(191l) Storage and Loss Modulus From Material Functions Applied to Inverse Emulsions

In the rheological characterization of fluids, the set of more “descriptive” experiments are those with a sinusoidal imposed stress (or strain) for determining the storage and the loss modulus (G’(w) and G’’(w)), due to the information they provide on the dynamic change of these key rheological variables. On the other hand, transient measures for determining material functions (creep compliance J(t) and relaxation modulus G(t)) which usually are easier experiments, are in most cases not considered as valuable by the wrong supposition that this parameters carry less information (maybe by its technical simplicity) despite the fact that theoretically both kind of experiments have the same information.^[1]

This work presents analytical and numerical methods for transforming material functions (creep compliance J(t) and relaxation modulus G(t)) into the dynamic functions (G’(w) and G’’(w)) with all the information that can be obtained from them. In the analytical treatment, expressions of J(t) and G(t) from classical and fractional constitutive models are adjusted from experimental data and the Fourier transform is employed for finding de complex modulus G*(w) (which is the Fourier transform of the first order derivate of G(t)).

For the numerical treatment, a mathematical procedure has been implemented relating the Laplace and Fourier transforms – by analytical continuation-- of the creep compliance measures with the complex modulus, then a numerical transform is found following the approach developed by Evans et al.^[2], in which a piecewise linear approximation is used.  Other numerical treatments (supported by integral transformation theory) have also been implemented avoiding complications in the Evans’ procedure. The experimental data used here are those of an inverse emulsion (W/O) at relatively high concentration. Material and dynamic functions were measured on a stress-controlled rheometer.

The results show that the information obtained from transient and periodic experiments is mathematically related, despite qualitative differences in their underlying physics, therefore from a rheological characterization standpoint, whichever of them can be employed. Our observations are in agreement with results reported by Evans et al.^[2], on polyisoprene melts, on the numerical transformation of creep compliance J(t) into the dynamic functions (G’(w) and G’’(w)).

References:

1. J. D. Ferry, Viscoelastic Properties of Polymers, 3^rd ed. (Wiley, New York, 1980).
2. R. M. L. Evans, M. Tassieri, D. Auhl, and T.A. Waigh. Direct conversion of rheological compliance measurements into storage and loss moduli. Physical Review 80, 012501 (2009).