Physical gel: Particles/polymers connected via reversible, non-covalent bonds forming a percolating network
Key features:
Bonds break/reform dynamically
Energy of a few to tens of k_BT
Self-healing possible
Thermoreversible
Examples:
Gelatin
Agarose
Yoghurt
Paint, inks
Cross-links form via dense local regions (microcrystalline/glassy) or specific junctions (helices in gelatin)
Physical vs Chemical Gels
Physical Gels
Reversible bonds
Energy of a few to tens of k_BT
Dynamic network
Thermoreversible
Chemical Gels
Irreversible covalent bonds
High energy barriers
Permanent network
Key transition:
Fluid → Gel occurs when a system-spanning cluster forms (percolation) and its bonds live long enough to make it permanent
Dramatic viscosity increase + elastic response
Gels are a second example of disordered solids (after glasses)
Percolation Theory
Problem: Square grid L\times L sites, randomly label N of them. When does a spanning cluster form?
p = N/L^2 = occupation probability
Percolation thresholdp_c
Below p_c: small clusters
Above p_c: giant spanning cluster
2D square lattice:p_c \approx 0.5927
Transition: Continuous (2nd order)
Percolation Simulation
Percolation Transition
Percolation Threshold: p_c Values
Non-universal: p_c depends on lattice type and dimension
Lattice
Dim
z
p_c^{site}
p_c^{bond}
Square
2
4
0.593
0.5
Triangular
2
6
0.5
0.347
Cubic
3
6
0.312
0.249
Hypercubic
4
8
0.197
0.160
Hypercubic
6
12
0.109
0.094
Higher dimensions → lower p_c (easier to percolate)
Critical Exponents: Universal
Universal: Critical exponents independent of lattice details
P_{\infty}(p)\propto (p-p_c)^{\beta}
Dimension d
\beta
2
5/36 ≈ 0.14
3
≈ 0.41
≥6
1 (mean-field)
Gels require percolation for mechanical stability!
Colloidal Gel Formation
MD simulation of dilute colloidal gel formation (largest cluster in red)
Percolating cluster → mechanical rigidity
Colloid-Polymer Mixtures
Depletion interactions like the Asakura-Oosawa model produce a very short ranged potential with strong attractive part.
W_{\mathrm{AO}}(r)=-\frac{4 \pi \rho_p k_B T}{3} R^3(1+q)^3\left[1-\frac{3}{4} \frac{r}{R(1+q)}+\frac{1}{16}\left(\frac{r}{R(1+q)}\right)^3\right], \quad 2R<r<2(R+R_p), \quad q=\dfrac{R_p}{R}
attractive + very short ranged → sticky interaction
Colloids cluster into nonequilibrium branched phase
Binodal becomes metastable to fluid-solid coexistence
Phase diagram of colloid-polymer mixtures for varying polymer-colloid size ratio q: small q makes the gas-liquid binodal metastable.
Simple Model Potentials
Square-well potential:
U(r) =
\begin{cases}
\infty & r < \sigma \\
-\epsilon & \sigma \leq r < \lambda \sigma \\
0 & r \geq \lambda \sigma
\end{cases}
The picture suggests the progressive freezing of coarsening upon spinodal decomposition: the gel keeps the spinodal length scale (a low-q peak in S(q)).
Colloid-polymer gel forming by arrested spinodal decomposition (video: Peter Lu; Lu et al., Nature453, 499 (2008))
Phase Diagram: Arrested Spinodal
Arrested spinodal scenario, see Zaccarelli et al 2007
Key lines: - Binodal (red): liquid-gas coexistence at equilibrium - Spinodal (dashed): limit of stability of the metastable phases. Instability (i.e. spinodal decomposition) starts inside - Percolation (purple): At some density, the clusters percolate (span the system) - Glass transition (blue): Dense phase becomes dynamically arrested (non-ergodic)
Gel path: deep quench → arrested dense phase + vapor
Summary: gels
Topic
Key result
Physical gels
reversible bonds (a few to tens of k_BT) forming a percolating network
