Virial Coefficient A2 Research Articles

Some Comments on the Analysis of the Third Virial Coefficient for Polymer Chains

An analysis is made of experimental data for the third virial coefficient A3 of atactic oligo- and poly(α-methylstyrene)s (a-PαMS) in the Θ solvent cyclohexane at 30.5 °C and in three good solvents, toluene, 4-tert-butyltoluene, and n-butyl chloride, at 25.0 °C on the basis of the helical wormlike (HW) chain model. It is found that A3 at Θ, which is denoted by A3,Θ, becomes a constant of 5.0×10−4 cm6 mol/g3 independent of the weight-average molecular weight Mw for Mw ≳ 104 and deviates from it for Mw ≲ 104 because of effects of chain ends. The observed dependence of A3,Θ on Mw may be well explained by the HW theory that takes account of the effects. For the three good-solvent systems, the behavior of the factor g defined by A3/[A2(HW)]2Mw is examined as a function of the cubed gyration-radius expansion factor αS3, where A2(HW) is the part of the second virial coefficient A2 without the effects of chain ends. It is found that the data points for a-PαMS samples with Mw ≳ 105 in all the three good solvents form nearly a single-composite curve, as predicted by the HW theory, and that they follow the Stockmayer–Casassa theoretical curve for αS3 ≳ 2 but deviate upward from it for αS3 ≲ 2 because of effects of three-segment interactions. Effects of chain stiffness are of minor importance for an explanation of this deviation. Some literature data for polystyrene are also examined.

Static and Dynamic Light Scattering of Polyelectrolyte/Surfactant Solutions: the Na‐Hyaluronate/(C10TAB) System

AbstractSummary: This work discusses the interactions between the anionic polyelectrolyte sodium hyaluronate (Na‐Hy) and the cationic surfactant decyltrimethylammonium bromide (C10TAB). Static and dynamic light scattering experiments were carried out in the single homogenous phases (region I: [C10TAB] < 0.040 M and region III: [C10TAB] > 0.350 M) in “salt‐free” and at different added salt concentrations (NaCl) in order to gain more understanding on the chain conformation and on the dynamics of polyelectrolyte/surfactant systems. In region I, the Na‐Hy “salt‐free” system exhibits the expected polyelectrolyte properties, i.e. the scattering intensity increases with the addition of surfactant or salt. As a consequence of this screening of the electrostatic interactions, a decrease in both the radius of gyration Rg and the second virial coefficient A2 is observed. This decrease is found more pronounced in the presence of surfactant C10TAB than in the case of added salt (Na‐Hy/NaCl). Consequently, the calculated apparent persistence length Lt, assuming a worm‐like chain at high ionic strength, is found about 58 Å in the case of surfactant and Lt = 71 Å in the case of added salt. This result demonstrates clearly that the binding of the surfactant to Na‐Hy is stronger than in the case of added salt and is at the origin of the phase separation that occurs at the concentration of C10TAB > 0.040 M. As far as the dynamic properties are concerned, the autocorrelation functions in “salt‐free” solutions measured in region I are better described by two relaxation modes (intermediate and slow) and by three relaxation modes (fast, intermediate and slow) in region III. The three relaxation modes reflect respectively the dynamics of free micelles (fast mode), association‐dissociation equilibrium of micelle‐like clusters (intermediate mode) and polymer‐micelles adsorption (slow mode). At high ionic strength (added salt) in region III, the micelle diameter increases (fast mode), the intermediate mode disappears and the polymer‐micelles adsorption diffusion coefficient increases due to the screening of the electrostatic interactions.Phase diagram of Na‐Hy polyelectrolyte in C10TAB surfactant and NaCl salt.imagePhase diagram of Na‐Hy polyelectrolyte in C10TAB surfactant and NaCl salt.

Second Virial Coefficient and Gyration-Radius Expansion Factor of Oligo- and Poly(α-methylstyrene)s near the ϑ Temperature

The second virial coefficient A2 was determined for atactic oligo- and poly(α-methylstyrene)s (a-PαMS) in cyclohexane below, at, and above ϑ (30.5 °C) in the range of weight-average molecular weigh...

Effect of Valence and Chemical Species of Added Electrolyte on Polyelectrolyte Conformations and Interactions

The effect of added electrolyte (AE) valence and species on polyelectrolytes was studied using automatic, continuous mixing (ACM), allowing measurements over composition gradients. The second virial coefficient A2 was decomposed into contributions from an electrostatically sensitive mean- square gyration radius 〈S2〉 and an external electrostatic field. 〈S2〉 matched predictions of combined electrostatic persistence length and excluded-volume theories, without adjustable parameters. A2 was modeled via an equivalent, soft-core, expansible, charged sphere. Whereas the scaling behavior of 〈S2〉, intrinsic viscosity ( ), and A2 was the same for all AE, the prefactors strongly depended on AE symmetry and valence. At given ionic strength 〈S2〉 ,( ), A2, effective linear charge density, and expansible sphere surface potential are less for divalent than monovalent AE. Different ion species for a given valence class showed second-order effects following a Hoffmeister series. ACM should also prove valuable for investigating AE-induced phase transitions in polyelectrolytes like DNA and proteins.

Synchrotron small angle X-ray scattering study on the conformation of polystyrene in compressed CO 2 -toluene mixture

Synchrotron small angle X-ray scattering (SAXS) was performed to investigate the effect of dissolved CO2 in toluene on the conformation of polystyrene (PS) in the solution. It has been found that the second virial coefficientA 2 and the radius of gyrationR g decrease with the increasing antisolvent CO2 pressure. The scattering intensity of PS chain followedI(h) ∞h −α under different antisolvent pressures (0, 0.6, 1.5, 2.4, 3.3, and 4.2 MPa), suggesting that the PS chain has self-similar structure behavior or a fractal structure in the presence of antisolvent CO2. All this reveals a large effect of antisolvent pressure or the solubility of CO2 in the solution on PS structure. The fractal dimensions increase with the increasing antisolvent pressure, indicating that the polymer chain changes from a swollen coil into a rather dense globule in the course of adding antisolvent CO2.

A Monte Carlo study of effects of chain stiffness and chain ends on dilute solution behavior of polymers. II. Second virial coefficient

A Monte Carlo (MC) study is made of the second virial coefficient A2 for polymers using two freely rotating chains, each of bond angle 109°, with the Lennard-Jones 6-12 intramolecular and intermolecular potentials between beads in a cutoff version for the number of bonds in the chain ranging from 6 to 1000 in the Θ and good-solvent conditions. It is found that effects of chain ends on A2 are appreciable for small molecular weight M, as was expected, and that the second virial coefficient A2,Θ at the Θ temperature, at which the ratio 〈S2〉/M of the mean-square radius of gyration 〈S2〉 to M becomes a constant independent of M for very large M, remains slightly negative even for such large (but finite) M where the effects of chain ends disappear. Such behavior of A2,Θ, which cannot be explained within the framework of the binary cluster theory, is shown to be understandable if possible effects of three-segment interactions are considered. The present MC data for A2 (along with the previous ones for 〈S2〉) may then be consistently explained by the existent theory based on the helical wormlike chain model only if a minor correction is made to the theoretical A2,Θ in almost the same range where the effects of chain ends are appeciable. The present MC data are also compared with experimental data, and it is shown that the latter may also be similarly explained.

Second Virial Coefficient of Oligo- and Poly(α-methylstyrene)s. Effects of Chain Stiffness, Chain Ends, and Three-Segment Interactions

The second virial coefficient A2 was determined from light scattering measurements for atactic oligo- and poly(α-methylstyrene)s (a-PαMS) with the fraction of racemic diads fr = 0.72 over a wide range of weight-average molecular weight Mw from 6.48 × 102 to 5.46 × 106 in cyclohexane at 30.5 °C (ϑ) and in toluene at 25.0 °C, and for Mw ≥ 2 × 105 in 4-tert-butyltoluene at 25.0 °C, in n-butyl chloride at 25.0 °C, and in cyclohexane at 35.0, 45.0, and 55.0 °C. It is shown that the observed characteristic features of A2 at ϑ, which is denoted by A2,ϑ, as a function of Mw in cyclohexane at ϑ, i.e., a sharp increase with decreasing Mw for small Mw and a rather long-tailed negative region for large Mw, may be well explained by the theory that takes account of the effects of chain ends and also of three-segment interactions. It is also shown that the observed dependence of A2 on Mw in toluene may be quantitatively explained by the theory that takes account of the effects of chain stiffness, local chain conformatio...

The effect of topological constraint on the theta temperature of a knotted polymer

Monte Carlo simulations were used to study the effect of topological constraints of knotted polymers on their theta temperatures. The theta temperatures were determined through two different definitions—the vanishing of the second virial coefficient A2=0, and the quasi-ideal behavior of the radius of gyration, 〈Rg2〉∼N. Prime knots with chain lengths from N=60 to 300 and with crossings from 31 to 91 were considered. For chains with finite lengths, it was found that the theta temperature determined from quasi-ideal condition of the knot increases, as the complexity of the knot increases. On the other hand, the topological complexity seemed to have no effect on the theta temperatures determined from the vanishing of the second virial coefficient. Also, our simulation results suggest that for chains with finite crossing numbers, as N→∞, theta temperatures for all knots obtained from two different approaches coincide and are equivalent to that of a linear polymer chain.

Solubilization of Alkylcyanobiphenyl Liquid Crystals in Aqueous Micellar Solutions of a Diblock Copolymer of Propylene Oxide and Ethylene Oxide Studied Using Dynamic and Static Light Scattering

The solubilization properties of P60E348 (P = oxypropylene, E = oxyethylene, subscripts = number of repeating units) diblock copolymer micelles containing liquid crystals have been studied by light scattering and UV spectroscopy for mainly dilute solutions from the following viewpoints: (i) the micellar solubilization rate J of a nematic liquid crystal (LC), predominantly pentylcyanobiphenyl, (ii) the amount of solubilized material at saturation, and (iii) the apparent size of the P60E348 micelles containing LC and their intermicellar interaction properties. It was shown, in experiments with solubilization temperature Ts = 50 °C, that the rate J is inversely proportional to the copolymer concentration cp in the range 2 g L-1 ≤ cp ≤ 10 g L-1 whereas J is constant at higher values of cp. It was also shown that J increases strongly with an increase in temperature. The micellar hydrodynamic radius Rh was independent of temperature in the range 30 °C ≤ T ≤ 60 °C, whereas there was a reduction in the hydrodynamic virial coefficient kD and in the thermodynamic second virial coefficient A2 as T was increased (for Ts = 40 °C). To probe the intermicellar interference, the excess scattered intensity was measured as a function of cp, showing an increasing deviation from linearity, reflecting the transition from a dilute to a concentrated system and the increasing predominance of intermicellar interactions.

Electrostatic and Association Phenomena in Aggregates of Polymers and Micelles

An automatic, continuous-mixing technique is introduced to measure light scattering and viscosity simultaneously over a wide range of solute concentrations in multicomponent systems. Behavior along arbitrary curves in component coordinates can be monitored. This technique is used to investigate the association and electrostatically based excluded-volume interactions that arise in solutions of interacting neutral polymers and charged micelles. In this study, the ionic surfactant sodium dodecyl sulfate (SDS) and a neutral polymer poly(vinylpyrrolidone) (PVP) were chosen. The SDS confers pseudopolyelectrolyte properties on the polymer/micelle aggregates (PSA), such as electrostatically enhanced second and third virial coefficients A2 and A3, respectively. The presence of a large A3 is directly observed as a well-defined maximum in scattering intensity versus [PVP] under saturating SDS conditions. The electrostatically enhanced three-body effects represented by A3 in light scattering are nearly 3 orders of magnitude greater than the corresponding three-body terms in viscosity. A viscosity effect reminiscent of the electroviscous effect seen for linear polyelectrolytes is observed, but it occurs under very different PSA concentration and solution ionic strength conditions.

Role of Competitive PEO−Water and Water−Water Hydrogen Bonding in Aqueous Solution PEO Behavior

The behavior of aqueous solutions of poly(ethylene oxide) (PEO) is studied theoretically by applying a mean field-like approach which includes the effect of the competition of PEO and water as proton acceptors in hydrogen bond formation. Accounting for this effect is of crucial importance for a correct description of all solution properties. We calculate the temperature and concentration dependence of the average fraction of hydrogen bonds between PEO and water and find very good agreement between our predictions and experimental or MD simulation data. We also make predictions concerning the temperature behavior of the second virial coefficient A2 and the effective interaction parameter χeff and compare it with experimental data. We found that the decrease of A2 with temperature is caused by the delicate balance of the opposing effects of water−PEO and water−water hydrogen bonding. The phase diagram for PEO of different molecular weights in water is calculated using experimentally reported data for the en...

Small-Angle X-ray Scattering by Dilute Solution of Bisphenol A Polycarbonate during Adding Antisolvent CO2

Small-angle X-ray scattering (SAXS) was used to investigate the dimension, shape, and microstructure of bisphenol A polycarbonate (PC) of narrow distribution in tetrahydrofuran (THF) in the course of adding gaseous antisolvent CO2 at 308.15 K and at pressures up to 2 MPa. The weight-average molecular weights (Mw) of the polymer samples were 1.4 × 104, 1.8 × 104, 2.4 × 104, and 2.9 × 104, and the concentration of the polymer in the solution ranged from 1.2 × 10-3 to 9.1 × 10-3 g/cm3. The second virial coefficient A2 and the radius of gyration 〈Rg2〉1/2 ≡ Rg were found to decrease with the increasing of the antisolvent pressure. The Mw of the PC chain was scaled to the Rg of the polymer chain as Mw ∝ RgdR (dR = 2.17, 2.38, 2.56, 2.74, and 2.95) for different antisolvent CO2 pressures (P = 0, 0.5, 1, 1.5, and 2 MPa), suggesting that the PC chain has a fractal structure in the presence antisolvent CO2. Self-similar structure behavior was also observed with a detailed analysis of the particle scattering factor P(q) (angular dependence of the scattered light), and the fractal dimensions df at 0, 0.5, 1.0, 1.5, and 2 MPa are 2.14, 2.35, 2.53, 2.70, and 2.92, respectively. All of these revealed a large effect of antisolvent pressure or the solubility of CO2 in the solution on PC structure. Increase of fractal dimension (dR or df) from 2.17 to 2.95 indicates that the polymer chain changes from a swollen coil to a rather dense globule in the course of adding antisolvent CO2.

Effect of Gas Antisolvent on Conformation of Polystyrene in Toluene: Viscosity and Small-Angle X-ray Scattering Study

Synchrotron radiation small-angle X-ray scattering (SAXS) and the viscosity technique were used to investigate the effect of dissolved CO2 in toluene on the conformation of polystyrene (PS) in the solution. The viscosity of PS solution decreases faster with increasing antisolvent CO2 pressure than that of the solvent in the absence of the polymer. The intrinsic viscosity [η] calculated using the well-known Huggins equation decreases with antisolvent pressure. It was found that the second virial coefficient A2 and the apparent mean-square radius of gyration 〈Rg2〉1/2 decreases with pressure of antisolvent CO2. All these phenomena can be attributed to the shrink of PS chain in the course of adding the gas antisolvent because the interaction between the polymer and solvent becomes weaker. The values 〈Rg2〉1/2 at different pressures obtained from SAXS data agree reasonably with those calculated from Flory theory using the viscosity data determined in this work. This implies that Flory theory, which has been use...

Ultracentrifugal Characterization of Poly(ethylene terephthalate) in Dilute Solutions

The well-fractionated poly(ethylene terephthalate) (PET) samples ranging in weight average molecular weight Mw from 0.9 to 16×104 have been studied by sedimentation velocity and equilibrium, and viscometry in hexafluoroisopropanol (HFIP) and m-cresol at 30°C. The second virial coefficient A2 determined in HFIP by the sedimentation equilibrium method is found to have high dependence on Mw expected for the helical wormlike chains in very good solvents. The constant Φ1/3P−1 in the Mandelkern-Flory equation is 2.54×106 for PET in HFIP and 2.57×106 in m-cresol, being in excellent agreement with that reported for several polymer-solvent systems. The dependences of sedimentation coefficient s0 on Mw agree with also those expected for good solvents. The Mark-Houwink-Sakurada relations established with HFIP and m-cresol as the solvents are [η]=2.95×10−4Mw0.72 in HFIP at 30°C (3.1×104<Mw) [η]=4.97×10−3Mw0.48 in HFIP at 30°C (3.1×104>Mw) [η]=4.04×10−4Mw0.68 in m-cresol at 30°C (2.4×104<Mw)The values of steric factor σ estimated from the unperturbed chain dimensions range from 1.3 to 1.5, irrespective of the solvent nature, and such small σ values suggest that the extent of steric hindrance to the internal rotation of PET chain is significantly low.

Characterization of cellulose in solvent mixtures with N-methylmorpholine-N-oxide by static light scattering

Cellulose was dissolved in a solvent mixture containing N-methylmorpholine-N-oxide (NMMNO), diethylenetriamine (DETA) and water. The weight-average molar masses Mw, radii of gyration RG, and second virial coefficients A2 of 9 different cellulose samples were determined by means of static light scattering (SLS) at room temperature. The influence of two different dissolution procedures was studied on four samples. The Mw results were compared with results obtained by gel permeation chromatography (GPC) with multi angle laser light scattering (MALLS) detection. The original samples and samples recovered from the solvent mixture by precipitation were tricarbanilated and dissolved in tetrahydrofuran for the GPC analysis. The SLS results led to the conclusion that cellulose was molecularly dispersed in the solvent mixture NMMNO/H2O/DETA. The Mw dependence of the radii of gyration showed a common power law behavior with an exponent which is identical to the theoretical value for flexible, linear chains in good solvents. The observed differences between SLS and GPC results, especially for the high-molecular samples, can be explained by a molecular degradation of cellulose during the dissolution in NMMNO/H2O/DETA. The different dissolution procedures and the use of potassium pyrophosphate as a stabilizer showed no significant effect on the determined results.

Small-Angle X-ray Scattering Study on the Conformation of Polystyrene in Compressed CO2−Tetrahydrofuran Mixtures

Cloud points of the compressed CO2 + tetrahydrofuran (THF) + polystyrene (PS) system was studied at temperature range of 298.15−328.15 K and at different pressures. The expansion of the solution caused by dissolved CO2 was also studied. Synchrotron radiation small-angle X-ray scattering (SAXS) was used to investigate the dissolved CO2 on the conformation of PS. The second virial coefficient A2 and the apparent mean-square radius of gyration 〈Rg2〉1/2 at different pressures were obtained. It was found that the PS chain shrinks with increasing the concentration of CO2.

Dynamics of a Polymer Solution in a Rigid Matrix. 2

Dynamic light scattering measurements of the diffusion coefficient D and the Rayleigh ratio Rθ are reported for dextran molecules confined in agarose gel networks of various concentrations. In this condition, the light scattered by the dextran is some 2 orders of magnitude less intense than that from the agarose. Three molecular weights of dextran were investigated: 7 × 104, 5 × 105, and 2 × 106 g mol-1. For the lowest molar mass it is confirmed that, below the dextran overlap concentration c*, the product DRθ is independent of the agarose concentration, showing that the reduction of the rate of diffusion inside the gel is the result of a decrease in the osmotic pressure in the confined geometry. For the higher molar masses, entanglement effects between the dextran and the network become noticeable in the more highly concentrated gels. The dynamic light scattering intensity measurements are also found to yield reasonable estimates of the molar mass Mw and radius of gyration RG of the trapped dextran molecules. The second virial coefficient A2 is positive, indicating that the agarose−water matrix acts as a good solvent for dextran, but the ratio of RG to the hydrodynamic radius is less than 1.5. These results are interpreted in terms of branching of the dextran molecule.

Structure Properties of Dextran. 2. Dilute Solution

Dilute solution properties of 13 dextrans with different molar masses in water and 0.5 M NaOH were investigated. Seven of these samples were products of Sigma prepared by bacteria from Leuconostoc mesenteroides, one was obtained by fractionation of the dextran with the highest molar mass, and five samples were obtained by degradation via controlled acid hydrolysis. Static and dynamic light scattering, viscometry and analysis of the reducing end group were applied. The molar mass dependencies of the radius of gyration Rg, the hydrodynamic radius Rh, the second virial coefficient A2 and the intrinsic viscosity [η] for the nonfractionated samples were determined. The results are compared with data from the literature. Size exclusion chromatography in on-line combination with multiple-angle laser light scattering and viscosity detection revealed different calibration curves Mi vs Ve for the studied samples. The molar mass dependencies of the radii and intrinsic viscosities could be measured. The data of the r...

Solution Properties of Cellulose Tris(phenyl carbamate). 1. Characterization of the Conformation and Intermolecular Interaction

The radius of gyration 〈S2〉1/2, hydrodynamic radius RH, intrinsic viscosity [η], and second virial coefficient A2 for cellulose tris(phenyl carbamate) (CTC) in tetrahydrofuran (THF) at 25 °C were determined in a wide range of molecular weights, 1.91 × 104 to 392 × 104, by static and dynamic light scattering and viscometry. Furthermore, the osmotic compressibility ∂c/∂Π for isotropic THF solutions of CTC samples was measured at 25 °C by static light scattering over wide ranges of the polymer concentration. Data of 〈S2〉1/2, RH, and [η] were analyzed by the established quasi-two parameter theory for the wormlike chain model to determine the molar mass ML per unit contour length and the persistence length q. On the other hand, results of A2 and ∂c/∂Π were analyzed on the basis of the scaled particle theory for wormlike spherocylinders combined with the quasi-two parameter theory for A2 to estimate the hard-core diameter d0 and strength δ of the soft dispersion force of the CTC chain in THF. The value of ML [=1040 (±70) nm-1] estimated is consistent with the 3/2 helical conformation of CTC in the crystal of the CTC-methyl ethyl ketone complex, while the hard-core diameter d0 [=1.4 (±0.1) nm] is slightly smaller than the interchain spacing (=1.66 nm) of CTC in the same crystal. The value of q [=10.5 (±1) nm] for CTC is comparable to literature values of q for some other cellulose derivatives reported so far.

Dendrigraft polystyrene initiated by poly(p-chloromethyl styrene): synthesis and properties

Dendrigraft polystyrene was prepared using poly(p-chloromethyl styrene) as the multicentre initiator by atom transfer radical polymerization (ATRP). The resulting polymer was characterized with light scattering, intrinsic viscosity, thermal analysis and atomic force microscopy (AFM) imaging. The AFM image showed there were two kinds of different structural dendrigraft polystyrenes in the product, ie the ‘coupling’ dendrigraft polystyrene and ‘free’ dendrigraft polystyrene. The ‘coupling’ structure might result from the coupling reaction of growing radical chain ends. The second virial coefficient A2 and intrinsic viscosity of the dendrigraft polystyrene were smaller than those of the relative linear and twelve-arm star polystyrene, and the thermal analysis showed that the initial decomposition temperature was a little higher than that of linear polystyrene. © 1999 Society of Chemical Industry