§ C-34 Colloidal Systems, Micelle Structure, Tyndall Cone & Schulze–Hardy Coagulation (Grade 9–11 · Colloid & Interface Science)
Chapter XXXIV. Rayleigh Light Scattering (I ∝ d⁶/λ⁴), Zeta Potential ζ & Coagulation by Multivalent Ions (z⁶). «Between the realm of true molecular solutions and coarse suspensions lies the world of neglected dimensions, the colloidal state.», Wolfgang Ostwald, 1914
Theory Summary & Scientific Mechanism
Colloidal sols contain dispersed nanoparticles or micelles ranging from 1 to 100 nm in diameter. Unlike true solutions (< 1 nm), colloids scatter an incident light beam into a visible Tyndall cone via Rayleigh scattering (Iₛcat ∝ N d⁶ / λ⁴).
Colloidal stability is governed by DLVO theory: micelles remain dispersed as long as electrostatic repulsion between their electrical double layers (measured by the zeta potential ζ) outweighs short-range van der Waals attraction.
Adding an electrolyte compresses the double layer and triggers coagulation (flocculation). According to the Schulze–Hardy rule, the critical coagulation concentration (CCC) drops inversely with the sixth power of the counter-ion charge: CCC ∝ z⁻⁶ (Na⁺ : Ca²⁺ : Al³⁺ = 1 : 64⁻¹ : 729⁻¹).
Everyday Analogy
A laser beam is invisible from the side in pure water, but in a colloidal sol it glows like a lightsaber, the Tyndall cone! To make those charged micelles clump together, a trivalent Al³⁺ ion is 3⁶ = 729 times more potent than Na⁺!
Interactive Question & Derivation
How can a colloidal sol (particle size 1–100 nm) be distinguished optically from a true solution, and why is Al³⁺ (z=3) hundreds of times more effective at coagulating a negatively charged sol than Na⁺ (z=1)?
- A: Colloidal micelles scatter a laser beam into a visible Tyndall cone (Iₛ ∝ d⁶ / λ⁴); by the Schulze–Hardy and DLVO rules, the critical coagulation concentration scales inversely with the 6th power of counter-ion charge (CCC ∝ z⁻⁶, so 1 : 3⁻⁶ = 1 : 729)
- B: True molecular solutions scatter light much more strongly than colloids, and counter-ion charge z has no effect on coagulation
- C: Adding electrolyte increases the Zeta potential ζ to infinity and prevents micelles from ever colliding
Study Hint: Colloidal micelles scatter light into a Tyndall cone (I ∝ d⁶/λ⁴), and critical coagulation concentration scales as 1/z⁶ (Schulze–Hardy rule).
Proof & Explanation: Colloidal micelles (1–100 nm, such as gold sol or Fe(OH)₃ sol) are stabilized against van der Waals attraction by electrostatic repulsion of their electrical double layers, characterized by the zeta potential ζ. When a laser beam passes through the sol, particles scatter light with Rayleigh intensity Iₛcat ∝ (N d⁶)/(λ⁴), forming a glowing Tyndall cone. Adding an electrolyte compresses the diffuse double layer; by the Schulze–Hardy rule (derived from DLVO theory), the critical coagulation concentration drops as CCC ∝ (1)/(z⁶) (Na⁺ : Ca²⁺ : Al³⁺ ≈ 1 : 64 : 729).
Key Equations & 3D Simulation Parameters
I_Tyndall ∝ I₀ (N d⁶)/(λ⁴), ζ = (σ κ⁻¹)/(εε₀), CCC ∝ (1)/(z⁶) (Schulze–Hardy)
- Colloidal micelle core diameter d: 2 … 80 nm (default: 35 nm)
- Coagulating electrolyte concentration C: 0 … 15 mmol/L (default: 2 mmol/L)
- Coagulating counter-ion charge z (1=Na⁺, 2=Ca²⁺, 3=Al³⁺): 1 … 3 (default: 1)
- Laser beam wavelength λ: 405 … 650 nm (default: 532 nm)