} + a_{1} l n (ρ_{1}) + a_{2} ρ_{1} + \frac{a_{3}}{T} + a_{4} l n (T) + a_{5} l n (y_{3}) + a_{6} l n (y_{3} ρ_{1} T)$	7	(Saadati Ardestani et al., 2020)
Garlapati–Madras semi-empirical	In 2010, the Garlapati–Madras equation was developed with seven constants, inspired by the model introduced by Jouyban et al. This equation is used to establish a relationship between the solubilities of high molecular weight solids in SC-CO₂, with or without Co-solvents, considering temperature, the density of SC-CO₂, and the mole fraction of Co-solvent.	7	(Garlapati and Madras, 2010)
Jouyban et al. Empirical	$\ln (y_{2}^{'}) = a_{0} + a_{1} y_{3} + a_{2} ρ_{1} + a_{3} P^{2} + a_{4} P T + \frac{a_{5} T}{P} + a_{6} l n ρ_{1}$	7	(Jouyban et al., 2002b)
Jouyban et al. Empirical	The training of the proposed model to predict the solubilities of organic solids in SC-CO₂, considering the presence of a Co-solvent, utilizes a minimum of six experimental data points from the collected data sets. To estimate solubility at various temperatures and pressures, an interpolation technique was employed. This correlation provides numerous benefits, such as a simple calculation procedure and higher accuracy in comparison to other empirical equations and equations of state.	7	(Jouyban et al., 2002b)

Table 5 Descriptions of the parameters in empirical and semi-empirical models used in this work.

Parameter	Description	System
$y_{2}$	Mole fraction (RXN + SC-CO₂)	Binary system
$y_{2}^{'}$	Mole fraction (RXN + SC-CO₂ + Ethanol)	Ternary system
$y_{3}$	The mole fraction of Co-solvent	Ternary system
$a_{0} - a_{5}$	Adjustable parameters	Binary system
$a_{0} - a_{6}$	Adjustable parameters	Ternary system
$ρ_{1}$	Density of SC-CO₂ (kg m⁻³)	Binary & Ternary system
$ρ_{r e f}$	Reference density (700 kg m⁻³)	Binary system
P_ref	Reference Pressure (0.1 MPa)	Binary & Ternary system
P	System Pressure (MPa)	Binary & Ternary system
T	System Temperature (K)	Binary & Ternary system

Empirical and semi-empirical models' constants were assessed using experimental information. Variable parameters were also fine-tuned using the simulated annealing (SA) algorithm in MATLAB software. The AARD% was also applied to evaluate the accuracy of the model.

(6)

AARD % = \frac{100}{N_{i} - Z} \sum_{i = 1}^{N_{i}} \frac{|y_{2}^{calc} - y_{2}^{\exp}|}{y_{2}^{\exp}}

Z and N_i represent the number of modifiable parameters for any demonstration and the number of information focuses in any set or model, respectively (Sajadian et al., 2022b). R_adj was also considered to compare different models (Jouyban et al., 2002a, Garlapati and Madras, 2010):

(7)

R_{adj} = \sqrt{|R^{2} - (\frac{Q (1 - R^{2})}{N - Q - 1})|}

Each set of data contained N data points. Moreover, Q denotes the number of self-determining

parameters of each equation. The R² correlation coefficient was also used to compare different models:

(8)

R^{2} = 1 - \frac{S S_{E}}{S S_{T}} = 1 - \frac{\sum {(y_{2 e x p} - y_{2})}^{2}}{{\sum (y_{2 e x p})}^{2} - \frac{{(Σ y)}^{2}}{N}}

Where

{S S}_{E}

represents the sum square error and

{S S}_{T}

denotes the total sum of squares (Sodeifian et al., 2019e).

3 Results and discussion

3.1

3.1 Solubility data systems- role of Co-solvent

The experimental solubilities of RXN in SC-CO₂ with and without ethanol Co-solvent (ternary and binary) were explored experimentally at 308–338 K and 12–30 MPa, as reported in Tables 6 and 7. The Span-Wanger EoS was used to determine the SC-CO₂ density (Span and Wagner, 1996). Additionally, each test was reassessed three times to enhance the precision, and the relative standard uncertainties fell below 5 %. The uncertainty of solubility was determined according to the guide of uncertainty measurement (GUM) proposed by the joint committee for guides in metrology (2008). Figs. 3 and 4 present the RXN mole fraction solubility vs. pressure and density at various temperatures for binary and ternary systems, respectively. In 2022, Sodeifian et al. studied RXN in binary mode. According to Table 6, the data difference is lower than 9 % (Sodeifian et al., 2023e). The supplementary information data consists of Tables S2 and S3, containing the tabulated information necessary for the calculation and analysis of ethanol (Co-solvent) content, mixture density, and CO₂ mass within the mixture.

Table 6 The experimental data of RXN solubility in SC-CO₂ based on distinct conditions.

Temperature^a (K)	Pressure^a (MPa)	Density^b (kg/m³)	Binary
Temperature^a (K)	Pressure^a (MPa)	Density^b (kg/m³)	y₂ × 10⁵ (Mole Fraction)	Experimental standard deviation, S(y) × (1 0 ⁵)^c	S × 10 (Solubility (g/l))	Expanded uncertainty of mole fraction (1 0 ⁵), Uc^d
308	12	768.42	0.405	0.007	0.031	0.011
	15	816.06	0.564	0.011	0.046	0.014
	18	848.87	0.745	0.018	0.063	0.020
	21	874.4	1.102	0.031	0.095	0.032
	24	895.54	1.248	0.041	0.111	0.043
	27	913.69	1.677	0.063	0.152	0.064
	30	929.68	1.924	0.080	0.177	0.081
318	12	659.73	0.256	0.006	0.017	0.011
	15	743.17	0.458	0.013	0.034	0.016
	18	790.18	0.676	0.022	0.053	0.023
	21	823.7	1.042	0.038	0.085	0.039
	24	850.1	1.389	0.059	0.117	0.060
	27	872.04	1.81	0.065	0.156	0.066
	30	890.92	2.163	0.087	0.191	0.088
328	12	506.85	0.171	0.003	0.009	0.012
	15	654.94	0.359	0.007	0.023	0.012
	18	724.13	0.593	0.014	0.043	0.017
	21	768.74	0.971	0.027	0.074	0.029
	24	801.92	1.57	0.054	0.125	0.056
	27	828.51	1.991	0.048	0.163	0.050
	30	850.83	2.396	0.069	0.202	0.071
338	12	384.17	0.102	0.003	0.004	0.016
	15	555.23	0.239	0.009	0.013	0.014
	18	651.18	0.48	0.019	0.031	0.021
	21	709.69	0.836	0.023	0.059	0.025
	24	751.17	1.71	0.055	0.127	0.056
	27	783.29	2.164	0.078	0.168	0.079
	30	809.58	2.572	0.007	0.206	0.019

aStandard uncertainty u are u(T) = 0.1 K; u(P) = 0.1 MPa.

bData from the Span–Wagner equation of state.

cThe experimental standard deviation and the experimental standard deviation of the mean (SD) were calculated by

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

and

S D (\bar{y}) = \frac{S (y_{k})}{\sqrt{n}}

respectively.

dThe relative combined standard uncertainty was obtained by

U_{combined} / y = \sqrt{\sum_{i = 1}^{N} {(P_{i} U (x_{i}) / x_{i})}^{2}}

. The expanded uncertainty (0.95 level of confidence) U is

k \times U_{combined}

Table 7 The experimental data of RXN solubility in SC-CO₂ – Ethanol based on distinct conditions. (3 mol % ethanol){, 2008 #1058}.

Temperature^a (K)	Pressure^a (MPa)	Density (kg/m³)	Ternary
Temperature^a (K)	Pressure^a (MPa)	Density (kg/m³)	y^'₂ × 10⁴ (Mole Fraction)	Experimental standard deviation, S (y ^') × 10^5b	Expanded uncertainty of mole fraction (1 0 ⁵), Uc ^c	Mass ethanol (gr)	e (Co-solvent effect)
308	12	769.05	0.439	0.0755	0.080	7.463	10.84
	15	815.18	0.582	0.1164	0.122	7.926	10.32
	18	846.85	0.753	0.1807	0.187	8.244	10.11
	21	871.44	0.999	0.2797	0.286	8.492	9.07
	24	891.764	1.087	0.3594	0.365	8.698	8.71
	27	909.18	1.208	0.451	0.457	8.874	7.20
	30	924.51	1.34	0.5539	0.56	9.029	6.96
318	12	663.14	0.328	0.0787	0.082	6.407	12.81
	15	744.53	0.528	0.1542	0.158	7.218	11.53
	18	790.14	0.763	0.2442	0.249	7.674	11.29
	21	822.57	1.031	0.3712	0.377	8.000	9.89
	24	848.04	1.208	0.5154	0.251	8.256	8.70
	27	869.17	1.441	0.5188	0.527	8.469	7.96
	30	887.32	1.603	0.6412	0.649	8.653	7.41
328	12	512.60	0.26	0.0458	0.050	4.923	15.20
	15	658.45	0.442	0.0884	0.093	6.361	12.31
	18	726.00	0.708	0.1699	0.176	7.033	11.94
	21	769.36	1.004	0.2811	0.288	7.466	10.34
	24	801.51	1.45	0.5027	0.510	7.788	9.24
	27	827.21	1.566	0.3758	0.388	8.047	7.87
	30	848.74	1.794	0.5191	0.531	8.263	7.49
338	12	390.45	0.191	0.0611	0.065	3.731	18.73
	15	560.44	0.387	0.1393	0.142	5.393	16.19
	18	654.77	0.652	0.2608	0.264	6.324	13.58
	21	711.93	1.079	0.3021	0.310	6.893	12.91
	24	752.30	1.635	0.5232	0.533	7.296	9.56
	27	783.47	1.846	0.6646	0.674	7.607	8.53
	30	808.92	2.022	0.8088	0.819	7.863	7.86

^b The experimental standard deviation and the experimental standard deviation of the mean (SD) were calculated by $S (y_{k}) = \sqrt{\frac{\sum_{j = 1}^{n} {(y_{i} - y)}^{2}}{n - 1}}$ and $S D (\bar{y}) = \frac{S (y_{k})}{\sqrt{n}}$ respectively.

aStandard uncertainty u are u(T) = 0.1 K; u(P) = 0.1 MPa.

cThe relative combined standard uncertainty was obtained by

U_{combined} / y = \sqrt{\sum_{i = 1}^{N} {(P_{i} U (x_{i}) / x_{i})}^{2}}

. The expanded uncertainty (0.95 level of confidence) U is

k \times U_{combined}

The RXN solubility in the binary system.

The RXN solubility in the ternary system.

In general, an increase in pressure enhances the density of SC-CO₂ and its solvation power. Consequently, the solubility of RXN in SC-CO₂ increases with pressure increment at constant temperature in both systems. Analysis of RXN solubility in binary and ternary systems indicates a significant increase in the solubility of RXN in the presence of a Co-solvent (ethanol). According to the generally recognized as safe (GRAS) designation, ethanol is an ideal Co-solvent for food applications (Güçlü-Üstündağ and Temelli, 2005). The maximum and minimum effects of the Co-solvent are 18.73 (338 K and 12 MPa) and 6.96 (308 K and 30 MPa), respectively, as determined by comparing the data and calculating the solubility enhancement (e) due to the efficacy of ethanol Co-solvent on the RXN solubility in SC-CO₂ (Eq. (9) (Araus et al., 2019, Sodeifian et al., 2021).

(9)

e = \frac{y_{2}^{'}}{y_{2}} * 100 = \frac{molefractionofTernary ({CO}_{2} + E t h a n o l)}{molefractionofbinary ({CO}_{2})} * 100

In addition, as illustrated in Fig. 5, the impact of ethanol on solubility is identified. The addition of a Co-solvent results in a solubility increase ranging from 7 to 19 % in the ternary system when compared to the binary system.

The influence of Co-solvent (ethanol) on solubility of RXN in SC-CO2. a. pressure, b. density.

CO₂ cannot be employed as a suitable solvent for most medicinal chemicals due to its low polarity. Moreover, hydrophobic and polar molecules are insoluble in SC-CO₂. Therefore,

variant Co-solvents have been introduced to enhance the solubility of drugs in SC-CO₂. Ethanol is miscible with SC-CO₂ and has shown a high dissolving capacity for numerous chemicals. Thus, ethanol can be employed as a Co-solvent in SC-CO₂ systems to improve the dissolving capability (Knez et al., 2017, Cheng et al., 2018).

Small concentrations of Co-solvents can be used to increase the solvation power of SC-CO₂. The effect of the Co-solvent was determined based on its concentration in the supercritical phase, which can be influenced by the phase and the treatment of the combination. To determine the influence of the Co-solvent, solvent-co-solvent combinations in a supercritical state (completely miscible) should be used (Güçlü-Üstündağ and Temelli, 2005, Pitchaiah et al., 2018). The key factor in the effect of the Co-solvent involves an increase in solvent density and intermolecular interactions. In multi-component systems, solubility can be improved either selectively or non-selectively. Selectivity does not increase in situations where the rise in insolubility is the result of an increment in the density of the solvent mixture. An increase is expected in both solubility and selectivity in the case of a specific intermolecular interaction between the Co-solvent and one of the solutes (such as H-bonding) (Güçlü-Üstündağ and Temelli, 2005, Cui et al., 2018, Li et al., 2018 ).

The impact of Co-solvent on solvent density, and therefore its contribution to the Co-solvent effect, varies depending on temperature and the specific Co-solvent. The inclusion of a Co-solvent increases the overall density of the supercritical fluid (SCF) by raising the density of the Co-solvent and causing SCF molecules to cluster around it. These density variations are particularly noticeable near the critical point of the solvent mixture, where the densities of both the Co-solvent and the SCF fluctuate the most, resulting in the highest degree of clustering. As the pressure increases, the density of the SCF mixture increases while the clustering decreases, causing the density isotherms to intersect. It should be noted that the addition of a Co-solvent, such as a hydrocarbon with a large molar volume, can enhance the solvation power of the SCF while reducing its molar density (Güçlü-Üstündağ and Temelli, 2005, Li et al., 2018).

A proper understanding of the Co-solvent outcome demands sufficient cognition of the intermolecular interactions between the solutes and solvents. The Co-solvent effect is mostly influenced by solute-co-solvent physical interactions such as dipole–dipole, dipole-induced dipole, and induced dipole-induced dipole (dispersion) interactions, as well as more specialized interactions such as H-bonding and charge transfer complexes (Prausnitz et al., 1999, Güçlü-Üstündağ and Temelli, 2005, Cui et al., 2018, Peyrovedin and Shariati, 2020, Matin et al., 2022 ).

H-bonding could be a donor–acceptor action and reaction, including H-bond- donating and accepting atoms. H-bonds are formed when the electronegativity of the H-bond donor is sufficiently high to draw electrons, partially exposing the protons. The acceptor atom has to possess lone-pairs or polarizable electrons to form a bond with the donor species. Functional groups could serve as acceptors (e.g., C = O), donors, or both (e.g., OH). As the most prevalent cases in essence and chemistry, moderate H-bonds are generated between neutral donors and acceptors like –OH and O = C (Jeffrey and Jeffrey, 1997). In the mixture of SC-CO₂ and ethanol, ethanol forms H-bonds for weak binding to CO₂ as a result of quadrupole-dipole interaction. Furthermore, the hydrogen bonding between RXN and this Co-solvent declined chemical potential, offering additional solute molecules to the supercritical phase (Güçlü-Üstündağ and Temelli, 2005, Araus et al., 2019, Ardestani et al., 2020 ).

Several articles have investigated crossover pressure and proposed some methods to predict the crossover pressure region (Adachi and Lu, 1983, Chimowitz et al., 1988, Del Valle and Aguilera, 1988, de Melo et al., 2009, Tabernero et al., 2010, Budkov et al., 2019, Kalikin et al., 2020, Kalikin et al., 2021 ). Correlation of the crossover pressure for the ternary system has been presented by Johnston et al.(Adachi and Lu, 1983) and Chimowitz et al.(Chimowitz et al., 1988). The crossover pressure was related to the enthalpy of sublimation and the partial molar enthalpy of the solute in the supercritical phase. The locations of the lower and upper crossover pressures were determined at the point where the partial molar enthalpy equals the negative of the enthalpy of sublimation. Johnston et al. (1987) applied the Peng-Robinson EoS with a binary interaction parameter regressed from a single experimental point to evaluate the partial molar enthalpy of the solute for determining the crossover points. Chimowitz et al. (1988) used a perturbed hard-sphere model EoS to correlate the crossover pressure for binary and ternary systems. Both of these methods require the P-y-T data to allow the prediction of the crossover point. Kalikin et al. (Kalikin et al., 2021) investigated the solubility of a set of poorly soluble drugs, which have been computed in a wide area of the phase diagram, based on the classical density functional theory. They found that the wider the temperature region of the experimental study, the more pronounced the effect of the crossover points drift. They also estimated solubility values using in situ IR spectroscopy and molecular dynamics simulations along the mentioned isochores and isotherms, respectively. Furthermore, they believed that the critical parameters, sublimation pressure, and molar volume of the compound play a crucial role in the determination of the crossover pressure (Tabernero et al., 2010). De Melo et al. investigated the Peng-Robinson-LCVM-UNIFAC equation and the effect of any uncertainty of some solid pure component properties on the upper crossover pressure. It is shown that the slope of the sublimation pressure curve plays a major role in the accuracy of the upper crossover pressure. To sum up, the crossover region depends on the critical properties of solutes, sublimation pressure, enthalpy of sublimation, partial molar enthalpy, and molar volume of the solute.

As suggested by the chemical structure of RXN (Table 3), its OH, NH, and C = O groups increase its polarity. Accordingly, the solubility of RXN in a high-density solvent will be higher than in a low-density solvent for both binary and ternary systems. At high pressures, SCF behaves like a liquid. Regarding the higher solvation power of liquids compared to gases, RXN is more soluble in solvents with higher density (see Fig. 4). Temperature is another key factor in the solubility of RXN. Based on Figs. 3 and 4, the solubility of RXN in both binary and ternary systems increased with constant pressure, due to temperature elevation. The binary and ternary systems showed crossover pressures. Concerning the solubility of solid materials in supercritical fluids, there is a well-known phenomenon called “retrograde vaporization” (RV), in which a rise in temperature at steady pressure causes a decline in solubility (Kalikin et al., 2021). The limits of this area are illustrated by two positions where all isotherms intersect and the plot of solubility vs. temperature shows extrema; the pressure values corresponding to these extrema are known as the lower and upper crossover pressures. In the case of binary and ternary systems, the crossover pressures roughly reside at 24 MPa and 21 MPa, respectively. The temperature of the system shows different impacts on RXN solubility at pressures higher and lower than the crossover pressure. The effects of temperature enhancement on solubility may vary due to the temperature dependence of the density of the solvent and the vapor pressure of the solute.

3.2

3.2 Analyzed solubility data correlation of two system

RXN solubility data were compared in two systems using ten empirical density-based models (Bian et al., Jouyban et al., Chrastil, Bartle et al., Mendez − Santiago − Teja (MST), González et al., Kumar, and Johnston (KJ), Sodeifian–Sajadian, Soltani-Mazloumi and Garlapati-Madras). The RXN solubility in SCF was correctly correlated by all correlations, as shown by the AARD% and Radj values. Using the calculated customizable parameters, the offered models may be employed to predict RXN solubility in binary and ternary modes at definite pressures and temperatures. The solubility data were correlated with high precision using the acquired adjustable parameters. Tables 8 and 9 list the parameters of the empirical and semi empirical model in the binary RXN-CO₂ system and the ternary RXN-Ethanol-CO₂ system, respectively. The correlation outcomes of each approach are displayed in Figs. 6 and 7. The AARD% of each model in both systems is shown in Fig. 8. Additionally, Jouyban et al. (AARD%=7.40 and R_adj = 0.993) is the most accurate model in the binary system. As seen, all models provide proper accuracy in ternary models, while the Garlapati-Madras (AARD%=6.16 and R_adj = 0.991) and Sodeifian-Sajadian (AARD%=6.13 and R_adj = 0.979) and Soltani-Mazloumi (AARD%=6.89 and R_adj = 0.987) models for the ternary system are the most accurate models.

Table 8 The correlation results of the RXN-CO₂ system provided by semi-empirical models.

Model	$a_{0}$	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	AARD%	R_adj
Chrastil	8.794	−5323.289	−43.942	–	–	–	16.75	0.989
KJ	−2.449	0.009	−5352.72	–	–	–	12.42	0.989
Bian et al.	−4.736	0.003	−2324.664	−3.713	17.185	–	8.05	0.986
Bartle et al.	16.928	−7878.695	0.014	–	–	–	16.79	0.986
MST	4.797	−1.182 $\times$ 10^-4	18.672	–	–	–	15.93	0.990
Jouyban et al.	–33.695	0.002	−0.004	9.689 $\times$ 10^-5	0.100	2.062	7.40	0.993

Table 9 The correlation results of the RXN-Ethanol-CO₂ system provided by the semi-empirical models.

Model	$a_{0}$	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	$a_{6}$	AARD%	R_adj
MST	−1920.282	3.510	16.855	−2.517 $\times$ 10^-5	–	–	–	12.37	0.986
González et al.	5.450	−2.867	−4.498 $\times$ 10⁴	−41.694	–	–	–	12.17	0.985
Sodeifian-Sajadian	−2.360	−0.482	0.018	0.695	–	–	–	6.13	0.979
Soltani-Mazloumi	4.608	−5194.273	1.885	−0.386	−0.495	–	–	6.89	0.987
Garlapati–Madras	−93.651	−6.345	0.011	−26.658	10.571	−8.462	3.027	6.16	0.991
Jouyban et al.	−50.732	−1.899	−0.007	−0.002	5.083 $\times$ 10^-4	0.008	6.70	9.39	0.989

Comparison of experimental (points) and calculated (line) solubilities of RXN in the binary system: (a) Chrastil, (b) KJ., (c) Bian et al (d) Bartle et al, (e) MST (f) Jouyban et al. models at various temperatures.

Comparison of experimental (points) and calculated (line) solubilities of RXN in the ternary system (3 mol % ethanol): (a) MST, (b) González et al., (c) Sodeifian-Sajadian, (d) Soltani-Mazloumi. (e) Garlapati–Madras and (f) Jouyban et al. models at various temperatures.

Comparing ARRD percentages between Binary and Ternary models.

4 Conclusion

The current research explored the RXN solubility in two systems (binary: SC-CO₂ and ternary: SC-CO₂ with Co-solvent (ethanol) at different pressures (12–30 MPa) temperatures (308–338 K). The RXN solubility in the binary and ternary systems ranged based on mole fraction from $1.0 \times 10^{- 6}$ to $2.57 \times 10^{- 5}$ and from $1.9 \times 10^{- 5}$ to $2.02 \times 10^{- 4}$ , respectively. The solubility values of RXN were correlated by various semi-empirical equations based on the corresponding equilibrium. Accordingly, the incorporation of ethanol Co-solvent considerably improved the RXN solubility due to dipole–dipole and dipole-induced dipole interactions between the Co-solvent and RXN. The highest Co-solvent effect (18.73) was identified at 12 MPa and 338 K, further confirming this hypothesis. The highest RXN solubility (y'₂ = 2.02 $\times$ 10^-4) at 30 MPa and 338 K, was recorded in a system with ethanol Co-solvent. Furthermore, according to the AARD% and R_adj values of the empirical and semi-empirical approaches, Jouyban et al. model for binary system and Garlapati-Madras, Sodeifian-Sajadian and Soltani-Mazloumi models can properly correlate the ternary system at examined temperatures and pressures.

CRediT authorship contribution statement

M.A.: Methodology, Writing- Original draft preparation, Data curation and Software, Reviewing and Editing. N.E.: Methodology, Writing- Original draft preparation, Conceptualization, Investigation, Validation, funding acquisition, Reviewing and Editing. B.H.: Validation, Methodology, Investigation, Reviewing. S.A.S.: Conceptualization, Project administration, Software, Supervision, Reviewing and Editing. A.A.: Investigation, Writing, Reviewing and Editing.

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.

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Appendix A

Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.arabjc.2024.105707.

Appendix A

Supplementary data

The following are the Supplementary data to this article:

Supplementary Data 1

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Introduction

Experiments

Materials
Experimental apparatus
Empirical and semi-empirical density-based models

Results and discussion

Solubility data systems- role of Co-solvent
Analyzed solubility data correlation of two system

Conclusion

CRediT authorship contribution statement

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Binary and ternary approach of solubility of Rivaroxaban for preparation of developed nano drug using supercritical fluid

Abstract

Keywords

Rivaroxaban

Solubility

Supercritical carbon dioxide

Semi-empirical modeling

Co-solvent

Nomenclature

Superscript

Subscripts

Abbreviations

Greek symbols

1 Introduction

2 Experiments

2.1 Materials

2.2 Experimental apparatus

2.3 Empirical and semi-empirical density-based models

3 Results and discussion

3.1 Solubility data systems- role of Co-solvent

3.2 Analyzed solubility data correlation of two system

4 Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

References

Appendix A

Supplementary data

Appendix A

Supplementary data

Supplementary Data 1

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