= Solution
Balancing buoyant weight with <Stokes drag> gives
$$
\frac{4\pi r^3}{3}\Delta\rho g=6\pi\eta rU_s,\qquad
\boxed{U_s=\frac{2\Delta\rho gr^2}{9\eta}.}
$$
This needs isolated spherical particles, low <Reynolds number> and negligible wall corrections. The supplied settling measurement gives $\Delta\rho\simeq99.1\,\mathrm{kg\,m^{-3}}$, hence \b[$\rho_p\simeq1.10\times10^3\,\mathrm{kg\,m^{-3}}$]. Its <Reynolds number> is about $7\times10^{-5}$, consistent with the approximation.
Compare gravitational work over one radius with <thermal energy>, or use $\mathrm{Pe}_g=U_sr/D$ with $D=k_BT/(6\pi\eta r)$. The <gravitational Peclet number> gives
$$
\mathrm{Pe}_g=\frac{4\pi\Delta\rho gr^4}{3k_BT},\qquad
\boxed{r_c\sim\left(\frac{3k_BT}{4\pi\Delta\rho g}\right)^{1/4}\simeq1.0\,\mathrm{\mu m}.}
$$
Below this <sedimentation-Brownian crossover radius> <Brownian motion> dominates settling on the particle-size scale. It does not eliminate gravity: the <sedimentation scale height> is $\ell_g=k_BT/[(4\pi r^3/3)\Delta\rho g]$. 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 has
$$
n_\pm=n e^{\mp e\psi/(k_BT)},\qquad
\nabla^2\psi=\frac{2en}{\epsilon}\sinh\frac{e\psi}{k_BT}.
$$
For $|e\psi|\ll k_BT$, linearization gives $\nabla^2\psi=\kappa^2\psi$, with \b[$\kappa^{-1}=\sqrt{\epsilon k_BT/(2ne^2)}$]. 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 is
$$
U(h)\sim C e^{-\kappa h}-\frac{A_Hr}{12h},
$$
with a short-range repulsive core needed outside the continuum model. Low salt can leave a barrier many times $k_BT$, 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.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2014/paper-1-dlvo.png]
{title=Original schematic DLVO potentials showing barrier reduction with increased screening}
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