On the theory of the second virial coefficient for polymer chains

1992 ◽  
Vol 25 (7) ◽  
pp. 1912-1916 ◽  
Author(s):  
Hiromi Yamakawa
Keyword(s):  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Polymer Chains

Collapse of Isolated Chains in a Network

1994 ◽  
Vol 376 ◽  
Author(s):  
R. M. Briber ◽  
X. Liu ◽  
B.J. Bauer
Keyword(s):  
Crosslink Density ◽  
Virial Coefficient ◽  
Linear Chain ◽  
Second Virial Coefficient ◽  
Polymer Chains ◽  
Network Density ◽  
Radius Of Gyration ◽  
Single Chain ◽  
Polystyrene Matrix ◽  
Wide Range

ABSTRACTIn this study we use small angle neutron scattering to investigate the conformation of linear deuterated polystyrene chains trapped in a crosslinked protonated polystyrene matrix. The second virial coefficient was obtained as a function of crosslink density for a wide range of crosslink density. It is shown that the second virial coefficient decreases with increasing crosslink density. By extrapolating the scattering to zero concentration of the linear chain at all values of q, the single chain scattering was obtained and radius of gyration was measured the function of network density. It was found that when the network density is low (NI < Nc where NI and Nc are the number of monomer units in the linear chain and the monomer units between crosslinks, respectively) the radius of gyration does not change. As the network density increases (NI > Nc ) radius of gyration decreases. In this region the inverse of the radius of gyration varies linearly with the inverse of Nc. When the crosslink density is very high (NI » Nc ), segregation of linear polymer chains occurs. These results are in agreement with prediction and computer simulation results of polymer chain conformation in a field of random obstacles where the crosslink junctions act as the effective obstacles.

Molecular weight dependence of the second virial coefficient for flexible polymer chains in two dimensions

1989 ◽  
Vol 22 (5) ◽  
pp. 2491-2496 ◽  
Author(s):  
D. Poupinet ◽  
R. Vilanove ◽  
F. Rondelez
Keyword(s):  
Molecular Weight ◽  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Two Dimensions ◽  
Polymer Chains ◽  
Flexible Polymer ◽  
Molecular Weight Dependence

Interpretation of Preferential Adsorption Using Random Phase Approximation Theory

1995 ◽  
Vol 60 (10) ◽  
pp. 1641-1652 ◽  
Author(s):  
Henri C. Benoît ◽  
Claude Strazielle
Keyword(s):  
Approximation Theory ◽  
Random Phase Approximation ◽  
Virial Coefficient ◽  
Random Phase ◽  
Second Virial Coefficient ◽  
Solvent Mixture ◽  
Gyration Radius ◽  
Thermodynamic Quantities ◽  
Phase Approximation ◽  
Preferential Adsorption

It has been shown that in light scattering experiments with polymers replacement of a solvent by a solvent mixture causes problems due to preferential adsorption of one of the solvents. The present paper extends this theory to be applicable to any angle of observation and any concentration by using the random phase approximation theory proposed by de Gennes. The corresponding formulas provide expressions for molecular weight, gyration radius, and the second virial coefficient, which enables measurements of these quantities provided enough information on molecular and thermodynamic quantities is available.

Scattering of anyons with hard-disk repulsion and the second virial coefficient

1991 ◽  
Vol 44 (19) ◽  
pp. 10731-10735 ◽  
Author(s):  
Akira Suzuki ◽  
M. K. Srivastava ◽  
R. K. Bhaduri ◽  
J. Law
Keyword(s):  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Hard Disk

GENERAL FORMULA FOR THE SECOND VIRIAL COEFFICIENT

1961 ◽  
Vol 39 (11) ◽  
pp. 1563-1572 ◽  
Author(s):  
J. Van Kranendonk
Keyword(s):  
Phase Shift ◽  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Resolvent Operators ◽  
Simple Derivation ◽  
Scattering Phase ◽  
Scattering Phase Shifts ◽  
Quantization Volume ◽  
Mechanical Expression ◽  
The Waves

A simple derivation is given of the quantum mechanical expression for the second virial coefficient in terms of the scattering phase shifts. The derivation does not require the introduction of a quantization volume and is based on the identity R(z)−R0(z) = R0(z)H1R(z), where R0(z) and R(z) are the resolvent operators corresponding to the unperturbed and total Hamiltonians H0 and H0 + H1 respectively. The derivation is valid in particular for a gas of excitons in a crystal for which the shape of the waves describing the relative motion of two excitons is not spherical, and, in general, varies with varying energy. The validity of the phase shift formula is demonstrated explicitly for this case by considering a quantization volume with a boundary the shape of which varies with the energy in such a way that for each energy the boundary is a surface of constant phase. The density of states prescribed by the phase shift formula is shown to result if the enclosed volume is required to be the same for all energies.

The Second Virial Coefficient near Absolute Zero

1953 ◽  
Vol 66 (9) ◽  
pp. 847-848
Author(s):  
D Ter Haar
Keyword(s):  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Absolute Zero

Theoretical Assessment of Compressibility Factor of Gases by Using Second Virial Coefficient

2018 ◽  
Vol 73 (2) ◽  
pp. 121-125
Author(s):  
Bahtiyar A. Mamedov ◽  
Elif Somuncu ◽  
Iskender M. Askerov
Keyword(s):  
Virial Coefficient ◽  
Second Virial Coefficient ◽  
Compressibility Factor ◽  
Analytical Approximation ◽  
Accurate Evaluation ◽  
Numerical Examples ◽  
Lennard Jones ◽  
Theoretical Assessment ◽  
Good Agreement

AbstractWe present a new analytical approximation for determining the compressibility factor of real gases at various temperature values. This algorithm is suitable for the accurate evaluation of the compressibility factor using the second virial coefficient with a Lennard–Jones (12-6) potential. Numerical examples are presented for the gases H2, N2, He, CO2, CH4 and air, and the results are compared with other studies in the literature. Our results showed good agreement with the data in the literature. The consistency of the results demonstrates the effectiveness of our analytical approximation for real gases.

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