Past exam of the soft condensed matter course of the University of Cambridge / 2015 / Paper 1 / 1

At fixed temperature and solvent permittivity, the Debye–Hückel screening length satisfies ${\lambda }_D^{-2}\propto {\sum }_i{c}_i{z}_i^2=2I$, where $I=\frac{1}{2}{\sum }_i{c}_i{z}_i^2$ is ionic strength. Fully dissociated sodium chloride has two monovalent ion species, giving $I_1=0.01$ M. Fully dissociated magnesium sulfate has two divalent species, givingThe multivalent-electrolyte screening-length ratio is thereforeThe squared valences are essential: counting only the total ion concentration misses a factor of four in the screening strength per formula unit. This is the comparable linear-screening estimate, assuming unchanged dielectric response and full dissociation. ion pairs and concentrated-electrolyte nonideality can make a real divalent-salt solution depart from it; no such correction is specified.

For a long ideal chain of $N$ statistical segments of length $b$, $R_G^2=N{b}^2/6$. At fixed repeat chemistry and statistical segment length, molar mass is proportional to $N$. The molecular-weight scaling of an ideal-chain gyration radius thus givesThe supplied size and ideal statistics are used as stipulated; an independently tabulated coil size would not replace the given baseline.

solvent quality changes the balance of segment–segment and segment–solvent contacts. In a good solvent, excluded-volume repulsion swells a sufficiently long isolated chain, with $R_G\propto N^{\nu }$, $\nu \simeq 0.588$ in three dimensions; the Flory approximation gives $\nu =3/5$. At theta conditions the effective two-segment excluded volume vanishes and the ideal exponent $1/2$ is recovered for long chains. A poor solvent favors a compact polymer globule, whose large-chain size scales approximately as $N^{1/3}$ at fixed internal density. solvent and temperature can also change local persistence length and hence the prefactor. These different conformations need not have the same radius at the same molar mass.

The independent fluid-mechanical relation iswhere $\rho$ is fluid mass density, $\eta$ dynamic viscosity and $\nu$ kinematic viscosity. For a generic rigid sphere, the convention must specify whether $L$ is its diameter or radius; their Reynolds numbers differ by a factor of two. The ratio compares advective inertia $\rho U^2/L$ with viscous stress gradients $\eta U/L^2$. Small Reynolds number supports a creeping-flow approximation when unsteady fluid inertia is also negligible. No organism-specific size, speed or propulsion calculation is made here.