The Concentration Dependence of Steady-State Compliance for Polymer Solutions

Abstract

The steady-state compliance J_e as a function of the molecular weight M and the concentration c for polymer solution can be expressed as:
J_e=αM/cRT for smaller values of c or M
=βM⁰/c^2∼3 for larger values of c and ^M
Here α and β are constants and RT is used in its usual meaning. The second behavior of J_e is characteristic of highly entangled system. In the concentration range where the transition from the first to the second behavior occurs, the J_e displays the maximum.
Either of the two relaxation spectra, A and B, are found to be compatible with these observed behaviors of J_e. These are:
(A) H_W(τ)=1/2cRT/M_e(τ_c/τ)^1/2 0<τ<τ_c
H_I(τ)=1/2(1-g_N)cRT/M_e(τ_c/τ)^1/2 τ_c<τ<τ₁
H_B(τ)=1/hg_NcRT/M_e τ_l<τ<τ_m
τ_c=τ₁E^-2, τ_m=τ_cE^3.5, τ_l=τ_me^h(1/E-1)
(B) H_W(τ)=1.75/3cRT/M_e(τ_c/τ)^2/3 0<τ<τ_c
H_Z(τ)=1.75/3(1-g_N)cRT/M_e(τ_c/τ)^2/3 τ_c<τ<τ₁
H_B(τ)=1/hg_NcRT/M_e τ_l<τ<τ_m
τ_c=τ₁E^-1.5, τ_m=τ_cE^3.5, τ_l=τ_me^{1.75h(1/E-1)/2}
The former corresponds to the Rouse theory, while the latter to the Zimm theory. Here, τ₁ represents the maximum relaxation time of the original Rouse or Zimm theory, M_e is entanglement spacing, E is the number of entanglement point in a molecule, hence E=M/M_e, and g_N and h are parameters representing the intensities of H_I and H_B, respectively.
The viscosity η, storage modulus G' and loss modulus G" are also calculated on the basis of these spectra. These results are qualitatively in agreement with the observed one, particularly in the case of high value of h and low value of g_N.