An algorithm for applying the Williams, Landel, and Ferry method of reduced variables

Abstract

The measurement of the dynamic complex modulus of viscoelastic materials on any apparatus is confined to a limited frequency and temperature range. Williams, Landel, and Ferry (WLF) [J. D. Ferry, Viscoelastic Properties of Polymers (Wiley, New York, 1980), 3rd ed. Chap. 11] reasoned that an increase in temperature produces an identical effect in a viscoelastic material as a decrease in frequency and vice versa. Empirically, then, there is a correspondence between temperature and frequency. The WLF method of reduced variables was proposed as a method of extending the effective frequency range of the dynamic modulus and loss factor. This technique involves measuring the complex modulus over a limited frequency range and a variety of temperatures. The data at different temperatures are then plotted versus frequency and shifted along the frequency axis until a smooth curve results. The effects are shifted modulus curves describing the complex modulus over a broad frequency range at some fixed temperature. Typically, the WLF shift requires several runs at temperatures close enough so that the modulus of one run overlaps the next. In this paper, the WLF method of reduced variables is reviewed and an algorithm for accomplishing the WLF shift is investigated. This algorithm shifts dynamic modulus data at various temperatures, the consecutive runs of which may not contain overlapping moduli.