🌡️ Sol-Gel Phase Transition

A polymer sol transitions to a gel as cross-link density p exceeds the percolation threshold p_c. Near p_c, the gel modulus G ~ (p - p_c)^t with universal exponent t ≈ 1.9.

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Left: Percolation lattice (connected = gel) · Right: G vs p curve with gel point · Adjust p slider

How it Works

This simulation models bond percolation on a 2D square lattice as an analogy for the sol-gel transition. Each bond between neighboring nodes represents a potential cross-link. At cross-link probability p, bonds are randomly placed. Below the percolation threshold p_c ≈ 0.5 (2D square lattice), only finite clusters exist (the sol). Above p_c, a spanning infinite cluster forms — the gel network.

The gel modulus grows from zero at p_c following the power law G = G₀·(p - p_c)^t, where t ≈ 1.9 in 3D (the displayed value for physical systems). The correlation length ξ ~ |p - p_c|^(-ν) diverges at p_c, meaning clusters become fractal at all length scales.

Gel modulus: G = G₀·(p − p_c)^t [for p > p_c]
Gel fraction: S∞ ~ (p − p_c)^β, β ≈ 0.41 (3D)
Correlation length: ξ ~ |p − p_c|^(−ν), ν ≈ 0.88
2D percolation threshold p_c ≈ 0.5 (bond, square lattice)

Frequently Asked Questions

What is a sol-gel transition?

A sol-gel transition (gelation) occurs when cross-links between polymer chains build up to a point where an infinite spanning network forms. The system transitions from a viscous sol (liquid-like) to an elastic gel (solid-like) at the gel point p_c.

What is percolation theory?

Percolation theory describes the formation of connected clusters on a lattice or in continuous space. At the percolation threshold p_c, the first spanning cluster appears. Near p_c, cluster size distribution follows power laws and the gel modulus scales as G ~ (p - p_c)^t.

What is the gel point?

The gel point is the cross-link density p_c at which the first continuous network spanning the entire sample forms. At the gel point, viscosity diverges and storage modulus G' first becomes non-zero. It is identified experimentally by the Winter-Chambon criterion.

What is the critical exponent t?

The critical exponent t governs how gel modulus grows above the gel point: G ~ (p - p_c)^t. In 3D, the universal value is t ≈ 1.9. This exponent is the same for many physical systems, reflecting universality in percolation theory.

What is the difference between chemical and physical gels?

Chemical gels have permanent covalent cross-links that cannot be reversed by heating. Physical gels form through non-covalent interactions (hydrogen bonds, hydrophobic associations) that are reversible with temperature.

What are fractal clusters near the gel point?

Near the gel point, polymer clusters have a fractal structure with fractal dimension d_f ≈ 2.5 in 3D. Their mass scales as M ~ R^d_f. This means clusters are self-similar across many length scales.

How does viscosity behave near the gel point?

Below p_c, viscosity diverges as η ~ (p_c - p)^(-s) with s ≈ 0.7 in 3D. Above p_c, the equilibrium modulus G_e grows as G ~ (p - p_c)^t. Exactly at p_c, G' and G'' follow power-law frequency dependence with the same exponent.

What is the Winter-Chambon criterion?

The Winter-Chambon criterion identifies the gel point by the condition that G' and G'' have identical frequency dependence: G' ∝ G'' ∝ ωⁿ at all frequencies. The loss tangent tan δ = G''/G' is independent of frequency at the gel point.

What is the sol-gel process in ceramics?

In ceramic processing, metal alkoxides (e.g., TEOS) undergo hydrolysis and condensation to form a colloidal sol that gels into a wet gel. Drying and sintering convert the gel to a dense ceramic, enabling fine microstructure control for optical glasses, membranes, and coatings.

What applications use sol-gel transitions?

Sol-gel transitions underlie: food gels (gelatin, agar), biopolymer hydrogels for drug delivery, silica sol-gel for optical coatings, polyacrylamide gels for electrophoresis, cement hydration, and thermoreversible polymer gels for biomedical scaffolds.

About this simulation

This simulator turns bond percolation on a 2D lattice into a live model of gelation. As you drag the cross-link probability slider, bonds snap into place at random, and once enough of them connect, a single cluster spans the entire grid — the moment a liquid sol becomes an elastic gel. The right-hand chart tracks the same event thermodynamically: gel modulus G stays pinned at zero until p crosses the percolation threshold p_c, then rises as a power law G = G₀·(p − p_c)^t, with the critical exponent t fixed near its universal 3D value of 1.9.

🔬 What it shows

Two synchronized views of the same transition: a bond-percolation lattice where connected clusters are color-coded (the spanning cluster glows amber once it forms), and a G vs p curve that stays flat at zero below p_c and bends upward as G₀·(p−p_c)^t above it.

🎮 How to use

Drag the p slider to add or remove cross-links, or move p_c and the exponent t to see how the gel point and steepness of the modulus curve shift. The G₀ prefactor slider rescales the modulus in pascals, and the Gel system dropdown loads presets for polyacrylamide, silica sol-gel, gelatin, and epoxy resin, each with its own p, p_c, t, and G₀.

💡 Did you know?

The critical exponent t ≈ 1.9 is universal — it shows up in gelling polymers, random resistor networks, and other systems that have nothing in common except crossing a percolation threshold, a striking example of universality in statistical physics.

Frequently asked questions

What does the cross-link probability p represent here?

p is the fraction of bonds between neighboring lattice sites that are randomly "switched on" as cross-links, ranging from 0 (no connections, pure sol) to 0.9 in the slider. It's the model's stand-in for how densely a real polymer sol has been chemically cross-linked.

Why does nothing happen until p passes p_c?

Below the percolation threshold p_c, only small, disconnected clusters of bonded sites exist — the lattice equivalent of finite polymer clusters floating in a viscous sol. There is no path spanning the grid, so the gel modulus G stays at exactly zero no matter how large g0 or t are set.

What does the power law G = G₀·(p − p_c)^t actually mean?

Once p exceeds p_c, the newly formed spanning cluster starts to bear mechanical stress, and its rigidity grows steeply: doubling the distance (p − p_c) multiplies the modulus by 2^t, roughly 3.7 times when t ≈ 1.9. That is why the curve on the right looks flat for a while near p_c and then curves upward sharply.

Why is the critical exponent t set near 1.9 by default?

t ≈ 1.9 is the accepted 3D universality-class value for how gel modulus scales near the percolation threshold, found in scaling theory and confirmed experimentally across chemically unrelated gelling systems. The simulator lets you drag t from 1.0 to 3.0 so you can see how a different exponent changes the shape of the G vs p curve.

What do the four gel system presets change?

Each preset (polyacrylamide, silica sol-gel, gelatin, epoxy resin) loads a different combination of p, p_c, exponent t, and prefactor G₀ that roughly reflects how that real material gels — for example gelatin uses a lower t (1.5) typical of physically cross-linked networks, while silica uses a higher t (2.1) and a larger G₀ reflecting its much stiffer covalent network.