Sol-Gel Percolation

What is modelled

Monomers in a liquid join into branched clusters as cross-links form. At low cross-link probability all clusters are small and the sample flows; at a critical probability one cluster reaches across the whole sample and the sample can carry load: a gel. The sketch uses a square lattice where every bond is made independently with probability p. A fixed random number per bond means raising p only adds bonds, as in real curing.

Equations

• Square-lattice bond percolation threshold pc = 1/2 (exact, by duality).
• Gel fraction near the threshold P ∝ (p − pc)β, β = 5/36 in two dimensions.
• Modulus stand-in: the sample is a random resistor network, voltage 1 on the top bus and 0 on the bottom bus, Kirchhoff Σ(Vi − Vj) = 0 solved by over-relaxation; relative conductance g = G/Gfull with Gfull = N/(N − 1).
• Near the threshold g ∝ (p − pc)t with t ≈ 1.3 in two dimensions (about 2.0 in three, as for gels).

Checks run in the page

• Union-find largest cluster against a flood fill.
• Fully linked lattice conductance equals 1 and layer currents agree (Kirchhoff).
• Spanning frequency at p = 1/2 on 300 random networks.
• Largest cluster never shrinks with p and the fraction of bonds made matches p.

Honest simplifications

A conductance network is used as a stand-in for the elastic modulus: real gels need bending and central-force rigidity, which has a different exponent. The lattice is small, so the transition is rounded and one network gives a noisy curve; the slope line is the textbook exponent drawn for comparison, not a fit. The two-dimensional exponents differ from the three-dimensional ones of the 3D page.

• Press Cure from sol to gel: the status changes from SOL to GEL close to p = 0.5.
• Press At the gel point and change the random network: the gel appears slightly earlier or later in each sample.
• Increase the lattice size to see the transition sharpen.