Studies on solubility measurement of codeine phosphate (pain reliever drug) in supercritical carbon dioxide and modeling

In this study, the solubilities of codeine phosphate, a widely used pain reliever, in supercritical carbon dioxide (SC-CO2) were measured under various pressures and temperature conditions. The lowest determined mole fraction of codeine phosphate in SC-CO2 was 1.297 × 10−5 at 308 K and 12 MPa, while the highest was 6.502 × 10−5 at 338 K and 27 MPa. These measured solubilities were then modeled using the equation of state model, specifically the Peng-Robinson model. A selection of density models, including the Chrastil model, Mendez-Santiago and Teja model, Bartle et al. model, Sodeifian et al. model, and Reddy-Garlapati model, were also employed. Additionally, three forms of solid–liquid equilibrium models, commonly called expanded liquid models (ELMs), were used. The average solvation enthalpy associated with the solubility of codeine phosphate in SC-CO2 was calculated to be − 16.97 kJ/mol. The three forms of the ELMs provided a satisfactory correlation to the solubility data, with the corresponding average absolute relative deviation percent (AARD%) under 12.63%. The most accurate ELM model recorded AARD% and AICc values of 8.89% and − 589.79, respectively.


OF
Objective function Q Equation (32)

S Saturation
The importance of supercritical fluids (SCFs) as solvents in various processes has been recognized for decades 1,2 .Significant applications of SCFs encompass particle sizing, extraction, reactions, and separations 3 .SCFs serve as solvents in all of these applications [3][4][5][6] .However, it is essential to note that while theoretically, all substances can attain a supercritical state, some necessitate exceedingly high pressures and temperatures to achieve this state, rendering it impractical and resource-intensive [7][8][9][10] .Carbon dioxide is a well-known substance that readily reaches its supercritical state with minimal effort [11][12][13] .Consequently, CO 2 as an SCF is extensively documented in the literature [14][15][16] .The sizing of drug particles, whether at the micro or nano level, primarily depends on their solubility [17][18][19][20] .The desired drug particle size can be achieved by rapidly expanding supercritical solutions (RESS) or anti-solvent processes [21][22][23] .The size of a drug particle can play a crucial role in treating various illnesses, as it significantly influences bioavailability [24][25][26][27][28] .Therefore, determining solubility is a fundamental step in micronization/nanonization.While recent literature reports the solubility of codeine phosphate in conventional solvents, information regarding its solubility in SCFs is notably absent [29][30][31] .Hence, this study focuses on measuring the solubility of codeine phosphate in supercritical carbon dioxide (SC-CO 2 ) under various conditions.A modeling task is also undertaken to facilitate the application of the acquired data.
Several methodologies are available in the literature for modeling solubility data; however, only three are considered user-friendly [32][33][34] .The first method involves the use of the Equation of State (EoS), which requires critical properties of both the solute (the drug) and the solvent (SC-CO 2 ).The second method relies on semiempirical models, often referred to as density-based models, which necessitate data on the density of the solvent, as well as temperature and pressure data.The final method is the solid-liquid equilibrium model, also known as the expanded liquid model (ELM), which requires information about the solute's enthalpy of melting and the solute's melting temperature [35][36][37][38] .To obtain the required properties such as critical temperature, critical pressure, acentric factor, molar volume, and sublimation pressures, standard group contribution techniques are employed [39][40][41] .However, there are instances where the application of group contribution methods becomes challenging due to the absence of functional group contributions, such as phosphate and sulfates.Applying EoS modeling and the solid-liquid equilibrium model can prove challenging [42][43][44][45] .Codeine phosphate, an analgesic drug, exemplifies such a compound where critical properties ( T c and P c ), molar volume(v 2 )and sublimation pressures are unavailable, and existing group contribution techniques cannot be applied due to the presence of phosphate in its structure.However, experimental data for the melting temperature (155 °C) and the heat of fusion (18.86 cal/g or 78.91 J/g or 31,358.83J/mol) of codeine phosphate are readily accessible [46][47][48] .The magnitude of codeine phosphate's solubility in SC-CO 2 determines the technique employed for drug micronization/ nanonization using SC-CO 2 .

Equipment details
Static equipment was employed for solubility measurements, as depicted in Fig. 1.Comprehensive equipment details can be found in our previous studies [49][50][51] .This section offers a concise explanation of the experimental setup and methodology.Thermodynamically, the measurement method falls under the category of isobaricisothermal methods 52 .Throughout the experiments, temperatures and pressures were rigorously controlled at the desired experimental conditions with a precision of ± 0.1 K for temperature and ± 0.1 MPa for pressure, respectively.Solubility measurements were conducted in triplicate for each data point.In each measurement, a known quantity of codeine phosphate drug (1 g) was utilized, and after reaching equilibrium, the saturated sample was collected through a 2-position 6-way port valve into a vial preloaded with demineralized water (DM water).After discharging 600 µL of saturated SC-CO 2 the port valve was rinsed with 1 ml of DM water, resulting in a total saturation solution volume of 5 ml.
The drug solubility values were measured by absorbance assays at max (281 nm) on a UNICO-4802 UV-Vis spectrophotometer with 1-cm pass length quartz cells and the solubility was calculated from the concentration of solute by using the calibration curve (with regression coefficient 0.999) and the UV-absorbance, Fig. 2.
For solubility calculations, the following equations were employed: Moreover, these quantities are defined as follows: In the above relations, C s is defined as the drug concentration in saturated sample vial in g/L.Also, the volume of the sampling loop and vial collection are expressed as V 1 (L) = 600 × 10 -6 m 3 and V s (L) = 5 × 10 -3 m 3 , respectively.The M s and M CO 2 are the molecular weights of the codeine phosphate drug (component 2) and CO 2 , respectively.
Solubility can be also expressed as: where, one can find the relation between S and y 2 as follows: Codeine phosphate's solubility was determined using a UV-visible spectrophotometer (Model UNICO-4802, double beam, with multipurpose software, USA), with DM water as the solvent.

Modeling
The solubility data obtained in this study were correlated with one equation of state (EoS), five density-based models, and three ELM models.we considered the Peng-Robinson (PR) EoS.In the case of density-based modeling, several well-known models, namely Chrastil, Mendez-Santiago, Teja (MT), Bartle et

EoS modeling
This model is an extension to the model framework suggested by Schmitt 53 and Reid and Estévez et al. 54 .PR EoS was used for the modeling.Solubility of codeine phosphate (solute, component 2) in SC-CO 2 (solvent, component 1) is expressed as 55 where P s 2 , φSC−CO 2

2
, φS 2 P , v 2 , T and R, are sublimation pressure, solid solute fugacity coefficient, saturation fugacity coefficient, system pressure, drug molar volume, system temperature and universal gas constant, respectively.The required equation for the solid solute fugacity coefficient in the SC-CO 2 ( φSC−CO 2

2
) is calculated using PR EoS.It is obtained from the following thermodynamic equation.
(1) For modeling tasks, critical temperature, critical pressure, centric factor, molar volume, and sublimation pressures of the codeine phosphate are required.Unfortunately, they are unavailable for this typical drug.Therefore, to overcome this drawback, the following assumptions are applied.

Assumption 2
The molar volume of solute ( v 2 ) is a function of SC-CO 2 (solvent) density ( ρ 1 ) 56 and in this work the following expression is used where K 1 , K 2 , K 3 have units are m 3 /mol, m 6 /mol kg, m 9 /mol kg 2 , respectively.

Assumption 3
The sublimation pressure of the solute is expressed as a function of temperature, and it is expressed as Eq. ( 11) 55 where β , γ and � sub δ are sublimation pressure expression coefficients.They are substituting Eqs. ( 9)-( 11), in Eq. ( 6), results in the solubility model based on PR EoS in terms of pressure, temperature, density, and some adjustable parameters.The adjustable parameters are a 2 , b 2 , β , γ,� sub δ,K 1 ,K 2 and K 3 .These parameters are treated as temperature-independent in the temperature range considered in the present work.The adjustable parameters are obtained by regression with experimental data.
For the data regression, the objective function, Eq. ( 12), is used 57 where y exp 2i is the experimental mole fraction of solute, and y calc 2i is the model predicted mole fraction of solute.

Density-based modeling
Chrastil model [58][59][60] Solute concentration and solvent density are related as follows: where c m is the mass concentration of solute,ρ m1 is the mass concentration of solvent, and κ,A 1 and B 1 are model constants.
Equation (1) can be rearranged to mole fraction as follows: where M ScF , M Solute and c m /M solute are molar mass of SCF, molar mass of solute, and molar concentration of solute ( c ), respectively.Also,ρ m1 /M ScF and κ are molar concentration of solvent(ρ 1 ), and association number, respectively.Furthermore, A 1 and B 1 are model constants.
Mole fraction ( y 2 ) and mole ratios are related as follows: (7) where κ,A 1 and B 1 are the model constants and their units are dimensionless, dimensionless and K, respectively.
Méndez-Santiago and Teja (MT) model 61 This model can generally be used for checking thermodynamic consistency.It is stated as Eq. ( 19) and when T ln y 2 P − C 2 T vs. ρ 1 is established, all data points fall around a single straight line where A 2 , B 2 and C 2 are the model constants and their units are K, K m 3 /kg and dimensionless, respectively Bartle et al. model 62 According to the model, the solubility is expressed as Eq. ( 20) where the pressure ( P ref ) and density for reference states ( ρ ref ) are considered 0.1 MPa, and 700 kg m −3 .Also, A 3 , B 3 and C 3 are the model constants and their units are dimensionless, K and m 3 /kg, respectively.From the constant B 3 , sublimation enthalpy can be obtained (i.e., sub H = −B 3 R J/mol).

Sodeifian et al. model 63
According to this model, the solubility is represented by Eq. ( 21) where A 4 , B 4 , C 4 , D 4 , E 4 and F 4 are the model constants and their units are dimensionless, K/MPa 2 , dimensionless, m 3 /Kg, 1/MPa and K, respectively.
Reddy-Garlapati model 64 According to the model, the solubility is expressed as Eq. ( 22) where A 5 B 5 , C 5 , D 5 , E 5 and F 5 are the model constants and all are dimensionless quantities;P r is reduced pressure and T r is reduced temperature.

Expanded Liquid Models (ELMs)
This section deals with models under the solid-liquid equilibrium model (also known as ELMs).It relies on the solution theory, where SC-CO 2 was considered an expanded liquid with infinite dissolved codeine phosphate.
The essential solubility expression is given by [65][66][67][68] where γ ∞ 2 is the activity coefficient of solute at infinite dilution, f S 2 , f L 2 are fugacity of codeine phosphate compound in the solid phase and expanded liquid phase, respectively.The basic equation for the fugacity ratio is represented by where C p implies the difference between the heat capacity of solid and expanded liquid states.When Eqs. (23  and 24) are combined, the solubility expression for ELM is obtained as Eq. ( 25) The solubility expression may be estimated with and without C p term.In the following section, three cases are presented.For all three cases, a unique expression for γ ∞ 2 used 23,69 was exp l 1 + l 2 (p/(RT) + l 3 (p/(RT)) 2 ).Case 1. C p = 0.The solubility expression for this case is written as Thus Eq. ( 26) has three maximum parameters (l 1 , l 2 and l 3 ).Case 2. C p = contant .Consider the constant C p is D 23 .The solubility expression for this case iswritten as Thus Eq. ( 27) has four maximum of parameters ( D , l 1 , l 2 and l 3 ) and respective units are J/mole K, dimensionless, J/mole MPa and J 2 /mole 2 MPa 2 , respectively.
Generally,C p it is a third-order polynomial equation in temperature equation; however, a recent study on solubility modeling shows that a good fit is achieved with the second-order polynomial.Thus, it is assumed that the C p quadratic function in temperature as Eq. ( 28) 70 Integral evaluation of Eq. ( 25) by substituting Eq. ( 28) results in Eq. ( 29) Thus, the solubility expression for this case is written as In Eq. ( 30), six parameters are there and they are β 1 ,β 2 ,β 3 , l 1 , l 2 and l 3 and their units are J/K kg, J/K 2 kg, J/ K 3 kg, dimensionless, J/mole MPa and J 2 /mole 2 MPa 2 , respectively.These parameters are optimally fitted to experimental solubility data by minimizing the error with the help of the objective function defined in Eq. (12).It is also important to note that all three expressions for solubility are explicit functions of composition.

Results and discussion
The present study reports the measured solubilities of codeine phosphate (C 18 H 21 NO 3 ) in supercritical carbon dioxide (SC-CO 2 ) at temperatures of 308, 318, 328, and 338 K, spanning a pressure range of 12-27 MPa.Three types of models mentioned in the previous section were used in data correlation.The correlation task was carried out in MATLAB 2019® using the inbuilt fminsearch algorithm.The optimization algorithm minimized the error and was used for parameter estimation for all the models mentioned in the previous section.The measured data are shown in Table 2.The solvent density was obtained from the NIST database 71 .Considering the order of magnitude of codeine phosphate solubility in SC-CO 2 , supercritical anti-solvent methods can be regarded an appropriate choice for producing fine particles of this drug.
The solubility of codeine phosphate in SC-CO 2 vs. pressure is depicted in Fig. 3. From the solubility plot, it is evident that a cross-over pressure is not observed for codeine phosphate.Since conducting experimental investigations at each required condition (pressure and temperature) is tedious, modeling becomes necessary.Therefore, modeling was performed in all three modes.Numerous equations of state (EoS) are available in the literature for modeling solubility data.However, the PR EoS was selected in this work due to its success in modeling the solubilities of solid substances in supercritical fluids (SCFs) [53][54][55]70 . Whe correlating the data, the PR EoS model parameters were treated as temperature-independent over 308-338 K.The objective function indicated in Eq. ( 12) was utilized for data correlation, and all the adjustable parameters were obtained through regression with experimental data.Table 3 presents the correlation constants of the PR EoS model.Sublimation enthalpies at 308, 318, 328, and 338 K were calculated from the vapor pressure expression constants using the following relation: Table 2. Solubility of crystalline codeine phosphatein SC-CO 2 at various temperatures and pressures.The (P i u(x i )/x i ) 2 .In this research, u(x i ) was considered as standard uncertainties of temperature, pressure, mole fraction, volumes and absorption.P i , sensitivity coefficients, are equal to the partial derivatives of y equation (Eq. 1) with respect to the x i .a Standard uncertainty u are u(T) = ± 0.1 K; u(p) = ± 1 bar.The value of the coverage factor k = 2 was chosen on the basis of the level of confidence of approximately 95 percent for calculating the expanded uncertainty.The estimated sublimation enthalpies are presented in Table 3.The correlating ability of the equation of state (EoS) method is depicted in Fig. 3.
When considering density-based models for data correlation, the Chrastil model (Eq.18), treated constants as independent variables, and their values were determined through regression with experimental data.The obtained constants are reported in Table 4.The correlating ability of the Chrastil model is illustrated in Fig. 4. Reasonable fit is observed when the data is represented as y 2 versus ρ 1 , this confirms the applicability of the Chrastil model to the solubility data 72,73 .From the parameters of the Chrastil model, the total enthalpy for codeine phosphate was derived, and its value is reported in Table 5.
The results for data fitting of the MT model (Eq.19) are presented in Fig. 6, and the corresponding parameters are reported in Table 4.The correlating ability of the MT model is evident in Fig. 5, where linear plots are observed when the data is plotted as T K ln y 2 • P − C 2 T versus ρ 1 (Fig. 6), this further confirms the suitability of the MT model for the solubility data 72,73 .Similarly, the model proposed by Bartle et al. (Eq.20) was correlated with solubility data, and the obtained results are reported in Table 4. Linear plots are also observed when the data is  Estimated Sublimation Enthalpies at T (K) Sublimation Enthalpy (kJ/mol) Estimated using    72,73 .From the parameters of the Bartle et al. model, the vaporization enthalpy was determined, and its value is reported in Table 5.
The solvation enthalpy was computed using the values of total and vaporization enthalpies, and the computed solvation enthalpy values are reported in Table 5. Notably, there is good agreement between the calculated average sublimation enthalpies from the PR EoS model (59.78 kJ/mol, as derived from Tables 3 and 5) and the calculated sublimation enthalpies from the Bartle et al. model (60.91 kJ/mol, as derived from Table 5).This suggests that using the PR EoS method in this study can yield meaningful correlation constants.However, the PR EoS accuracy decreases as the temperature increases from 308 to 338 K, possibly due to the temperature dependency of adjustable parameters.Figure 8 depicts the data fitting achieved using the Sodeifian and Reddy-Garlapati models.www.nature.com/scientificreports/Three forms of expanded liquid models, precisely Eqs.(26), and (30), underwent evaluation with experimental data using the objective function mentioned in Eq. ( 12).Among these models, Eq. ( 30), which possesses the highest number of parameters, strongly agrees with the experimental data.Table 6 presents all the parameters associated with the expanded liquid models, and Fig. 9 shows the data correlation capabilities of these models.
The quality of data fit is contingent upon the number of parameters employed in the model.The Akaike Information Criterion (AIC) and the corrected AIC (AICc) are utilized to discern the optimal model.AIC c is computed based on AIC [74][75][76][77] , mathematical criteria commonly employed for assessing the compatibility of a solubility model with the corresponding solubility data.In statistics, these criteria compare solubility models and determine whichbest fits the data.AIC is appropriate when the data set comprises more than 40 data points, whereas AIC c is preferred when the data set contains fewer than 40 data points 75,76 .The following is relation between AIC and AIC c .Additionally, the adjustable or mode parameters may be determined by different algorithms or methods such as nonlinear regression models 78,79 .
In Eq. ( 32), N represents the number of experimental data points, Q denotes the adjustable constants of the model, and AIC is defined as the sum of N ln SSE N &2Q , where SSE stands for the sum of squared error.Table 7 displays all the computed values, revealing that Eq. ( 30) exhibits the lowest AIC c value, establishing it as the most suitable model for the given data.
The best model has the lowest AIC c value.The six-parameter ELM model is identified as the optimal choice, while based on AIC c , the Chrastil model exhibits a weaker correlation than the other models considered in this study.

2 n− 1 .
experimental standard deviation was obtained by S y k = n j=1 (yj−y) Expanded uncertainty (U) and the relative combined standard uncertainty (u combined /y) are defined, respectively, as follows: (U) = k*u combined (k = 2) and u combined y = N i=1

Table 1 .
Molecular structure and physiochemical properties of used materials.

Formula Structure M W (g/mol) λ max (nm) CAS number Minimum purity Mass fraction
where n drug and n CO 2 represent the moles of codeine phosphate and CO 2 , respectively.

Table 3 .
Correlation constant of EoS model.

Table 4 .
Correlation constant of density-based models.

Table 5 .
Thermodynamic parameters of codeine phosphate-SC-CO 2 system.d Obtained as a result of