This is a 2D cross-section through the same well-mixed suspension: a slice plane through the container shows micelle cores as circles and drug molecules as dots undergoing a genuine 2D random walk (not a projection of 3D motion — the step statistics, capture geometry and packing are computed natively in the plane). Alongside it, a live parameter diagram plots the Langmuir binding isotherm itself — solubilized fraction versus log P — so you can see the whole equilibrium curve, not just today's slider value.
P = C_core / C_water, log P shown by the slider
θ = P·[D_free] / (1 + P·[D_free])
S = N_micelles × capacity (total core sites)
Solve for free drug f: P·f² + (1 + P·S − N·P)·f − N = 0
Each frame, every free drug molecule performs a clamped 2D random walk (Brownian motion, reflecting off the slice boundary). When it drifts within capture range of a micelle whose core is not full, it binds with probability 1 − e−k_on·dt, where k_on scales with 10log P; a bound molecule can also escape back into the water with probability 1 − e−k_off·dt. The isotherm panel on the right solves the same quadratic equilibrium for every log P from −1 to 3 (holding the current micelle concentration and capacity fixed) and traces it as a curve, with a marker at the current log P showing both the theoretical prediction and the running simulated value converging onto it.
- Partition coefficient (log P) — how strongly the drug prefers the hydrophobic core over water.
- Micelle concentration — number of micelles in the cross-section (above the critical micelle concentration, more micelles means more available core sites).
- Core loading capacity — how many drug molecules a single micelle core can physically hold before it saturates.
- Diffusion rate — how fast drug molecules random-walk through the aqueous phase (a proxy for temperature).
Real-world relevance: this is the mechanism used to formulate poorly water-soluble drugs (paclitaxel, cyclosporine, many oncology payloads) inside amphiphilic block-copolymer or surfactant micelles for intravenous delivery.