LaMer Nucleation & Size-Focusing (2D) — Colloidal Nanocrystal Synthesis

Synthesis Conditions

Precursor feed rate 30

Reaction temperature 500 K

Ligand (surfactant) concentration 40

Batch State

Regime

Induction

C(t) monomer conc.0.00

Mean radius

0.0 nm

Dispersion (CV)

0.0%

Nanocrystal count

Batch time

0.0 s

How it works

The LaMer model of colloidal nanocrystal synthesis tracks free monomer concentration C(t) against two thresholds: the solubility C_s (red mark) and the nucleation threshold C_nuc (grey mark, raised by higher ligand coverage and lowered by higher temperature).

dC/dt = feed − Σ(2πR·dR/dt)·k_g − N_nuc·k_n

Nucleation (C > C_nuc):  rate ∝ (C − C_nuc)²
Growth (diffusion-limited): dR/dt = k_kin·D(T)·(C − C_eq(R)) / R
Gibbs–Thomson:  C_eq(R) = C_s·exp(α_GT/R)

Induction — feed raises C toward C_nuc with no solid phase yet.
Burst nucleation — once C > C_nuc, many nuclei form almost simultaneously, crashing C back down (a hot-injection burst forces this instantly).
Size-focusing growth — while C stays well above C_eq(R) for every particle, the 1/R term in the growth law makes small crystals grow faster than large ones, narrowing the size distribution (falling CV%).
Ostwald ripening (defocusing) — once C decays toward C_s, curvature raises C_eq for the smallest particles above the bulk C, so they dissolve while larger particles keep growing — the population broadens again.

Feed rate sets how fast monomer is supplied. Temperature raises diffusive growth kinetics and lowers the nucleation barrier. Ligand concentration caps kinetic growth (steric hindrance) and lowers surface energy, damping the Gibbs–Thomson curvature term that drives ripening — this is why capping ligands (oleate, TOPO, thiols) are what let real colloidal syntheses (CdSe, PbS, Au nanocrystal quantum dots) stay monodisperse.

The concentration-vs-time chart below is the classic "LaMer curve": a rising ramp during induction, a sharp drop at the nucleation burst, then a slow decay through the growth/ripening phases. The histogram tracks the actual simulated radius of every live particle — its narrowing during focusing and re-broadening during ripening are read directly off the particle population, not scripted.

Below mean radius Near mean radius Above mean radius

LaMer curve — C(t)

Particle size distribution