Stress Relaxation Following Non-Newtonian Flow

The Slowest Relaxation Mechanism of Na-Polyacrylate Solutions

Abstract

It is assumed that the substances described in the present paper have elements of viscoelastic mechanism which are non-linear at large stresses, but become linear at small stresses in behaviors of flow and internal deformation. In this case the generalized Maxwell formula (1) may be reduced to Eq. (2) at the terminal region of stress relaxation following non-Newtonian flow.
dσ(t)/dt=σ^m_j=1dσ_j(t)/dt=σ^m_j=1{-σ_j(t)/τ_j} (1)
dσ(t)^*/dt=σ^m_j=1dσ_j(t)^*/dt=σ^m_j=1{-σ_j(t)^*/τ_j°} (2)
The asterisk in the equation refers to small stresses. τ_j°is the relaxation time of the jth mechanism in its ordinary sense at small stresses. By integrating Eq. (2),
σ(t)^*=σ^m_j=1σ_j(0)^*_e^-t/τ_j° (3)
where σ_j(0)^*, as an integration constant, is the extrapolated value of the stress to t=0, under which the viscoelastic mechanisms obey Newtonian and Hookean laws. Dividing Eq. (3) by γ:
σ(t)^*/γ=σ^m_j=1η_j(0)^*_e^-t/τ_j° (4)
where η_j(0)^*=σ_j(0)^*/γ. On the analogy of the linear viscoelastic theory η_j(0)^* seems to be equal to η_j°=τ_j°G_j°which may prevail in the viscoelastic behaviors under small stresses. If t>>τ_m°(the slowest relaxation time), Eq. (4) will become:
σ(t)^*/γ=η_m(0)^*e^-t/τ_m° (5)
where the mechanisms of smaller relaxation time than τ_m°are neglected. According to the theory presented here, the relaxation curves plotted by log{σ(t)/γ}vs. t must become a single straight line at the final process of relaxations, independent of γ values under the steady flow.
However, the experimental curves for the materials investigated here become almost straight lines in parallel after the initial rapid relaxations. This phenomenon may be interpreted by the following formula.
η_m(0)^*≡τ_m°G_m(0)^*≠τ_m°G_m°≡η_m° (6)
Therefore, the effects of non-Newtonian flow seem to have influence on the whole processes of the relaxation and to be characterized as follows; the maximum relaxation time is not altered, but the distribution function G_m°is depressed with increase of the shearing rate under the flow. These phenomena must be derived from the shear break-down of secondary structures under the shear, where the secondary structures seem to contribute mainly to the slowest relaxation mechanism. The evidence of structural break-down of the samples under the flow is certified by the method of Jobling et al..
By comparison of dependencies of η_m(0)^* and η_a on γ, the following relations are expected:
lim_γ→0η_m(0)^*=η_m°≅η=lim_γ→0η_a (7)
log η_a vs. logγ and log η_m(0)^* vs. logγ curves at various concentrations seems to be subject to reduction to the master curve, respectively, by the identically reduced variables for both cases.
τ_m°increases exponentially with concentration in Na-PAA solutions.