Cross Virial Coefficients Research Articles

Thermodynamics of NaCl in dense water vapor via cross virial coefficients

Sodium chloride in the fluid phase is important for the chemistry of natural hydrothermal processes in the Earth’s crust and derived industrial utilizations, where thermodynamics is an essential part that can also to some extent fill in missing experimental data. The solubility of sodium chloride in crystalline form (halite) in water vapor is modeled using a novel combination of an accurate equation of state for pure water and a truncated virial equation for water mixed with sodium chloride. In the resulting Helmholtz energy expression, interactions between H2O and NaCl are incorporated through the second (B12), third (C112), and fourth (D1112) cross virial coefficients. The temperature dependence of the coefficients has been determined by regression to experimental data. Specifically, the temperature dependence of B12 is assumed to follow that of a square-well gas, whereas C112 and D1112 are calculated using purely exponential functions of inverse temperature. The temperature dependence of B12 is supported by independent calculations of ab initio interaction energies between H2O and NaCl. For water itself, all terms in the virial equation are collectively modeled by the IAPWS-95 equation of state. For sodium chloride, interactions between two or more NaCl molecules are disregarded. The model shows quantitative agreement with available solubility data in the vapor–halite coexistence region, which implies that the water vapor density is approximately less than 130kgm−3. Its theoretical basis ensures that predictions are reliable in regions where experimental data are missing. The direct calculation of cross virial coefficients is overall useful for modeling geochemical fluids with low solute concentrations, including systems with multiple solvents.

Impact of the solvent properties on molecular interactions and phase behaviour of alginate-gelatin systems

Alginate-gelatin systems are used in a wide range of applications, e.g., foods, cosmetics, drugs, and medicine. To target the techno-functionalities of alginate-gelatin systems such as texture, stability, or appearance it is necessary to understand the structure formation of gel systems, which is influenced by the molecular interactions and the phase behaviour. Therefore, the aim of this study was to characterise the molecular interactions and phase behaviour of dilute alginate-gelatin systems. For this purpose, we analysed alginate, gelatin A and their mixtures at pH 7. First, we characterised the effect of the sodium chloride concentration on the single polymer solutions by measuring the critical overlap concentration, the zeta potential, intrinsic viscosity, the Huggins coefficient as well as the molecular weight and the second virial coefficient. To gain a better understanding of the molecular interactions between alginate and gelatin, we determined the Huggins coefficient and the interaction parameter by Wolf as a function of mixing ratio, NaCl and urea concentration. Additionally, we calculated the second cross-virial coefficient. Lastly, we characterised the influence of the interactions on the secondary structure of gelatin by Fourier-transform-infrared-spectroscopy. The attractive interactions between alginate and gelatin potentially enabling the formation of soluble complexes, were demonstrated as deviations from additivity of intrinsic viscosity and negative cross-virial coefficients. Interestingly, the Wolf interaction parameter was affected by the NaCl concentration and the mixing ratio, whereas the presence of urea showed no impact. This indicates that the attractive interactions of alginate and gelatin A at pH 7 is rather caused by electrostatic interactions than hydrogen bonds. The present study clarifies the relevance of hydrogen bonds and electrostatic interactions as well as the impact of mixing ratio and solvent properties on the phase behaviour of alginate-gelatin systems. The insights contribute to a better understanding of the structure formation of concentrated alginate-gelatin systems, levelling the path to specific modifications of functionality.

Meta-analysis of critical points to determine second virial coefficients for binary biopolymer mixtures

A method is introduced to extract virial coefficients from phase diagrams. Previous work demonstrated that the critical point of a phase diagram (two parameters) of a binary polymer mixture provides too little information to allow a full description of the phase diagram when using the Edmond-Ogston model (which requires three parameters). In this paper, it is shown that simultaneous analysis of multiple phase diagrams of binary polymer mixtures with shared components does allow the extraction of virial coefficients. The method is illustrated by analyzing phase diagrams for various polymer mixtures directly available in literature. The obtained virial coefficients for the polyethylene oxide/polyethylene glycol (PEO) and dextran polymers are compared to values as obtained by membrane osmometry and shown to be in satisfactory agreement. These virial coefficients are shown to follow a power-law dependence on the molecular weight. In addition, it is observed that the cross-virial coefficient can be approximately described by a simple mixing rule. This rule makes it possible to estimate the phase diagrams for polymer mixtures, where the virial coefficients of the pure components are known, but the cross-virial coefficient is unknown. The method to extract virial coefficients is also applied to a number of mixtures containing agarose, bovine serum albumin or gelatin, mixed with either PEO or dextran. The virial coefficients obtained in a simultaneous fit involving fitting the critical points of multiple phase diagrams are used to calculate the full original individual phase diagrams. Generally, there is good agreement between such a calculated phase diagram and the original experimental phase diagrams.

Phase-Separating Binary Polymer Mixtures: The Degeneracy of the Virial Coefficients and Their Extraction from Phase Diagrams.

The Edmond–Ogston model for phase separation in binary polymer mixtures is based on a truncated virial expansion of the Helmholtz free energy up to the second-order terms in the concentration of the polymers. The second virial coefficients (B11, B12, B22) are the three parameters of the model. Analytical solutions are presented for the critical point and the spinodal in terms of molar concentrations. The calculation of the binodal is simplified by splitting the problem into a part that can be solved analytically and a (two-dimensional) problem that generally needs to be solved numerically, except in some specific cases. The slope of the tie-lines is identified as a suitable parameter that can be varied between two well-defined limits (close to and far away from the critical point) to perform the numerical part of the calculation systematically. Surprisingly, the analysis reveals a degenerate behavior within the model in the sense that a critical point or tie-line corresponds to an infinite set of triplets of second virial coefficients (B11, B12, B22). Since the Edmond–Ogston model is equivalent to the Flory–Huggins model up to the second order of the expansion in the concentrations, this degeneracy is also present in the Flory–Huggins model. However, as long as the virial coefficients predict the correct critical point, the shape of the binodal is relatively insensitive to the specific choice of the virial coefficients, except in a narrow range of values for the cross-virial coefficient B12.

Correlation and prediction of thermodynamic properties of dilute solutes in water up to high T and P. II. Normal fluids CO2, C2H4, C2H6, C3H8, n-C4H10, i-C4H10, functional groups CH3 and CH2

Fugacity coefficients of dilute aqueous solutions of six species belonging to normal fluids (CO2, C2H4, C2H6, C3H8, n-C4H10, i-C4H10) as well as functional groups CH3 and CH2 have been evaluated from ambient conditions (298 K, 0.1 MPa) to high temperatures (up to 2000 K) and water densities (up to 1500 kg m−3). The basis of the approach is a correlation developed for the function, A12∞=V2∞/κTRT, that includes known constraints: second virial coefficients at low water densities, hard sphere behavior at high water densities, near-critical principles, and corresponding-states relations. A comparison with literature data at subcritical (Henry's and vapor-liquid distribution constants) and supercritical temperatures (fugacity coefficients) is made. For aqueous CO2, the agreement with literature data is generally satisfactory. It is shown that for other solutes existing disagreements can be traced mostly to the values of the second cross virial coefficients employed in the literature models, suggesting the room for improvement of existing recommendations.

Osmometric and viscometric study of levan, β-lactoglobulin and their mixtures

The current study investigated the molecular interactions of the exopolysaccharide levan (low and high molecular weight), β-lactoglobulin and their mixtures at different NaCl concentrations (2.5 mM and 100 mM, pH 7) using viscometry and membrane osmometry. Both methods were used to predict the phase behavior of the mixed polymer systems. A positive cross-virial coefficient indicates repulsive pair interactions between levan and β-lactoglobulin due to the excluded volume effect at both salt concentrations. A higher molecular weight of levan seemed to enhance the repulsive forces, while the NaCl concentration had only a minor influence. For the phase behavior, the charge of the β-lactoglobulin and therefore the NaCl concentration plays an important role. At low salt concentrations the repulsive, electrostatic forces between the individual β-lactoglobulin molecules counteract the concentration of the protein in one phase. Therefore, both polymers seem to be co-soluble regardless of the molecular weight of levan. At 100 mM NaCl, the electrostatic repulsion between the β-lactoglobulin molecules is screened and attractive protein - protein interactions predominated. This facilitates the formation of two phases and segregative phase separation due to the excluded volume effect becomes more likely.

Determination of Protein-Protein Interactions in a Mixture of Two Monoclonal Antibodies.

The coformulation of monoclonal antibody (mAb) mixtures provides an attractive route to achieving therapeutic efficacy where the targeting of multiple epitopes is necessary. Controlling and predicting the behavior of such mixtures requires elucidating the molecular basis for the self- and cross-protein-protein interactions and how they depend on solution variables. While self-interactions are now beginning to be well understood, systematic studies of cross-interactions between mAbs in solution do not exist. Here, we have used static light scattering to measure the set of self- and cross-osmotic second virial coefficients in a solution containing a mixture of two mAbs, mAbA and mAbB, as a function of ionic strength and pH. mAbB exhibits strong association at a low ionic strength, which is attributed to an electrostatic attraction that is enhanced by the presence of a strong short-ranged attraction of nonelectrostatic origin. Under all solution conditions, the measured cross-interactions are intermediate self-interactions and follow similar patterns of behavior. There is a strong electrostatic attraction at higher pH values, reflecting the behavior of mAbB. Protein-protein interactions become more attractive with an increasing pH due to reducing the overall protein net charges, an effect that is attenuated with an increasing ionic strength due to the screening of electrostatic interactions. Under moderate ionic strength conditions, the reduced cross-virial coefficient, which reflects only the energetic contribution to protein-protein interactions, is given by a geometric average of the corresponding self-coefficients. We show the relationship can be rationalized using a patchy sphere model, where the interaction energy between sites i and j is given by the arithmetic mean of the i-i and j-j interactions. The geometric mean does not necessarily apply to all mAb mixtures and is expected to break down at a lower ionic strength due to the nonadditivity of electrostatic interactions.

Thermodynamic properties of dilute hydrogen in supercritical water

A thermodynamic model is developed to calculate the fugacity coefficients and partial molar volumes of hydrogen at infinite dilution in water at 647.1–2000 K and pure water densities between 0 and 1500 kg m−3. The model is based on the predicted values of DCFI (the dimensionless integral of the infinite dilution hydrogen - water direct correlation). Values of DCFI at low water densities are calculated from accurately known second cross virial coefficients; at high water densities predictions are based on the relations from the theory of a mixture of hard spheres; DCFI values at intermediate water densities are interpolated using a variant of corresponding-states correlation. Predicted values of the hydrogen fugacity coefficients at infinite dilution in water are compared with experimental data and results of the literature equations of state. The included Excel spreadsheet allows calculations provided that values of T, P, ρ1∗, and κT for pure water are entered by a user.

Interpolation of intermolecular potentials using Gaussian processes.

A procedure is proposed to produce intermolecular potential energy surfaces from limited data. The procedure involves generation of geometrical configurations using a Latin hypercube design, with a maximin criterion, based on inverse internuclear distances. Gaussian processes are used to interpolate the data, using over-specified inverse molecular distances as covariates, greatly improving the interpolation. Symmetric covariance functions are specified so that the interpolation surface obeys all relevant symmetries, reducing prediction errors. The interpolation scheme can be applied to many important molecular interactions with trivial modifications. Results are presented for three systems involving CO2, a system with a deep energy minimum (HF-HF), and a system with 48 symmetries (CH4-N2). In each case, the procedure accurately predicts an independent test set. Training this method with high-precision ab initio evaluations of the CO2-CO interaction enables a parameter-free, first-principles prediction of the CO2-CO cross virial coefficient that agrees very well with experiments.

Second virial coefficients: a route to combining rules?

ABSTRACTThe effect of combining rules on the second virial coefficient is systematically examined for a series of n-alkanes up to octane with the aim to assess the impact of the individual site–site interactions and to find potential general regularities. To describe the interactions of n-alkanes, we use the TraPPE-UA force field and examine the effect of deviations from the Lorentz–Berthelot combining rule, both separately for the energy and length scaling parameters of the Lennard-Jones potential, and for their combined change. The results reveal the different impacts of both site–site interactions and distance and energy scaling parameters on the cross virial coefficient, which can potentially be used in force field development. An attempt is also made to find a scaling that could provide a uniform result.

Protein partition coefficients can be estimated efficiently by hybrid shortcut calculations

The extraction of therapeutic proteins like monoclonal antibodies in aqueous two-phase systems (ATPS) is a suitable alternative to common cost intensive chromatographic purification steps within the downstream processing. Thereby the protein partitioning can be selectively changed using a displacement agent (additional salt) in order to allow for a successful purification of the target protein. Within this work a new shortcut strategy for the calculation of protein partition coefficients in polymer-salt ATPS is presented. The required protein-solute (phase-forming component, displacement agent) interactions are covered by the cross virial coefficient B23 measured by composition gradient multi-angle light scattering (CG-MALS). Using this shortcut calculation allows for an efficient determination of the partition coefficients of the target protein immunoglobulin G (IgG) and the impurity human serum albumin (HSA) within PEG-citrate and PEG-phosphate ATPS independently on the protein concentration. We demonstrate that the selection of a suitable displacement agent allowing for a selective purification of IgG from HSA is accessible by B23. Based on the determination of the protein–protein interactions via CG-MALS covered by the second osmotic virial coefficient B22 a further optimization of ATPS preventing protein precipitation is enabled. The results show that our approach contributes to an efficient downstream processing development.

QSPR-Modeling for the Second Virial Cross-Coefficients of Binary Organic Mixtures

The second virial cross-coefficient is an important characteristic of the pair intermolecular interactions that describes solely the heterogeneous interactions. In the current study, the authors made the first attempt to develop rigorous QSPR models for analysis and prediction of the second virial cross-coefficient. Novel descriptors to describe pair intermolecular interactions were implemented. Statistical characteristics of the obtained models showed high performance. Prediction errors are comparable to the errors of data. Theoretically predicted values of the second virial cross-coefficient may be used to derive PVT-properties of mixtures at the different temperatures as well as to calculate intermolecular pair potential.

Osmotic Virial Coefficients as Access to the Protein Partitioning in Aqueous Two-Phase Systems

A promising alternative to state of the art chromatographic separations of therapeutic proteins is the extraction of the target protein using an aqueous two-phase system (ATPS). The use of an additional salt working as a displacement agent can influence the protein partitioning behavior in ATPS and thus enable a selective purification of the target protein. The selection of a suitable ATPS for protein extraction requires information concerning the protein-protein interactions (second osmotic virial coefficient B22 ) as well as the interactions between protein and solute (displacement agent and phase-forming components) (cross virial coefficient B23 ). In this work, the partitioning behavior and the precipitation affinity of immunoglobulin G (IgG) is considered within a polyethylene glycol (PEG)-phosphate ATPS. The influence on IgG partitioning upon addition of NaCl and (NH4)2 SO4 was investigated. In order to access the IgG precipitation affinity and the IgG partitioning behavior, the B22 and B23 values were determined for several combinations of solute [PEG, phosphate buffer, NaCl, and (NH4)2 SO4 ] and IgG based on static light scattering measurements. A qualitative estimation of the IgG precipitation affinity and the suitability of a solute as potential displacement agent within the PEG-phosphate ATPS on the basis of the measured B22 and B23 values is presented.

Interactions in protein mixtures. Part II: A virial approach to predict phase behavior

A virial theory was used to relate molecular interactions (in terms of second virial coefficients, B′) and molecular size ratios to liquid–liquid phase separation. Application of the virial theory to binary hard sphere mixtures (additive and non-additive) confirmed the applicability of this simple approach towards predicting phase behavior based on two-particle interactions.Experimentally, second cross virial coefficients were obtained for dextran/gelatin, whey protein isolate (WPI)/gelatin mixtures and whey protein aggregate (WPA)/gelatin mixtures using membrane osmometry at varying ionic strength. From this, solvent conditions where interactions between proteins are dominated by electrostatics and solvent conditions where interactions are dominated by hard body interactions could be determined.Using experimentally obtained second virial coefficients, the liquid–liquid phase separation for gelatin/dextran mixtures was successfully predicted. Second cross virial coefficients for gelatin/whey protein isolate and for gelatin/whey protein aggregate could be related to the absence of phase separation in these mixtures. This could be related to a similar size of the proteins and their non-additive behavior at conditions where they mainly interact via hard body interactions.

Calculation of intermolecular potentials for H2[sbnd]H2 and H2[sbnd]O2 dimers ab initio and prediction of second virial coefficients

The intermolecular interaction potentials of the dimers H2H2 and H2O2 were calculated from quantum mechanics, using coupled-cluster theory CCSD(T) and correlation-consistent basis sets aug-cc-pVmZ (m=2,3); the results were extrapolated to the basis set limit aug-cc-pV23Z. The interaction energies were corrected for the basis set superposition error with the counterpoise scheme. For comparison also Møller–Plesset perturbation theory (at levels 2–4) with the basis sets aug-cc-pVTZ were considered, but the results proved inferior. The quantum mechanical results were used to construct analytical pair potential functions. From these functions the second virial coefficients of hydrogen and the cross virial coefficients of the hydrogen–oxygen system were obtained by integration; in both cases corrections for quantum effects were included. The results agree well with experimental data, if available, or with empirical correlations.

Influence of macromolecular precipitants on phase behavior of monoclonal antibodies.

For the successful application of protein crystallization as a downstream step, a profound knowledge of protein phase behavior in solutions is needed. Therefore, a systematic screening was conducted to analyze the influence of macromolecular precipitants in the form of polyethylene glycol (PEG). First, the influence of molecular weight and concentration of PEG at different pH-values were investigated and analyzed in three-dimensional (3-D) phase diagrams to find appropriate conditions in terms of a fast kinetic and crystal size for downstream processing. In comparison to the use of salts as precipitant, PEG was more suitable to obtain compact 3-D crystals over a broad range of conditions, whereby the molecular weight of PEG is, besides the pH-value, the most important parameter. Second, osmotic second virial coefficients as parameters for protein interactions are experimentally determined with static light scattering to gain a deep insight view in the phase behavior on a molecular basis. The PEG-protein solutions were analyzed as a pseudo-one-compartment system. As the precipitant is also a macromolecule, the new approach of analyzing cross-interactions between the protein and the macromolecule PEG in form of the osmotic second cross-virial coefficient (B23 ) was applied. Both parameters help to understand the protein phase behavior. However, a predictive description of protein phase behavior for systems consisting of monoclonal antibodies and PEG as precipitant is not possible, as kinetic phenomena and concentration dependencies were not taken into account.

Osmotic second virial cross‐coefficient measurements for binary combination of lysozyme, ovalbumin, and α‐amylase in salt solutions

Interactions measurement is a valuable tool to predict equilibrium phase separation of a desired protein in the presence of unwanted macromolecules. In this study, cross-interactions were measured as the osmotic second virial cross-coefficients (B23 ) for the three binary protein systems involving lysozyme, ovalbumin, and α-amylase in salt solutions (sodium chloride and ammonium sulfate). They were correlated with solubility for the binary protein mixtures. The cross-interaction behavior at different salt concentrations was interpreted by either electrostatic or hydrophobic interaction forces. At low salt concentrations, the protein surface charge dominates cross-interaction behavior as a function of pH. With added ovalbumin, the lysozyme solubility decreased linearly at low salt concentration in sodium chloride and increased at high salt concentration in ammonium sulfate. The B23 value was found to be proportional to the slope of the lysozyme solubility against ovalbumin concentration and the correlation was explained by preferential interaction theory.

CCSD(T) potential energy and induced dipole surfaces for N2–H2(D2): Retrieval of the collision-induced absorption integrated intensities in the regions of the fundamental and first overtone vibrational transitions

The present work aims at ab initio characterization of the integrated intensity temperature variation of collision-induced absorption (CIA) in N(2)-H(2)(D(2)). Global fits of potential energy surface (PES) and induced dipole moment surface (IDS) were made on the basis of CCSD(T) (coupled cluster with single and double and perturbative triple excitations) calculations with aug-cc-pV(T,Q)Z basis sets. Basis set superposition error correction and extrapolation to complete basis set (CBS) limit techniques were applied to both energy and dipole moment. Classical second cross virial coefficient calculations accounting for the first quantum correction were employed to prove the quality of the obtained PES. The CIA temperature dependence was found in satisfactory agreement with available experimental data.

Thermodynamics of Si(OH)4 in the vapor phase of water: Henry’s and vapor–liquid distribution constants, fugacity and cross virial coefficients

The fugacity coefficients of Si(OH)4 are evaluated from solubilities of solid phases of SiO2 in the vapor phase of water. The virial equation of state, truncated at the third virial coefficient, is employed to describe the fugacity coefficients of Si(OH)4. The temperature dependencies of the second, B12, and the third, C112, cross virial coefficients for H2O–Si(OH)4 interactions are approximated by empirical relations. It is found that silica–water interactions in the vapor phase are significantly more non-ideal compared to water–water interactions. Knowledge of B12 and C112 allows calculation of solubilities of quartz (Q) and amorphous silica (AS) in steam up to the density of 200kgm−3 in satisfactory agreement with available data, and should provide reasonable solubility values at temperatures where no experimental results exist. The calculated values of the solubility of Q and AS in saturated vapor up to the critical temperature of water, Tc, are tabulated.The partial molar properties of dilute solutes close to the critical point of water are governed by the Krichevskii parameter, the value of which for Si(OH)4 is evaluated from available data (mainly vapor–liquid distribution constants for silica) to be equal to −187±10MPa. The knowledge of the thermodynamic properties of Si(OH)4 in the ideal gas state and in the state of the standard solution in liquid water allows calculating Henry’s constant, kH, for Si(OH)4 up to 623.15K at water saturation pressure P1∗. The theoretically-based equation, containing the Krichevskii parameter, allows extrapolating kH values all the way toward the critical temperature of water. This, in turn, makes possible calculation of the solubility of quartz and amorphous silica in liquid water at P1∗ at all temperatures up to Tc. The presented results should be useful for modeling solid–liquid–vapor, solid–vapor and liquid–vapor equilibria in the H2O–SiO2 system at steam densities up to 200kgm−3.

Thermodynamics of B(OH) 3 in the vapor phase of water: Vapor–liquid and Henry's constants, fugacity and second cross virial coefficients

Available experimental data are employed to determine: (1) the enthalpy of formation, Δ f H°, of B(OH) 3, in the ideal gas state; (2) the value of the Krichevskii parameter, ( ∂ P / ∂ x 2 ) T , V , x 2 = 0 c for boric acid in water; (3) values of the vapor–liquid distribution constants and Henry's constants for H 3BO 3 at infinite dilution in water for temperatures between 273.15 K and T c . The obtained information allows calculation of the fugacity coefficients for boric acid in the vapor phase of water and second cross virial coefficients for interactions between molecules of H 2O and B(OH) 3 at temperatures between 450 and T c .