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A useful microscopic screening estimate compares thermal energy with the buoyant gravitational potential energy over one particle radius. Write $\Delta \rho =50\mathrm{kg}{\mathrm{m}}^{-3}$ and $m_b=(4\pi /3)a^3\Delta \rho$. The sedimentation length is ${\ell }_g=k_{B}T/(m_{b}g)$, and the gravitational Peclet number isUsing $T=298\mathrm{K}$ and requiring ${\mathrm{Pe}}_g\lesssim 1$ givesEquivalently, combining Stokes law speed $v_s=2\Delta \rho g{a}^2/(9\eta )$ and diffusion coefficient $D=k_{B}T/(6\pi \eta a)$ gives $v_{s}a/D={\mathrm{Pe}}_g$. Buoyancy, not the full polystyrene density, is essential to the estimate. Taking a diameter rather than radius as the comparison length changes the numerical cutoff by an order-one factor.

There is no universal largest particle that stays uniformly suspended indefinitely. If uniformity over a vessel height $H$ is required, use ${\ell }_g\gtrsim H$, not merely ${\ell }_g\gtrsim a$. For example $H=1\mathrm{cm}$ gives a much smaller radius, about 58 nm. Observation time, density matching, convection, particle polydispersity and shape also matter. Independent of sedimentation, electrostatic/steric barriers must prevent aggregation; low ionic strength, suitable surface chemistry and clean solvent can be as important as size.

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.