Some graphical methods for the analysis of mechanical and dielectric relaxation data

Abstract

Three types of coordinate systems for the analysis of relaxation data are examined. When the data can be described by a single relaxation time, the graphs are straight lines. The slopes and intercepts of these lines can be used to evaluate the parameters which characterize the process, such as relaxation time. The advantages and disadvantages of displaying data with each of these types of coordinate systems are known and can be applied here. When more than one relaxation time is present, the graphs are curves. Nevertheless, the limiting slopes and intercepts can still be used to estimate the relaxation parameters. In this case various average relaxation times, 〈τ〉 p , are obtained, $$\left\langle \tau \right\rangle _p = {{\sum\limits_{i = 1}^m {\Delta _i } \tau _i^p } \mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^m {\Delta _i } \tau _i^p } {\sum\limits_{i = 1}^m {\Delta _i } }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{i = 1}^m {\Delta _i } }}$$ wherem is the number of relaxation processes,τ i andΔ i are the time and magnitude of thei-th process andp is any integer between −3 and +4. These averages are useful both as a means of characterizing the distribution of relaxation times and determining other parameters describing the relaxation process. The application of these graphical methods is illustrated in two specific areas: the frequency dependence of the shear modulus and ultrasonic longitudinal wave propagation. For these two examples the expressions for the intercepts and limiting slopes are evaluated. With these expressions the graphs of experimental data can be used to estimate the relevant parameters. A section discussing the special case of a system with two relaxation times is also included. This section illustrates how intuitive generalization from a case with one relaxation time can be misleading.