(112h) Jim Swan and Rheometry: Medium Amplitude Parallel Superposition (MAPS) Rheology

Jim Swan was a master at looking at vexing problems through a fresh new lens. We shared a mutual interest in the rheometric technique of Large Amplitude Oscillatory Shear (LAOS) and in integrating this data-rich protocol into a systematic and automated tool that could perhaps be used to “learn” the constitutive response of a new material. We were both aware of the standard mathematical representation for nonlinear viscoelasticity in terms of a Volterra series expansion of the shear stress as a functional of the shear strain. Though formally elegant, previous attempts to use this representation in complex fluids characterization were heavily constrained by the computational complexity and the scourge of experimental noise in the extended temporal data that needs to be collected. The unique insight Jim brought to the problem was to shift the description to frequency space, which leverages both the algebraic simplifications of the Fourier transform and the associated spectral noise reduction in frequency domain transforms of experimental time series. Together with a graduate student, Kyle Lennon, this allowed us to develop a unique theoretical and experimental framework that we called medium amplitude parallel superposition (MAPS) rheology. This novel approach reveals an exotic material property, the third order complex modulus, that describes completely the weak, time-dependent nonlinearities of the shear stress within a homogeneously-sheared viscoelastic material. This three-dimensional material function is a unifying super-set of the response functions measured in medium amplitude oscillatory shear (MAOS) and parallel superposition (PS) experiments. Unlike the MAOS and PS transfer functions, the third order complex modulus can be used to construct the weakly nonlinear stress response to an arbitrary deformation history. This material function completely characterizes the nonlinear viscoelasticity of unknown materials at third order, and even for relatively simple constitutive models possesses startlingly rich and distinctive features. An experimental protocol was also developed that enabled direct measurement of the third order complex modulus using existing commercial rheometers and their associated control software. This enabled us to demonstrate the technique’s potential by comprehensively mapping the weakly nonlinear viscoelastic response of a common wormlike micellar fluid and rigorously comparing the data with the predictions of several proposed nonlinear constitutive models.