Animations for the Molecules and Formulas of the used Principle
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Model for Relaxation Modulus
Real polymer relaxation occurs in two phases: first, Rouse (1953) relaxation and later
in second phase with reptation, collisions,
and entangling in longitudinal modes, all of which we refer to here as entanglement relaxation.
Molecular entanglements will open
and relax after deformation at times t >0.001 s,
where most G(t) observations are made.
Fig. 1. Schematic models of the relaxation modulus.
a. The standard normal rheologically effective distribution (RED), w(t), plotted on a logarithmic scale with tc = 1 s.
Its characteristic relaxation according to Eq. (5) is very close to the classical Maxwellian single-element relaxation, with t = 100 s (dashed line).
b. Relaxation by Eq. (7) using the log-normal distributions w(log t) and w(log t/R), with constants tc = 10-6 s, P' and P'' set to 0.1,
and R = 104. Both normalized relaxation moduli G(t)/G0 and logarithmic log G(t) with illustrated G0=10Pa are shown.
Typical similar real polymer has average molecular weight M in range 100,000, and the polydispersity index, M/Mn below 1.2.
We need to convert distribution scales between molecular weight scale M and rheologically effective scale as a function of time t
by introducing the melt calibration,
which is the relation between rheological properties and the molecular weight.
Fig. 2. Relation model for melts, and conversions from the RED to the MWD.
a. In a relaxed polymer, a test point in the melt after a small shear deformation at t0
is equivalent to a statistical nonrelaxed sphere of volume V0,
which shrinks as a function of radius r(t) as the farthest molecules or the ends of individual molecules relax first, resulting in less deformation
relative to the test point.
b. Melt calibration curve M(t) for PS, and the respective typical universal calibration curves for different SEC columns marked by dashed lines.
c. RED w(log t) from the time scale converted to inverse molecular weight scale w (top) and MWD w(log M) using Eq. (9).
Model for Viscosity and Dynamic Moduli
We have developed a principle for using complex viscosity data obtained from a dynamic rheometer measured in the frequency-sweep mode.
Fig. 3. Schematic model of the viscosity and dynamic moduli.
General Model for Viscosity and dynamic moduli
Fig. 4. General Model for Viscosity and dynamic moduli