Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section A(iii)

Let $\Phi (x)$ be the electrostatic potential in volts, relative to the bulk value, and $x$ the distance normal to the surface. The positive quantity $e$ is the elementary charge; the monovalent ions carry $\pm e$. The bulk number density $n_0$ refers to each ionic species in a symmetric electrolyte, so the total bulk ion density is $2n_0$. ${ϵ}_0$ is vacuum permittivity, ${ϵ}_r$ relative permittivity, $k_B$ the Boltzmann constant and $T$ absolute temperature. The hyperbolic sine acts on the dimensionless potential $u=e\Phi /(k_{B}T)$.

The printed sign is inconsistent with physical Poisson–Boltzmann screening under these usual definitions. Indeed,and Poisson's equation gives ${\Phi }^{\prime \prime }=-\rho /({ϵ}_0{ϵ}_r)=+2e{n}_0\sinh u/({ϵ}_0{ϵ}_r)$, with a plus sign. The literal printed equation and the physical screened solution must therefore be distinguished.

The dimensionally meaningful small-potential condition is $|e\Phi |\ll k_{B}T$, not a comparison of volts directly with energy. Expanding $\sinh u=u+O(u^3)$ gives the physical linearized equationFor a single plane with bulk boundary condition $\Phi (\infty )=0$, $\Phi (x)={\Phi }_0{e}^{-x/{\lambda }_D}$. At fixed temperature and permittivity, ${\lambda }_D\propto n_0^{-1/2}$: the curve decreases with increasing density, diverging in the ideal zero-salt limit.

Literal expansion of the printed minus-sign equation instead gives ${\Phi }^{\prime \prime }=-{\kappa }_D^2\Phi$, with sine/cosine solutions. A nonzero such solution cannot decay to zero at infinity. That literal equation does not describe a Debye length. The figure and following ion profiles use the physically derived plus-sign equation, with this correction explicit.

Take a positive surface potential ${\Phi }_0>0$, so positive ions are co-ions and negative ions counterions. With $u_0=e{\Phi }_0/(k_{B}T)\ll 1$,The counterion density is slightly enhanced at the surface, the co-ion density slightly depleted, and both approach $n_0$ over a Debye length. To first order their sum is $2n_0$, while their difference supplies the screening charge. Reverse the surface sign and the ionic roles interchange. These are number-density profiles, not trajectories of individual ions.

For large positive dimensionless surface potential $u_0=e{\Phi }_0/(k_{B}T)\gtrsim 1$, use the nonlinear Boltzmann distribution rather than their linear expansion:The counterions become strongly enriched near the wall and co-ions strongly depleted. A planar mean-field solution isIt approaches the linear screening form far from the wall. Nonlinearity preserves positive densities and produces a much stronger near-wall asymmetry; an extrapolated linear co-ion profile could incorrectly become negative. The total local ion density is $2n_0\cosh u$, no longer constant.

The original figure compares weak- and strong-potential profiles. At very large local concentrations, finite ion size, specific adsorption and ion correlations limit this ideal point-ion mean-field model; it does not predict unlimited physically realizable counterion packing.

Interpret the given noise correlation as the three-dimensional dot-product correlation. Each Cartesian component then has $⟨{\xi }_i(t){\xi }_j(t^{\prime })⟩=2\gamma k_{B}T{\delta }_{ij}\delta (t-t^{\prime })$. For the usual Gaussian thermal-noise idealization, each coordinate is an Ornstein-Uhlenbeck process. With $\tau =\gamma /k$,For a particle initially at ${\mathbf{x}}_0=0$,Hence its long-time displacement about the trap/elastic equilibrium hasThis also follows from equipartition theorem: each of three quadratic coordinates contributes $k_{B}T/2$.

A different common experimental quantity is the stationary time-lag displacement. It obeyswith long-lag RMS $\sqrt{6k_{B}T/k}$. The factor of two distinguishes fluctuations about a mean position from differences between two decorrelated positions. The observation convention must be stated rather than silently interchanged.

For a no-slip spherical tracer of radius $a$ in an isotropic incompressible elastic continuum, a small translation produces restoring force $6\pi aG\mathbf{x}$. ThusCombining this with the equilibrium three-dimensional positional variance givesEquivalently, if $P_{\infty }$ is the stationary three-dimensional displacement-MSD plateau, $G=k_{B}T/(\pi a{P}_{\infty })$. For a single measured coordinate, $⟨x_i^2⟩=k_{B}T/k$ and $G=k_{B}T/(6\pi a⟨x_i^2⟩)$.

This is passive microrheology. Calibrate bead radius, position sensitivity, temperature and localization noise; subtract drift and test that a plateau is resolved. The continuum must be meaningful on the bead scale, the bead must couple without slip, and thermal equilibrium must hold. Boundaries, compressibility, active motion, probe attachment and unresolved relaxation can change the inferred response. An imposed optical trap adds stiffness and must be removed or independently calibrated rather than mistaken for material elasticity.