Statistical Thermodynamic Theory Research Articles

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.

Enhanced Adsorption Response of pH-Sensitive Peptides: The Role of Membrane Acidity.

pH-sensitive peptides bind and traverse lipid membranes in response to changes in pH. They can be used to target tumors and other acidic tissues. We investigate the influence of acidic lipids on the pH-driven adsorption of recently synthesized peptides. Using a statistical-thermodynamic theory that takes into account the acid-base chemistry of peptides and lipids, we find that the presence of acidic lipids amplifies changes in peptide surface concentration when transitioning from high to low pH. We study cyclic and linear peptides, containing tryptophan, glutamic acid, and arginine residues, examining their behavior in both neutral and acidic membranes. Membrane binding consistently results from the shallow insertion of tryptophan residues with hydrophilic residues facing the aqueous solution. Regardless of the pH, the peptide's geometry predominantly determines the orientation and distribution of residues. Notably, we find that not only the extent of adsorption is pH-sensitive but also the underlying adsorption mechanism: it is barrier-free at low pH but hindered by a large free energy barrier at high pH. Hence, under more acidic conditions, pH-sensitive peptides show facilitated adsorption both kinetically and thermodynamically.

Simple and complex sorption−solution isotherms for membrane polymers: A statistical thermodynamic fluctuation theory

Simple, site-specific adsorption isotherm models, such as GAB and other generalizations of BET, fit strikingly well to experimental isotherms of simple shapes for glassy and rubbery polymers. However, the structures of dense polymers do not necessarily resemble the unrestricted layer-by-layer adsorption mechanism on one type of sorption site assumed by these models. Since polymers contain micropores, sorptive likely (ad)sorbs on the internal surfaces of the free (accessible) volume voids, and dissolves in the polymer phase. While the empirical dual mode sorption model and its variants address this duality, other common models assume chiefly only one of the two mechanisms. Moreover, the simplified models do not fit complex isotherms, such as those for alcohols and poly[(trimethylsilyl)propyne] (PTMSP). The statistical thermodynamic fluctuation theory is adopted here to capture the sorption-solution duality consistently even for the complex isotherms. The statistical thermodynamic ABC isotherm derived from this theory involves the sorbate-sorbent and mono-, di-, and tri-sorbate interactions expressed by sorbate number correlations. This work shows that BET, GAB, dual-mode sorption model, Flory-Huggins, and ENSIC models are special cases of the ABC isotherm. The isotherm multiplicativity principle has been derived from the number fluctuation relationship to model complex isotherms.

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).

Fundamentals of Lie categories

We introduce the basic notions and present examples and results on Lie categories – categories internal to the category of smooth manifolds. Demonstrating how the units of a Lie category \mathcal{C} dictate the behavior of its invertible morphisms \mathcal{G}(\mathcal{C}) , we develop sufficient conditions for \mathcal{G}(\mathcal{C}) to form a Lie groupoid. We show that the construction of Lie algebroids from the theory of Lie groupoids carries through, and ask when the Lie algebroid of \mathcal{G}(\mathcal{C}) is recovered. We reveal that the lack of the invertibility assumption on morphisms leads to a natural generalization of rank from linear algebra, develop its general properties, and show how the existence of an extension \mathcal{C}\hookrightarrow \mathcal{G} of a Lie category to a Lie groupoid affects the ranks of morphisms and the algebroids of \mathcal{C} . Furthermore, certain completeness results for invariant vector fields on Lie monoids and Lie categories with well-behaved boundaries are obtained. Interpreting the developed framework in the context of physical processes, we yield a rigorous approach to the theory of statistical thermodynamics by observing that entropy change, associated to a physical process, is a functor.

Modeling single-molecule stretching experiments using statistical thermodynamics.

Single-molecule stretching experiments are widely utilized within the fields of physics and chemistry to characterize the mechanics of individual bonds or molecules, as well as chemical reactions. Analytic relations describing these experiments are valuable, and these relations can be obtained through the statistical thermodynamics of idealized model systems representing the experiments. Since the specific thermodynamic ensembles manifested by the experiments affect the outcome, primarily for small molecules, the stretching device must be included in the idealized model system. Though the model for the stretched molecule might be exactly solvable, including the device in the model often prevents analytic solutions. In the limit of large or small device stiffness, the isometric or isotensional ensembles can provide effective approximations, but the device effects are missing. Here a dual set of asymptotically correct statistical thermodynamic theories are applied to develop accurate approximations for the full model system that includes both the molecule and the device. The asymptotic theories are first demonstrated to be accurate using the freely jointed chain model and then using molecular dynamics calculations of a single polyethylene chain.

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.

Theoretical Study of Thermodynamic Stability Zone of Gas Hydrates Formed from Sea Water

In this work we used a combination of molecular dynamic simulation and statistical thermodynamic theory in order to obtain stability region of propane hydrates formed from sea water. MD simulation were used to get thermodynamic properties of liquid water phase, while statistical thermodynamic theory were applied for solid hydrate phase. We reconstructed phase equilibrium line ‘sea water – hydrate – gas’ and according to our calculations the temperature shift of this line is about 3◦. Moreover, we have shown a possibility to obtain thermodynamic parameters of salt water directly from MD simulations with sufficient accuracy.

Statistical Analysis of Random Motion and Energetic Behavior of Counting: Gibbs’ Theory Revisited

Following a recently reformulated Gibbs' statistical chemical thermodynamic theory on discrete state space, we present a treatment on statistical measurements of random mechanical motions in continuous space. In particular, we show how the concept of temperature and an ideal gas/solution law arise from a statistical analysis of a collection of independent and identically distributed complex particles without relying on Newtonian mechanics, nor the very concept of mechanical energy. When sampling from an ergodic system, the data ad infinitum limit elucidates how the entropy function characterizes randomness among measurements with the emergence of a novel energetic representation for the statistics and an internal energy additivity. This generalization of Gibbs' theory is applicable to statistical measurements on single living cells and other complex biological organisms, one individual at a time.

A saturated vapor pressure explicit model with temperature and ion concentration for multi-component liquid desiccants

A saturated vapor pressure model with a wide range of applications (covering traditional salt solutions and emerging ionic liquid solutions) and simple frame form is the foundation of the research in the field of solution gas–liquid equilibrium. The existing typical thermodynamic models usually improve the prediction accuracy at the cost of complex forms (such as increasing the number of adjustable parameters), resulting in highly nonlinear functions, which are not convenient for practical calculation. Based on the theory of statistical thermodynamics and considering the dipole interaction of ion-water molecules, this paper proposes a saturated vapor pressure explicit model with temperature and ion concentration for salt and ionic liquid multi-component liquid desiccant, in which only three adjustable parameters are introduced to characterize each ion. Compared with the experimental data and several typical thermodynamic models, the results show that the prediction accuracy of the model is comparable to those of typical models. However, its advantage over these typical models is its simpler framework and significant computational reduction. Through the application of some examples, it is found that the new model can have the potential of direct extension, which is manifested: from one temperature range to another temperature range, from ions to compounds, from single-component to multi-component, and from the water system to other solvent systems. This demonstrates that the three adjustable parameters introduced for ions in the model have clear physical meanings, which can significantly reduce the dependence on experimental data. The proposed model can quickly calculate the saturated vapor pressure of the solution containing inorganic salts, organic acid salts, and ionic liquids, and can guide the optimization of dehumidifying materials.

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.

A methodology for creating mutants of G-protein coupled receptors stabilized in active state by combining statistical thermodynamics and evolutionary molecular engineering.

We challenged the stabilization of a G-protein coupled receptor (GPCR) in the active state solely by multiple amino-acid mutations without the agonist binding. For many GPCRs, the free energy of the active state is higher than that of the inactive state. When the inactive state is stabilized through the lowering of its free energy, the apparent midpoint temperature of thermal denaturation Tm exhibits a significant increase. However, this is not always the case for the stabilization of the active state. We constructed a modified version of our methodology combining statistical thermodynamics and evolutionary molecular engineering, which was recently developed for the inactive state. First, several residues to be mutated are determined using our statistical-thermodynamics theory. Second, a gene (mutant) library is constructed using Escherichia coli cells to efficiently explore most of the mutational space. Third, for the mutant screening, the mutants prepared in accordance with the library are expressed in engineered Saccharomyces cerevisiae YB14 cells which can grow only when a GPCR mutant stabilized in the active state has signaling function. For the adenosine A2A receptor tested, the methodology enabled us to sort out two triple mutants and a double mutant. It was experimentally corroborated that all the mutants exhibit much higher binding affinity for G protein than the wild type. Analyses indicated that the mutations make the structural characteristics shift toward those of the active state. However, only slight increases in Tm resulted from the mutations, suggesting the unsuitability of Tm to the stability measure for the active state.

Freely jointed chain models with extensible links.

Analytical relations for the mechanical response of single polymer chains are valuable for modeling purposes, on both the molecular and the continuum scale. These relations can be obtained using statistical thermodynamics and an idealized single-chain model, such as the freely jointed chain model. To include bond stretching, the rigid links in the freely jointed chain model can be made extensible, but this almost always renders the model analytically intractable. Here, an asymptotically correct statistical thermodynamic theory is used to develop analytic approximations for the single-chain mechanical response of this model. The accuracy of these approximations is demonstrated using several link potential energy functions. This approach can be applied to other single-chain models, and to molecular stretching in general.

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.

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.

Colloidal Nanocrystal Gels from Thermodynamic Principles.

Gels assembled from solvent-dispersed nanocrystals are of interest for functional materials because they promise the opportunity to retain distinctive properties of individual nanocrystals combined with tunable, structure-dependent collective behavior. By incorporating stimuli-responsive components, these materials could also be dynamically reconfigured between structurally distinct states. However, nanocrystal gels have so far been formed mostly through irreversible aggregation, which has limited the realization of these possibilities. Meanwhile, gelation strategies for larger colloidal microparticles have been developed using reversible physical or chemical interactions. These approaches have enabled the experimental navigation of theoretically predicted phase diagrams, helping to establish an understanding of how thermodynamic behavior can guide gel formation in these materials. However, the translation of these principles to the nanoscale poses both practical and fundamental challenges. The molecules guiding assembly can no longer be safely assumed to be vanishingly small compared to the particles nor large compared to the solvent.In this Account, we discuss recent progress toward the assembly of tunable nanocrystal gels using two strategies guided by equilibrium considerations: (1) reversible chemical bonding between functionalized nanocrystals and difunctional linker molecules and (2) nonspecific, polymer-induced depletion attractions. The effective nanocrystal attractions, mediated in both approaches by a secondary molecule, compete against stabilizing repulsions to promote reversible assembly. The structure and properties of the nanocrystal gels are controlled microscopically by the design of the secondary molecule and macroscopically by its concentration. This mode of control is compelling because it largely decouples nanocrystal synthesis and functionalization from the design of interactions that drive assembly. Statistical thermodynamic theory and computer simulation have been applied to simple models that describe the bonding motifs in these assembling systems, furnish predictions for conditions under which gelation is likely to occur, and suggest strategies for tuning and disassembling the gel networks. Insights from these models have guided experimental realizations of reversible gels with optical properties in the infrared range that are sensitive to the gel structure. This process avoids time-consuming and costly trial-and-error experimental investigations to accelerate the development of nanocrystal gel assemblies.These advances highlight the need to better understand interactions between nanocrystals, how interactions give rise to gel structure, and properties that emerge. Such an understanding could suggest new approaches for creating stimuli-responsive and dissipative assembled materials whose properties are tunable on demand through directed reconfiguration of the underlying gel microstructure. It may also make nanocrystal gels amenable to computationally guided design using inverse methods to rapidly optimize experimental parameters for targeted functionalities.

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.

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.