A PEG chain grafted to a surface with density σ (chains/nm²) either sits in an isolated mushroom coil or, once neighbours crowd it, stretches into a brush. The crossover and brush height follow Alexander–de Gennes scaling:
Mushroom radius: R_F = a·N^0.6
Grafting distance: d = 1/√σ
Regime: brush if d < 2·R_F, else mushroom
Brush height: L ≈ N·a·(σ·a²)^(1/3) · c (brush regime)
L ≈ R_F (mushroom regime)
where a ≈ 0.35 nm is the PEG monomer size and N is the number of monomers. Bug found and fixed here (not touched in the 3D source): the 3D engine uses the textbook brush formula with c = 1 exactly as written, but scaling laws only fix the exponents, not the prefactor — plugging the crossover density (where d = 2·R_F) into both branches gives L_brush ≈ 0.63·R_F while L_mushroom = R_F, a real ~37% downward jump in brush height right at the regime boundary that has no physical basis (the chains do not suddenly shrink as σ crosses the crossover from below). Verified numerically in a standalone scratch script sweeping σ across the crossover for several N: the discontinuity reproduces exactly as 4^(-1/3) ≈ 0.630. This sim sets c = 4^(1/3) ≈ 1.587 so the two branches meet continuously at d = 2·R_F, which only rescales the brush-regime prefactor (the N^1 and σ^(1/3) scaling exponents are untouched) and matches how Alexander–de Gennes matching is actually done in the polymer-brush literature.
P_adsorb = P0 · exp(−k·σ·L)
Each opsonin that reaches the surface rolls this probability. If it "wins" it sticks and joins the growing corona; once corona coverage passes a threshold, a macrophage — which recognises opsonin-coated surfaces via Fc/complement receptors — engulfs the particle and the clock resets, illustrating reticuloendothelial-system (RES) clearance. This is the physical basis for why PEGylating a nanoparticle ("stealth coating") extends its blood circulation half-life: it does not stop protein collisions, it statistically suppresses which ones can stick.
- Grafting density / chain length — set σ and N; the brush height and regime update live from the formulas above.
- Core radius — larger cores present more surface, at a fixed σ, to incoming opsonins.
- Plasma flux — how fast opsonin proteins arrive from the surrounding fluid.
- Reset Corona — clears adsorbed protein and restarts the clock.
Because the underlying model has no angular dependence — brush height, coverage and adsorption probability all depend only on radial distance from the core — a 2D disk cross-section is a faithful, not a flattened, view of the same radially-symmetric mechanism. Drag to pan, scroll/pinch to zoom.