Formulating rationally via statistical thermodynamics

Gas and Liquid Isotherms: The Need for a Common Foundation

Sorption isotherms for gases and liquids have long been formulated separately. There is a fundamental problem with this approach: the popular isotherm models (such as Langmuir, BET, and GAB) for gases cannot be applied straightforwardly to sorption from solution. This contrasts with the theory of liquid solutions, where solute-solute interaction, mediated by the solvent, is captured as the potential of mean force, providing powerful interpretive tools (e.g., virial expansion) founded on the gas-liquid analogy. This analogy will be extended to sorption by adopting sorbate numbers and their fluctuations as the common foundation. This enables the gas and liquid isotherm equations to have an analogous mathematical form with a universal language for interfaces and liquid solutions.

Temperature Dependence of Hydrotropy

The solubility of hydrophobic solutes increases dramatically with the temperature when hydrotropes are added to water. In this paper, the mechanism of this well-known observation will be explained via statistical thermodynamics through (i) enhanced enthalpy-hydrotrope number correlation locally (around the solute) that promotes the temperature dependence and (ii) hydrotrope self-association in the bulk solution that suppresses the temperature dependence. The contribution from (i), demonstrated to be dominant for urea as a hydrotrope, signifies the weakening of interaction energies around the solute (local) than in the bulk that accompanies incoming hydrotrope molecules. Thus, studying hydrotropic solubilization along the temperature and hydrotrope concentration provides complementary information on the local-bulk difference: the local accumulation of hydrotropes around the solute, driven by the enhanced local hydrotrope self-association, is also accompanied by the overall local weakening of energetic interactions, reflecting the fluctuational nature of hydrotrope association and the mediating role of water molecules.

Synergistic Solvation as the Enhancement of Local Mixing

Mixing two solvents can sometimes make a much better solvent than expected from their weighted mean. This phenomenon, called synergistic solvation, has commonly been explained via the Hildebrand and Hansen solubility parameters, yet their inability in other solubilization phenomena, most notably hydrotropy, necessitates an alternative route to elucidating solubilization. While, recently, the universal theory of solubilization was founded on the statistical thermodynamic fluctuation theory (as a generalization of the Kirkwood-Buff theory), its demand for experimental data processing has been a hindrance for its wider application. This can be overcome by the solubility isotherm theory, which is founded on the fluctuation theory yet reduces experimental data processing significantly to the level of isotherm analysis in sorption. The isotherm analysis identifies the driving force of synergistic solvation as the enhancement of solvent mixing around the solute, opposite in behavior to hydrotropy (characterized by the enhancement of demixing or self-association around the solute). Thus, the fluctuation theory, including its solubility isotherms, provides a universal language for solubilization across the historic subcategorization of solubilizers, for which different (and often contradictory) mechanistic models have been proposed.

Sorption Hysteresis: A Statistical Thermodynamic Fluctuation Theory

Hysteresis is observed commonly in sorption isotherms of porous materials. Still, there has so far been no unified approach that can both model hysteresis and assess its underlying energetics. Standard approaches, such as capillary condensation and isotherms based on interfacial equations of state, have not proved to be up to the task. Here, we show that a statistical thermodynamic approach can achieve the following needs simultaneously: (i) showing why adsorption and desorption transitions may be sharp yet continuous; (ii) providing a simple (analytic) isotherm equation for hysteresis branches; (iii) clarifying the energetics underlying sorption hysteresis; and (iv) providing macroscopic and nanoscopic perspectives to understanding hysteresis. This approach identifies the two pairs of parameters (determinable by fitting experimental data) that are required to describe the hysteresis: the free energy per molecule within the pore clusters and the cluster size in the pores. The present paper focuses on providing mechanistic insights to IUPAC hysteresis types H1, H2(a), and H2(b) and can also be applied to the isotherm types IV and V.

Replacing the Langmuir Isotherm with the Statistical Thermodynamic Fluctuation Theory

In the age of all-atom simulations, primitive isotherm models, such as Langmuir, BET, and GAB, are still used widely for analyzing experimental data. However, their routine applications to complex materials are not in line with their underlying assumptions (i.e., statistically independent adsorption sites with no interfacial structural changes), which manifests as the temperature dependence of the monolayer capacity. Our proposal is to replace these models with the statistical thermodynamic fluctuation theory because the ABC isotherm derived from it (i) contains these primitive models as its special cases, (ii) is applicable to any interfacial geometry, and (iii) is linked to molecular distribution functions, sharing the same language as simulations. Rectifying the inability of the primitive isotherm models to handle attractive and repulsive interactions consistently leads to a reconsideration of how physical interpretations should be attributed to the isotherms of empirical origin (e.g., Freundlich).

Interactions between the protein barnase and co-solutes studied by NMR

Protein solubility and stability depend on the co-solutes present. There is little theoretical basis for selection of suitable co-solutes. Some guidance is provided by the Hofmeister series, an empirical ordering of anions according to their effect on solubility and stability; and by osmolytes, which are small organic molecules produced by cells to allow them to function in stressful environments. Here, NMR titrations of the protein barnase with Hofmeister anions and osmolytes are used to measure and locate binding, and thus to separate binding and bulk solvent effects. We describe a rationalisation of Hofmeister (and inverse Hofmeister) effects, which is similar to the traditional chaotrope/kosmotrope idea but based on solvent fluctuation rather than water withdrawal, and characterise how co-solutes affect protein stability and solubility, based on solvent fluctuations. This provides a coherent explanation for solute effects, and points towards a more rational basis for choice of excipients.

Actual Amount Adsorbed as Estimated from the Surface Excess Isotherm

The amount of adsorption at equilibrium is commonly used for reporting solid/solution isotherms, despite the admonishment by the International Union of Pure and Applied Chemistry (IUPAC) against equating the surface excess (i.e., the measurable quantity for sorption, signifying the competitive sorption of adsorbate and solvent) with the actual amount adsorbed. The consensus, more generally stated, is that the surface excess cannot be divided into individual isotherms for sorbate and solvent unless simplifying model assumptions are introduced. Here we show, contrary to the IUPAC report, that there exists a simple method for assigning the total isotherm to the sorbate's actual amount adsorbed and to the individual solute isotherm. This requires a combination of isotherm and volumetric measurements. For dilute sorbates, we establish criteria to show if the total isotherm is dominated by the amount of sorption at the interface, in agreement with the common assumption in the practical literature. In the absence of the volume data, we propose an approximate yet more versatile method based on the specific surface area to carry out order-of-magnitude analysis to examine whether the actual amount adsorbed dominates surface excess. Application of our methods to the adsorption of sodium decyl sulfate on polystyrene latex, malachite green on activated carbons, and thiophenes on a metal-organic framework all demonstrated the dominance of the actual amount adsorbed, significantly simplifying isotherm analysis in terms of the underlying interactions (i.e., surface-sorbate and net self-interactions at the interface), eliminating the need for excess surface quantities. Analysis of fully miscible solvent-sorbate isotherms (e.g., the mixtures of organic solvents adsorbed on mesoporous silica and carbonaceous adsorbents) indicates the contributions from both sorbate and solvent isotherms.

Sorption from Solution: A Statistical Thermodynamic Fluctuation Theory

Given an experimental solid/solution sorption isotherm, how can we gain insight into the underlying sorption mechanism on a molecular basis? Classifying sorption isotherms, for both completely and partially miscible solvent/sorbate systems, has been useful, yet the molecular foundation of these classifications remains speculative. Isotherm models, developed predominantly for solid/gas sorption, have been adapted to solid/solution isotherms, yet how their parameters should be interpreted physically has long remained ambiguous. To overcome the inconclusiveness, we establish in this paper a universal theory that can be used for interpreting and modeling solid/solution sorption. This novel theory shares the same theoretical foundation (i.e., the statistical thermodynamic fluctuation theory) not only with solid/gas sorption but also with solvation in liquid solutions and solution nonidealities. The key is the Kirkwood-Buff χ parameter, which quantifies the net self-interaction (i.e., solvent-solvent and sorbate-sorbate interactions minus solvent-sorbate interaction) via the Kirkwood-Buff integral in the same manner as the solvation theory and, unlike the Flory χ, is not limited to the lattice model. We will demonstrate that the Kirkwood-Buff χ is the key not only to isotherm classification but also to generalizing our recent statistical thermodynamic gas (vapor) isotherm, which is capable of fitting most of the solid/solution isotherm types.

Understanding Sorption Mechanisms Directly from Isotherms

Currently, more than 100 isotherm models coexist for the six IUPAC isotherm types. However, no mechanistic insights can be reached when several models, each claiming a different mechanism, fit an experimental isotherm equally well. More frequently, popular isotherm models [such as the site-specific models like Langmuir, Brunauer-Emmett-Teller (BET), and Guggenheim-Anderson-de Boer (GAB)] have been applied to real and complex systems that break their basic assumptions. To overcome such conundrums, we establish a universal approach to model all isotherm types, attributing the difference to the sorbate-sorbate and sorbate-surface interactions in a systematic manner. We have generalized the language of the traditional sorption models (such as the monolayer capacity and the BET constant) to the model-free concepts of partitioning and association coefficients that can be applied across the isotherm types. Through such a generalization, the apparent contradictions, caused by applying the site-specific models alongside with cross-sectional area of sorbates for the purpose of surface area determination, can be eliminated straightforwardly.

Dichloromethane replacement: towards greener chromatography via Kirkwood-Buff integrals

Dichloromethane (DCM) is a useful and advantageous solvent used in pharmaceutical development due to its low cost, miscibility with other organic solvents, high volatility, and ability to solubilize drug molecules of variable polarities and functionalities. Despite this favourable behaviour, efforts to identify safer and more sustainable alternatives to hazardous, halogenated solvents is imperative to the expansion of green chemistry. In this work, bio-derived esters tert-butyl acetate, sec-butyl acetate, ethyl isobutyrate, and methyl pivalate are experimentally identified as safe and sustainable alternatives to directly replace DCM within thin-layer chromatography (TLC) in the analysis of small, common drug molecules. To elucidate the intermolecular interactions influencing retardation factors (R_f) a statistical thermodynamic framework, which quantifies the driving molecular interactions that yield empirical TLC measurements, is presented. Within this framework, we are able to deduce R_f dependence on polar eluent concentration, in the presence of a low-polar mediating solvent, between the stationary and mobile phases. The strength of competitive analyte-eluent (and analyte-solvent interactions) are quantified through Kirkwood-Buff integrals (KBIs); resulting KBI terms at the dilute eluent limit provide a theoretical foundation for the observed suitability of alternative green solvents for the replacement of dichloromethane in TLC.

Cooperative Sorption on Heterogeneous Surfaces

Heterogeneous adsorbents, those composed of multiple surface and pore types, can result in stepwise isotherms that have been difficult to model. The complexity of these systems has often led to appealing to empirical equations without physical insights, unrealistic assumptions with many parameters, or applicability limited to a particular class of isotherms. Here, we present a statistical thermodynamic approach to model stepwise isotherms, those consisting of either an initial rise followed by a sigmoid or multiple sigmoidal steps, founded on the rigorous statistical thermodynamic theory of sorption. Our only postulates are (i) the finite ranged nature of the interface and (ii) the existence of several different types of microscopic interfacial subsystems that act independently in sorption. These two postulates have led to the superposition scheme of simple surface (i.e., Langmuir type) and cooperative isotherms. Our approach has successfully modeled the adsorption on micro-mesoporous carbons, gate-opening adsorbents, and hydrogen-bonded organic frameworks. In contrast to the previous models that start with a priori assumptions on sorption mechanisms, the advantages of our approach are that it can be applied universally under the above two postulates and that all of the fitting parameters can be interpreted with statistical thermodynamics, leading to clear insights on sorption mechanisms.

Surface Area Estimation: Replacing the Brunauer-Emmett-Teller Model with the Statistical Thermodynamic Fluctuation Theory

Surface area estimation using the Brunauer-Emmett-Teller (BET) analysis has been beset by difficulties. The BET model has been applied routinely to systems that break its basic assumptions. Even though unphysical results arising from force-fitting can be eliminated by the consistency criteria, such a practice, in turn, complicates the simplicity of the linearized BET plot. We have derived a general isotherm from the statistical thermodynamic fluctuation theory, leading to facile isotherm fitting because our isotherm is free of the BET assumptions. The reinterpretation of the monolayer capacity and the BET constant has led to a statistical thermodynamic generalization of the BET analysis. The key is Point M, which is defined as the activity at which the sorbate-sorbate excess number at the interface is at its minimum (i.e., the point of strongest sorbate-sorbate exclusion). The straightforwardness of identifying Point M and the ease of fitting by the statistical thermodynamic isotherm have been demonstrated using zeolite 13X and a Portland cement paste. The adsorption at Point M is an alternative for the BET monolayer capacity, making the BET model and its consistency criteria unnecessary. The excess number (i) replaces the BET constant as the measure of knee sharpness and monolayer coverage, (ii) links macroscopic (isotherms) to microscopic (simulation), and (iii) serves as a measure of sorbate-sorbate interaction as a signature of sorption cooperativity in porous materials. Thus, interpretive clarity and ease of analysis have been achieved by a statistical thermodynamic generalization of the BET analysis.

Cryogenic electron tomography to determine thermodynamic quantities for nanoparticle dispersions

Here we present a method to extract thermodynamic quantities for nanoparticle dispersions in solvents. The method is based on the study of tomograms obtained from cryogenic electron tomography (cryoET). The approach is demonstrated for gold nanoparticles (diameter < 5 nm). Tomograms are reconstructed from tilt-series 2D images. Once the three-dimensional (3D) coordinates for the centres of mass of all of the particles in the sample are determined, we calculate the pair distribution function g(r) and the potential of mean force U(r) without any assumption. Importantly, we show that further quantitative information from 3D tomograms is readily available as the spatial fluctuation in the particles' position can be efficiently determined. This in turn allows for the prompt derivation of the Kirkwood-Buff integrals with all their associated quantities such as the second virial coefficient. Finally, the structure factor and the agglomeration states of the particles are evaluated directly. These thermodynamic quantities provide key insights into the dispersion properties of the particles. The method works well both for dispersed systems containing isolated particles and for systems with varying degrees of agglomerations.

The Impact of Size and Shape in the Performance of Hydrotropes: A Case-Study of Alkanediols

Inspired by the recently proposed cooperative mechanism of hydrotropy, where water molecules mediate the aggregation of hydrotrope around the solute, this work studies the impact of apolar volume and polar...

Temperature Dependence of Sorption

Understanding how sorption depends on temperature on a molecular basis has been made difficult by the coexistence of isotherm models, each assuming a different sorption mechanism and the routine application of planar, multilayer sorption models (such as Brunauer-Emmett-Teller (BET) and Guggenheim-Anderson-de Boer (GAB)) beyond their premises. Furthermore, a common observation that adsorption isotherms measured at different temperatures fall onto a single "characteristic curve" when plotted against the adsorption potential has not been given a clear explanation, due to its ambiguous foundation. Extending our recent statistical thermodynamic fluctuation theory of sorption, we have generalized the classical isosteric theory of sorption into a statistical thermodynamic fluctuation theory and clarified how sorption depends on temperature. We have shown that a characteristic curve exists when sorbate number increment contributes purely energetically to the interface, whereas the correlation between sorbate number and entropy drives the temperature dependence of an isotherm. This theory rationalizes the opposite temperature dependence of water vapor sorption on activated carbons with uniform versus broad pore size distributions and can be applied to moisture sorption on starch gels. The adsorption potential is a convenient variable for sorption in its ability to unify sorbate-sorbate fluctuation and the isosteric thermodynamics of sorption.

Cooperative Sorption on Porous Materials

The functional shape of a sorption isotherm is determined by underlying molecular interactions. However, doubts have been raised on whether the sorption mechanism can be understood in principle from analyzing sorption curves via a range of competing models. We have shown recently that it is possible to translate a sorption isotherm to the underlying molecular interactions via rigorous statistical thermodynamics. The aim of this paper is to fill the gap between the statistical thermodynamic theory and analyzing experimental sorption isotherms, especially of microporous and mesoporous materials. Based on a statistical thermodynamic approach to interfaces, we have derived a cooperative isotherm, as a generalization of the Hill isotherm and our cooperative solubilization model, without the need for assumptions on adsorption sites, layers, and pore geometry. Instead, the statistical characterization of sorbates, such as the sorbate-interface distribution function and the sorbate number distribution, as well as the existence of statistically independent units of the interface, underlies the cooperative sorption isotherm. Our isotherm can be applied directly to literature data to reveal a few key system attributes that control the isotherm: the cooperative number of sorbates and the free energy of transferring sorbates from the saturated vapor to the interface. The sorbate-sorbate interaction is quantified also via the Kirkwood-Buff integral and the excess numbers.

Sorption: A Statistical Thermodynamic Fluctuation Theory

Can the sorption mechanism be proven by fitting an isotherm model to an experiment? Such a question arises because (i) multiple isotherm models, with different assumptions on sorption mechanisms, often fit an experimental isotherm equally well, (ii) some isotherm models [such as Brunauer-Emmett-Teller (BET) and Guggenheim-Anderson-de Boer (GAB)] fit experimental isotherms that do not satisfy the underlying assumptions of the model, and (iii) some isotherms (such as Oswin and Peleg) are empirical equations that do not have a well-defined basis on sorption mechanisms. To overcome these difficulties, we propose a universal route of elucidating the sorption mechanism directly from an experimental isotherm, without an isotherm model, based on the statistical thermodynamic fluctuation theory. We have shown that how sorbate-sorbate interaction depends on activity is the key to understanding the sorption mechanism. Without assuming adsorption sites and planar layers, an isotherm can be derived, which contains the Langmuir, BET, and GAB models as its special cases. We have constructed a universal approach applicable to adsorption and absorption, solid and liquid sorbents, and vapor and liquid sorbates and demonstrated its efficacy using the humidity sorption isotherm of sucrose from both the solid and liquid sides.

Using Microemulsions: Formulation Based on Knowledge of Their Mesostructure

Microemulsions, as thermodynamically stable mixtures of oil, water, and surfactant, are known and have been studied for more than 70 years. However, even today there are still quite a number of unclear aspects, and more recent research work has modified and extended our picture. This review gives a short overview of how the understanding of microemulsions has developed, the current view on their properties and structural features, and in particular, how they are related to applications. We also discuss more recent developments regarding nonclassical microemulsions such as surfactant-free (ultraflexible) microemulsions or ones containing uncommon solvents or amphiphiles (like antagonistic salts). These new findings challenge to some extent our previous understanding of microemulsions, which therefore has to be extended to look at the different types of microemulsions in a unified way. In particular, the flexibility of the amphiphilic film is the key property to classify different microemulsion types and their properties in this review. Such a classification of microemulsions requires a thorough determination of their structural properties, and therefore, the experimental methods to determine microemulsion structure and dynamics are reviewed briefly, with a particular emphasis on recent developments in the field of direct imaging by means of electron microscopy. Based on this classification of microemulsions, we then discuss their applications, where the application demands have to be met by the properties of the microemulsion, which in turn are controlled by the flexibility of their amphiphilic interface. Another frequently important aspect for applications is the control of the rheological properties. Normally, microemulsions are low viscous and therefore enhancing viscosity has to be achieved by either having high concentrations (often not wished for) or additives, which do not significantly interfere with the microemulsion. Accordingly, this review gives a comprehensive account of the properties of microemulsions, including most recent developments and bringing them together from a united viewpoint, with an emphasis on how this affects the way of formulating microemulsions for a given application with desired properties.

Cholinium-based ionic liquids as bioinspired hydrotropes to tackle solubility challenges in drug formulation

Hydrotropy is a well-established strategy to enhance the aqueous solubility of hydrophobic drugs, facilitating their formulation for oral and dermal delivery. However, most hydrotropes studied so far possess toxicity issues and are inefficient, with large amounts being needed to achieve significant solubility increases. Inspired by recent developments in the understanding of the mechanism of hydrotropy that reveal ionic liquids as powerful hydrotropes, in the present work the use of cholinium vanillate, cholinium gallate, and cholinium salicylate to enhance the aqueous solubility of two model drugs, ibuprofen and naproxen, is investigated. It is shown that cholinium vanillate and cholinium gallate are able to increase the solubility of ibuprofen up to 500-fold, while all three ionic liquids revealed solubility enhancements up to 600-fold in the case of naproxen. Remarkably, cholinium salicylate increases the solubility of ibuprofen up to 6000-fold. The results obtained reveal the exceptional hydrotropic ability of cholinium-based ionic liquids to increase the solubility of hydrophobic drugs, even at diluted concentrations (below 1 mol·kg^-1), when compared with conventional hydrotropes. These results are especially relevant in the field of drug formulation due to the bio-based nature of these ionic liquids and their low toxicity profiles. Finally, the solubility mechanism in these novel hydrotropes is shown to depend on synergism between both amphiphilic ions.

Cooperativity in micellar solubilization

Sudden onset of solubilization is observed widely around or below the critical micelle concentration (CMC) of surfactants. It has also been reported that micellization is induced by the solutes even below CMC and the solubilized solute increases the aggregation number of the surfactant. These observations suggest enhanced cooperativity in micellization upon solubilization. Recently, we have developed a rigorous statistical thermodynamic theory of cooperative solubilization. Its application to hydrotropy revealed the mechanism of cooperative hydrotropy: hydrotrope self-association enhanced by solutes. Here we generalize our previous cooperative solubilization theory to surfactants. We have shown that the well-known experimental observations, such as the reduction of CMC in the presence of the solutes and the increase of aggregation number, are the manifestations of cooperative solubilization. Thus, the surfactant self-association enhanced by a solute is the driving force of cooperativity and a part of a universal cooperative solubilization mechanism common to hydrotropes and surfactants at low concentrations.

The impact of the counterion in the performance of ionic hydrotropes

The efficiency of an ionic hydrotrope is shown to increase with the hydrophobicity of its counterion, challenging the common view that ionic hydrotropes should possess a small, densely charged counterion such as sodium or chloride.

Salt-induced LCST-type thermal gelation of methylcellulose: quantifying non-specific interactions via fluctuation theory

What drives the phase separation of water-soluble polymers in the presence of electrolytes was quantified on a molecular scale via statistical thermodynamic fluctuation theory. Quantifying polymer-water and polymer-salt interactions enabled us to identify the dominant interaction for phase separation. As a model system, the lower critical solution temperature (LCST) type thermal gelation of methylcellulose (MC) in aqueous salt solutions was chosen. The Kirkwood-Buff integrals for intermolecular interactions, calculated from the published calorimetric and volumetric data, showed that (1) the accumulation of salts around MC molecules (favourable interaction between salts and MC) inhibits thermal gelation and the depletion of salts from MC (unfavourable interaction between salts and MC) promotes the gelation, and (2) this salt-MC interaction is the dominant factor (50-100 times stronger than the water-MC interaction). This insight from the fluctuation theory is at odds with the age-old consensus regarding the driving force of thermal gelation: water structure change in the presence of salts induces the promotion or inhibition of thermal gelation. However, our conclusion is founded upon the ability of the fluctuation theory to quantify water-MC and salt-MC interaction independently via the Kirkwood-Buff integrals. Flory-Huggins (FH) theory, on the contrary, could not separate these two interactions owing to the lack of a thermodynamic degree of freedom because the lattice solution is assumed to be fully packed. In addition, the dominant contribution from salt depletion poses difficulty for the χ parameter, which is essentially the difference of contact energies. Our approach, requiring calorimetric and volumetric data alone as input, provides a simple and versatile method towards elucidating the effect of cosolvents on biopolymer phase separation of physiological importance.