Simple joint inversion localized formulae for relaxation spectrum recovery

In oscillatory shear experiments, the values of the storage and loss moduli, G′(ω) and G″(ω), respectively, are only measured and recorded for a number of values of the frequency ω in some well-defined finite range [ω_min,ω_max]. In many practical situations, when the range [ω_min,ω_max] is sufficiently large, information about the associated polymer dynamics can be assessed by simply comparing the interrelationship between the frequency dependence of G′(ω) and G″(ω). For other situations, the required rheological insight can only be obtained once explicit knowledge about the structure of the relaxation time spectrum H(τ) has been determined through the inversion of the measured storage and loss moduli G′(ω) and G″(ω). For the recovery of an approximation to H(τ), in order to cope with the limited range [ω_min,ω_max] of the measurements, some form of localization algorithm is required. A popular strategy for achieving this is to assume that H(τ) has a separated discrete point mass (Dirac delta function) structure. However, this expedient overlooks the potential information contained in the structure of a possibly continuous H(τ). In this paper, simple localization algorithms and, in particular, a joint inversion least squares procedure, are proposed for the rapid recovery of accurate approximations to continuous H(τ) from limited measurements of G′(ω) and G″(ω).