Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section A / A3

A linear viscoelastic fluid responds linearly to small imposed strain histories but retains memory of earlier deformation. For shear strain $\gamma (t)$ and a relaxed initial state, Boltzmann superposition givesHere $G(t)$ is the causal relaxation modulus, the stress following a unit step strain. Doubling a sufficiently small strain history doubles the stress, and responses to added histories add. Elasticity stores deformation energy; viscosity dissipates it. Linearity is an amplitude approximation, not an assertion that the material has no relaxation times.

The Maxwell model has a spring of modulus $G_0$ in series with a dashpot of viscosity $\eta$. Their stresses are equal and strains add, giving $\dot{\gamma }=\dot{\sigma }/G_0+\sigma /\eta$. After a step strain ${\gamma }_0$, the strain rate is zero for $t>0$, so $\dot{\sigma }=-\sigma /\tau$ with $\tau =\eta /G_0$. Initially the spring takes the step, $\sigma (0^{+})=G_0{\gamma }_0$. Hence

For the convention $\gamma =\mathrm{Re}({\gamma }_0{e}^{i\omega t})$, the complex viscoelastic modulus is $G^{\ast }=i\omega {\int }_0^{\infty }G(t)e^{-i\omega t}dt$. ThusThe storage modulus exceeds the loss modulus for $\omega \tau >1$, so the response is predominantly solid-like above the crossoverAt low frequency $G^{\ast }\simeq i\omega \eta$; at high frequency $G^{\ast }\to G_0$.

The creep compliance is the strain per unit step stress: if $\sigma (t)={\sigma }_{0}H(t)$, then $\gamma (t)={\sigma }_{0}J(t)$ for $t>0$. A Newtonian fluid obeys $\sigma =\eta \dot{\gamma }$, so under constant stress its initially zero strain grows as $\gamma ={\sigma }_{0}t/\eta$. Therefore

Passive microrheology uses thermal bead motion as a probe of mechanical compliance. The fluctuation-dissipation theorem relates equilibrium displacement fluctuations to the same linear response measured by a small applied force. The stated thermal stress scale has units of energy per bead volume; dividing the mean-square displacement by $a^2$ provides a dimensionless effective strain. With that prescribed analogy,The displacement is relative to the starting position, and the variance here is the sum over three spatial coordinates. For a Newtonian fluid, substitution gives $⟨\Delta r^2⟩=k_{B}Tt/(\pi \eta a)=6Dt$, exactly the Stokes–Einstein relation $D=k_{B}T/(6\pi \eta a)$.

The continuum interpretation assumes a dilute no-slip spherical probe, thermal equilibrium, linear response and a homogeneous incompressible medium on the bead scale. The stated stress estimate is an effective scale for the analogy; a literal instantaneous stress variance would also require a microscopic or temporal averaging prescription.

Under a step stress, the Maxwell spring extends immediately by ${\sigma }_0/G_0$, while the dashpot adds ${\sigma }_{0}t/\eta$. Hence $J(t)=G_0^{-1}+t/\eta$, and the microrheology of a Maxwell fluid relation givesThis idealized overdamped relation describes an elastic plateau followed by long-time diffusion. Its nonzero extrapolated intercept is not the actual displacement variance at exactly zero lag, which is zero. Inertia or fast solvent relaxation must resolve the initial rise; those processes were omitted from the ideal spring-dashpot model.

An entropic network has approximately one thermal energy of elastic cost per mesh volume, giving $G_0\sim k_{B}T/{\xi }^3$. At $T=300\mathrm{K}$, the supplied mesh size and viscosity giveThe intermediate-time plateau in the three-dimensional mean squared displacement isAfter the omitted fast initial transient, the log-log sketch is approximately flat for $t\ll \tau$, crosses over near $\tau$, and grows linearly for $t\gg \tau$. The figure shows the Maxwell compliance prediction over this range.

An ordinary 25–30 frame-per-second recording samples intervals of about $0.03$–$0.04\mathrm{s}$, comparable to $\tau$. It therefore has little access to a clear plateau at lags well below the crossover, even if its positional precision resolves a displacement of a few tenths of a micrometre. Higher frame rates and calibrated localization noise would improve the modulus measurement. The bead radius is only twice the mesh size, so treating the network as a homogeneous continuum is also an approximation.