Estimation of third virial coefficients at low reduced temperatures

We use the Clausius–Clapeyron equation to calculate third virial coefficients at low reduced temperatures. This procedure gives an alternative to predict third virial coefficients in a region where the third virial coefficient is difficult to measure. We compare the results of this method with published third virial coefficient data. Calculated third virial coefficients have average percentage deviations within 5% of the experimental values at reduced temperatures between 0.8 and 1.0.     

Precision measurements of osmotic pressure in concentrated polymer solutions—II

Precision measurements of osmotic pressure have been carried out with several polymers (polystyrene, polymethylmethacrylate and polyvinyl acetate) in organic solvents. The concentration dependence of the osmotic pressure over a wide range of concentrations (up to 0·2 g/cm³) was established and the osmotic virial coefficients were evaluated. From the temperature dependence of the osmotic pressure, the partial molar heats and entropies of mixing were also determined. In the system polystyrene-cyclohexane, measurements were carried out near the θ-temperature; variation of the virial coefficients with temperature and molecular weight was studied. It was found that the conventionally defined θ-temperature is molecular weight dependent. At the θ-temperature, the third virial coefficient is positive and tends to a nonzero limit when the molecular weight tends to infinity.     

Ab initio virial equation of state for argon using a new nonadditive three-body potential

An ab initio nonadditive three-body potential for argon has been developed using quantum-chemical calculations at the CCSD(T) and CCSDT levels of theory. Applying this potential together with a recent ab initio pair potential from the literature, the third and fourth to seventh pressure virial coefficients of argon were computed by standard numerical integration and the Mayer-sampling Monte Carlo method, respectively, for a wide temperature range. All calculated virial coefficients were fitted separately as polynomials in temperature. The results for the third virial coefficient agree with values evaluated directly from experimental data and with those computed for other nonadditive three-body potentials. We also redetermined the second and third virial coefficients from the best experimental pρT data utilizing the computed higher virial coefficients as constraints. Thus, a significantly closer agreement of the calculated third virial coefficients with the experimental data was achieved. For different orders of the virial expansion, pρT data have been calculated and compared with results from high quality measurements in the gaseous and supercritical region. The theoretically predicted pressures are within the very small experimental errors of ±0.02% for p ≤ 12 MPa in the supercritical region near room temperature, whereas for subcritical temperatures the deviations increase up to +0.3%. The computed pressure at the critical density and temperature is about 1.3% below the experimental value. At pressures between 200 MPa and 1000 MPa and at 373 K, the calculated values deviate by 1% to 9% from the experimental results.     

Statistical thermodynamics of polymer solutions. VI. Comblike branched molecules with branches placed randomly along the backbone

Thermodynamic interactions between arbitrarily branched flexible “comb” molecules (with branches of uniform length affixed to backbones of uniform length but with the number of branches on a molecule and their placement arbitrary) are treated by Zimm's perturbation method to obtain results valid to the double contact approximation. Thus, bimolecular cluster configurations with one and two intermolecular contacts, but not more, are correctly accounted for. This general result is applied to “random” combs (with the number of branches per comb uniform, but the placement random) and to “heterogeneous” combs (with both the number of branches on a molecule and their placement random). Results for the random combs are very similar to those reported earlier for “symmetrical” combs (with f uniform branches disposed at points along the backbone chain so as to subdivide it into f + 1 equal sections), but the second virial coefficient for the random model is slightly the larger when the number of branches per chain is small and the fraction of the molecule in the backbone is at least a few per cent. Since random combs are uniform in mass, the osmotic pressure and light-scattering second virial coefficients are alike; but for a system of heterogeneous combs the virial coefficients differ from one another, and from the random comb result, when the mean number of branches f? per chain is small and they are of appreciable length in comparison to the backbone. This behavior reflects the heterogeneity in both molecular mass and dimensions existing under these circumstances. As f? increases and/or the mean fraction of segments in the chain backbone becomes large, coincidence with the random comb result is approached. The double-contact second virial coefficients for both random and heterogeneous combs are obtained in closed analytical expression that are much more easily reduced to numerical results than the expression for symmetrical combs given earlier. These two models also correspond better than the symmetrical combs to materials actually obtainable.     

Sensitivity of the thermal and acoustic virial coefficients of argon to the argon interaction potential

Second, third, and fourth thermal and acoustic virial coefficients between 100 and 1000 K are computed for different argon interaction models derived from combinations of accurate two- and three-body potentials. Differences between the various interaction models tested mirror the presumed order in the accuracy of these models, but are not well captured at the level of the lowest-order contributions in the virial expansion: While the second- and third-order virial coefficients are found to be rather insensitive to small variations in the two- and three-body potentials, more pronounced differences in higher-order coefficients are currently of limited use in assessing the accuracy of the interaction potential due to difficulties in the unambiguous experimental determination of these higher-order coefficients. In contrast, pressure-volume and speed-of-sound data - both of which are experimentally known to highest accuracies - are found to be insensitive to small variations in the interaction model. All but the least accurate models reproduce experimental pressure-volume and speed-of-sound data near-quantitatively in regions where the (fourth-order) virial expansions apply. All quantities considered are found to be completely unaffected by a non-vanishing quadruple-dipole four-body potential.     

Path-integral calculation of the third virial coefficient of quantum gases at low temperatures

We derive path-integral expressions for the second and third virial coefficients of monatomic quantum gases. Unlike previous work that considered only Boltzmann statistics, we include exchange effects (Bose-Einstein or Fermi-Dirac statistics). We use state-of-the-art pair and three-body potentials to calculate the third virial coefficient of (3)He and (4)He in the temperature range 2.6-24.5561 K. We obtain uncertainties smaller than those of the limited experimental data. Inclusion of exchange effects is necessary to obtain accurate results below about 7 K.     

Notes on the theory of high polymer solutions

We have noted some correspondences in the theories of high polymer solutions. This article is divided in two independent sections: (I) Osmotic pressure. The second virial coefficients for polymers of the pearl-necklace type are given and discussed in two limiting cases—the dumbbell and the infinitely polymerized molecule having a definite length. The variation of the coefficients against the polymerization is predictable from these limiting cases (Fig. 1). Attention must be paid to the manner of taking the (abscissa). The case of infinite polymerization can be considered as corresponding to polymers of a compact single-body type such as globular proteins. Thus, an interesting correspondence in the second virial coefficients can be judged for the pearl-necklace and the compact single-body type of polymers. The general feature of the coefficient is also discussed. (II) Intrinsic viscosity. The relation between the two methods (Kirkwood-Riseman and Debye-Bueche) for intrinsic viscosity of high polymer solutions is discussed. Because the respective fundamental equations are equivalent to each other, we may choose whichever of these two methods we like.     

Synthesis and liquid-crystalline behaviour of liposaccharide based comb polymers

In order to investigate the relationship between the liquid-crystalline state and the biological properties of polymers, we synthesized and studied liquid-crystalline comb polymers with a polyacrylamide main chain and three types of lipo-saccharidic side chains. These comb polymers were synthesized in four steps: first the polymerizable group was linked to the amino end of an α,ω-aliphatic amino acid; in the second step the ω-carboxylic acid function was activated in the form of an N-hydroxysuccinimidylester; in the third step the active ester was aminolysed by the amine function of the saccharide (N-methyl-D-glucamine, 2-amino-2-deoxy-β-D-glucose or 1-amino-1-deoxy-β-D-galactose) and a polymerizable liposaccharide was obtained; finally the liposaccharides were transformed into comb polymers by free-radical polymerization. Comb polymers exhibit mesophases in concentrated aqueous, ethanol or dimethylsulphoxide solution, and their mesomorphic character remains after the slow evaporation of the solvent. X-ray diffraction studies have shown that the mesophases have smectic or nematic ordering, depending upon the nature of the saccharidic residues of the liposaccharidic side chains.     

The second and the third virial coefficients of athermal polymer solutions

The second and the third virial coefficients in the lattice model of athermal mixtures of molecules of different sizes are calculated. All computations have been done for two- and three-dimensional simple square and simple cubic lattices.     

Generalised virial equation of state for natural gas systems

In this work we present a generalised virial equation of state for natural gas systems under custody transfer conditions. The model is based on corresponding states expressions for the second and third virial coefficients with argon as the reference fluid. These functional forms involve 12 adjustable coefficients. For the extension to mixtures we propose a one-fluid mixture model with binary interaction parameters in the combining rules for the mixture critical temperature and density. We obtained overall average absolute deviations (AAD) of 0.04 and 0.08% in pure-fluid compression factors and speeds of sound; AADs of 0.07 and 0.19% in compression factors and speeds of sound, respectively, of binary mixtures and AADs of 0.047, and 0.13% in natural gas compression factors and speeds of sound, respectively. These results compare favourably with equivalent calculations with other generalised virial coefficient models.     

Corrections to scaling and crossover from good- to theta-solvent regimes of interacting polymers

We exploit known properties of universal ratios, involving the radius of gyration R(g), the second and third virial coefficients B(2) and B(3), and the effective pair potential between the centers of mass of self-avoiding polymer chains with nearest-neighbor attraction, as well as Monte Carlo simulations, to investigate the crossover from good- to theta-solvent regimes of polymers of finite length L. The scaling limit and finite-L corrections to scaling are investigated in the good-solvent case and close to the theta temperature. Detailed interpolation formulas are derived from Monte Carlo data and results for the Edwards two-parameter model, providing estimates of universal ratios as functions of the observable ratio A(2)=B(2)/R(g) (3) over the whole temperature range, from the theta point to the good-solvent regime. The convergence with L (L< or =8000) is found to be satisfactory under good-solvent conditions, but longer chains would be required to match theoretical predictions near the theta point, due to logarithmic corrections. A quantitative estimate of the universal ratio A(3)=B(3)/R(g) (6) as a function of temperature shows that the third virial coefficient remains positive throughout, and goes through a pronounced minimum at the theta temperature, which goes to zero as 1/ln L in the scaling limit.     

Alternative biofuels

This study presents the experimental results for the dimethyl ether (DME) + propane system obtained using the Burnett method. The apparatus was calibrated using helium. PVTx measurements were taken for four isotherms (344, 354, 364, and 375 K), performing 16 Burnett expansions in pressures ranging from about 3,000 to 70 kPa. The second and third virial coefficients were derived from experimental results. The experimental uncertainty in the second and third virial coefficients was estimated to be within ±5 cm³/mol and ±1,000 cm⁶/mol², respectively.     

Mayer-sampling Monte Carlo calculations of uniquely flexible contributions to virial coefficients

We present methods for computing contributions to the virial coefficients uniquely associated with molecular flexibility, and we demonstrate their use with application to the third, fourth, and fifth virial coefficients of united-atom models of linear alkanes and methanol belonging to the suite of transferrable potentials for phase equilibria (TraPPE-UA). We find that these uniquely flexible contributions are more difficult to compute than the remainder of the coefficient, especially for the conditions at which they appear to be most important. The significance of these contributions relative to the full virial coefficient grows with the number of sites (the size of the molecule), the number of molecules, and, to a certain extent, the temperature. The nature of the site-site interactions is of great importance: the significance of the uniquely flexible contribution at third and fourth order is orders of magnitude larger for TraPPE-UA methanol, which has Coulombic interactions, than for TraPPE-UA propane, which does not, even though both models have three sites per molecule and comparable bending potentials. While the uniquely flexible contribution of TraPPE-UA propane has a negligible impact on its third-order virial-equation-of-state estimate of the critical point, the uniquely flexible contribution of TraPPE-UA methanol increases this estimate of its critical pressure by about 5%.     

Second and third virial coefficients for (methane + nitrogen)

The compression factors Z of (methane  +  nitrogen) of various compositions were measured using a direct method in the temperature range 308.15 K to 333.15 K and at pressures up to 12.0 MPa. Unlike-interaction second virial coefficients were calculated from the data and compared with previous experimental values and values predicted by the Tsonopoulos and GERG equations. Unlike-interaction third virial coefficients were also calculated and compared with estimates provided by both the GERG correlations and the method given by Orbey and Vera.     

Excess virial coefficients with application in data analysis

This paper presents the general expression for excess virial coefficients of binary mixtures in the form of Bartlett's law. Imposing this expression in data reduction to determine mixture virial coefficients provides thermodynamically consistent evaluations of cross virial coefficients. Standard data reduction procedures do not provide values of the mixture and cross virial coefficients that are both consistent and accurate. We illustrate this new technique by analyzing Burnett data and direct density measurements. In addition, we present the general expression for cross virial coefficients under model solution behavior.     

Rescaled density expansions and demixing in hard-sphere binary mixtures

The demixing transition of a binary fluid mixture of additive hard spheres is analyzed for different size asymmetries by starting from the exact low-density expansion of the pressure. Already within the second virial approximation the fluid separates into two phases of different composition with a lower consolute critical point. By successively incorporating the third, fourth, and fifth virial coefficients, the critical consolute point moves to higher values of the pressure and to lower values of the partial number fraction of the large spheres. When the exact low-density expansion of the pressure is rescaled to higher densities as in the Percus-Yevick theory, by adding more exact virial coefficients a different qualitative movement of the critical consolute point in the phase diagram is found. It is argued that the Percus-Yevick factor appearing in many empirical equations of state for the mixture has a deep influence on the location of the critical consolute point, so that the resulting phase diagram for a prescribed equation has to be taken with caution.     

Thermodynamic studies of molecular interactions in aqueous alpha-cyclodextrin solutions: application of McMillan-Mayer and Kirkwood-Buff theories

Osmotic vapor pressure and density measurements were made for aqueous alpha-cyclodextrin (alpha-CD) solutions in the temperature range between 293.15 and 313.15 K. The experimental osmotic coefficient data were used to determine the corresponding activity coefficients and the excess Gibbs free energy of solutions. Further, the activity data obtained at different temperatures along with the enthalpies of dissolution (reported in the literature) were processed to obtain the excess enthalpy and excess entropy values for the solution process. The partial molar entropies of water and of alpha-cyclodextrin were calculated at different temperatures and also at different concentrations of alpha-CD. Using the partial molar volume data at infinite dilution, the solute-solvent cluster integrals were evaluated which yielded information about solute-solvent interactions. The application of McMillan-Mayer theory of solutions was made to obtain osmotic second and third virial coefficients which were decomposed into attractive and repulsive contributions to solute-solute interactions. The second and third osmotic virial coefficients are positive and show minimum at 303.15 K. The Kirkwood-Buff (KB) integrals G(ij), defined by the equation G(ij) = f(infinity)0 (g(ij)- 1)4pir(2) dr, have been evaluated using the experimental osmotic coefficient (and hence activity coefficient) and partial molar volume data. The limiting values of KB integrals, G(ij)(0) are compared with molecular interaction parameters (solute-solute i.e., osmotic second virial coefficient) obtained using McMillan-Mayer theory of solutions. We found an excellent agreement between the two approaches.     

Effect of solvent quality on the conformations of a model comb polymer

The effect of solvent quality on the equilibrium structure of a densely branched comb polymer is investigated based on the structure factor analyses by off-lattice Monte Carlo simulations. First, theta temperature (theta(infinity)) must be determined to identify the solvent condition. We locate the characteristic temperature theta(A)(N) at which the second virial coefficient vanishes and the transition temperature theta(R)(N) at which radius of gyration R(g) of the chain varies most rapidly with temperature, i.e., d(2)R(g)/dT(2)|(theta(R)) = 0. N represents the total number of monomers of a comb. As N --> infinity, theta(A) and theta(R) coincide to a point that is identified as the true theta temperature (theta(infinity)). The structure factors of the main chain, the side chain, and the whole polymer are calculated, respectively. It is found that at T = theta(infinity), the structural factors S(qR(g)) for the overall comb polymers match quite well with those of their Gaussian counterparts. When T< theta(infinity), the overall comb polymer assumes collapsed conformations, similar to a homogeneous sphere. However, the structure factor of the side chain indicates that it always remains in an expanded state regardless of the solvent condition. It is attributed to the strong interactions between side chains. The same effect leads to enhanced rigidity of the main chain in comparison to the linear chain, as clearly observed from the rescaled Kratky plot.     

Equation of state of nonadditive d-dimensional hard-sphere mixtures

An equation of state for a multicomponent mixture of nonadditive hard spheres in d dimensions is proposed. It yields a rather simple density dependence and constitutes a natural extension of the equation of state for additive hard spheres proposed by us [A. Santos, S. B. Yuste, and M. Lopez de Haro, Mol. Phys. 96, 1 (1999)]. The proposal relies on the known exact second and third virial coefficients and requires as input the compressibility factor of the one-component system. A comparison is carried out both with another recent theoretical proposal based on a similar philosophy and with the available exact results and simulation data in d=1, 2, and 3. Good general agreement with the reported values of the virial coefficients and of the compressibility factor of binary mixtures is observed, especially for high asymmetries and/or positive nonadditivities.     

Up to fourth virial coefficients from simple and efficient internal-coordinate sampling: application to neon

A simple and efficient internal-coordinate importance sampling protocol for the Monte Carlo computation of (up to fourth-order) virial coefficients ?B(n) of atomic systems is proposed. The key feature is a multivariate sampling distribution that mimics the product structure of the dominating pairwise-additive parts of the ?B(n). This scheme is shown to be competitive over routine numerical methods and, as a proof of principle, applied to neon: The second, third, and fourth virial coefficients of neon as well as equation-of-state data are computed from ab initio two- and three-body potentials; four-body contributions are found to be insignificant. Kirkwood-Wigner quantum corrections to first order are found to be crucial to the observed agreement with recent ab initio and experimental reference data sets but are likely inadequate at very low temperatures.