Fluctuation Solution Theory Research Articles

Partial molar volumes of 1–1 electrolytes at high T and P: correlations and predictions

Knowledge of the partial molar volumes of aqueous ions allows accurate calculation of the pressure dependence of equilibrium constants, solubility of minerals, etc., thus being useful for thermodynamic modeling of hydrothermal processes. This study analyzed methods to correlate and predict the values of the partial molar volumes at infinite dilution, V2o, for 1–1 electrolytes and singly charged ions at elevated T and P. Since the precise experimental values of the dielectric constant of water are measured only up to 873 K, we were interested only in non-electrostatic ways to correlate V2o data. First of all, we compiled the V2o values at T > 373 K for the following 1–1 electrolytes: HCl, LiCl, LiI, LiNO3, LiOH, NaF, NaCl, NaBr, NaI, NaNO3, NaOH, NaHCO3, NaClO4, NaH2PO4, NaTr (Tr stands for triflate), KF, KCl, KBr, KI, KNO3, KOH, CsBr, and NH4Cl. Relations, following from the “density” model and from the Fluctuation Solution Theory (FST) were employed to analyze data. It was concluded that at the current state of knowledge of V2o the FST-relations for electrolytes are recommended mainly to reject strongly deviating experimental outliers. However, the “density” model provides a simple and fairly accurate way to describe the compiled set of data with only two parameters for each ion, n and Vhc, values of which were evaluated for the following singly-charged ions: H+, Li+, Na+, K+, Cs+, NH4+, F-, Cl-, Br-, I-, OH–, NO3–, H2PO4-, HCO3–, ClO4-, Tr- (Tr = triflate). Following Mesmer et al. (1988), we consider the fitting parameters Vhc and n to be related to the intrinsic volume of the ion and to the number of water molecules transferred from the bulk water to a hydration shell around the ion, respectively.Although the values of n and Vhc were fitted from data at temperatures from 373 to 673 K and water densities, do, from 0.5 to 1.1 g cm−3, apparently, they predict realistic V2o values at temperatures up to 1600 K and do ranges from 0.2 to 1.6 g cm−3.

Molecular correlation function integrals for condensed systems with solid-liquid-liquid equilibria

Fluctuation Solution Theory (FST) is an exact statistical mechanical formulation for solution thermodynamic properties and their derivatives via integrals of direct correlation functions (DCFI) and total correlation functions (TCFI). This work tabulates DCFIs and TCFIs for a series of strongly nonideal systems involving condensed-phase equilibria (LLE, SLE and SLLE) over wide ranges of temperature, pressure, and composition. The values obtained can be used for FST modeling of such systems as well as to test the accuracy of other models such as equations of state.

Ensemble transformation in the fluctuation theory

Interactions in complex solutions that consist of multiple components can be quantified via number correlations observed within an isochoric subsystem. The fluctuation solution theory carries out a conversion between experimental data (usually measured under isobaric conditions) and the isochoric number correlations through cumbersome thermodynamic variable transformations. In contrast, we have recently demonstrated heuristically that direct transformation of statistical variables (i.e., those variables fluctuating in statistical ensembles such as volume and particle numbers) can lead to equivalent results via simple algebra. This paper reveals the geometrical basis of fluctuation and the invariants underlying the equivalence between thermodynamic and statistical variable transformations. Based on the quasi-thermodynamic fluctuation theory and the postulate that concentration and its fluctuation are invariant under ensemble transformation, we show that the thermodynamic and statistical variable transformations correspond to the change of basis on the Hessian matrix and the vectors whose elements are the deviations of statistical variables from their mean values, respectively, under which the quadratic form of fluctuation is invariant. Statistical variable transformation can also be used in cases when a set of experimental data do not belong to the same ensemble. When combined with the order-of-magnitude analysis, our formalism shows that the quasi-thermodynamic formalism of fluctuation at the thermodynamic limit is valid only for extensive variables and cannot be applied to intensive variables.

Gas or Liquid? The Supercritical Behavior of Pure Fluids.

By definition, the distinction between a gas and a liquid ceases to exist beyond the critical point for pure fluids. Nevertheless, there remains a strong desire to attribute gas-like or liquid-like behavior to fluids corresponding to different parts of the supercritical region, especially as this becomes important for understanding and designing the properties of supercritical fluids. Here, we use a combination of fluctuation solution theory and accurate equation of state data to elucidate an easily accessible dividing line and corresponding transition regime between liquid-like and gas-like behavior in the supercritical region of all pure fluids. Liquid-like behavior in the supercritical region is characterized by a negative skewness in the particle number distribution for an equivalent open system, indicating that particle deletion is favored for liquids, whereas gas-like behavior is characterized by a positive skewness, indicating that particle insertion is favored for gases. Identical behavior is observed either side of the liquid-vapor line. The possible consequences for the behavior of fluids at the critical point are also discussed.

Volumetric properties and phase behavior of sulfur dioxide, carbon disulfide and oxygen in high-pressure carbon dioxide fluid

Acid gas re-injection or carbon capture and storage are two technologies now implemented for management of produced CO2 or reducing the industrial emission of CO2. For both technologies, impurities such as H2O, H2S, O2, and SO2 can find a way into a CO2 injectate stream. In addition to unwanted reactions, these impurities can alter the phase behavior and volumetric properties of the injectate stream. Measuring accurate densities associated with mixing various impurities and CO2 fluid is one method to optimize mixing coefficients for calculating reliable phase behavior and chemical activities for these systems. In this work, the volumetric influence and/or phase behavior of SO2, CS2, or O2 impurities in a dense CO2 phase were studied through high-precision density measurements. Densities of the CO2 mixtures were measured with a custom vibrating tube densimeter with an average uncertainty of ±0.07 kg m−3 for temperatures of T = 50 to 125 °C and pressures at p ≤ 35 MPa. The densities were used in calculating the apparent molar volumes, which were subsequently used to (i) calibrate or verify mixing coefficients for reference quality reduced Helmholtz energy equations-of-state, and (ii) provide a fitted equation based on fluctuation solution theory which can be used to calculate fugacity coefficients and partial molar volumes of CS2 at infinite dilution. Validating our optimized mixing coefficients for the CO2 + SO2 mixture showed more reliable vapor-liquid equilibrium phase behavior when compared to those previously estimated in the literature.

Statistical thermodynamic foundation for mesoscale aggregation in ternary mixtures.

Ternary solvent mixtures with two mutually miscible and one immiscible solvent pairings often exhibit persistent scattering profiles corresponding to mesoscale structure formation. Despite the morphological information on such mesostructures via extensive scattering measurements and simulation, the origin of these mesostructures, why they persist over a wide composition range, and why they appear around the plait point have remained a mystery. Here we answer all these questions through constructing a fundamental molecular thermodynamic theory, by utilizing thermodynamic stability, scattering and the fluctuation solution theory. The plait point condition, when interpreted via differential geometry, is shown to be the origin of the large structure factor persistent over a wide composition range.

Experimental investigation of triplet correlation approximations for fluid water

Triplet correlations play a central role in our understanding of fluids and their properties. Of particular interest is the relationship between the pair and triplet correlations. Here we use a combination of Fluctuation Solution Theory and experimental pair radial distribution functions to investigate the accuracy of the Kirkwood Superposition Approximation (KSA), as given by integrals over the relevant pair and triplet correlation functions, at a series of state points for pure water using only experimental quantities. The KSA performs poorly, in agreement with a variety of other studies. Several additional approximate relationships between the pair and triplet correlations in fluids are also investigated and generally provide good agreement for the fluid thermodynamics for regions of the phase diagram where the compressibility is small. A simple power law relationship between the pair and triplet fluctuations is particularly successful for state points displaying low to moderately high compressibilities.

Simulated pressure denaturation thermodynamics of ubiquitin

Simulations of protein thermodynamics are generally difficult to perform and provide limited information. It is desirable to increase the degree of detail provided by simulation and thereby the potential insight into the thermodynamic properties of proteins. In this study, we outline how to analyze simulation trajectories to decompose conformation-specific, parameter free, thermodynamically defined protein volumes into residue-based contributions. The total volumes are obtained using established methods from Fluctuation Solution Theory, while the volume decomposition is new and is performed using a simple proximity method. Native and fully extended ubiquitin are used as the test conformations. Changes in the protein volumes are then followed as a function of pressure, allowing for conformation-specific protein compressibility values to also be obtained. Residue volume and compressibility values indicate significant contributions to protein denaturation thermodynamics from nonpolar and coil residues, together with a general negative compressibility exhibited by acidic residues.

Gaussian and non-Gaussian fluctuations in pure classical fluids

The particle number, energy, and volume probability distributions in the canonical, isothermal-isobaric, grand canonical, and isobaric-isenthalpic ensembles are investigated. In particular, we consider Gaussian and non-Gaussian behavior and formulate the results in terms of a single expression valid for all the ensembles employing common, experimentally accessible, thermodynamic derivatives. This is achieved using Fluctuation Solution Theory to help manipulate derivatives of the entropy. The properties of the distributions are then investigated using available equations of state for fluid water and argon. Purely Gaussian behavior is not observed for any of the state points considered here. A set of simple measures, involving thermodynamic derivatives, indicating non-Gaussian behavior is proposed. A general expression, valid in the high temperature limit, for small energy fluctuations in the canonical ensemble is provided.

Fluctuation solution theory of pure fluids

Fluctuation Solution Theory (FST) provides an alternative view of fluid thermodynamics in terms of pair fluctuations in the particle number and excess energy observed for an equivalent open system. Here we extend the FST approach to provide a series of triplet and quadruplet particle and excess energy fluctuations that can also be used to help understand the behavior of fluids. The fluctuations for the gas, liquid, and supercritical regions of three fluids (H2O, CO2, and SF6) are then determined from accurate equations of state. Many of the fluctuating quantities change sign on moving from the gas to liquid phase and, therefore, we argue that the fluctuations can be used to characterize gas and liquid behavior. Further analysis provides an approach to isolate contributions to the excess energy fluctuations arising from just the intermolecular interactions and also indicates that the triplet and quadruplet particle fluctuations are related to the pair particle fluctuations by a simple power law for large regions of the phase diagram away from the critical point.

The partial molar volumes for water dissolved in high-pressure carbon dioxide from T = (318.28 to 369.40) K and pressures to p = 35 MPa

Post-compression CO2 delivered to sub-surface reservoirs often contains various impurities including sub-saturated water, where understanding phase behavior or chemical reactions involving water requires volumetric information. A new optically coupled vibrating tube densimeter has been developed for volumetric measurements of CO2 injectate fluids for compression conditions and injectate transportation (temperatures less than 423K and pressures less than 35MPa). Details regarding the design, operation and calibration of the instrument are reported. Density differences for sub-saturated H2O in CO2 from T=(318 to 369)K and up to p=35MPa (m=(0.053 to 0.065)mol·kg−1) are reported and used to calculate molar volumes. To our knowledge, these are the first sub-saturated measurements to be reported at the conditions typical for CO2 compression and transportation. As expected, the apparent molar volumes become significantly negative near the CO2 supercritical conditions. Using these molar volumes, we have (i) provided a fitted equation based on Fluctuation Solution Theory which can be used to calculate H2O fugacity coefficients at infinite dilution, (ii) found the Krichevskii parameter to be AKr=−(17.7±3.0)MPa and (iii) used the volumetric data to optimize reduced Helmholtz energy mixing parameters which can be used with high-accuracy equations-of-state (γv,12=1.03070 and γT,12=0.81776). Comparisons to previously reported data for the H2O-rich phase and the recent mixing model of Gernert and Span show the resulting corresponding state mixing parameters can be utilized for a much larger range of H2O concentrations, temperatures up to T=673K and pressures up to p=100MPa.

Particle and Energy Pair and Triplet Correlations in Liquids and Liquid Mixtures from Experiment and Simulation.

Recent advances in fluctuation solution theory (FST) have provided access to information concerning triplet fluctuations and integrals, in addition to the established pair fluctuations and integrals, for liquids and liquid mixtures using both experimental and simulation data. Here, FST is used to investigate pair and triplet correlations for (i) pure water as provided by experiment and simulation using both polarizable and nonpolarizable water models, (ii) liquid mixtures of methanol and water as provided by experiment and simulation, and (iii) native and denatured states of proteins as provided by simulation. The last application is particularly powerful, as it provides exact equations for the volume, enthalpy, compressibility, thermal expansion, and heat capacity of a single protein form provided by a single simulation. In addition, a discussion of the quality of the integrals obtained from experiment and simulation is provided. The results clearly illustrate that FST can be a powerful tool for the analysis and interpretation of both experimental and simulation data in complex liquid mixtures, including biomolecular systems, and that current simulation protocols can provide reliable values for the pair and triplet correlations and integrals.

Experimental triplet and quadruplet fluctuation densities and spatial distribution function integrals for liquid mixtures.

Kirkwood-Buff or Fluctuation Solution Theory can be used to provide experimental pair fluctuations, and/or integrals over the pair distribution functions, from experimental thermodynamic data on liquid mixtures. Here, this type of approach is used to provide triplet and quadruplet fluctuations, and the corresponding integrals over the triplet and quadruplet distribution functions, in a purely thermodynamic manner that avoids the use of structure factors. The approach is then applied to binary mixtures of water + methanol and benzene + methanol over the full composition range under ambient conditions. The observed correlations between the different species vary significantly with composition. The magnitude of the fluctuations and integrals appears to increase as the number of the most polar molecule involved in the fluctuation or integral also increases. A simple physical picture of the fluctuations is provided to help rationalize some of these variations.

Experimental triplet and quadruplet fluctuation densities and spatial distribution function integrals for pure liquids.

Fluctuation solution theory has provided an alternative view of many liquid mixture properties in terms of particle number fluctuations. The particle number fluctuations can also be related to integrals of the corresponding two body distribution functions between molecular pairs in order to provide a more physical picture of solution behavior and molecule affinities. Here, we extend this type of approach to provide expressions for higher order triplet and quadruplet fluctuations, and thereby integrals over the corresponding distribution functions, all of which can be obtained from available experimental thermodynamic data. The fluctuations and integrals are then determined using the International Association for the Properties of Water and Steam Formulation 1995 (IAPWS-95) equation of state for the liquid phase of pure water. The results indicate small, but significant, deviations from a Gaussian distribution for the molecules in this system. The pressure and temperature dependence of the fluctuations and integrals, as well as the limiting behavior as one approaches both the triple point and the critical point, are also examined.

Hydrotropy: binding models vs. statistical thermodynamics

Hydrophobic drugs can often be solubilized by the addition of hydrotropes. We have previously shown that preferential drug-hydrotrope association is one of the major factors of increased solubility (but not "hydrotrope clustering" or changes in "water structure"). How, then, can we understand this drug-hydrotrope interaction at a molecular level? Thermodynamic models based upon stoichiometric solute-water and solute-hydrotrope binding have long been used to understand solubilization microscopically. Such binding models have shown that the solvation numbers or coordination numbers of the water and hydrotrope molecules around the drug solute is the key quantity for solute-water and solute-hydrotrope interaction. However, we show that a rigorous statistical thermodynamic theory (the fluctuation solution theory originated by Kirkwood and Buff) requires the total reconsideration of such a paradigm. Here we show that (i) the excess solvation number (the net increase or decrease, relative to the bulk, of the solvent molecules around the solute), not the coordination number, is the key quantity for describing the solute-hydrotrope interaction; (ii) solute-hydrotrope binding is beyond the reach of the stoichiometric models because long-range solvation structure plays an important role.

Kirkwood-Buff Coarse-Grained Force Fields for Aqueous Solutions.

We present an approach to systematically coarse-grain liquid mixtures using the fluctuation solution theory of Kirkwood and Buff in conjunction with the iterative Boltzmann inversion method. The approach preserves both the liquid structure at pair level and the dependence of solvation free energies on solvent composition within a unified coarse-graining framework. To test the robustness of our approach, we simulated urea-water and benzene-water systems at different concentrations. For urea-water, three different coarse-grained potentials were developed at different urea concentrations, in order to extend the simulations of urea-water mixtures up to 8 molar urea concentration. In spite of their inherent state point dependence, we find that the single-site models for urea and water are transferable in concentration windows of approximately 2 M. We discuss the development and application of these solvent models in coarse-grained biomolecular simulations.

Molecular Thermodynamic Modeling of Mixed Solvent Solubility

A method based on statistical mechanical fluctuation solution theory for composition derivatives of activity coefficients is employed for estimating dilute solubilities of 11 solid pharmaceutical solutes in nearly 70 mixed aqueous and nonaqueous solvent systems. The solvent mixtures range from nearly ideal to strongly nonideal. The database covers a temperature range from 293 to 323 K. Comparisons with available data and other existing solubility methods show that the method successfully describes a variety of observed mixed solvent solubility behaviors using solute−solvent parameters from global regression of ternary data as well as predictions based on pure solvent solubilities with an average error of about 10% on mole fractions.

Correlation of phase equilibria and liquid densities for gases with ionic liquids

We describe a method for correlating and estimating gas solubilities and liquid densities of ionic liquids based on a corresponding-states form for direct correlation function integrals (DCFIs) from fluctuation solution theory, as previously applied to nonionic systems. A quantitative description is obtained for the volumetric behavior of pure liquids at elevated pressures. Solubilities and partial molar volumes are given over wide ranges of conditions using only pure component information and a binary parameter for hydrogen, carbon monoxide, oxygen and methane in the ionic liquids 1-butyl-3-methyl-imidazolium hexafluorophosphate, [bmim][PF 6], and 1-hexyl-3-methyl-imidazolium bis(trifluoromethylsulfonyl)imide, [hmim][Tf 2N]. Comparisons are made with other literature treatments and a predictive approach is described.

Method for predicting solubilities of solids in mixed solvents

AbstractA method is presented for predicting solubilities of solid solutes in mixed solvents, based on excess Henry's law constants. The basis is statistical mechanical fluctuation solution theory for composition derivatives of solute/solvent infinite dilution activity coefficients. Suitable approximations are made for a single parameter characterizing solute/solvent interactions. Comparisons with available data show that the method is successful in describing a variety of observed mixed solvent solubility behavior, including nearly ideal systems with small excess solubilities, systems with solute‐independent excess solubilities, and systems deviating from these simple rules. Successful predictions for new solvent mixtures can be made using limited data from other mixtures. © 2009 American Institute of Chemical Engineers AIChE J, 2009

Total correlation function integrals and isothermal compressibilities from molecular simulations

Generation of thermodynamic data, here compressed liquid density and isothermal compressibility data, using molecular dynamics simulations is investigated. Five normal alkane systems are simulated at three different state points. We compare two main approaches to isothermal compressibilities: (1) fluctuation solution theory analysis of trajectories obtained from simulations to yield total correlation function integrals; and (2) the more commonly used fluctuation formula. The results show that the two approaches yield consistent values and consistent uncertainties. Also, the computations converge in approximately the same amount of time. This suggests that computation of total correlation function integrals is a route to isothermal compressibility, as accurate and fast as well-established benchmark techniques. A crucial step is the integration of the radial distribution function. To obtain sensible results, this integral must be evaluated quite accurately. A previously employed integration method fits a radial distribution function expression to data. A modification is suggested for improving the accuracy, so that more accurate (individual) total correlation function integrals are obtained.