Virial Coefficient Research Articles

Prediction of the liquid-crystal phase behavior of hard right triangles from fourth-virial density-functional theories.

We have used an extended scaled-particle theory that incorporates four-body correlations through the fourth-order virial coefficient to analyze the orientational properties of a fluid of hard right isosceles triangles. This fluid has been analyzed by computer simulation studies, with clear indications of strong octatic correlations present in the liquid-crystal phase, although the more symmetric order tetratic phase would seem to be the most plausible candidate. Standard theories based on the second virial coefficient are unable to reproduce this behavior. Our extended theory predicts that octatic correlations, associated to a symmetry under global rotations of the oriented fluid by 45^{∘}, are highly enhanced, but not enough to give rise to a thermodynamically stable phase with strict octatic symmetry. We discuss different scenarios to improve the theoretical understanding of the elusive octatic phase in this intriguing fluid.

Gaseous speed of sound and virial coefficients for 2,3,3,3-tetrafluoropropene (R1234yf) + 1,1,1,2-tetrafluoroethane (R134a) binary mixture

The binary mixture 2,3,3,3-tetrafluoropropene (R1234yf) + 1,1,1,2-tetrafluoroethane (R134a) is commercially interesting and is widely studied in thermodynamics properties, but except for gaseous speed of sound. In this work, we modified the experimental procedure of the cylindrical fixed-path acoustic resonance method for the sound speed measurement of mixtures. Then we obtained the first two sets of gaseous sound speed data at the molar fraction of about (0.207/0.793) and (0.580/0.420) with a combined relative expanded uncertainty (k = 2) of 0.03%. The accurate molar fractions along each isotherm line were determined by extrapolating the experimental data with the absolute expanded uncertainty (k = 2) of 0.003. The interaction virial coefficients were derived from the experimental sound speed data and were used to establish a hard-core square-well molecular potential model. A previous virial coefficient correlation was modified to predict the virial coefficients for mixtures. The experimental results and model calculations agree well with each other, confirming the correctness of the experimental method. The data could provide references to test and develop the mixing model.

Experimental support for reclassification of the light scattering second virial coefficient from macromolecular solutions as a hydrodynamic parameter.

This investigation examines the source of the disparity between experimental values of the light scattering second virial coefficient [Formula: see text] (mL.mol/g2) for proteins and those predicted on the statistical mechanical basis of excluded volume. A much better theoretical description of published results for lysozyme is obtained by considering the experimental parameters to monitor the difference between the thermodynamic excluded volume term and its hydrodynamic counterpart. This involves a combination of parameters quantifying concentration dependence of the translational diffusion coefficient obtained from dynamic light scattering measurements. That finding is shown to account for observations of a strong correlation between [Formula: see text] (mL/g), where M2 is the molar mass (molecular weight) of the macromolecule and the diffusion concentration parameter [Formula: see text] (mL/g). On the grounds that [Formula: see text] is regarded as a hydrodynamic parameter, the same status should be accorded the light scattering second virial coefficient rather than its current incorrect thermodynamic designation as [Formula: see text] (mL.mol/g2), or just B, the osmotic second virial coefficient for protein self-interaction.

Second virial coefficients for a Fermi gas trapped in an anisotropic harmonic potential

Second virial coefficients for a Fermi gas trapped in an anisotropic harmonic potential

Target searches of interacting Brownian particles in dilute systems.

We study the target searches of interacting Brownian particles in a finite domain, focusing on the effect of interparticle interactions on the search time. We derive the integral equationfor the mean first-passage time and acquire its solution as a series expansion in the orders of the Mayer function. We analytically obtain the leading order correction to the search time for dilute systems, which are most relevant to target search problems and prove a universal relation given by the particle density and the second virial coefficient. Finally, we validate our theoretical prediction by Langevin dynamics simulations for the various types of the interaction potential.

Influence of solvent quality on the swelling and deswelling and the shear modulus of semi-dilute solution cross-linked poly(vinyl acetate) gels.

We systematically examine the influence of varying temperature (T) over a large range in model poly(vinyl acetate) gels swollen in isopropyl alcohol. The theta temperature Θ, at which the second virial coefficient A2 vanishes, is found to be equal to within numerical uncertainty to the corresponding high molecular mass polymer solution value without cross-links, and we quantify the swelling and deswelling of our model gels relative to their size at T = Θ, as customary for individual flexible polymer chains in solutions. We also quantify the "solvent quality" dependence of the shear modulus G relative to G(T = Θ) and compare to the hydrogel swelling factor, α. We find that all our network swelling and deswelling data can be reduced to a scaling equation of the same general form as derived from renormalization group theory for flexible linear polymer chains in solutions so that it is not necessary to invoke either the Flory-Huggins mean field theory or the Flory-Rehner hypothesis that the elastic and mixing contributions to the free energy of network swelling are separable to describe our data. We also find that changes of G relative to G(T = Θ) are directly related to α. At the same time, we find that classical rubber elasticity theory describes many aspects of these semi-dilute solution cross-linked networks, regardless of the solvent quality, although the prefactor clearly reflects the existence of network defects whose concentration depends on the initial polymer concentration of the polymer solution from which the networks were synthesized.

Influence of Molecular Structures of Organic Foulants on the Antifouling Properties of Poly(2-methoxyethyl acrylate) and Its Analogs: A Molecular Dynamics Study.

Elucidating the fouling phenomena of polymer surfaces will facilitate the molecular design of high-performance biomedical devices. Here, we investigated the remarkable antifouling properties of two acrylate materials, poly(2-methoxyethyl acrylate) (PMEA) and poly(3-methoxypropionic acid vinyl ester) (PMePVE), which have a terminal methoxy group on the side chain, via molecular dynamics simulations of binary mixtures of acrylate/methacrylate trimers with n-pentane or 2,2-dimethylpropane (neopentane), that serve as the nonpolar organic probe (organic foulants). The second virial coefficient (B2) was determined to assess the aggregation/dispersion properties in the binary mixtures. The order of the B2 values for the trimer/pentane mixtures indicated that the terminal methoxy group of the side chain plays an important role in enhancing the fouling resistance to nonpolar organic foulants. Here, we hypothesized that the antifouling properties of PMEA/PMePVE surfaces originate from the resistance. To evaluate the molecular-level accessibility of organic foulants to acrylate/methacrylate materials, we examined the radial distribution functions (RDFs) of the terminal methyl groups of neopentane around the main chains of the acrylate/methacrylate trimers. As a result, the third distinct RDF peaks are observed only for the methacrylate trimers. The peaks are attributed to the hydrophobic interactions between the methyl group of neopentane and that of the main chain of the trimer. Accordingly, the methyl group of the main chain of methacrylate materials, such as poly(2-hydroxyethyl methacrylate) and poly(2-methoxyethyl methacrylate), unfavorably induces fouling with organic foulants. In this study, we clarify that preventing hydrophobic interactions between an organic foulant and polymeric material is essential for enhancing the antifouling property. Our approach has great potential for evaluating the molecular-level affinities of organic foulant with polymer surfaces for the molecular design of excellent antifouling polymeric materials.

High Pressure Light Scattering of Therapeutic Proteins To Probe Aggregation and Protein-Protein Interactions.

There is interest in the direct in situ measurement of protein aggregation and reversible protein-protein interactions at high pressure as a means to assess protein stability. This is currently limited by the availability of in-house analytical methods. High-pressure (HP) scattering instrumentation (using either neutrons, X-rays, or light sources) are relatively rare, due to extensive development hurdles and lack of standardization. This report focuses on design, operation, and application of a new HP light scattering apparatus based on commercially available equipment with a view to wider applications. HP static light scattering results were obtained for two monoclonal antibodies (MAbs) that exhibit different extents of unfolding and aggregation at these conditions. Aggregation that was observed during in situ pressure incubations varied by MAb and total ionic strength of solution. This was conducted in tandem with ex situ measurements on MAb solutions that were incubated under pressure, where monomer loss was measured with size exclusion chromatography. Pressure cycling was also used to assess the extent of pressure-induced reversible and irreversible aggregation. Finally, the ability of the HP light scattering apparatus to assess the influence of pressure on reversible protein-protein interactions in the canonical sense of second osmotic virial coefficients was assessed using lysozyme, a relatively well-characterized protein under hydrostatic pressure. The method offers a convenient and reproducible capability that complements current small angle neutron/X-ray instrumentation, providing measurements that can be used to optimize the planning and interpretation of scattering data from synchrotron or neutron research facilities. Our results address a growing demand to characterize protein aggregates and aggregation-prone partially unfolded intermediates.

Thermodynamic Modeling of Methane Hydrate Phase Equilibria in Chloride and Bromide Solutions

The formation of hydrates is a major issue in the oil and gas industry. Injection of electrolytes is considered one of the best ways to prevent its formation. A thermodynamic model is developed from the Klauda and Sandler model, PRSV EoS, and Pitzer’s correlation to study the phase equilibria of methane hydrate in chloride and bromide electrolyte solutions. The model employs the parameters reported by Pitzer and Mayorga to calculate the Debye–Huckel coefficient and the second and third virial coefficients at 25 °C. The %AAD in predicted equilibrium pressures from experiments is 3.70 when the hydrate formed in NaCl, KCl, MgCl2, and CaCl2 solutions, whereas it is 2.33 in NaBr, KBr, MgBr2, CaBr2, and ZnBr2 solutions. The inhibition effect of any electrolyte on methane hydrate formation at 1 wt % is small, slowly increases at 3 wt %, and then pronounces at 5, 10, and 15 wt %. The cage occupancies have been found to increase and the activity of water has been found to decrease with the increase in electrolytes concentration. The freezing point depression of water in the presence of the strongest inhibitor MgCl2 has been observed to be −12.82 °C, whereas in the presence of the weakest inhibitor ZnBr2 it is −4.22 °C, both at 15 wt % concentration. Based on the phase equilibrium results and the freezing point depression of water, the order of the methane hydrate inhibition effect among all of the electrolytes studied here is MgCl2 > NaCl > CaCl2 > KCl > NaBr > ZnCl2 > MgBr2 > KBr > CaBr2 > ZnBr2.

Sorption property, excess enthalpy, and solvatochromic parameters of choline lactate

Sorption properties, including isotherm shape, adsorption energy, maximum uptake, and cyclic hydrothermal stability, have a significant influence on the optimization of absorbents for diverse desiccant air conditioning systems. The sorption behavior of the water/choline lactate ([ch][lac]) pair was examined using a dynamic vapor sorption (DVS) analyzer. Data analysis identified type-III isotherms. The Dubinin-Astakhov (D-A) formulation, defined by a temperature-independent Polani potential, was successfully applied to the sorption equilibrium curve. The Kamlet-Taft polarity parameters were tested using solvatochromic probes with a UV/Vis spectrophotometer. Multivariate regression analysis was performed to correlate the water affinity coefficient of the D-A model with the Kamlet-Taft polarity parameters using linear solvation energy relationships (ISERs). The solvatochromic parameter of dipolarity/palarizability (π*) had a positive contribution to the water affinity coefficient of ionic liquid (IL) desiccants, and the general trend of influence was hydrogenbondacidity(α)>π*>hydrogenbondbasicity(β). Excess enthalpies were correlated well with solvatochromic parameters. The differential enthalpies and entropies, first virial coefficient (A0), and isosteric enthalpies of sorption at zero surface coverage were also discussed. The enthalpy of activation at zero surface coverage was deduced from sorption kinetics. The hydrothermal cyclic stabilities and ambient regenerabilities were determined by cycle stress tests.

Gaseous speed of sound, ideal-gas heat capacity, and virial coefficients for cis-1,1,1,4,4,4-hexafluoro-2-butene (R1336mzz(Z)) and cis-1,2,3,3,3-pentafluoroprop-1-ene (R1225ye(Z))

The emerging low-global-warming-potential refrigerants cis-1,1,1,4,4,4-Hexafluoro-2-butene (R1336mzz(Z)) and cis-1,2,3,3,3-pentafluoroprop-1-ene (R1225ye(Z)) are showing high application potential in refrigeration and organic Rankine cycle, but the necessary thermophysical property data are too scarce to support their commercial application. Among the thermophysical properties, the speed of sound is essential for engineering design and scientific research. In this study, the speed of sound of gaseous R1336mzz(Z) and R1225ye(Z) was determined experimentally by the acoustic resonance method with the help of a fixed-path cylindrical cavity. The sound speed data of R1336mzz(Z) covers the temperature range of (323 to 363) K and pressure range of (57 to 383) kPa, and that of R1225ye(Z) covers (298 to 363) K and (40 to 1022) kPa. The relative uncertainties of these measurements are within a few parts in 10−4, allowing accurate analysis to derive the ideal-gas heat capacity and the 2nd acoustic density virial coefficients. Further, the parameters of the hard-core square-well intermolecular potentials are fitted to the acoustic data, allowing us to infer an estimate of the 2nd density virial coefficient of these substances. Quantitative calculation of uncertainties is also performed for these derived thermophysical parameters. Comparison of all these results with other data in the existing literature and the predictions of dedicated equations of state shows good agreement, supporting the claimed uncertainty of the experimental results.

Characteristic curves of the Mie fluid

Brown's characteristic curves of the Mie n,m fluid were investigated using molecular dynamics (MD) simulation and a molecular-based equation of state. Both, the 0th-order (Zeno) and three 1st-order characteristic curves (Amagat, Boyle, and Charles) were studied. The study was carried out using a hybrid approach combining MD simulation and molecular theory. Thereby, exact values for the zero-density limit of the characteristic curves were obtained using the second virial coefficient route. The focus was on the variation of the repulsive exponent of the Mie potential with n=8,10,11,12,13,14,16,20,36,48 (dispersive exponent constant at m=6). Also, the influence of the dispersive exponent on the characteristic curves was studied with m=4,5,6,7,8 (repulsive exponent constant at n=12). In total, 14 Mie fluids were studied. Based on the results, the influence of the shape of the intermolecular potential on the macroscopic properties, i.e. the characteristic curves, is elucidated. Interestingly, the Amagat curve is found to exhibit a peculiar dependency on the repulsive exponent. However, the postulates of Brown regarding the form and regularities of the characteristic curves are confirmed for all studied Mie fluids. Furthermore, the applicability of the classical corresponding states principle is assessed using the characteristic curve data. Important differences are obtained for the two exponents: The corresponding states principle captures reasonably well the variation of the dispersive exponent, but fails for describing the influence of the repulsive exponent. The results from this work are compared with predictions from a molecular-based equation of state, which is found to perform well in almost all studied cases.

An intermolecular potential for hydrogen: Classical molecular simulation of pressure-density-temperature behavior, vapor-liquid equilibria, and critical and triple point properties.

An intermolecular potential is reported for molecular hydrogen that combines two-body interactions from abinitio data with three-body interactions. The accuracy of the two-body potential is validated by comparison with experimental second virial coefficient data. Experimental pressure-density-temperature data are used to validate the addition of three-body interactions, often yielding very accurate predictions. Classical Monte Carlo simulations that neglect quantum effects are reported for the vapor-liquid equilibria (VLE), critical properties, and the triple point. A comparison with experimental data indicates that the effect of quantum interactions is to narrow the VLE phase envelope and to lower the critical temperature. The three-body interactions have a considerable influence on the phase behavior, resulting in good agreement with the experimental density. The critical properties of the two-body + three-body potential for hydrogen provide an alternative set of input parameters to improve the accuracy of theoretical predictions at temperatures above 100K. In the vicinity of the critical point, the coexistence densities do not obey the law of rectilinear diameters, which is a feature that has largely been overlooked in both experimental data and reference equations of state.

Speed of Sound Measurements in Helium at Pressures from 15 to 100 MPa and Temperatures from 273 to 373 K.

The speed of sound in helium was measured along five isotherms in a temperature range from 273 to 373 K at pressures from 15 to 100 MPa with a relative expanded uncertainty (k = 2) from 0.02 to 0.04%. A dual-path pulse-echo system was utilized to conduct these measurements. The data were compared with the reference equation of state developed by Ortiz Vega et al. At pressures up to 50 MPa, relative deviations were within the uncertainty of our measurements, while, at higher pressures, increasing negative deviations were observed up to -0.26%. We also compared the results with predictions based on the virial equation of state correct to the seventh virial coefficient, using the ab initio virial coefficients reported recently by Gokul et al., finding agreement to within the experimental uncertainty at all investigated states.

What Does the Geometry of the Hβ BLR Depend On?

We combine our dynamical modeling black-hole mass measurements from the Lick AGN Monitoring Project 2016 sample with measured cross-correlation time lags and line widths to recover individual scale factors, f, used in traditional reverberation-mapping analyses. We extend our sample by including prior results from Code for AGN Reverberation and Modeling of Emission Lines (caramel) studies that have utilized our methods. Aiming to improve the precision of black-hole mass estimates, as well as uncover any regularities in the behavior of the broad-line region (BLR), we search for correlations between f and other AGN/BLR parameters. We find (i) evidence for a correlation between the virial coefficient and black-hole mass, (ii) marginal evidence for a similar correlation between and black-hole mass, (iii) marginal evidence for an anticorrelation of BLR disk thickness with and , and (iv) marginal evidence for an anticorrelation of inclination angle with , , and . Last, we find marginal evidence for a correlation between line-profile shape, when using the root-mean-square spectrum, , and the virial coefficient, , and investigate how BLR properties might be related to line-profile shape using caramel models.

Physically based equation of state for Mie ν-6 fluids.

We develop a physically based equation of state that describes Mie ν-6 fluids with an accuracy comparable to that of state-of-the-art empirical models. The equation of state is developed within the framework of the uv-theory [T. van Westen and J. Gross, J. Chem. Phys. 155, 244501 (2021)], which is modified by incorporating the third virial coefficient B3 in the low-density description of the model. The new model interpolates between a first-order Weeks-Chandler-Andersen (WCA) perturbation theory at high densities and a modified first-order WCA theory that recovers the virial expansion up to B3 at low densities. A new algebraic equation for the third virial coefficient of Mie ν-6 fluids is developed-other inputs are taken from previous work. Predicted thermodynamic properties and phase equilibria are compared to a comprehensive database of molecular simulation results from the literature, including Mie fluids of repulsive exponents 9 ≤ ν ≤ 48. The new equation of state is applicable to states with densities up to ρ*(T*)⪅1.1+0.12T* and temperatures T* > 0.3. For the Lennard-Jones fluid (ν = 12), the performance of the model is comparable to that of the best empirical equations of state available. As compared to empirical models, the physical basis of the new model provides several advantages, however: (1) the new model is applicable to Mie fluids of repulsive exponents 9 ≤ ν ≤ 48 instead of only ν = 12, (2) the model leads to a better description of the meta-stable and unstable region (which is important for describing interfacial properties by classical density functional theory), and (3) being a first-order perturbation theory, the new model (potentially) allows an easier and more rigorous extension to non-spherical (chain) fluids and mixtures.

An alternative approach to the osmotic second virial coefficient of protein solutions and its application to liquid-liquid phase separation.

The osmotic second virial coefficient B2 is an important parameter to describe the interactions and phase behavior of protein solutions, including colloidal systems and macromolecular solutions. Another key parameter to describe the driving force of the nucleation of a new phase is the supersaturation, which is used in the classical nucleation theory framework and is connected with the favorable contribution in the Gibbs free energy in the bulk solution. In this article, we establish a connection between B2 calculated from small angle x-ray scattering (SAXS) data and the values of B2 obtained from supersaturation measurements using thermodynamics considerations. The values of the second virial coefficient calculated employing this method agree with those determined via SAXS in the region near the liquid-liquid phase separation border for human serum albumin and bovine serum albumin. The general relations adopted are shown to be useful for the estimation of the second virial coefficient B2 for globular proteins, in the proximity of the binodal biphasic coexistent region.

Role of Strong Localized vs Weak Distributed Interactions in Disordered Protein Phase Separation.

Interaction strength and localization are critical parameters controlling the single-chain and condensed-state properties of intrinsically disordered proteins (IDPs). Here, we decipher these relationships using coarse-grained heteropolymers comprised of hydrophobic (H) and polar (P) monomers as model IDPs. We systematically vary the fraction of P monomers XP and employ two distinct particle-based models that include either strong localized attractions between only H-H pairs (HP model) or weak distributed attractions between both H-H and H-P pairs (HP+ model). To compare different sequences and models, we first carefully tune the attraction strength for all sequences to match the single-chain radius of gyration. Interestingly, we find that this procedure produces similar conformational ensembles, nonbonded potential energies, and chain-level dynamics for single chains of almost all sequences in both models, with some deviations for the HP model at large XP. However, we observe a surprisingly rich phase behavior for the sequences in both models that deviates from the expectation that similarity at the single-chain level will translate to a similar phase-separation propensity. Coexistence between dilute and dense phases is only observed up to a model-dependent XP, despite the presence of favorable interchain interactions, which we quantify using the second virial coefficient. Instead, the limited number of attractive sites (H monomers) leads to the self-assembly of finite-sized clusters of different sizes depending on XP. Our findings strongly suggest that models with distributed interactions favor the formation of liquid-like condensates over a much larger range of sequence compositions compared to models with localized interactions.

Origins of the Failure of the Activity Virial Series

We examine the development of the virial equation of state when expressed as a series in the activity with coefficients labeled bn. Using the one-dimensional hard-rod model as a prototype, we consider steps in its development that introduce inaccuracy that leads it to form a divergent series. We discuss the role of volume dependence of the virial coefficients and present expressions and calculations for volume-dependent coefficients bn(V) for the hard-rod model up to n = 200. We examine alternative methods for computing properties from the bn. We recommend that further efforts be made to compute volume-dependent virial coefficients as a means to understand better the virial equation of state and to make it more robust in applications.

The Relationship between Protein–Protein Interactions and Liquid–Liquid Phase Separation for Monoclonal Antibodies

Being able to predict and control concentrated solution properties for solutions of monoclonal antibodies (mAbs) is critical for developing therapeutic formulations. At higher protein concentrations, undesirable solution properties include high viscosities, opalescence, particle formation, and precipitation. The overall aim of this work is to understand the relationship between commonly measured dilute solution parameters, the reduced osmotic second virial coefficient b22 and the diffusion interaction parameter kD and liquid-liquid phase separation, which occurs at higher protein concentrations. For globular proteins such as lysozyme or γB-crystallin, the location of the liquid-liquid coexistence curve is controlled by the net protein-protein interaction, which is related to b22. Because many mAbs undergo reversible self-association due to forming highly directional interactions, it is not known if b22 can be used as a reliable predictor for LLPS since increasing the anisotropy in the interaction potential causes phase separation to occur at much stonger net protein-protein attractions or lower values of b22. Here, we map the coexistence curves for three mAbs, referred to as COE-01, COE-07, and COE-19, in terms of b22 and kD values. The measurements are carried out at a low salt condition near the pI, where protein-protein interactions are expected to be anisotropic due to the presence of electrostatic attractions, and under salting-out conditions at high ammonium sulfate concentrations, which is expected to reduce the anisotropy by screening electrostatic interactions. We also show that deviations from a linear correlation between b22 and kD can be used as an indicator of reversible self-association. Each of the mAbs under salting-out conditions follows the correlation supporting the hypothesis that protein-protein interactions are nonspecific, while deviations from the correlation occur for COE-01 and COE-19 under low salt conditions indicating the mAbs undergo reversible self-association. For five out of the six conditions, the onset of phase separation, as reflected by the reduced virial coefficient at the critical point b22c occurs in a small window -1.6 > b22c > -2.3, which is similar to what has been observed for lysozyme and for bovine γB-crystallin. Under low salt conditions, b22c ≈ -5.1 for COE-19, which we previously showed to self-associate into small oligomers. Our findings suggest that under conditions where mAb interactions are weakly anisotropic, such as occur at high salt conditions, phase separation will begin to occur in a small window of b22. Deviations from the window can occur when mAbs undergo reversible self-association, although this is not always the case and likely depends upon whether or not highly directional interactions are passivated in the oligomer formation. We expect fitting LLPS measurements to simplified interaction models for mAbs will provide additional insight into the nature of the protein-protein interactions and guide their development for calculating concentrated solution properties.