Techniques in oscillatory shear rheology - Indian Institute of ...

Techniques in oscillatory shear rheology 152.2 General response in LAOSWhen a material is subjected to strain as given in Eq. 1, the general stress responseof the material can be written as,σ(t) =N∑1σ n sin(nωt + δ n ) , (19)where σ n and δ n are amplitude and phase lag of the n th harmonic. In this equation,σ, a function of time, is being expressed as a Fourier series. In case of linearviscoelasticity or SAOS, only the first term of this series exists (Eq. 2).The expression for stress can also be written as a series using complex notation,σ =N∑1G ∗ nγ n . (20)For linear viscoelastic response, only the first term of the summation exists(Eq. 7). Similar to storage and loss modulus in case of linear viscoelastic response,we can define a series of storage and loss moduli G ′ n and G ′′n, corresponding to n thharmonic in the series in Eq. 20. These moduli can also be written in terms of σ nand phase δ n from Eq. 19.For isotropic liquids and given that viscosity is not a function of direction ofstrain rate, it can be shown that only the odd harmonics of the above series are nonzero.This is also the case for dielectric properties, where dielectric constants arenot functions of the direction of electric field. The overall number of terms, N canbe arbitrarily large. However, most measurements have been made upto the fifthharmonic due to experimental limitations. The issues related to detection of higherharmonics are discussed later in Sect. 2.4.The measurement of stress, through the torque or force sensors of the rheometer,are done discretely at a finite sample interval (t s ). Based on the sampling frequency(N s = 1/t s , number of data points per second), we obtain a time series of discretemeasurements. This time domain series can be denoted as,(a)(b)Fig. 8 (a) Time domain signal and the corresponding (b) Fourier transform [25]