De Gennes has shown that the properties of an isolated polymer in a solution (a chain with excluded volume) can be deduced within the framework of a Lagrangian theory for a zero component field in the absence of an external field. This result in generalized to the case of polymer solutions at intermediate concentrations. It is shown that a grand ensemble of polymers can be described by using a Lagrangian theory for a zero component field coupled to an external field. The concentrations Cp of polymers (chains) and C m of monomers (links) are fixed by two chemical potentials. It is shown that the osmotic pressure obeys a scaling law of the form (P/KTCp) = F(Cp N3ν) where N is the mean number of monomers per polymer (N = Cp/C m) and ν the critical index defining the size of a long isolated polymer. The function F(λ) can be expanded in powers of λ and it is given implicitly by the generating functional of the zero-momentum vertex functions derived from the Lagrangian theory. The results seem to be in good agreement with experiments
Neutron coherent scattering techniques have been used for the determination of the conformation of polymer in bulk and experimental details are given about the application of this method to the study of polymeric systems. Measurements have been made for small and intermediate momentum ranges on a series of eight monodisperse deuterated polystyrenes of molecular weight ranging from 21,000 to 1,100,000. The results lead to the concluson that in amorphous state the conformation of the polymer molecule is indistinguishable from that in solvent and that the Debye scattering function which is valid for unperturbed chains applies for q ~1 as low as 10 Á.
The ground-state and the spin-wave states of the Hamiltonian, H= ∑ i (SixSi+1x+SiySi+1y+ρSizSi+1z),are studied for all values of ρ, and analytical expressions are given for their energies. On the other hand, by using a canonical transformation which changes H(ρ) into -H(- ρ), the states of highest energy can also be obtained. The ground state is ferromagnetic for ρ ≤ − 1 and antiferromagnetic for ρ ≥ −1. For ρ = ±1, the energy has singularities, but it remains continuous. For ρ = 1, all its derivatives are also continuous. In the range − 1 ≤ ρ ≤ 1, the spin-wave states of given momentum are degenerate but for ρ ≥ 1; this degeneracy is removed, and an energy gap G(ρ) appears.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.