## Mechanical response of colloidal system ### Parameters and linear behaviors For Newtonian fluid, $\tau = \eta \frac{d\gamma}{dt} =\eta \frac{du}{dy} = \eta \dot{\gamma}$ This $\dot{\gamma}$ is the shear strain rate, people typically call it shear rate. $\tau$ is the shear stress. For viscoelastic materials, people use $G=G'+i G''$ as the dynamic modulus. The $G'$ is *storage modulus* for elastic behavior, and $G''$ is the *loss modulus* for viscous behavior. This gives the stress strain relation as $\sigma = G \gamma = (G'+iG'')\gamma$ And by oscillatory experiments with frequency $\omega$, the loss modulus can be lined to viscosity as $\frac{G''}{\omega} = \eta$ For low frequency oscillation, the elastic response dominant so $G'>G''$. And we have $G'(\omega) \sim \omega^{2} ,\ G''(\omega) \sim \omega$ This can be obtained from a modified [Maxwell model](https://en.wikipedia.org/wiki/Maxwell_material), i.e., $G'=\frac{G_{M}(\omega \tau)^{2}}{1+(\omega \tau)^{2}}$ $G''=\frac{G_{M}\omega \tau}{1+(\omega \tau)^{2}}+\omega\eta'_{\infty}$ $G_{M}$ is a constant storage modulus, $\eta'_{\infty}$ is a constant limiting high frequency viscosity, $\tau$ is relation time. To characterize a viscoelastic fluid one most often uses the linear dynamic moduli $G'(\omega)$ and $G''(\omega)$, and the steady state properties $\eta(\dot{\gamma})$ and $\Psi_{1}(\dot{\gamma})$, the normal stress coefficient. ([[introduction-to-colloid-science-and-rheology.pdf#page=30&selection=4,0,35,16|intro-to-rheology, page 30]]) $\lim_{\dot{\gamma} \rightarrow 0} \eta(\dot{\gamma}) = \lim_{\omega\rightarrow 0} \frac{G''}{\omega}$ $\lim_{\dot{\gamma} \rightarrow 0} \Psi_{1}(\dot{\gamma}) = 2\lim_{\omega\rightarrow 0} \frac{G'}{\omega^{2}}$ The dynamic moduli, yield stress and critical strain values like $\gamma_{lin}$, $\gamma_{y}$ would also affected by the volume fraction of gel and local structures, see [[Colloidal attractions and flocculated dispersions]] for the power scaling of the volume fraction and change with parameters. As for models between pure liquid and pure solid , besides the previously mentioned Maxwell model and modified Maxwell model, one may also use the [Herschel-Bulkley fluid](https://en.wikipedia.org/wiki/Herschel%E2%80%93Bulkley_fluid) , or for simpler case, [[Mechanical response of colloidal system#Bingham plastic|Bingham plastic]] to characterize the response of materials. ^653ab5 >[!Question] >Since the storage and loss modules are also real and imaginary variables, and the forms are very similar to refractive index. Will they follow [[Kramers–Kronig relations]] also? ### Herschel-Bulkley fluid The expression is $\begin{align} & \dot{\gamma} = 0, &\text{if } \tau<\tau_{0} \\ & \tau = \tau_{0} + k \dot{\gamma}^{n}, &\text{if } \tau \geq \tau_{0} \end{align}$ Below the critical stress $\tau_{0}$, the material does not flow and behaves like elastic solid; above the critical stress, flow get initialized and show an non-Newtonian response with a consistency index $k$. This can also be written as a tensor form. See the wiki links below. ### Bingham plastic Just like Herschel-Bulkley fluid, it will flow only after a critical stress, but after that it behaves like Newtonian fluid. $\begin{align} & \dot{\gamma} = 0, &\text{if } \tau<\tau_{0} \ & \tau = \tau_{0} + \mu \dot{\gamma}, &\text{if } \tau \geq \tau_{0} \end{align}$ In this case, we also have the same viscosity defined for Newtonian fluid. >[!Info] >See more on: [Herschel–Bulkley fluid](https://en.wikipedia.org/wiki/Herschel%E2%80%93Bulkley_fluid) and [Bingham plastic](https://en.wikipedia.org/wiki/Bingham_plastic).