Rheometer

A rheometer imposes a controlled deformation or stress and measures the material response to determine a constitutive equation.

Concentric cylinders impose shear while torque and rotation measure a material's response. The displayed calibration holds for Newtonian laminar flow, inner-cylinder rotation and a stationary outer cylinder. In a narrow gap, the rate is approximately $\Omega R_i/(R_o-R_i)$ and inner-wall stress is $\mathcal{T}/(2\pi L{R}_i^2)$. End effects, wall slip and inertial secondary flow require control.

A small-angle cone rotating above a stationary plate imposes approximately uniform shear rate, since local gap and wall speed both scale with radius. Integrating the uniform shear stress with moment arm r over the plate gives the displayed torque. The leading approximation neglects edge effects and inertia.

A parallel-plate rheometer shears a sample in a thin gap between coaxial disks. Rotation at angular speed $\Omega$ gives local shear rate $\dot{\gamma }(r)=\Omega r/h$, so differentiating the measured torque with respect to rim shear rate can recover a generalized-Newtonian viscosity.

Small-amplitude oscillatory shear probes linear viscoelastic response by imposing $\gamma ={\gamma }_0\sin \omega t$ and resolving the stress into components in phase with strain and strain rate.

Measurements at different frequencies while retaining an amplitude in the linear regime. They compare observation time with material relaxation times.

Measurements at fixed frequency and varying oscillation amplitude. Amplitude-independent moduli identify a possible linear regime; dependence on amplitude signals nonlinear response or an experimental artifact.

The dynamic modulus is the complex stress-amplitude to strain-amplitude ratio in steady oscillatory response. With the convention $e^{i\omega t}$, the real part is the storage modulus and the imaginary part is the loss modulus. For $ϵ={ϵ}_0\cos \omega t$, $\sigma ={ϵ}_0(G^{\prime }\cos \omega t-G^{\prime \prime }\sin \omega t)$. The average supplied power is $\omega {ϵ}_0^2{G}^{\prime \prime }/2$, fixing the loss-sign convention. The elastic coefficient is a shear or extension modulus according to the deformation measured; it is unrelated to the complex modulus meaning absolute value of a complex number.

A series spring of modulus $G$ and dashpot of viscosity $\eta$ have $\tau =\eta /G$ and $\dot{\gamma }=\dot{\sigma }/G+\sigma /\eta$. With $e^{i\omega t}$ amplitudes, $G^{\prime }=G(\omega \tau {)}^2/[1+(\omega \tau {)}^2]$ and $G^{\prime \prime }=G\omega \tau /[1+(\omega \tau {)}^2]$. Low frequencies are fluid-like; high frequencies retain an elastic plateau. The loss peaks at $\omega \tau =1$.

A thin uniform circular rod of radius $r$ and length $\ell$ undergoing a small end twist of complex amplitude $\Theta$ has shear-strain amplitude $\gamma (\rho )=\rho \Theta /\ell$. A complex viscoelastic modulus $G^{\ast }(\omega )$ gives shear-stress amplitude $G^{\ast }\rho \Theta /\ell$. Integrating the restoring torque givesThe area integral is the polar second moment of the cross-section, not a mass moment of inertia. For a disc of moment of inertia $I$ and negligible rod inertia, harmonic angular motion satisfies $[-I{\omega }^2+K^{\ast }(\omega )]\Theta =0$ in a free-mode calculation, or equals the external torque amplitude in a driven measurement. The complex stiffness is a frequency-domain constitutive representation; a general real-time response has memory and cannot be described by putting an arbitrary frequency-dependent complex number into an instantaneous real equation.

The rubbery plateau is a frequency or time range in which a flexible polymer network has an approximately constant elastic storage modulus dominated by chain entropy. A permanently crosslinked elastomer retains a nonzero long-time modulus. An entangled uncrosslinked polymer melt can have a transient plateau but ultimately reaches terminal fluid flow at sufficiently low frequency. The plateau therefore must be interpreted with the network lifetime and observation window specified.

The storage modulus is the in-phase coefficient $G^{\prime }$ in $\tau ={\gamma }_0(G^{\prime }\sin \omega t+G^{\prime \prime }\cos \omega t)$. It measures elastic energy storage.

The loss modulus is the quadrature coefficient $G^{\prime \prime }$ in oscillatory shear. It measures viscous energy dissipation.

For strain $\gamma ={\gamma }_0\cos \omega t$ and stress $\sigma ={\gamma }_0(G^{\prime }\cos \omega t-G^{\prime \prime }\sin \omega t)$, integrating supplied power $\sigma \dot{\gamma }$ over one period gives the displayed dissipated energy per unit volume. The storage term integrates to zero; passivity requires nonnegative loss modulus for positive frequency in this convention.

The loss tangent $\tan \delta =G^{\prime \prime }/G^{\prime }$ compares dissipative and elastic response. Its crossover through one often identifies a dominant viscoelastic relaxation time.

A viscoelastic relaxation time is the characteristic time over which stored stress decays after deformation. A single-mode material commonly crosses from viscous to elastic response around $\omega \lambda =1$.