A novel interfacial thermodynamic model for predicting solubility of nanoparticles coated by stabilizers

Kai Ge ,Yuanhui Ji, *,Xiaohua Lu

1 Jiangsu Province Hi-Tech Key Laboratory for Biomedical Research,School of Chemistry and Chemical Engineering,Southeast University,Nanjing 211189,China

2 Key Laboratory of Material and Chemical Engineering,Nanjing Tech University,Nanjing 211816,China

Keywords:Nanoparticles Solubility Gibbs energy of the interface PC-SAFT Molecular parameters

ABSTRACT To improve the stability of nanoparticles in aqueous solution,polymer or surfactant,etc.are often added in solutions during the preparation process of nanoparticles,which can induce new interfaces that influence the solubility of nanoparticles.In this work,a novel interfacial thermodynamic model for describing the Gibbs energy of the nanoparticles coated by stabilizers was proposed to predict the solubility of nanoparticles.Within the developed model,the activity coefficient of nano metal system was determined by Davies model and that of nano drug system by Perturbed-Chain Statistical Associating Fluid Theory(PC-SAFT).The Gibbs energy of the interface was established as a function of molecular parameters via the application for nano metal system.Furthermore,the model was further used to predict the solubility of nano drugs itraconazole,fenofibrate,and griseofulvin.It was found that the Gibbs energy of the interface plays an important role especially when the radius of nano metal is less than 40 nm,and the developed model can predict the solubility of nano drug with high accuracy in comparison with the experimental data as well as predict the changing trend of solubility of nano drugs that increases as the particle size decreases.Meanwhile,the stabilization mechanism of stabilizers on nano drugs was studied which provided theoretical guidance for the selection of polymer or surfactant stabilizer.These findings showed that the developed model can provide a reliable prediction of the solubility of nanoparticles and help to comprehend the stabilization mechanism of the stabilizers on nano drugs with different particle sizes,which is expected to provide important information for the design of nano drugs formulations.

1.Introduction

Recent advancements in the special properties of nanoparticles,such as large specific surface area,excellent optical properties,great catalytic performance,and high solubility,result in an increased number of researches in materials,catalysis,and medicine [1–3].The melting point of nanoparticles with particle size effect and substrate effect has been carefully studied to control the stability of the nanocatalysts [4].However,the solubility of nanoparticles has not been studied in detail,which is an important property that has been of great concern in many applications.The supersaturation calculated by the solubility can provide guidances for the selection of preparation conditions of nanomaterial prepared by precipitation method [5,6].In the field of biomedicine,nanoparticles are widely used as tools to treat and detect diseases.For example,nano metal(e.g.Au)and nano metal oxide(e.g.MnO2,SiO2) were applied in drug delivery systems [7,8]while magnetic nanoparticles (e.g.Fe3O4,MnO) were used in magnetic resonance contrast agents [9].The release of these nanoparticles in vivo remains unclear and may have a cytotoxicity effect,which is in relation to solubility significantly.Furthermore,researchers focus on the nanocrystallization of poorly water-soluble drugs [10],which can greatly improve their solubility,dissolution rate,and bioavailability because of decreased particle size and increased surface area.Nevertheless,the experimental determination of the solubility of nanoparticles faces more severe challenges compared with that of the micronized or larger particles.While the method is widely used that the solubility is determined by separating dissolved and undissolved components,its applicability for nanosystems is limited since the insoluble nanoparticles often cannot be completely separated,the dissolution equilibrium cannot be confirmed,and the results are difficult to repeat[11].In-situ measurement methods with second-derivative UV spectroscopy have been proposed,which can obtain the dissolution rate accurately but overestimate the equilibrium concentration [12].Light scattering and turbidity are reliable methods with many application limitations that it is necessary to eliminate the scattering produced by the stabilizer and keep the sample pure [13].Consequently,it is essential to develop a reliable theoretical model to predict the solubility of nanoparticles.

The most famous equation for predicting the increase in the saturation solubility of nanoparticles is the Ostwald-Freundlich equation,which originates from the Gibbs-Thomson effect,as shown in Eq.(1).This equation attributes the increase in solubility to the curvature of the particles,which cannot explain the special phenomena related to nanofilms and nanowires in a sense due to the nonexistent curvature.Therefore,Kaptay proposed that the size-effect could be attributed to the high specific surface area instead of the high curvature of phase [16].

where Cs,rand Cs,∞represent the solubility of particles with the radius r and bulk phase,respectively;γ is the interfacial tension between the solid phase and surrounding medium;Vmis the molar volume of the solid phase.R and T represent the molar gas constant and the temperature in Kelvin,respectively.

However,the Ostwald-Freundlich equation was used for qualitative predictions of the increased tendency of saturation solubility with decreasing particle size in most cases,while few quantitative predictions were carried out.Van Eerdenbrugh et al.accurately measured the solubility of itraconazole,loviride,phenytoin,and naproxen nanosuspensions by light scattering and turbidity methods as well as compared the results with those of the Ostwald-Freundlich equation [17],which was found that the experimental results are consistent with the predicted results.However,the interfacial tension between drugs and water they adopted ignored the role of surfactant,and the influence of the adsorption interface was beyond consideration.Murdande et al.also proposed the same opinion despite the surface tension of water was used to replace the interface tension between solid and liquid in the calculation,which would theoretically overestimate the improvement of the solubility of nanoparticles [18].In addition,Eqs.(2),(3) and (4)were developed to describe the equilibrium constant of nano silver system [19],which has also been applied to nano zinc oxide systems [20].

Nevertheless,polymer or surfactant,etc.are often added in solutions during the preparation process of nanoparticles to improve the stability and biocompatibility [21,22].It has been experimentally proved that solid stabilizers attach on the surface of the nanoparticles.Zhang et al.[23]found that the polyvinyl pyrrolidone -sodium dodecyl sulfate complex formed a solid-toliquid equilibrium between the bonded and unbonded forms on the surface of nano probucol by the method of13C solution-state NMR.Hasegawa et al.[24]evaluated the molecular states of piroxicam/poloxamer nanosuspension,finding a core-shell structure that crystals and amorphous piroxicam coexist and are enclosed by poloxamer.The detailed structure of nanosuspension was further investigated by Kojima et al.[25],and they concluded that the drug core particles were surrounded by a semi-solid phase consisting of drug and stabilizer which is in equilibrium with the solution phase.All the experiment researches have proved the generation of a new interface,which may cause additional Gibbs energy of the interface and affect the solubility of nanoparticles significantly.Previous research by Wu et al.[4]has demonstrated that the new interface has a great influence on the melting point of the metal,revealing the important role of the Gibbs energy of the interface.However,none of the above solubility prediction theoretical models have involved the interface effect.In addition,in these reported models,the activity coefficients of substances are all ignored,which may induce a significant impact on the solubility prediction results.

Therefore,in this work,a novel interfacial thermodynamic model for describing the Gibbs energy of the nanoparticles coated by stabilizers was proposed to provide a reliable prediction of the solubility of nanoparticles.Within the developed model,the activity coefficient of nano metal system was determined by Davies model and that of nano drug system by Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT).The Gibbs energy of the interface was established as a function of molecular parameters via the application for nano metal systems.Furthermore,the model was further used to predict the solubility of nano drugs itraconazole,fenofibrate,and griseofulvin.

2.Model Description

2.1.Interfacial thermodynamic model

According to the principle of solid-liquid phase equilibrium,the chemical potential of the component B with large particles in the solid phase is equal to its chemical potential in the saturated liquid solution phase.

The chemical potentials of the pure solid B and Bnanoare equal to their molar Gibbs energy,respectively.

From the perspective of classical thermodynamics,the molar Gibbs energy of the pure solid B and Bnanoare defined as the following equations.

Therefore,the Ostwald-Freundlich equation was obtained due to the consideration of the surface energy and the neglect of the activity coefficient,as shown in Eq.(1).However,when dealing with the molar Gibbs energy of the nanoparticles coated by stabilizers,the Gibbs energy of the interface needs to be considered and thus the Eq.(12) should be modified in the following form.

2.2.Corresponding states theory analysis of

According to the work performed by Gubbins et al.[28],the interactions between the nanoparticles and the coated stabilizer can be expressed using a 10-4-3 potential of Steele [29],in which the shape and the surface roughness of the coated stabilizer are ignored.

where the subscripts (nano and stab) denote the nanoparticle and stabilizer,respectively;z is the distance between the nanoparticle and stabilizer;σnano-staband unano-stabare the usual parameters of size and energy well depth,which is assumed to be the parameters of the Lennard-Jones model;ρstabis the number of solid atoms per unit stabilizer volume;Δ is the distance between the molecular layers of solid stabilizer.

The corresponding states theory analysis was further performed to obtain the function of the grand free energy.

2.3.Thermodynamic calculation

Within the developed model,the activity coefficient of nano metal system was determined by Davies model,and that of nano drug systems by PC-SAFT.The Davies model is an extended form of Debye-Hückel equation (ionic strength I ≤0.5),as shown in Eq.(21).

Table 1 Physical properties of the studied metals

3.Model Application for Nano Metal Systems

As mentioned in the model description,the relationship between the solubility enhancement,surface energy,and interface energy was successfully established.In this session,the solubility data of nano metal coated by stabilizer,which was taken from the reference[37–40],was analyzed to obtain a general expression of the interfacial Gibbs energy.The dissolution of nano silver can last up to three months,and the dissolution temperature has not been reported in the literature,which is therefore assumed as 298.15 K.The dissolution temperature of nano copper was determined as 298.15 K according to the reference [40].The physical properties of the studied metals were summarized in Table 1.

3.1.Equilibrium constant of metal-stabilizer systems in bulk phase

According to Eq.(18),an approach to derive the equilibrium constant of metal-stabilizer systems in bulk phase should be proposed as the stabilizer may affect its solubility of bulk phase.It can be seen in Fig.1 that a linear relationship between the logarithm of the dissolution equilibrium constant of Ag-PEGSH,Ag-Citrate,and Cu-PVP systems and the reciprocal of the particle size was found with great R squared.Therefore,the intercept of the linear relationship is regarded as their equilibrium constant in bulk phase by extrapolating 1/r to 0.Meanwhile,a similar linear relationship was also observed in the Ag-PVP system with an R squared of 0.9073.However,it was found that the equilibrium constant of Ag-PVP system at the particle size of 40 nm is far less than the intercept of the linear relationship of Ag-PVP system,revealing this intercept of the linear relationship of Ag-PVP system cannot represent its equilibrium constant in bulk phase.Consequently,the equilibrium constant of Ag-PVP at particle size of 40 nm was determined as the equilibrium constant in bulk phase.

Fig.1.The relationship between the equilibrium constant of metal-stabilizer systems and particle size.

3.2.Energy contributions and proportion of interface energy

3.3.Generalized interfacial Gibbs energy relationship and model prediction

Fig.2.Thechangeof energycontributions with the particle size;Green squares,red circles,and blue triangles represent GEE,-,and ,respectively;Yellow stars represent the proportion of toGEE.

As discussed in Section 3.2 that a functional relationship between the Gibbs energy of the interface and the particle size was proposed,the analysis of this relationship was further performed in this section.Referring to the expression of surface energy(Eq.(14)),a linear regression on/Vm/γ and 1/r was performed.During the regression process,19 nm Ag-PVP was found to be multitwinned icosahedral particles by TEM in the literature,which may impact the surface energy and the solubility [38].Therefore,19 nm Ag-PVP is not included in the regression.According to the results of the regression shown in Fig.3,the great linear relationship is verified by the R square of 0.9163,and the relationship is concluded in Eq.(35).The results revealed thatis related to Vm,γ,and 1/r,which is in good accordance with the enhancement of the interfacial Gibbs energy with decreasing particle size.

Meanwhile,the predictions of GEE of nano metal coated by different stabilizers were implemented based on the generalized interfacial Gibbs energy relationship developed in this work and the results are illustrated in Fig.4.As observed in Fig.4,the predicted GEE of four systems had a good agreement with the experimental value,where no apparent deviation from the contour was detected.Moreover,the average relative deviations(ARD)of GEE of nano metal were calculated,which were expressed by the following Eq.(36).The prediction performance of the Gibbs energy enhancement of nanoparticles relative to large particles was also verified by the low ARDs (11.75% for Ag-PEGSH,19.06% for Ag-PVP,24.93% for Ag-Citrate,and 5.04% for Cu-PVP).

Fig.3.Regression of the generalized relationship between the Gibbs energy of the interface and the particle size.

Fig.4.The prediction of the GEE of nanoparticles coated with different stabilizers by the developed model.

3.4.Generalized interfacial Gibbs energy relationship based on molecular parameters

4.Solubility Prediction of Nano Drugs

According to the increased solubility data of the nano metals in Table 5,the Gibbs energy of the interface was established as a function of molecular parameters (Eq.(43)).Inspired by the excellent prediction of nano metals,the model is further used to predict the solubility of nano drugs (Itraconazole (ITZ),Fenofibrate (FFB),and Griseofulvin (GRI)).PC-SAFT model was utilized to calculate the activity coefficients of drugs to obtain the activity ratio of various systems,where the binary interaction parameters were set to 0 in the prediction work.

Table 2 PC-SAFT pure-component parameters of Ag,Cu,and Citrate

Table 3 PC-SAFT pure-component parameters of PEGSH and PVP

Table 4 The molecular parameters of metals and stabilizers

Fig.5.Linear relationship between molecular parameters of four metal-stabilizer systems.

The physical parameters and molecular parameters of drugs and related stabilizers were summarized in Tables 6 and 7,respectively.

Subsequently,based on the Eq.(44)and the molecular parameters in Tables 6 and 7,the solubility of nano drug coated by stabilizers was predicted and the results are shown in Table 8.The Ostwald-Freundlich equation was also used to predict the solubility for comparison.It was observed from Table 8 that the developed model in this work accurately predicted the solubility of nano drugs coated by stabilizers compared with the experimental data,which is more accurate than that by the Ostwald-Freundlich equation.For example,the experimental value and predicted value of the ITZ-TPGS system achieve a perfect match,indicating that it is vital to consider the interface energy when calculating the solubility of nano drug coated by stabilizers.In conclusion,this work demonstrated that the developed interfacial Gibbs energy model could well describe the Gibbs energy of nanoparticles coated by stabilizers and accurately predict their solubilities.

Meanwhile,Fig.6 further shows the change of solubility with particle size calculated by the novel developed model,which implies that the solubility of nano drugs increases as the particle size decreases.It was found that the particle size of each system that can fulfill the solubility enhancement ratio of 1.3 is different,such as 53 nm for ITZ-TPGS,43 nm for FBB-HPC(SL),and 57 nm for GRI-HPC(SL).Furthermore,based on the molecular parameters in Tables 4 and 7,the proportion of interfacial Gibbs energy of four ITZ-stabilizer systems with the particle size of 220 nm relative to pure ITZ with the particle size of 220 nm was calculated,as shown in Fig.7.The interface effect caused by stabilizers results in the great improvement of the Gibbs energy compared with the Gibbs energy of pure ITZ.In particular,HCP(SL) was found to cause the maximum improvement of Gibbs energy,suggesting that HCP(SL)is the most suitable stabilizer for ITZ because it can provide the maximum solubility enhancement.Overall,these results implied that the developed model could predict the solubilizing effect and stabilization mechanism of the stabilizers on nano drugs with different particle sizes,which may provide theoretical guidance for the selection of polymer or surfactant stabilizer.

5.Conclusions

A novel interfacial thermodynamic model for describing the Gibbs energy of the nanoparticles coated by stabilizers was proposed to predict the solubility of nano drugs and to investigate the stabilization mechanism of different stabilizers on nano drugs with different particle sizes.Within the developed model,the activity coefficients of nano metals were determined by the Daviesmodel and that of nano drug systems by PC-SAFT.The Gibbs energy of the interface was established as a function of molecular parameters via the application for nano metal system.

Table 5 The solubility of nano drugs taken from literature

Table 6 The physical parameters of drugs and stabilizers

Table 7 The molecular parameters of drugs and stabilizers

Table 8 The solubility of nano drug coated by stabilizers predicted by the Ostwald-Freundlich equation and the developed interfacial Gibbs energy model and relative deviation between the predicted and experimental value

Fig.6.The change of the solubility of nano drugs coated by stabilizers with particle size calculated by the developed interfacial Gibbs energy model.

Additionally,the analysis of Gibbs energy contributions of surface item and interface item implies that the interface effect played an important role in Gibbs energy of nanoparticles coated by stabilizers,especially when the radius of the nano metal is less than 40 nm.The developed model was further used to predict the solubility of nano drugs coated by different stabilizers.It was found that the developed model can predict the solubility of nano drugs with high accuracy in comparison with the experimental data and also predict that the solubility of nano drugs increased as the particle size decreased.Meanwhile,the stabilization mechanism of stabilizers to nano drugs was studied,which provided theoretical guidance for the selection of polymer or surfactant stabilizer.These findings showed that the developed model can provide a reliable prediction of the solubility of nanoparticles coated by stabilizers and help to comprehend the stabilization mechanism of the stabilizers on nano drugs with different particle sizes.This work is expected to provide important information for the design of nano drug formulations.

Fig.7.The proportion of interfacial Gibbs energy of four ITZ-stabilizer systems with particle size of 220 nm relative to pure ITZ with particle size of 220 nm.

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.

Acknowledgements

This research received funding from the National Natural Science Foundation of China (21776046,21978047),the Fundamental Research Funds for the Central Universities(2242020K40033),and the Six Talent Peaks Project in Jiangsu Province (XCL-079).