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Formation of most nuclei within an interval short compared with subsequent particle growth. A common birth period can help narrow a particle size distribution if mixing and later growth are sufficiently uniform and secondary nucleation is suppressed.

The size parameter $x=2\pi r/\lambda$, complex refractive index and particle shape determine scattering strength and direction. In the small-sphere Rayleigh scattering regime, scattering cross-section scales as $r^6{\lambda }^{-4}|(m^2-1)/(m^2+2){|}^2$; the Mie scattering regime is needed when particle size approaches the wavelength. Particle number density and particle size distribution, aerosol hygroscopicity and the angular fraction returned to space control an ensemble's radiative effect. High scattering does not imply strong cooling if most light is scattered forward or the particle also absorbs strongly; single-scattering albedo distinguishes scattering from absorption.

A La Mer diagram plots the concentration of growth species against time. Label the equilibrium solubility $C_s$ and a higher effective nucleation threshold $C_n$. Generate growth species until the concentration briefly exceeds $C_n$; a short burst nucleation interval then creates seeds. Consumption brings the concentration below $C_n$ while leaving it above $C_s$, allowing existing particles to grow without substantial continuing nucleation. Rapid mixing, a short seed-formation interval and a common growth period help produce a narrow particle size distribution. Suppress secondary nucleation and aggregation, then quench or consume the remaining growth species when the desired size is reached. The threshold is an operational kinetic scale, not a sharp new equilibrium phase boundary.

Reducing the solid-liquid interfacial tension lowers the classical barrier, which scales as ${\gamma }^3$ at fixed supersaturation. More seeds can then form at a smaller supersaturation; at fixed total precipitated mass, more seeds generally mean a smaller mean final particle size. A narrower distribution is not guaranteed: if nucleation continues throughout growth, particles have different ages and the distribution can broaden. A brief well-separated burst and uniform subsequent growth are the controlling requirements, not low interfacial tension alone. The historical model concerns separated nucleation/growth in sulfur hydrosols, rather than a universal demonstrated mechanism for every nanoparticle synthesis.^[ref]

Prepare size-characterized gold-particle ensembles, heat them on a calibrated in-situ electron diffraction stage, and follow the loss of sharp fcc diffraction features and appearance of liquid-like diffuse scattering. Measure the particle size distribution rather than treating a polydisperse ensemble as one diameter. Temperature-dependent weakening of a diffraction peak alone is insufficient: thermal vibrations, particle motion, coalescence, evaporation and beam damage can also reduce intensity. Low-dose repeats, heating/cooling runs, size checks before and after heating, a bulk-like reference and support-dependent controls help distinguish melting from those effects. Electron diffraction measurements on gold particles down to about 2 nm are a classic demonstration.^[ref]

The physical origin is the large interface contribution to free energy at small radius. A curved solid–liquid interface raises the solid's thermodynamic equilibrium chemical potential relative to the flat-interface case, so coexistence occurs below the bulk melting temperature. Reduced coordination and surface disorder provide a microscopic interpretation, but the quantitative continuum relation is a free energy balance, not simply a claim that every surface atom has a universal lower melting point.

In terms of diameter $d$, the model isThus melting temperature rises toward the bulk value as diameter increases, with a depression proportional to $1/d$. The analytic sketch below includes the calculated point. It deliberately does not extrapolate the continuum expression down to a zero or negative melting temperature.

Facet-dependent energies, non-spherical shapes, size-dependent interfacial energies and latent heat, structural changes or magic cluster sizes can modify the trend. Premelting can broaden the transition. A substrate or encapsulating matrix changes interface energetics and can stabilize or destabilize the solid. Ligands, impurities, strain, particle size distributions, coalescence during heating and nonequilibrium temperature/beam effects can shift the measured transition. These possibilities motivate separate thermal, structural and size controls rather than a fit to diameter alone.

Flocculation indicates that the attraction/kinetic balance has changed: electrolyte may screen a double-layer barrier; pH may reduce surface charge; multivalent ions can promote aggregation; added polymer can bridge particles or cause depletion force; poor solvent conditions can collapse stabilizing layers. Rapid evaporation concentrates both particles and salts and can force contacts faster than they anneal. Aggregates then sediment much faster because their effective buoyant size has grown.

Check ionic strength, pH, surface charge and the particle size distribution before and during assembly. Dialyse or reduce salt where appropriate, restore surface charge, add a nonbridging steric stabilization with a compatible solvent, and slow the concentration step. Compare untreated and deliberately salt-screened controls. Restoring repulsion after irreversible aggregation may not redisperse the particles, so prevention is preferable to assuming all settled material can be recovered.