Chain length effects in the second virial coefficient of linear polymer molecules

A generalized perturbation theory is presented for the second virial coefficient of linear and branched polymer systems. Results have been computed for linear chains having two to five hundred statistical segments. These are found to differ significantly from the long-chain asymptotic results of Zimm. A semi-empirical modification of the Flory--Orofino theory is also suggested.     

Pressure dependence of the second virial coefficient of dilute polystyrene solutions

The pressure derivatives of the second virial coefficients [dA₂/dP; 0.1 ≤ P (MPa) ≤ 35.0] for dilute polystyrene (PS) solutions in good, θ, and poor solvents were measured with static light scattering. The solvent quality improved (dA₂/dP > 0) in the good and poor solvents that we investigated (toluene, chloroform; and methylcyclohexane) but deteriorated (dA₂/dP < 0) in θ solvents (cyclohexane and 50‐50 cis,trans‐decalin). The effects of temperature [22 < T (°C) < 45] and molecular weight [25 × 10³ < weight‐average molecular weight (amu mol^?1) < 900 × 10³] on dA₂/dP for PS/cyclohexane solutions were examined. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 3070–3076, 2003     

Influence of polymolecularity on the second virial coefficient of polymers

Scaling theory is applied to derive expressions describing the influence of polymolecularity on the second virial coefficient, A₂, as obtained from osmotic pressure and light scattering measurements. Numerical values of polymolecularity correction factors are calculated for Schulz-Zimm and logarithmic normal distributions of the molecular weight, different qualities of the solvent and several ratios of the weight-average and the number-average molecular weights M̄_w/M̄_n. It is found that in the equation $ A_2 = K_{A_2 } \cdot M_{{\rm av}}^{a_{A_2 } } $ the weight-average molecular weight is a good approximation for M_av if A₂ is measured via light scattering, while the number-average molecular weight can be inserted for M_av if A₂ stems from osmotic pressure measurements.     

The segment-cloud model of the second virial coefficient of polymer solutions: effects of chain length and branching

The segment-cloud model for polymer molecules has been used, and the second virial coefficient A₂ obtained as a function of the interaction parameter z for linear and branched chains having different values of n. It is observed that the chain length effect, though much smaller than in the perturbation theory, increases as the degree of branching increases. Also, the branching parameter g is found to be a better correlating parameter than the segment density distribution for A₂. This is in contrast to earlier results for the perturbation theory of the excluded volume.     

Influence of polymolecularity on the second virial coefficient of polymers, 2

It is shown that the recently developed scaling theory for the second virial coefficient, A₂, of dilute solutions of polymers with non-uniform molecular weight distributions in a good solvent is able to explain in principle all seemingly contradictory observations reported in the literature, in contrast to all other theories proposed so far.     

The scaling behavior of the second virial coefficient of linear and ring polymer

The scaling behavior of the second virial coefficient of ring polymers at the theta temperature of the corresponding linear polymer(θ_L) is investigated by off-lattice Monte Carlo simulations. The effects of the solvents are modeled by pairwise interaction between polymer monomers in this approach. Using the umbrella sampling, we calculate the effective potential U(r) between two ring polymers as well as the second virial coefficient A_2 of ring polymers at θ_L, which results from a combination of 3-body interactions and topological constraints. The trend in the strength of the effective potential with respect to chain length shows a non-monotonic behavior, differently from that caused only by topological constraints. Our simulation suggests that there are three regimes about the scaling behavior of A_2 of ring polymers at θ_L: 3-body interactions dominating regime, the crossover regime, and the topological constraints dominating regime.     

Calculation of the binary diffusion second virial coefficient

Calculations of the first density correction to the binary diffusion coefficient are presented for several mixtures. These calculations are based on the classical kinetic theory for mixtures developed by Bennett and Curtiss. The theoretical predictions agree well with experimental data.     

Intermolecular potential and second virial coefficient of the water-hydrogen complex

We construct a rigid-body (five-dimensional) potential-energy surface for the water-hydrogen complex using scaled perturbation theory (SPT). An analytic fit of this surface is obtained, and, using this, two minima are found. The global minimum has C2v symmetry, with the hydrogen molecule acting as a proton donor to the oxygen atom on water. A local minimum with Cs symmetry has the hydrogen molecule acting as a proton acceptor to one of the hydrogen atoms on water, where the OH bond and H2 are in a T-shaped configuration. The SPT global minimum is bound by 1097 microEh (Eh approximately 4.359744 x 10(-18) J). Our best estimate of the binding energy, from a complete basis set extrapolation of coupled-cluster calculations, is 1076.1 microEh. The fitted surface is used to calculate the second cross virial coefficient over a wide temperature range (100-3000 K). Three complementary methods are used to quantify quantum statistical mechanical effects that become significant at low temperatures. We compare our results with experimental data, which are available over a smaller temperature range (230-700 K). Generally good agreement is found, but the experimental data are subject to larger uncertainties.     

Intermolecular potential and second virial coefficient of the water-nitrogen complex

The authors construct a rigid-body (five-dimensional) potential energy surface for the water-nitrogen complex using the systematic intermolecular potential extrapolation routine. The intermolecular potential is then extrapolated to the limit of a complete basis set. An analytic fit of this surface is obtained, and, using this, the global minimum energy is found. The minimum is located in an arrangement in which N2 is near the H atom of H2O, almost collinear with the OH bond. The best estimate of the binding energy is 441 cm-1 (1 cm-1 approximately 1.986 43x10(-23) J). The extrapolated potential is then used to calculate the second cross virial coefficient over a wide temperature range (100-3000 K). These calculated second virial coefficients are generally consistent with experimental data, but for the most part the former have smaller uncertainties.     

The second virial coefficient of argon at low temperatures

We present experimental data of second virial coefficients of argon at temperatures of 77.3, 87.2 and 90.2 K. The results are in excellent agreement with theoretical values.     

Counterion adsorption theory of dilute polyelectrolyte solutions: apparent molecular weight, second virial coefficient, and intermolecular structure factor

Polyelectrolyte chains are well known to be strongly correlated even in extremely dilute solutions in the absence of additional strong electrolytes. Such correlations result in severe difficulties in interpreting light scattering measurements in the determination of the molecular weight, radius of gyration, and the second virial coefficient of charged macromolecules at lower ionic strengths from added strong electrolytes. By accounting for charge-regularization of the polyelectrolyte by the counterions, we present a theory of the apparent molecular weight, second virial coefficient, and the intermolecular structure factor in dilute polyelectrolyte solutions in terms of concentrations of the polymer and the added strong electrolyte. The counterion adsorption of the polyelectrolyte chains to differing levels at different concentrations of the strong electrolyte can lead to even an order of magnitude discrepancy in the molecular weight inferred from light scattering measurements. Based on counterion-mediated charge regularization, the second virial coefficient of the polyelectrolyte and the interchain structure factor are derived self-consistently. The effect of the interchain correlations, dominating at lower salt concentrations, on the inference of the radius of gyration and on molecular weight is derived. Conditions for the onset of nonmonotonic scattering wave vector dependence of scattered intensity upon lowering the electrolyte concentration and interpretation of the apparent radius of gyration are derived in terms of the counterion adsorption mechanism.     

Effects of chain flexibility on polymer conformation in dilute solution studied by lattice monte carlo simulation

Effects of chain flexibility on the conformation of homopolymers in good solvents have been investigated by Monte Carlo simulation. Bond angle constraint coupled with persistence length of polymer chains has been introduced in the modified eight-site bond fluctuation simulation model. The study about the effects of chain flexibility on polymer sizes reveals that the orientation of polymer chains under confinement is driven by the loss of conformation entropy.The conformation of polymer chains undergoing a gradual change from spherical iso-diametric ellipsoid to rodlike iso-diametric ellipsoid with the decrease of polymer chain flexibility in a wide region has been clearly illustrated from several aspects. Furthermore, a comparison of the freely jointed chain (FJC) model and the wormlike chain (WLC) model has also been made to describe the polymer sizes in terms of chain flexibility and quasi-quantitative boundary toward the suitability of the models.     

Determination of unlike-pair potential parameters from second virial coefficient data

Using available interaction virial coefficient data for forty-five non-polar binary systems, potential parameters for unlike interactions have been obtained by an analytical procedure for the Lennard—Jones (12-6) and Kihara intermolecular potentials.     

Interdependence of enthalpic and entropic contributions to the second osmotic virial coefficient

The interdependence of the enthalpic contribution A_{2, H} and the entropic contribution A_{2, s} to the second osmotic virial coefficient for a given polymer-solvent system has been investigated from the experimental and the theoretical point of view. Experimentally, the following common facts were observed for various systems at temperatures and pressures below the critical values for the solvent. Both the isobaric and isothermal dependences can be approximated over relatively wide ranges of A_{2, H} by linear relations with a slope deviating only slightly, but in a characteristic manner from a value of ?1. When the temperature is increased at constant pressure one moves along an isobar towards higher A_{2, H}; when the pressure is increased at constant temperature, one moves along an isotherm in the opposite direction, i.e., towards lower A₂, _H. Theoretically this behavior can be described in a qualitative manner, starting from a relation derived by Patterson and Delmas on the basis of the Prigogine corresponding-states theory. The reasons for the lack of quantitative agreement are discussed.     

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.     

Various contributions to the osmotic second virial coefficient in protein-water-cosolvent solutions

An analysis of the cosolvent concentration dependence of the osmotic second virial coefficient (OSVC) in water-protein-cosolvent mixtures is developed. The Kirkwood-Buff fluctuation theory for ternary mixtures is used as the main theoretical tool. On its basis, the OSVC is expressed in terms of the thermodynamic properties of infinitely dilute (with respect to the protein) water-protein-cosolvent mixtures. These properties can be divided into two groups: (1) those of infinitely dilute protein solutions (such as the partial molar volume of a protein at infinite dilution and the derivatives of the protein activity coefficient with respect to the protein and water molar fractions) and (2) those of the protein-free water-cosolvent mixture (such as its concentrations, the isothermal compressibility, the partial molar volumes, and the derivative of the water activity coefficient with respect to the water molar fraction). Expressions are derived for the OSVC of ideal mixtures and for a mixture in which only the binary mixed solvent is ideal. The latter expression contains three contributions: (1) one due to the protein-solvent interactions B2(p-s), which is connected to the preferential binding parameter, (2) another one due to protein/protein interactions (B2(p-p)), and (3) a third one representing an ideal mixture contribution (B2(id)). The cosolvent composition dependencies of these three contributions were examined for several water-protein-cosolvent mixtures using experimental data regarding the OSVC and the preferential binding parameter. For the water-lysozyme-arginine mixture, it was found that OSVC exhibits the behavior of an ideal mixture and that B2(id) provides the main contribution to the OSVC. For the other mixtures considered (water-Hm MalDH-NaCl, water-Hm MalDH-(NH4)2SO4, and water-lysozyme-NaCl mixtures), it was found that the contribution of the protein-solvent interactions B2(p-s) is responsible for the composition dependence of the OSVC on the cosolvent concentration, whereas the two remaining contributions (B2(p-p)) and B2(id)) are almost composition independent.     

Potential energy surface and second virial coefficient of methane-water from ab initio calculations

Six-dimensional intermolecular potential energy surfaces (PESs) for the interaction of CH4 with H2O are presented, obtained from ab initio calculations using symmetry-adapted perturbation theory (SAPT) at two different levels of intramonomer correlation and the supermolecular approach at three different levels of electron correlation. Both CH4 and H2O are assumed to be rigid molecules with interatomic distances and angles fixed at the average values in the ground-state vibration. A physically motivated analytical expression for each PES has been developed as a sum of site-site functions. The PES of the CH4-H2O dimer has only two symmetry-distinct minima. From the SAPT calculations, the global minimum has an energy of -1.03 kcal/mol at a geometry where H2O is the proton donor, HO-H...CH4, with the O-H-C angle of 165 degrees, while the secondary minimum, with an energy of -0.72 kcal/mol, has CH4 in the role of the proton donor (H3C-H...OH2). We estimated the complete basis set limit of the SAPT interaction energy at the global minimum to be -1.06 kcal/mol. The classical cross second virial coefficient B12(T) has been calculated for the temperature range 298-653 K. Our best results agree well with some experiments, allowing an evaluation of the quality of experimental results.     

Influence of chain conformation on translational diffusion of poly(2-vinyl pyridine) in dilute solution

A comparative photon correlation spectroscopy study is reported of the concentration-dependent translational diffusion coefficient D_t of atactic poly(2-vinyl pyridine) in tetrahydrofuran and in aqueous solution, in the form of the poly(2-vinyl pyridinium)chloride salt (α = 0.4). The limiting Stokes radius of the polymer is observed to be identical within experimental error in tetrahydrofuran (THF) at temperatures below 30°C and in aqueous solutions at high ionic strength. This numerical value is comparable to expectation for an unperturbed atactic vinyl polymer chain and indicates a compact, possibly micellar, conformation. Raising the temperature in THF above 30°C and decreasing the ionic strength or increasing the ionization above α = 0.4 in aqueous solvents causes a discontinuous cooperative transition to a more expanded structure. The effect of the conformational change is also manifest in the concentration dependence of D_t. Using experimental estimates of the second osmotic virial coefficients obtained by total scattered intensity measurements, the experimental data for dD_t/dc are compared with prediction based on hydrodynamic theory. Substantial disagreement is found between theory and experiment, especially in the aqueous system. In 0.01M NaCl, decrease in polyion concentration induces the transition from the compact form to a highly extended structure. Angle-dependent quasielastic light scattering data from the expanded state provides information about the intramolecular chain dynamics.     

Classical and quantal calculations of the dimerization constant and second virial coefficient for argon

Summary Computations of the second virial coefficient and thermodynamic equilibrium constant for the dimerization of argon are reported. These are based on accurate analytic representations of the Ar-Ar interaction energy. Calculations have been made using classical and quantal statistical mechanics and for the second virial coefficient the JWKB series as well.     

Dynamics of branched polymer molecules in dilute solution

Theoretical formulas for the intrinsic viscosity and viscoelastic properties of some model branched molecules in dilute solution are calculated by means of the normal coordinate method of Rouse modified to include hydrodynamic interactions. The calculations are exact except for the usual approximation of the hydrodynamic interactions by the Kirkwood-Riseman formula. The ratio of the intrinsic viscosity of a branched molecule to that of a linear molecule of the same weight is found to vary almost as the square root of the ratio of the mean square radii, instead of as the latter ratio to three-halves power, as has been postulated before. It is proposed that this square root relation is applicable in general to branched molecules of all types. Several sets of experimental data in the literature are shown to agree well with this hypothesis.