On Modeling the Temperature Effects on Biopolymers and Foods Undergoing Glass Transition without the WLF Equation

Abstract

Traditionally, the effect of temperature on the rate of biochemical reactions and biological processes in foods, and on the mechanical properties of biopolymers including foods, has been described by the Arrhenius equation which has a single adjustable parameter, namely the “energy of activation.” During the last three decades, this model has been frequently replaced by the WLF equation, borrowed from Polymer Science, which has two adjustable parameters and hence better fit to experimental data. It is demonstrated that the WLF model (and hence also the VTF model) is identical to an expanded version of the Arrhenius equation where the absolute temperature is replaced by an adjustable reference temperature. Both versions imply that the curve describing a process or reaction’s rate rise with temperature or the viscosity or modulus drop with temperature must have the same characteristic upper concavity above and below the glass transition temperature, T_g, however it is defined and determined. Nevertheless, at least some reported experimental data recorded at or around the transition regime suggest otherwise and in certain cases even show concavity direction inversion. The mathematical description of such relationships requires different kinds of temperature-dependence models, and two such alternatives are described. Also suggested are two different ways to present the temperature as a dimensionless independent variable which enables to lump and compare different transition patterns in the same graph. The described approach is purely formalistic; no fit considerations are invoked and neither model is claimed to be exclusive.

Appendix

The expanded Arrhenius and WLF equations are the same model.

From [13]

$${Log}_{10}\left[\frac{\mu (T)}{{\mu }_{Tref}}\right]=\frac{1}{Ln[10]}Ln\left[\frac{\mu (T)}{{\mu }_{Tref}}\right]=\frac{a}{Ln[10]}\left(\frac{1}{T+b}-\frac{1}{Tref+b}\right)=-\frac{a}{Ln\left[10\right]} \left(\frac{T-Tref}{\left(T+b\right)\left(Tref+b\right)}\right)= -\frac{a}{Ln\left[10\right]} \left(\frac{T-Tref}{\left(Tref+b\right)T+\left(Tref+b\right)b}\right)=- \left(\frac{a}{Ln\left[10\right] (Tref+b)}\right)\left(\frac{T-Tref}{T+b}\right)=-\left(\frac{a}{Ln\left[10\right] (Tref+b)}\right)\left(\frac{T-Tref}{T-Tref+b+Tref}\right)=-\left(\frac{a}{Ln\left[10\right] (Tref+b)}\right)\left(\frac{T-Tref}{b+Tref+T-Tref}\right)=-\frac{{C}_{1}(T-Tref}{{C}_{2}+(T-Tref)}$$

where ${C}_{1}=\frac{a}{Ln\left[10\right] (Tref+b)}$ and ${C}_{2}=Tref+b$