Higher-order Virial Coefficients Research Articles

Measurements of virial coefficients of Helium, Argon and Nitrogen for the needs of static expansion method

<p>Generally, there are three primary methods in vacuum metrology: mercury manometer, static expansion method, and continuous expansion method. For the pressure below 10 Pa, the idea of the primary standard is that the gas is measured precisely at a pressure as high as possible, and then the gas is expanded to the bigger volumes; this allows to calculate the expanded pressure. An important parameter that needs to be taken care of in primary vacuum calibration methods is the compressibility factor of the working gas. The influence of virial coefficients on the realization of primary standards in vacuum metrology, especially in the realization of the static expansion method is very important. In this paper we will present the measured data for virial coefficients of three gases Helium, Argon and Nitrogen measured at room temperature and a pressure range from 3 kPa to 130 kPa. The dominating term due to real gas properties arises from the second virial coefficient. The influence of higher orders of virial coefficients drops rapidly with lower pressure, particularly for gas pressures values lower than one atmosphere. Hence, in our calculation, the series of real gas was used for the first and second virial coefficients but not for higher-order virial coefficients.</p>

Reexamining equationsof state of oblate hard ellipsoids of revolution: Numerical simulation utilizing a cluster Monte Carlo algorithm and comparison to virial theory.

We provide highly accurate equation-of-state data determined by means of cluster Monte Carlo simulations for the isotropic phase of oblate hard ellipsoids of revolution. Both equation-of-state data and phase boundaries of the isotropic phase are obtained from relatively large ensembles with typically 1000 particles. The comparison of simulation data with a virial approach gives evidence for the importance of high-order so-far-unknown virial coefficients and therewith many-particle interactions in dense, isotropic systems of anisotropic particles. While a virial approach with a rescaled Carnahan-Starling correction for the unknown, higher-order virial coefficients reproduces the simulation data of moderately anisotropic particles with high accuracy, we suggest for highly anisotropic shapes a simple, heuristic equationof state as a suitable approach.

Toward an Automated-Algebra Framework for High Orders in the Virial Expansion of Quantum Matter

The virial expansion provides a non-perturbative view into the thermodynamics of quantum many-body systems in dilute regimes. While powerful, the expansion is challenging as calculating its coefficients at each order n requires analyzing (if not solving) the quantum n-body problem. In this work, we present a comprehensive review of automated algebra methods, which we developed to calculate high-order virial coefficients. The methods are computational but non-stochastic, thus avoiding statistical effects; they are also for the most part analytic, not numerical, and amenable to massively parallel computer architectures. We show formalism and results for coefficients characterizing the thermodynamics (pressure, density, energy, static susceptibilities) of homogeneous and harmonically trapped systems and explain how to generalize them to other observables such as the momentum distribution, Tan contact, and the structure factor.

On Using the BMCSL Equation of State to Renormalize the Onsager Theory Approach to Modeling Hard Prolate Spheroidal Liquid Crystal Mixtures

Modifications to the traditional Onsager theory for modeling isotropic–nematic phase transitions in hard prolate spheroidal systems are presented. Pure component systems are used to identify the need to update the Lee–Parsons resummation term. The Lee–Parsons resummation term uses the Carnahan–Starling equation of state to approximate higher-order virial coefficients beyond the second virial coefficient employed in Onsager’s original theoretical approach. As more exact ways of calculating the excluded volume of two hard prolate spheroids of a given orientation are used, the division of the excluded volume by eight, which is an empirical correction used in the original Lee–Parsons resummation term, must be replaced by six to yield a better match between the theoretical and simulation results. These modifications are also extended to binary mixtures of hard prolate spheroids using the Boublík–Mansoori–Carnahan–Starling–Leland (BMCSL) equation of state.

Liquid-liquid transition as a perturbation on the van der Waals' equation

In this work, we will show that liquid-liquid transition can be understood as a perturbation on the classical van der Waals’ state equation. We will propose a perturbed form for the van der Waals' state equation that, no matter how tiny the perturbation is, it will be enough to describe a new critical point similar to that observed in liquids at extreme conditions of pressure and density: the liquid-liquid transition. We will also show that the precise position of this critical point is given by high order virial coefficients. As they are not presently known, the measured position of the liquid-liquid transition can be combined with the results of this paper to estimate the values of these virial coefficients.

Protein Translational Diffusion and Intermolecular Interactions of Globular and Intrinsically Unstructured Proteins.

The study of intermolecular interactions of proteins has been an important problem for many years. This paper presents an approach to analyze different levels of protein interactions in solutions through a set of the second- and higher-order virial coefficients. The proposed approach is based on the diversified analysis of protein translational collective diffusion and self-diffusion obtained by dynamic light scattering and the pulsed-field gradient NMR (PFG NMR) spectroscopy experimental data. The experimental results were analyzed within the theoretical approach based on Vink's frictional formalism of nonequilibrium thermodynamics and the standard Derjaguin-Landau-Verwey-Overbeekb (DLVO) theory of interactions of colloid particles in electrolyte solutions. The second- and higher-order virial coefficients were obtained to estimate the pairwise and many-body intermolecular interactions in the solutions of globular α-chymotrypsin and intrinsically unstructured αS-casein. The second virial coefficients were calculated from the model of the protein-protein potential of mean force. The description of protein-protein interactions includes a set of interaction potentials: the attractive charge-dipole, dipole-dipole, the dispersion Hamaker, the mean force osmotic-attraction, and the repulsive charge-charge ones. It has been found that the major contribution to the intermolecular αS-casein interactions is made by the repulsive charge-charge potential, whereas for the case of α-chymotrypsin, the contributions from other types of interaction are of importance. It was determined that the model was well suited to describe the interactions of both globular and intrinsically disordered proteins. The suggested combination of Vink's approach and the DLVO theory is novel and holds much promise to make a profound analysis of the processes in systems containing various types of protein molecules.

Calculation of high-order virial coefficients for the square-well potential.

Accurate virial coefficients B_{N}(λ,ɛ) (where ɛ is the well depth) for the three-dimensional square-well and square-step potentials are calculated for orders N=5-9 and well widths λ=1.1-2.0 using a very fast recursive method. The efficiency of the algorithm is enhanced significantly by exploiting permutation symmetry and by storing integrands for reuse during the calculation. For N=9 the storage requirements become sufficiently large that a parallel algorithm is developed. The methodology is general and is applicable to other discrete potentials. The computed coefficients are precise even near the critical temperature, and thus open up possibilities for analysis of criticality of the system, which is currently not accessible by any other means.

Asymptotic expression of the virial coefficients for hard sphere systems

We evidence via a computation in the reciprocal space the asymptotic behavior of the high order virial coefficients for a hard sphere system. These coefficients, if their order is high enough, are those of a geometric series. We thus are able to give an explicit expression of the equation of state of the hard sphere system at high density when the fluid phase is no longer the stable one; in the disordered phase this equation of state exhibits a simple pole at the random close packing density. We can then estimate the packing densities of the freezing point of the disordered phase and also of the melting point of the fcc ordered phase. The results are compared with those of the numerical simulations.

Equations of state of hard-body fluids: a new proposal

A new and accurate equation of state for hard-body fluids is proposed. For linear molecules, the equation only requires the second virial coefficient, which is known exactly for convex as well as for many non-convex hard-body fluids, to provide very good agreement with simulations even for quite elongated molecules. For non-linear molecules, some of the higher order virial coefficients might be needed, for which reasonable approximations would probably be sufficient to obtain satisfactory results.

Модифікація методу вибірки Майера для розрахунку віріальних коефіцієнтів вищих порядків

A technique for the calculation of high-order virial coefficients, which combines the quadrature integration and the Mayer sampling Monte Carlo method (MSMC), is proposed. Unlike the original MSMC, this technique does not require to know the reference coefficients for the hard-sphere potential and can be used in a wide range of temperatures and for various interaction potentials. In addition, the proposed method has a higher accuracy at lower computational costs. It has been used to obtain some new data on the seventh virial coefficient of the Lennard-Jones (12-6) model.

Computation of high-order virial coefficients in high-dimensional hard-sphere fluids by Mayer sampling

The Mayer sampling method was used to compute the virial coefficients of high-dimensional hard-sphere fluids. The first 64 virial coefficients for dimensions 12 < D ⩽ 100 were obtained to high precision, and several lower dimensional virial coefficients were computed. The radii of convergence of the virial series in 13, 15, 17 and 19 dimensions agreed well with the analytical results from the Percus–Yevick closure.

Virial expansion around thes-wave Feshbach resonance in mass-imbalanced Fermi gases

We derive the expressions for the virial coefficients up to the third order for two-species fermion mixtures, and investigate the fugacity of the species and the interaction energy density around the $s$-wave Feshbach resonance in $^{6}\mathrm{Li}$-$^{40}\mathrm{K}$ mixtures. The theoretical results show that the magnitude of the virial coefficients depends on the mass ratio in a mass-imbalanced system. We find that the contribution of two-body bound states to high-order virial coefficients is very small at temperatures of microkelvin order. With the decrease of $T/{T}_{F}$, the positions of the maxima of fugacity and interaction energy density shift away from the resonance position in the third-order virial expansion calculation. The mass ratio determines the relevance between the fugacity and number density of heavy species. Our conclusion is also effective for other mass-imbalanced systems.

Application of Levin's transformations to virial series

A new method of estimating high-order virial coefficients for fluids composed of equal three-dimensional rigid spheres is proposed. The predicted $B_{11}$ and $B_{12}$ values are in good agreement with reliable estimates previously reported. A new application of the Levin's transformations is developed, as well as a new way of using Levin's transformations is suggested. For the virial series of packing factor powers, this method estimates the $B_{13}$ value near 173.

Generalized Onsager theory of liquid crystals

The Onsager theory is known to be inaccurate in its prediction of the critical transition density for small aspect ratio hard rods. In this paper we generalize the Onsager theory in two dimensions by taking into account the short-range order as well as the higher-order virial coefficients, up to the fourth order. By carrying out molecular dynamics (MD) simulations on "molecules" comprising linked hard disks with an aspect ratio ℓ ranging from 5 to 13, we show that the generalized theory is much improved as compared to the traditional theory, with its predictions of the transition density agreeing well with the simulation results. This indicates the importance of short-range order considerations (in conjunction with steric repulsion) for molecules with ℓ≤10, a group which includes the most commonly encountered thermotropic liquid crystals. MD simulations further yield evidence for hexagonal order for molecules with ℓ≤8, indicating an intermediate hexagonal phase before solidifying at higher densities.

Calculation of high-order virial coefficients with applications to hard and soft spheres.

A virial expansion of fluid pressure in powers of the density can be used to calculate a wealth of thermodynamic information, but the Nth virial coefficient, which multiplies the Nth power of the density in the expansion, becomes rapidly more complicated with increasing N. This Letter shows that the Nth virial coefficient can be calculated using a method that scales exponentially with N in computer time and memory. This is orders of magnitude more efficient than any existing method for large N, and the method is simple and general. New results are presented for N = 11 and 12 for hard spheres, and N = 9 and 10 for soft spheres.

A generic equation of state for liquid crystalline phases of hard-oblate particles

A generic equation of state (EoS) is developed for the hard cylindrical disc model to describe the isotropic phase of hard cut-sphere particles introducing a correction parameter to incorporate the negative contributions from higher-order virial coefficients. The isotropic–nematic–columnar phase diagram of hard cut-sphere fluids is investigated combining the new EoS with a scaled Onsager free energy for the nematic phase and an extended cell theory for columnar phase. By mapping the virial coefficients of an oblate spherocylinder on to those of the cylindrical disc (which are known algebraically), the new generic EoS is used to describe the isotropic and nematic phases of hard oblate spherocylinder particles. The predictions of the generic EoS are compared with available simulation data.

Integral-equation theories and Mayer-sampling Monte Carlo: a tandem approach for computing virial coefficients of simple fluids

Mayer-sampling Monte Carlo (MSMC) has enabled computation of higher-order virial coefficients than previously possible for a variety of potential models, but it is not required for computation of the entire virial coefficient for models that are spherically symmetric: approximations that result from the hypernetted-chain (HNC) or Percus–Yevick (PY) integral-equation theories in conjunction with the compressibility equation (c) or virial equation (v) can be computed quickly by fast Fourier transforms. For the fourth and fifth virial coefficients of the Lennard–Jones potential (with parameters σ and ε), we demonstrate that the corrections to each of the four approximations (HNC(c), HNC(v), PY(c), and PY(v)) are faster to compute to a desired precision by MSMC than the full coefficient itself, with the exception of the PY(v) correction at fifth order, and that the optimal decomposition with regard to precision can be identified using a fraction of the steps required to obtain precise virial coefficients. At reduced temperatures kT/ε greater than 4, the PY(c) correction is fastest to compute by MSMC at both fourth and fifth orders. For lower temperatures, the HNC(v) decomposition is most efficient at fourth order, while the HNC(c) decomposition is most efficient at fifth order. These results are specific to the Lennard–Jones potential, but the method for determining the optimal decomposition is applicable to any spherically symmetric potential.

Impact of short range hydrophobic interactions and long range electrostatic forces on the aggregation kinetics of a monoclonal antibody and a dual-variable domain immunoglobulin at low and high concentrations

The purpose of this work was to determine the nature of long and short-range forces governing protein aggregation kinetics at low and high concentrations for a monoclonal antibody (IgG1) and a dual-variable-domain immunoglobulin (DVD-Ig™). Protein–protein interactions (PPI) were studied under dilute conditions by utilizing the methods of static (B22) and dynamic light scattering (kD). PPI in solutions containing minimal ionic strengths were characterized to get detailed insights into the impact of ionic strength on aggregation. Microcalorimetry and susceptibility to denature at air–liquid interface were used to assess the tertiary structure and quiescent stability studies were conducted to study aggregation characteristics. Results for IgG1 showed that electrostatic interactions governed protein aggregation kinetics both under dilute and concentrated conditions (i.e., 5mg/mL and 150mg/mL). For DVD-Ig™ molecules, on the other hand, although electrostatic interactions governed protein aggregation under dilute conditions, hydrophobic forces clearly determined the kinetics at high concentrations. This manuscript shows for the first time that short-range hydrophobic interactions can outweigh electrostatic forces and play an important role in determining protein aggregation at high concentrations. Additionally, results show that although higher-order virial coefficients become significant under low ionic strength conditions, removal of added charges may be used to enhance the aggregation stability of dilute protein formulations.

A systematic formulation of the virial expansion for nonadditive interaction potentials

A new formulation of the virial expansion for a classical gas is derived without the restriction to pairwise-additive interaction potentials. Explicit expressions up to the eighth virial coefficient, suitable for numerical evaluation, are given in the form of integrals over sums of cluster diagrams. Compared with previous approaches, the number of cluster diagrams increases more moderately with increasing order of the virial coefficient. Thus, the new formulation should be particularly useful for the computation of high-order virial coefficients.

Self-organized stiffness in regular fractal polymer structures

We investigated membrane-like polymer structures of fractal connectivity such as Sierpinski gaskets and Sierpinski carpets applying the bond fluctuation model in three dimensions. Without excluded volume (phantom), both polymeric fractals obey Gaussian elasticity on larger scales determined by their spectral dimension. On the other hand, the swelling effect due to excluded volume is rather distinct between the two polymeric fractals: Self-avoiding Sierpinski gaskets can be described using a Flory-type mean-field argument. Sierpinski carpets having a spectral dimension closer to perfect membranes are significantly more strongly swollen than predicted. Based on our simulation results it cannot be excluded that Sierpinski carpets in athermal solvent show a flat phase on larger scales. We tested the self-consistency of Flory predictions using a virial expansion to higher orders. From this we conclude that the third virial coefficient contributes marginally to Sierpinski gaskets, but higher order virial coefficients are relevant for Sierpinski carpets.