A regularization-free method for the calculation of molecular weight distributions from dynamic moduli data

Abstract

There are several models for the determination of molecular weight distributions (MWDs) of linear, entangled, polymer melts via rheometry. Typically, however, models require a priori knowledge of the critical molecular weight, the plateau modulus, and parameters relating relaxation time and molecular weight (e.g., k and α in τ=kM^α). Also, in an effort to obtain the most general MWD or to describe certain polymer relaxation mechanisms, models often rely on the inversion of integral equations via regularization. Here, the inversion of integral equations is avoided by using a simple double-reptation model and assuming that the MWD can be described by an analytic function. Moreover, by taking advantage of dimensionless variables and explicit analytic relations, we have developed an unambiguous and virtually parameter-free methodology for the determination of MWDs via rheometry. Unimodal MWDs have been determined using only a priori knowledge of the exponent α and dynamic moduli data. In addition, the uncertainty in rheological MWD determinations has been quantified, and it is shown that the reliability of the predictions is greater for the high-molecular-weight portion of the distribution.

Appendix

If all the chains in a given polymer sample are too short to entangle, the relaxation modulus can be described solely in terms of Rouse processes, e.g. [2]

$$ \frac{{G(t)}} {{G_{\text{N}}^{\text{o}} }} = \frac{{\pi ^2 M_{\text{C}} }} {{12}}\int\limits_0^\infty {\frac{{\exp [ - t/\tau_{\text{R}} (M)]}} {M}W(M)\frac{{dM}} {M},} $$

(26)

where the Rouse relaxation time τ_R is proportional to M². Equation 26 can be combined with [5]

$$ G(t) = \int\limits_0^\infty {h_{\text{R}} (\tau_{\text{R}})\exp \left[ { - t/\tau_{\text{R}} } \right]\frac{{d\tau_{\text{R}} }} {{\tau_{\text{R}} }}} = 2\int\limits_0^\infty {\tilde h_{\text{R}} (M)\exp \left[ { - t/\tau_{\text{R}} (M)} \right]\frac{{dM}} {M}} $$

(27)

to obtain

$$ \int\limits_0^\infty {\left[ {2\tilde h_{\text{R}} (M)\exp \left[ { - t/\tau_R (M)} \right] - \frac{{\pi ^2 M_{\text{C}} }} {{12}}G_{\text{N}}^{\text{o}} \frac{{\exp [ - t/\tau_{\text{R}} (M)]}} {M}W(M)} \right]\frac{{dM}} {M}} = 0. $$

(28)

Equation 28 is satisfied when

$$ \frac{{2\tilde h_{\text{R}} (M)}} {{G_{\text{N}}^{\text{o}} }} = \frac{{\pi ^2 M_{\text{C}} }} {{12}}\frac{{W(M)}} {M}. $$

(29)

To obtain Eq. 14, let $H_{\text{R}} (N) = 2\tilde h_{\text{R}} (mN)/G_{\text{N}}^{\text{o}} .$ In general, as in Eqs (12) and (13), both reptation and Rouses processes need to be considered.