On scaling theories of polymer solutions

Abstract

We present a rigorous derivation of a scaling theory of the thermodynamics of polymer solutions at finite concentrations. The derivation proceeds directly in 3(2, 4, etc.) dimensions from the expression for the partition function for a solution of monodisperse continuous Gaussian chains with excluded volume. There is no need for the use of renormalization group methods or for the extrapolation of results from calculations near four-dimensions. Nethertheless, the importance of four dimensions emerges directly from the scaling theory as does a proof that only the binary excluded volume parameter appears in the equation of state, average chain dimensions, etc., in the limit of long enough chains. The irrelevance of the range of the binary segment interaction and the sufficiency of the two parameter theory is thus established by the scaling theory in a straightforward and elementary fashion. We demonstrate that the power law dependence, ν in 〈R2〉∝L2ν, of the mean square end-to-end distance 〈R2〉 on the chain length L in good solutions can be calculated directly in three (two, etc.) dimensions from the scaling relations. Applications are presented to good solutions at finite concentrations, to confined chains in good solvents, and to the properties of solutions of block copolymers.

Scaling behavior of dilute polymer solutions in elongational flows

“…The numerically obtained values of the scaling exponents agree with Eqs. (9) and (14), as may be expected from the fact that it is only the (Gaussian) tail of the span distribution that makes a major contribution to the free energy of the polymer at the large spans ( » N°- 5 ) that minimize it in the good-solvent case.…”

Self-Repelling Fractional Brownian Motion - A Generalized Edwards Model for Chain Polymers

“…((4.13)) (Here and in the following we label formulas from -or analogous to those in -the paper [14] by their numbers in that article, in double brackets.) The derivation of this formula is specific to the Brownian motion case and does not hold for general υ H .…”

Separation of viable and nonviable animal cell using dielectrophoretic filter

“…Employing a Hamiltonian, which incorporates both the connectivity of the chain and extra interactions, the ideal size ρ ∼ N 1/2 due to the connectivity becomes proportional to ρ ∼ N ν , with ν different from 1/2 because of the extra interactions. Example of extra interactions due to excluded volume effects are given in ref 18, 19. Effects from other interactions like the attractions between the monomeric units in a poor solvent or the interactions in a polyelectrolyte in the presence of a salt, which straightly affect the size ρ of the chain and the exponent ν , can also be incorporated.…”