Virial Coefficient 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.

Generic low-density corrections to the equation of state of chain molecules with repulsive intermolecular forces

Molecular-based equations of state for describing the thermodynamics of chain molecules are often based on mean-field like arguments that reduce the problem of describing the interactions between chains to a simpler one involving only nonbonded monomers. While for dense liquids such arguments are known to work well, at low density they are typically less appropriate due to an incomplete description of the effect of chain connectivity on the local environment of the chains’ monomer segments. To address this issue, we develop three semi-empirical approaches that significantly improve the thermodynamic description of chain molecules at low density. The approaches are developed for chain molecules with repulsive intermolecular forces; therefore, they could be used as reference models for developing equations of the state of real fluids based on perturbation theory. All three approaches are extensions of Wertheim’s first-order thermodynamic perturbation theory (TPT1) for polymerization. The first model, referred to as TPT1-v, incorporates a second-virial correction that is scaled to zero at liquid-like densities. The second model, referred to as TPT1-y, introduces a Helmholtz-energy contribution to account for correlations between next-nearest-neighbor segments within chain molecules. The third approach, called TPT-E, directly modifies TPT1 without utilizing an additional Helmholtz energy contribution. By employing TPT1 at the core of these approaches, we ensure an accurate description of mixtures and enable a seamless extension from chains of tangentially bonded hard-sphere segments of equal size to hetero-segmented chains, fused chains, and chains of soft repulsive segments (which are influenced by temperature). The low-density corrections implemented in TPT1 are designed to preserve these good characteristics, as confirmed through comparisons with novel molecular simulation results for the pressure of various chain fluids. TPT1-v exhibits excellent transferability across different chain types, but it relies on knowing the second virial coefficient of the chain molecules, which is non-trivial to obtain and determined here using Monte Carlo simulation. The TPT1-y model, on the other hand, achieves comparable accuracy to TPT1-v while being fully predictive, requiring no input besides the geometry of the chain molecules.

Accurate neural-network-based fitting of full-dimensional N2 –Ar and N2–CH4 two-body potential energy surfaces aimed at spectral simulations

We describe the development of machine-learned (ML) potentials for flexible, weakly interacting monomers. A recently suggested permutationally invariant polynomial neural network (PIP-NN) approach is utilised to represent the full-dimensional two-body component of the molecular pair energy. To ensure the asymptotic zero-interaction limit, a tailored subset of the full invariant polynomial basis set is utilised, and their variables are modified to achieve a better fit of the correct asymptotic behaviour at a long range. This new technique is used to build full-dimensional potentials for the two-body N 2 –Ar and N 2 –CH 4 interactions by fitting databases of ab initio energies calculated at the coupled-cluster level of theory. The second virial coefficient, fully accounting for molecular flexibility, is then calculated within the classical framework using the obtained PIP-NN potential surfaces. A trajectory-based simulation of the N 2 –Ar collision-induced absorption is conducted, covering both the far- and mid-infrared ranges.

Combining Scattering Experiments and Colloid Theory to Characterize Charge Effects in Concentrated Antibody Solutions.

Charges and their contribution to protein-protein interactions are essential for the key structural and dynamic properties of monoclonal antibody (mAb) solutions. In fact, they influence the apparent molecular weight, the static structure factor, the collective diffusion coefficient, or the relative viscosity, and their concentration dependence. Further, charges play an important role in the colloidal stability of mAbs. There exist standard experimental tools to characterize mAb net charges, such as the measurement of the electrophoretic mobility, the second virial coefficient, or the diffusion interaction parameter. However, the resulting values are difficult to directly relate to the actual overall net charge of the antibody and to theoretical predictions based on its known molecular structure. Here, we report the results of a systematic investigation of the solution properties of a charged IgG1 mAb as a function of concentration and ionic strength using a combination of electrophoretic measurements, static and dynamic light scattering, small-angle X-ray scattering, and tracer particle-based microrheology. We analyze and interpret the experimental results using established colloid theory and coarse-grained computer simulations. We discuss the potential and limits of colloidal models for the description of the interaction effects of charged mAbs, in particular pointing out the importance of incorporating shape and charge anisotropy when attempting to predict structural and dynamic solution properties at high concentrations.

Modelling bi-specific antibodies in aqueous solution

This study presents theoretical results for physico-chemical properties of system of molecules modeling bi-specific antibodies, such as, dual-variable-domain monoclonal antibodies (DVD-Ig) and Fabs-In-Tandem Immunoglobulin (FIT-Ig). These molecules are representatives of the engineered proteins that combine the function and specificity of two monoclonal antibodies. Individual molecules are here depicted as an assembly of nine (or in case of the Fit-Ig eleven) hard spheres, organized to resemble the Y-shaped object. The effects of the increased size, asymmetry, and flexibility of individual molecules on measurable properties of such systems of molecules are investigated. We examined the liquid-liquid phase separation, the second virial coefficient B2, and viscosity under various experimental conditions. The calculations are compared with the data for regular monoclonal antibodies and discussed in view of the experimental results for DVD-Ig solutions available in literature.

Electrical conductivity of hydrogen plasmas: Low-density benchmarks and virial expansion including e–e collisions

An improved virial expansion for the low-density limit of the electrical conductivity σ(T,n) of hydrogen as the simplest ionic plasma is presented. Quantum statistical methods provide exact values for the lowest virial coefficients, which serve as a benchmark for analytical approaches to electrical conductivity as well as for numerical results from density functional theory-based molecular dynamics simulations (DFT-MD) or path-integral Monte Carlo simulations. The correction factor introduced by Reinholz et al. [Phys. Rev. E 91, 043105 (2015)] is applied to describe the inclusion of electron–electron collisions in DFT-based calculations of transport coefficients. As a benchmark, the first virial coefficient is correctly described with this approach. The value of the second virial coefficient is discussed, and questions about its value according to DFT-MD simulations are addressed.

Influence of solvent quality on the swelling and shear modulus of polymer gels chemically cross-linked in solution

AbstractThe effect of temperature (T) is studied on the swelling of model poly(vinyl acetate) (PVAc) gels in isopropyl alcohol. The theta temperature Θ of these gels, at which the second osmotic virial coefficient A2 vanishes, is close to that of the corresponding high molecular mass polymer solution without cross-links so that solvent quality may be defined in the same way as the corresponding precursor polymer solution. We quantified the swelling and deswelling of PVAc gels relative to their size at the theta temperature, and also determined the effect of T on the shear modulus of these gels. It was found that both swelling and deswelling data could be reduced to a universal scaling equation of the same general form as derived from renormalization group (RG) theory for flexible linear polymer chains in solutions. Graphical abstract Variaton of the volumetric swelling factor as a function of the reduced temperature for PVAc gels swollen in isopropyl alcohol.

Line shapes of isotropic and depolarized CILS spectra of neon: Interaction potential and pair polarizability

The present work proposes three interaction potentials between two gaseous neon atoms and applies them to the study of models representing the anisotropy and trace of the induced pair polarizability tensor. The proposed empirical interatomic potentials for the gaseous Ne-Ne interaction are developed by fitting the Improved Lennard-Jones (ILJ), Modified Tang-Toennies (MTT) and Barker et al. (BFW) potentials simultaneously to the spectroscopic and transport properties over a wide temperature range. The validity of the proposed interatomic potentials is verified by comparing the calculated and experimental second pressure virial coefficient over a wide temperature range. In addition to this, the results show satisfactory accordance between the experimental and calculated vibrational energy levels using the deduced pair potential parameters compared to those reported for neon gas. These comparisons exhibit acceptable accuracy when the present work potential parameters are inserted to reproduce the transport and thermo-physical properties, besides the spectral moments and the spectral line-shape intensities for neon gas. The isotropic and depolarized collision-induced light scattering (CILS) spectra are produced and analyzed as functions of the potential considered. They exhibit sensitivity to both the attractive part of the potential and the short-range values of the polarizability anisotropy or mean at intermediate and high frequencies. Mean and anisotropy are deduced directly from spectral line shapes using a new inversion method.

Impact of Combination Rules, Level of Theory, and Potential Function on the Modeling of Gas- and Condensed-Phase Properties of Noble Gases.

The systems of noble gases are particularly instructive for molecular modeling due to the elemental nature of their interactions. They do not normally form bonds nor possess a (permanent) dipole moment, and the only forces determining their bonding/clustering stems from van der Waals forces─dispersion and Pauli repulsion, which can be modeled by empirical potential functions. Combination rules, that is, formulas to derive parameters for pair potentials of heterodimers from parameters of corresponding homodimers, have been studied at length for the Lennard-Jones 12-6 potentials but not in great detail for other, more accurate, potentials. In this work, we examine the usefulness of nine empirical potentials in their ability to reproduce quantum mechanical (QM) benchmark dissociation curves of noble gas dimers (He, Ne, Ar, Kr, and Xe homo- and heterodimers), and we systematically study the efficacy of different permutations of combination relations for each parameter of the potentials. Our QM benchmark comprises dissociation curves computed by several different coupled cluster implementations as well as symmetry-adapted perturbation theory. The two-parameter Lennard-Jones potentials were decisively outperformed by more elaborate potentials that sport a 25-30 times lower root-mean-square error (RMSE) when fitted to QM dissociation curves. Very good fits to the QM dissociation curves can be achieved with relatively inexpensive four- or even three-parameter potentials, for instance, the damped 14-7 potential (Halgren, J. Am. Chem. Soc. 1992, 114, 7827-7843), a four-parameter Buckingham potential (Werhahn et al., Chem. Phys. Lett. 2015, 619, 133-138), or the three-parameter Morse potential (Morse, Phys. Rev. 1929, 34, 57-64). Potentials for heterodimers that are generated from combination rules have an RMSE that is up to 20 times higher than potentials that are directly fitted to the QM dissociation curves. This means that the RMSE, in particular, for light atoms, is comparable in magnitude to the well-depth of the potential. Based on a systematic permutation of combination rules, we present one or more combination rules for each potential tested that yield a relatively low RMSE. Two new combination rules are introduced that perform well, one for the van der Waals radius σij as and one for the well-depth ϵij as . The QM data and the fitted potentials were evaluated in the gas phase against experimental second virial coefficients for homo- and heterodimers, the latter of which allowed evaluation of the combination rules. The fitted models were used to perform condensed phase molecular dynamics simulations to verify the melting points, liquid densities at the melting point, and the enthalpies of vaporization produced by the models for pure substances. Subtle differences in the benchmark potentials, in particular, the well-depth, due to the level of theory used were found here to have a profound effect on the macroscopic properties of noble gases: second virial coefficients or the bulk properties in simulations. By explicitly including three-body dispersion in molecular simulations employing the best pair potential, we were able to obtain accurate melting points as well as satisfactory densities and enthalpies of vaporization.

Virial equation of state for a granular system.

The equation of state for an ideal gas is simple, which is . In the case of imperfect gases where mutual interactions among the constituents are important, pressure P can be expressed as the series expansion of density n with appropriate coefficients, known as virial coefficients . In this paper, we have obtained the first four virial coefficients for a model interaction potential using multidimensional Monte-Carlo integration and importance sampling methods. Next, we perform molecular dynamics simulations with the same for a many-particle system to obtain P as a function of T and n. We compare our numerical data with the virial equation of state.

Analytical Solution of Non-Linear Oscillator and Thermal Properties of Interacting System of Real Polyatomic Molecules

The study examined the solution of nonlinear oscillator and thermal properties of interacting system of real polyatomic molecules. A nonlinear oscillator potential was substituted into the radial part of the Schrodinger wave equation and solved using the Frobenius method to obtain the energy eigenvalues of the system. The partition function and the energy eigenvalues are used to determine three thermal properties for four polyatomic molecules which showed that added energy is required in the absence of the nonlinear term as the orbital and magnetic quantum numbers are increased and degeneracy completely removed. Other findings also reveal that as the virial coefficients of the four polyatomic molecules vary, the additive property of entropy is observed as well as the enhancement of the mean energy approximately proportional to the volume of the molecules. The four polyatomic molecules under consideration are environmentally impactful and are indispensable to man.

Effect of counterion size on polyelectrolyte conformations and thermodynamics.

We present a theoretical model to study the effect of counterion size on the effective charge, size, and thermodynamic behavior of a single, isolated, and flexible polyelectrolyte (PE) chain. We analyze how altering counterion size modifies the energy and entropy contributions to the system, including the ion-pair free energy, excluded volume interactions, entropy of free and condensed ions, and dipolar attraction among monomer-counterion pairs, which result in competing effects challenging intuitive predictions. The PE self-energy is calculated using the Edwards-Muthukumar Hamiltonian, considering a Gaussian monomer distribution for the PE. The condensed ions are assumed to be confined within a cylindrical volume around the PE backbone. The dipolar and excluded volume interactions are described by the second and third virial coefficients. The assumption of freely rotating dipoles results in a first-order coil-globule transition of the PE chain. A more realistic, weaker dipolar attraction, parameterized in our theory, shifts it to a second-order continuous transition. We calculate the size scaling-exponent of the PE and find exponents according to the relative dominance of the electrostatic, excluded volume, or dipolar effects. We further identify the entropy- and energy-driven regimes of the effective charge and conformation of the PE, highlighting the interplay of free ion entropy and ion-pair energy with varying electrostatic strengths. The crossover strength, dependent on the counterion size, indicates that diminishing sizes favor counterion condensation at the expense of free ion entropy. The predictions of the model are consistent with trends in simulations and generalize the findings of the point-like counterion theories.

Impact of Glycosylation on Protein-Protein Self-Interactions of Monoclonal Antibodies.

Protein self-interactions measured via second osmotic virial coefficients (B22) and dynamic light scattering interaction parameter values (kD) are often used as metrics for assessing the favorability of protein candidates and different formulations during monoclonal antibody (MAb) product development. Model predictions of B22 or kD typically do not account for glycans, though glycosylation can potentially impact experimental MAb self-interactions. To the best of our knowledge, the impact of MAb glycosylation on the experimentally measured B22 and kD values has not yet been reported. B22 and kD values of two fully deglycosylated MAbs and their native (i.e., fully glycosylated) counterparts were measured by light scattering over a range of pH and ionic strength conditions. Significant differences between B22 and kD of the native and deglycosylated forms were observed at a range of low to high ionic strengths used to modulate the effect of electrostatic contributions. Differences were most pronounced at low ionic strength, indicating that electrostatic interactions are a contributing factor. Though B22 and kD values were statistically equivalent at high ionic strengths where electrostatics were fully screened, we observed protein-dependent qualitative differences, which indicate that steric interactions may also play a role in the observed B22 and kD differences. A domain-level coarse-grained molecular model accounting for charge differences was considered to potentially provide additional insight but was not fully predictive of the behavior across all of the solution conditions investigated. This highlights that both the level of modeling and lack of inclusion of glycans may limit existing models in making quantitatively accurate predictions of self-interactions.

Advancing Molecular Weight Determination of Lignin by Multi-Angle Light Scattering.

Due to the complexity and recalcitrance of lignin, its chemical characterization is a key factor preventing the valorization of this abundant material. Multi-angle light scattering (MALS) is becoming a sought-after technique for absolute molecular weight (MW) determination of polymers and proteins. Lignin is a suitable candidate for MW determination via MALS, yet further investigation is required to confirm its absolute MW values and molecular size. Studies aiming to break down lignin into a variety of renewable products will benefit greatly from a simple and reliable determination method like MALS. Recent pioneering studies, discussed in this review, addressed several key challenges in lignin's MW characterization. Nevertheless, some lignin-specific issues still need to be considered for in-depth characterization. This study explores how MALS instrumentation manages the complexities of determining lignin's MW, e.g., with simultaneous fractionation and fluorescence interference mitigation. Additionally, we rationalize the importance of a more detailed light scattering analysis for lignin characterization, including aspects like the second virial coefficient and radius of gyration.

Virial coefficients of the uniform electron gas from path-integral Monte Carlo simulations.

The properties of plasmas in the low-density limit are described by virial expansions. Analytical expressions are known from Green's function approaches only for the first three virial coefficients. Accurate path-integral Monte Carlo (PIMC) simulations have recently been performed for the uniform electron gas, allowing the virial expansions to be analyzed and interpolation formulas to be derived. The exact expression for the second virial coefficient is used to test the accuracy of the PIMC simulations and the range of validity of the interpolation formula of Groth etal. [Phys. Rev. Lett. 119, 135001 (2017)0031-900710.1103/PhysRevLett.119.135001], and we discuss the fourth virial coefficient, which is not exactly known yet. Combining PIMC simulations with benchmarks from exact virial expansion results would allow us to obtain more accurate representations of the equationof state for parameter ranges of conditions which are of interest, e.g., for helioseismology.

Versatile Capillary Cells for Handling Concentrated Samples in Analytical Ultracentrifugation.

In concentrated macromolecular dispersions, far-from-ideal intermolecular interactions determine the dispersion behaviors including phase transition, crystallization, and liquid-liquid phase separation. Here, we present a novel versatile capillary-cell design for analytical ultracentrifugation-sedimentation equilibrium (AUC-SE), ideal for studying samples at high concentrations. Current setups for such studies are difficult and unreliable to handle, leading to a low experimental success rate. The design presented here is easy to use, robust, and reusable for samples in both aqueous and organic solvents while requiring no special tools or chemical modification of AUC cells. The key and unique feature is the fabrication of liquid reservoirs directly on the bottom window of AUC cells, which can be easily realized by laser ablation or mechanical drilling. The channel length and optical path length are therefore tunable. The success rate for assembling this new cell is close to 100%. We demonstrate the practicality of this cell by studying: (1) the equation of state and second virial coefficients of concentrated gold nanoparticle dispersions in water and bovine serum albumin (BSA) as well as lysozyme solution in aqueous buffers, (2) the gelation phase transition of DNA and BSA solutions, and (3) liquid-liquid phase separation of concentrated BSA/polyethylene glycol (PEG) droplets.

Temperature Dependence of Membrane Permeability Parameters for Five Cell Types Using Nonideal Thermodynamic Assumptions to Mathematically Model Cryopreservation Protocols.

Cryopreservation is the process of preserving biological matter at subzero temperatures for long-term storage. During cryopreservation, cells are susceptible to various injuries that can be mitigated by controlling the cooling and warming profiles and cryoprotective agent (CPA) addition and removal procedures. Mathematical modeling of the changing cell volume at different temperatures can greatly reduce the experiments needed to optimize cryopreservation protocols. Such mathematical modeling requires as inputs the cell membrane permeabilities to water and CPA and the osmotically inactive fraction of the cell. Since the intra- and extracellular solutions are generally thermodynamically nonideal, our group has been incorporating the osmotic virial equation to model the solution thermodynamics that underlie the cell volume change equations, adding the second and third osmotic virial coefficients of the grouped intracellular solute to the cell osmotic parameters that must be measured. In our previous work, we reported methods to obtain cell osmotic parameters at room temperature by fitting experimental cell volume kinetic data with equations that incorporated nonideal solution thermodynamics assumptions. Since the relevant cell volume excursions occur at different temperatures, the temperature dependence of the osmotic parameters plays an important role. In this work, we present a new two-part fitting method to obtain five cell-type-specific parameters (water permeability, dimethyl sulfoxide permeability, osmotically inactive fraction, and the second and third osmotic virial coefficients of the intracellular solution) from experimental measurements of equilibrium cell volume and cell volume as a function of time at room temperature and 0 °C for five cell types, namely, human umbilical vein endothelial cells (HUVECs), H9c2 rat myoblasts, porcine corneal endothelial cells (PCECs), the Jurkat T-lymphocyte cell line, and human cerebral microvascular endothelial cells (hCMECs/D3 cell line). The fitting method in this work is based on both equilibrium and kinetic cell volume data, enabling us to solve some technical challenges and expand our previously reported measurement technique to 0 °C. Finally, we use the measured parameters to model the cell volume changes for a HUVEC cryopreservation protocol to demonstrate the impact of the nonideal thermodynamic assumptions on predicting the changing cell volume during freezing and thawing.

Cross Second and Third Virial Coefficients and Dilute Gas Transport Properties of the (H2O + Ar) System from First-Principles Calculations

Cross Second and Third Virial Coefficients and Dilute Gas Transport Properties of the (H₂O + Ar) System from First-Principles Calculations

Virialized equation of state for warm and dense stellar plasmas in proto-neutron stars and supernova matter

ABSTRACT We present microscopic molecular dynamics simulations including the efficient Ewald sum procedure to study warm and dense stellar plasmas consisting of finite-size ion charges immerse in a relativistic neutralizing electron gas. For densities typical of Supernova matter and crust in a proto-neutron star, we select a representative single ion composition and obtain the virialized equation of state (vEoS). We scrutinize the finite-size and screening corrections to the Coulomb potential appearing in the virial coefficients B2, B3, and B4 as a function of temperature. In addition, we study the thermal heat capacity at constant volume, CV, and the generalized Mayer’s relation i.e. the difference CP − CV with CP being the heat capacity at constant pressure, obtaining clear features signalling the onset of the liquid–gas phase transition. Our findings show that microscopic simulations reproduce the discontinuity in CV, whose value lies between that of idealized gas and crystallized configurations. We study the pressure isotherms marking the boundary of the metastable region before the gaseous transition takes place. The resulting vEoS displays a behaviour where effective virial coefficients include extra density dependence showing a generalized density–temperature form. As an application we parametrize pressure as a function of density and temperature under the form of an artificial neural network showing the potential of machine learning for future regression analysis in more refined multicomponent approaches. This is of interest to size the importance of these corrections in the liquid–gas phase transition in warm and dense plasma phases contributing to the cooling behaviour of early Supernova phases and proto-neutron stars.

Comprehensive quantum calculation of the first dielectric virial coefficient of water.

We present a complete calculation, fully accounting for quantum effects and for molecular flexibility, of the first dielectric virial coefficient of water and its isotopologues. The contribution of the electronic polarizability is computed from a state-of-the-art intramolecular potential and polarizability surface from the literature, and its small temperature dependence is quantified. The dipolar polarizability is calculated in a similar manner with an accurate literature dipole-moment surface; it differs from the classical result both due to the different molecular geometries sampled at different temperatures and due to the quantization of rotation. We calculate the dipolar contribution independently from spectroscopic information in the HITRAN2020 database and find that the two methods yield consistent results. The resulting first dielectric virial coefficient provides a complete description of the dielectric constant at low density that can be used in humidity metrology and as a boundary condition for new formulations for the static dielectric constant of water and heavy water.