Colloidal stability
Important practical questions: - Does dispersion change when T, P or... is changed? - If T, P or... is changed and the dispersion phase separates, what are then the final products? - How fast does the phase separation take place? Examples: - Food processing - Protein crystal growing - Rain makers - Cataract (grauer Star) - Synthesis of monodisperse colloidal particles - Percipitation of CaCO 3
Dutch example: Cheese Depletion of Wey proteins by Extracellular Poly-Saccharides ρ(wey proteins) low ρ(wey proteins) high
Phase separation: System with concentration ρ 0 decomposes in to a phase with a higher concentration ρ and a phase with a lower concentration ρ - Two types of phase separation, two types of kinetics : Liquid-solid or Gas-Solid Gas-liquid
Molecular systems hard to study: - Phase separation is too fast - Particles are too small to be observed with a microscope - No control on/knowledge of the interaction potential Use Colloids: - Mesoscopic Brownian particles dispersed in solvent instead of molecules dispersed in vacuum consider Π instead of P. Advantages: - Potential can be tuned by choice of particles and solvent - Dynamics is slow - Particles are detectable by microscopy and light scattering
Simple estimation of spinodal When B becomes sufficiently negative demixing takes place ( Π v / kt) c c φ = 1 ( B / v ) φ... Estimation spinodal: φ spinodal v = c B c HS B ( ϕ ) /B 5 0 15 10 5 0 0 0.5 1 1.5 ϕ
Colloid Dispersion with mediated by depletion forces q=a FOS /R a FOS R
Second virial coefficient B can be computed from depletion potential: ( r) ω B = π r 1 d e kt r 0 for r < σ ω ( r) = Π V () r for σ < r < σ δ p overlap 0 for r > σ δ σ δ Π V () r p ov B = 4V π r 1 exp d r HS kt σ
Demixing concentrations estimated from B De Hek and Vrij, 1980: 1 wt% HS-like silica a=46 nm mixed with PS in CHX ( a ) ( b ) 4 c PS (%) 3 1 0 4 5 6 10 log (M/g mol 1 )
Mao, Cates, Lekkerkerker JCP 97 Sphere Rod Mixtures R = 3 μm c TRIS = 1. mm κ -1 = 1.4 nm - - - - - - - - - - - - - - - - - - - - - φ(h)/k B T 1 10 8 6 4 0 - -4 ( h) Δφ Geff h φdepl = Bexp{ κh} kt kt kt B B B -6 0 00 400 600 800 1000 h / nm 41 fn B=1400 k B T κ -1 = 13. nm L=880 nm φ 3 depl N A h = π c RL 1 kt B 3 M fd L
Example: stearyl alcohol-coated silica in cyclohexane Tunable attraction, hard core repulsion R V T V r sphere r sphere r R V Potential of colloidal particles similar to non-ideal gas van der Waals-like behavior of osmotic pressure Π.
Unstable region Temperature quench Π Unstable region: < 0 Π ρ Π or > 0 v Π v Connection to free energy:π = A V N V/N = v = 1 ρ d ( A/ N) Unstable region: < 0 d v A/N v
Phase separation? Minimize Free Energy A/N A 0 A A - A/N A 0 A A - 1/c 1/ c0 1/c v 1/c 1/ c 1/c 0 v A 0 >A A - A 0 <A A - d ( A/ N) < d v Concave, : YES 0 Convex, d ( A/ N) d v > 0 : NO
Spinodal A/N Π Π V/N d ( A dv / N ) = 0 v
Meta-stable region A/N A A -- A 0 1 / c 1/ c0 1/c A A -- <A 0 Lower free energy for big fluctuations But small probability
Unstable: Spinodal decomposition Meta-stable: Nucleation and growth A/N A/N v - Small fluctuations grow - Instantaneous start of phase separation - Phase separation starts throughout the sample v - Only big fluctuations grow - Induction time before phase separation - Phase separation only local
Where does the phase separtion end up?
Concentrations of lowest energy: d ( A / N ) d ( A / N ) Π( c ) = ( ) dv end = Π c dend v C Equality of chemical potential end F / C end N = Π ρμ A/N A 0 A end A -end μ 1 / c end 1/ c 1/c 0 end The line of minimum energy We found the binodal points (which depend on T or )
Spinodal and Binodal A/N Critical point Π Π( c end ) = Π( c end ) Π V/N d ( A dv / N ) = 0 v
Small density fluctuation δρ Attractive potential between particles
gas-liquid Lennard-Jones interaction: f ( a ) polymer concentration inversed temperature! α β ϕ
Simulation of the spinodal decomposition followed by domain coarsening End states after demixing Cates and Wagner, Nov. 000
Attractive potential between particles Small density fluctuation δρ Hard sphere
Fluid-solid f ( b ) fluid-solid α β ϕ No attraction, so: 1) very simple phase diagram. ) use equation of state for fluid and FCC crystal
Nucleation and growth Colloidal hard spheres P. N. Pusey Glass (see Zorn) heterogeneous homogeneous
Shallow quench Small density fluctuation δρ N x N1 Big density fluctuation δρ nucleation
Very deep quench nucleation Δt
Nucleation and growth Concentrated hard spheres, initial stage: nuclei dissolve Gasser et al, Science 9, April 001
Nucleation and growth Full picture: Gasser et al, Science 9, April 001
Hard spheres freely overlapping spheres Depletion potential: W dep (r) = Π V overlap (r) larger chains longer range of W range. more chains depth, more attractive W depth at c=c*, a HS /a FOS =5/3, W min =.5 kt
f Phase behaviour from f gas-liquid ( a ) f ( b ) fluid-solid α β ϕ α β ϕ f close ( c ) to a triple point f Metastable ( d ) gas-liquid fluid-solid α βγ δ α β ϕ ϕ
Compare to atomic/molecular fluids with Lennard-Jones interaction: Lennard-Jones interaction: polymer concentration inversed temperature!
Compare to atomic/molecular fluids with Lennard-Jones interaction: Lennard-Jones fluid n=1: polymer mixture: colloid-
Is dv ρ 1 true? Π = ( V ct V red ) α( N / V )