 Solution

Balancing buoyant weight with Stokes drag givesThis needs isolated spherical particles, low Reynolds number and negligible wall corrections. The supplied settling measurement gives $\Delta \rho \simeq 99.1\mathrm{kg}{\mathrm{m}}^{-3}$, hence ${\rho }_p\simeq 1.10\times 10^3\mathrm{kg}{\mathrm{m}}^{-3}$. Its Reynolds number is about $7\times 10^{-5}$, consistent with the approximation.

Compare gravitational work over one radius with thermal energy, or use ${\mathrm{Pe}}_g=U_{s}r/D$ with $D=k_{B}T/(6\pi \eta r)$. The gravitational Peclet number givesBelow this sedimentation-Brownian crossover radius Brownian motion dominates settling on the particle-size scale. It does not eliminate gravity: the sedimentation scale height is ${\ell }_g=k_{B}T/[(4\pi r^3/3)\Delta \rho g]$. For negligible concentration gradients across a sample height $H$, the stronger condition is ${\ell }_g\gg H$. No universal permanently nonsedimenting size follows without a length/time criterion.

Brownian encounters bring small particles into contact; van der Waals attraction can then bind them. Surface charge can create a repulsive barrier through Debye screening of colloid repulsion. For a monovalent salt reservoir with number density $n$ of each ionic species, Poisson–Boltzmann theory hasFor $|e\psi |\ll k_{B}T$, linearization gives ${\nabla }^2\psi ={\kappa }^2\psi$, with ${\kappa }^{-1}=\sqrt{ϵk_{B}T/(2n{e}^2)}$. If $n$ instead means total ion density, the factor two is absent. The planar potential decays as $e^{-\kappa h}$; a sphere has an additional $1/r$ factor. This mean-field approximation neglects specific adsorption, ion correlations and finite ion size.

DLVO theory adds screened repulsion to dispersion attraction. At small surface separation $h$ for equal spheres of radius $r$, a useful schematic form iswith a short-range repulsive core needed outside the continuum model. Low salt can leave a barrier many times $k_{B}T$, giving kinetic stability. Higher salt shortens the repulsion and can lower or remove the barrier, allowing coagulation into a deep primary minimum. The primary and secondary DLVO minima have different consequences: a shallow secondary minimum permits looser, sometimes reversible aggregation. The surface boundary condition affects $C$; screening length alone is not a complete measured interaction law. Non-DLVO hydration or steric forces can matter at contact.