Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2017 / Paper 2 / Section B / 23B

Brownian motion is the fluctuating motion of a suspended particle produced by thermal molecular collisions. Using the standard Brownian displacement conventions, in a uniform fluid without external drift, mean displacement is zero but mean-square displacement grows with time. For overdamped diffusion, $⟨\Delta x^2⟩=2Dt$ in one coordinate and $⟨|\Delta \mathbf{r}{|}^2⟩=6Dt$ in three dimensions, with $D=k_{B}T/\zeta$ and Stokes friction $\zeta =6\pi \eta a$ for a no-slip sphere.

Brownian transport brings nanoparticles into encounters and allows them to explore configurations during self-assembly. Reversible interactions permit rearrangement toward low-free-energy structures; interactions that are too strong can instead trap disordered aggregates. Thermal fluctuations also oppose localization in shallow traps. The symbol $\zeta$ avoids confusing friction with an unrelated chemical potential.

For radius $a=10\mathrm{nm}$ and gold density $2.0\times 10^4\mathrm{kg}{\mathrm{m}}^{-3}$, $m=8.38\times 10^{-20}\mathrm{kg}$. The supplied Stokes model gives $\zeta =3.96\times 10^{-12}\mathrm{kg}{\mathrm{s}}^{-1}$ and $D=1.05\times 10^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ at $300\mathrm{K}$. Therefore the standard one-second diffusive rms distances are $45.7\mu \mathrm{m}$ in one coordinate and $79.2\mu \mathrm{m}$ in three dimensions.

Using the source's supplied shorthand literally with $m⟨v^2⟩=3k_{B}T$ instead gives $\sqrt{3k_{B}Tt/\zeta }=56.0\mu \mathrm{m}$. That shorthand lacks the factor two in the standard long-time mean-square displacement; it should not be identified with either of the preceding standard rms conventions. None of these rms values is a cumulative path length.

An ideal drag-free fall from rest gives $g{t}^2/2=4.91\mathrm{m}$. Actual gravitational settling in the assumed viscous model is much smaller: ${\tau }_p=m/\zeta =2.12\times 10^{-8}\mathrm{s}$, $u_t=mg/\zeta =0.208\mu \mathrm{m}{\mathrm{s}}^{-1}$, and displacement in one second is approximately $0.208\mu \mathrm{m}$. Thus diffusion overwhelms settling, even though ideal vacuum fall exceeds the rms diffusion distance. For rarefied-gas nanoparticle drag, at a 20-nm size in argon, continuum no-slip drag may require a rarefaction/slip correction; pressure and gas mean-free-path data are absent, so these numerical transport estimates are conditional on the supplied Stokes model.

For fluid density ${\rho }_f$, speed $U$ and characteristic length $d$, inertial kinetic energy scales as ${\rho }_f{d}^3{U}^2$ while viscous work over distance $d$ scales as $\eta U{d}^2$. Their ratio is the Reynolds number,Smaller particles at a fixed speed have smaller Re, so viscous stresses dominate fluid inertia when $\mathrm{Re}\ll 1$. Increasing speed increases Re and eventually makes inertia more important, rather than making viscous dominance stronger.

Particle inertia is a separate issue. Its particle velocity-relaxation time is ${\tau }_p={\rho }_p{d}^2/(18\eta )$; comparing it with a travel time $d/U$ gives a Stokes number ${\tau }_{p}U/d$. This uses particle density ${\rho }_p$, whereas the fluid Reynolds number uses fluid density ${\rho }_f$.

Interpreting the nominal size as diameter $d=10\mathrm{nm}$, fluid viscous dominance requireswith ${\rho }_f$ in $\mathrm{kg}{\mathrm{m}}^{-3}$. The liquid density is not supplied, so the conventional fluid-Reynolds crossover is not numerically fixed. As an explicitly additional illustrative assumption, ${\rho }_f=870\mathrm{kg}{\mathrm{m}}^{-3}$ gives $\mathrm{Re}=1$ near $69\mathrm{m}{\mathrm{s}}^{-1}$; ordinary much slower nanoparticle motion is well within the viscous regime.

If instead the supplied gold density is inserted into the dimensionless ratio ${\rho }_{p}Ud/\eta$, that ratio reaches one at $3.0\mathrm{m}{\mathrm{s}}^{-1}$. It is not the fluid Reynolds number. Including the spherical Stokes prefactor, the particle relaxation criterion ${\tau }_{p}U/d=1$ occurs at $54\mathrm{m}{\mathrm{s}}^{-1}$. These quantify different inertia comparisons. The wording's speed direction is reversed: viscous dominance is obtained by making speed sufficiently small. If 10 nm denotes radius, the length-based thresholds halve.

A tightly focused laser produces a strong intensity gradient. A small polarizable particle experiences a gradient force toward increasing intensity when the real polarizability has the appropriate positive sign; radiation pressure from scattering and absorption pushes predominantly along the beam. A high-numerical-aperture objective creates lateral confinement and a sufficiently strong opposing axial gradient for a three-dimensional optical trap. Equilibrium is generally shifted from the intensity maximum by radiation pressure.

For a Rayleigh sphere, $\alpha =4\pi {ϵ}_m{a}^3({ϵ}_p-{ϵ}_m)/({ϵ}_p+2{ϵ}_m)$ in an absolute-permittivity convention, and ${\mathbf{F}}_{\mathrm{grad}}\simeq \frac{1}{4}\mathrm{Re}\alpha \nabla |E{|}^2$ for peak complex field amplitude. Near a stable point, $F_x\simeq -k_{x}x$; the trap depth must exceed thermal fluctuations sufficiently to retain the particle. Viscous drag opposes motion, Brownian forcing produces fluctuations, and gravity/buoyancy are normally small at this scale.

For absorbing gold, use its complex wavelength-dependent permittivity: scattering, absorption and heating can frustrate trapping. The existence of an ideal dielectric gradient-force trap does not guarantee stable trapping of every metal nanoparticle at every wavelength or power.^[ref]