A colloid is a mixture in which particles between roughly 1 nm and 1 µm are dispersed in a continuous medium — examples include milk, paint, blood plasma, and aerosol sprays. Whether those particles clump together (flocculate) or remain evenly spread (form a stable sol) is governed by DLVO theory, named after Derjaguin, Landau, Verwey, and Overbeek, which balances two competing forces: an attractive van der Waals interaction that pulls particles together at short range, and an electrostatic double-layer repulsion that pushes them apart. The real-world stakes are high: DLVO failure modes cause cement to set, vaccines to aggregate, and wastewater to clarify.
In this simulation, particles undergo Brownian motion with a step size obeying the Stokes–Einstein relation D = k_B T / (6πηr). You can tune particle count, temperature, viscosity, and radius, then push the system over its stability threshold by raising the ionic strength (salt concentration), which screens the electrostatic barrier and triggers diffusion-limited cluster aggregation. A fractal dimension readout and cluster statistics track the transition in real time.
What is DLVO theory?
DLVO theory (1941–1948) describes the total interaction energy between two colloidal particles as the sum of an attractive van der Waals potential (V_vdW ∝ –A/6h, where A is the Hamaker constant and h is surface separation) and a repulsive electrostatic double-layer potential that decays exponentially with the Debye screening length κ⁻¹. The resulting energy curve typically has a primary minimum at very close range, a repulsive maximum (the "energy barrier"), and a shallow secondary minimum at larger separations.
Why does adding salt cause particles to aggregate?
Each colloidal particle is surrounded by a cloud of counter-ions (the electric double layer). Increasing salt concentration compresses this cloud, reducing the Debye screening length κ⁻¹ ∝ (I)^(–1/2), where I is ionic strength. As κ⁻¹ shrinks, the repulsive energy barrier lowers until particles can jump across it by thermal fluctuation, leading to rapid irreversible aggregation — a phenomenon called coagulation.
What is Brownian motion and how is it modelled here?
Brownian motion is the random jiggling of small particles caused by constant collisions with the surrounding solvent molecules. The diffusion coefficient is given by the Stokes–Einstein equation D = k_B T / (6πηr), so smaller particles, higher temperatures, and lower viscosities all lead to faster diffusion. In the simulation each particle takes a random displacement each frame with variance proportional to 2D·Δt, reproducing the correct statistical behaviour.
When the energy barrier has been removed by salt, every collision between particles or clusters results in permanent sticking. The resulting growth process is called diffusion-limited cluster aggregation (DLCA). Clusters that form by DLCA are fractal objects with a characteristic fractal dimension d_f ≈ 1.8 in 2D, which means they are far more open and tenuous than compact spheres (d_f = 2). This fractal structure can be measured from the slope of log(cluster mass) vs log(cluster radius).
Temperature appears in two competing places: higher T increases the Brownian kick (D ∝ T), making particles explore faster and collide more frequently, but it also adds k_B T of thermal energy that helps particles surmount the DLVO energy barrier. In most systems raising temperature reduces stability, though in charge-stabilised colloids where the zeta potential increases with T the effect can be reversed.
The Hamaker constant A characterises how strongly two materials attract each other via van der Waals forces; typical values range from 10⁻²¹ J (weak, e.g. polystyrene in water) to 10⁻¹⁹ J (strong, e.g. metals in vacuum). The stickiness slider in this simulation scales the depth of the primary minimum — increasing it makes particles adhere more readily once they come into close contact, mimicking a larger Hamaker constant.
The zeta potential ζ is the electric potential at the slipping plane of the double layer surrounding a particle, typically measured in millivolts. Particles with |ζ| > 30 mV are generally considered stable because the repulsive barrier is high enough to prevent aggregation at room temperature. The zeta potential can be measured by electrophoresis and is routinely used in pharmaceutical and materials quality control to predict shelf life.
Gravity pulls denser particles towards the bottom at a rate governed by Stokes' law: v = 2r²(ρ_p − ρ_f)g / (9η). For particles smaller than about 100 nm in water, Brownian motion resists sedimentation and the suspension remains kinetically stable for months. Larger particles or high-density materials sediment quickly. When gravity is enabled in the simulation, clusters settle at a rate that scales with their size, creating visible concentration gradients.
The CCC is the salt concentration at which the energy barrier exactly vanishes and rapid aggregation sets in. According to the Schulze–Hardy rule, the CCC scales with the inverse sixth power of counter-ion valence — so replacing monovalent Na⁺ with divalent Ca²⁺ can lower the CCC by a factor of 64. This rule explains why sea water (with Mg²⁺ and Ca²⁺) flocculates river sediment rapidly when the two meet at an estuary.
The fractal dimension d_f describes how cluster mass M scales with radius R: M ∝ R^(d_f). A compact disc has d_f = 2; a straight line has d_f = 1. DLCA clusters sit around d_f ≈ 1.78 in 2D, indicating a ramified, tenuous structure. As you increase salt and watch clusters grow, you can observe this value emerge from the statistics and verify that aggregation is proceeding in the diffusion-limited regime.
Virtually every liquid consumer product exploits colloidal science: ink-jet inks must not aggregate in the cartridge but must coalesce on paper; sunscreen suspensions of ZnO or TiO₂ nanoparticles must stay dispersed uniformly; injectable drug nanoparticles must survive blood plasma long enough to reach a tumour. Industrial processes such as water treatment deliberately trigger coagulation to remove suspended clay and organic matter before filtration.