Dispersion systems. Dispersion system = dispersed phase in a continuum phase (medium) s/l, l/l,... According to the size of the dispersed phase:

Transcription

1 Dispersion systems 1/20 Dispersion system = dispersed phase in a continuum phase (medium) s/l, l/l,... According to the size of the dispersed phase: coarse dispersion (suspension), > 1 µm colloid 1 µm 1 nm heterogeneous (micelles, precipitates,... ) homogeneous (solutions of macromolecules) solutions Examples: polymer solutions, asphalt concrete, starch, milk, fresh precipitate...

2 Properties of dispersions 2/20 Coarse dispersions: turbidity (look as a haze), milk-like size λ: white/gray (if particles are not colored) size λ: Tyndall phenomenon, blue is scattered more (red Sun in dust) size λ: Rayleigh scattering, blue is scattered more elastic scattered photon energy does not change Colligative properties measurable only in fine dispersions Brownian motion, diffusivity decreases as the size increases (D = k B T/6πηR) Viscosity bigger than the fluid medium, often non-newton plasticity = a minimum stress needed to flow, viscosity decreases with the shear strain dilatancy = viscosity increases with flow speed/shear strain (starch + water) Density in between both phases Surface tension often lower credit: weeks/squishy/

3 Dispersion systems 3/20 credit: wikipedia

4 Classification 4/20 Shape of particles: globular, isometric particles (r x r y r z ) laminar, anisometric particles (r x r y r z ) fibrillar, anisometric particles (r x r y r z ) Interactions: lyophilic dispersions (medium wets the particles, θ < 90 ) in water: hydrophilic lyophobic dispersions (does not wet) in water: hydrophobic

5 Preparation of dispersions 5/20 polymerization oversaturation of a solution (ouzo, anise drink) oversaturation of a micellar colloid (over CMC = critical micellar concentration) mechanically (grinding, ultrasound) electrically (arch, cathodic sputtering) precipitation reactions insoluble product (AgBr in a photographic emulsion ) Microcrystals of a precipitate may be aggregated (flocculated), because the el. double layer is thin in a concentrated ionic solution (according to the DLVO theory), rinsing out the ions stabilizes he colloid (peptization). Aggregation caused by weak forces: free disperged particles flocculation peptization weakly bound aggregates

6 Distribution functions [show/fraktaly.sh] 6/20 E.g., mass (differential) distribution function F w (m): ratio (prob = count/(total number)) of particles of masses in interval (m, m + dm) is F w (m)dm. Normalization: 0 F w (m)dm = 1 Cumulative (integrated) distrib. function = ratio of particles of masses < m: I w (m) = m 0 F w (m )dm, Q w (m) = Similar: Distribution function of particle volumes,... m F w (m )dm = 1 I w (m) Monodisperse system all particles of the same size (peak on F w (m)); may form crystals Spheres: fcc 74%, random close packing 64% Polydisperse systems: e.g. asphalt concrete (example of random fractal) try to guess the fraction of mineral filling Fraction = group of particles of approx. the same size (filtering,... )

7 Sedimentation Potential of external force (e.g., gravity) = U External force: F = grad U Forces of gravity (acceleration = g): 7/20 asphalt concrete: 95 % mineral filling 5 % bitumen F = mg, U = mhg Forces in a centrifuge: F = mrω 2, U = 1 2 m(rω)2, ω = 2πν = 2π τ ω = angular (circular) frequency ν = frequency (in Hz or RPM, 1 RPM = 1 60 Hz) R = radius of rotation τ = period

8 Sedimentation velocity 8/20 Density of particles = ρ 1 Dispersion medium density = ρ Particle volume = V 1 Friction coefficient = f Sedimentation velocity = v In the field of gravity: The force must be corrected for the buoyant force (Archimedes principle): F = V 1 (ρ 1 ρ)g, v = F f = V 1(ρ 1 ρ)g f Spherical particles: V 1 = 4 3 πr3, f = 6πηr (Stokes) v = 2r2 9η (ρ 1 ρ)g In a centrifuge: use Rω 2 instead of g (typical 1000g 10, 000g, ultracentrifuge up to 10 6 g) Small particles sediment slowly. Molecules sediment, too (albeit very slowly): uranium enrichment by centrifugation of UF 6 (g)

9 Sedimentation equilibrium 9/20 From Boltzmann probability Ideal solution: concentration Boltzmann probability ( ) U( r) c( r) = c 0 exp k B T In a gravitational field this is the barometric formula: ( ) V1 (ρ c(h) = c(0) exp 1 ρ)gh k B T In a centrifuge of angular frequency ω = 2πν: ( 12 V 1 (ρ 1 ρ)(rω) 2 ) c(r) = c(0) exp k B T

10 Sedimentation equilibrium 10/20 From the speeds of sedimentation and diffusion v sedimentation = F f = U f v diffusion = J c = D c c = D ln c = k BT f ln c v sedimentation + v diffusion = 0 c = c 0 exp ( U k B T Let µ be per particle (not mole), infinite dilution approximation: Thus is equivalent to µ = µ 0 + k B T ln(c/c st ) v diffusion = 1 f µ v sedimentation + v diffusion = 0 U + µ = const Some time ago, we used the assumption of U + µ = const to derive D = k B T/f. )

11 Examples [xcat centrifuge.evu] 11/20 Example. The equilibrium concentration of monodisperse oil droplets in a cuvette 10 cm tall is twice as large near the surface than at the bottom. Calculate the diameter of oil droplets. The temperature is 25 C, the density of water is g cm 3, the density of oil is g cm nm Example. Consider a globular protein of molecular weight of 20 kda. What is the speed of sedimentation in a centrifuge of RPM at a point 5 cm from the axis? The density of the protein is 1.35 g cm 3, the viscosity of water is mpa s. V1 = m 3, r = 1.8 nm, v = 0.32 mm h 1, a = 32200g

12 Stability of dispersions Dispersions are thermodynamically metastable (large interface) sedimentation ( ), creaming ( ) flocculation (reversible), coagulation (irreversible) coalescence (of droplets) Ostwald ripening (small large, Kelvin equation) Stabilization: 12/20 by electric double layer DLVO theory (Derjaguin, Landau, Verwey, Overbeek) steric (adsorption of macromolecules in a good solvent) depletion (macromolecules in between particles) credit: electrosteric kinetic (in a very viscous medium)

13 DLVO theory 13/20 Drjagin (Der gin, Derjaguin, Deryagin) + Landau (Landau), Verwey + Overbeek. repulsion of charged surfaces (screened by a Gouy Chapman (diffusion) layer) stabilizes a colloid attractive dispersion (London) forces tries to stick particles together Stability is a result of a competition between both forces surface charge increases stability dispersion forces decrease stability

14 DLVO theory: Electrostatic repulsion Diffusion (Gouy Chapman) layer, 1:1 electrolyte, small φe/k B T: 14/20 Surface charge = layer charge σ = 0 For small φ(x)e k B T : φ = φ 0 e x/λ, where λ = (ρ + ρ )dx = σ 0 0 cf ɛrt 2cF 2 { [ exp φ(x)e ] exp k B T 2cF φ(x)e k B T dx = 2λcFφ e 0 k B T = ɛ λ φ 0 [ ]} φ(x)e dx k B T ɛ/λ = capacity of the Gouy Chapman double layer (per unit area) Energy (per unit area) of the surface charge σ in potential φ 0 is (in the superposition approximation, that s why multiplied by 2) E elst = 2σφ 0 e d/λ = 2 λσ2 ɛ e d/λ = 2 ɛφ2 0 λ e d/λ Formulas for curved interfaces are more complex, however, the leading term is always e d/λ

15 van der Waals forces 15/20 For 2 molecules r apart (r overlap of orbitals), the energy decays as 1/r 6 : charge charge: u 1/r u(r) = C r 6 dipole charge: u 1/r 2 (fixed dipole orientation) dipole dipole: u 1/r 3 Freely rotating dipole freely rotating dipole: u 1/r 6 Dipole induced dipole: u 1/r 6 London (dispersion) force (fluctuating dipole fluctuation dipole): fluctuation dipole el.field 1/r 3 induced dipole 1/r 3 u 1/r 6 Usually the most pronounced London force for r 1µm: u 1/r 7.

16 DLVO theory: van der Waals forces 16/20 u(r) = C r 6 Contributions from atom pairs are considered as independent (valid to about %) For two bodies we make a sum (integral); e.g., for a slit: Point surface (half-space) first: Then a column of area da: u wall (d) = NC 2π 12d 3 N = N A n/v = number density E vdw = πn2 C 12d 2 = A 12πd 2 (per unit area) A = (πn) 2 C = Hamaker constant of given substance, [A] = J Two R-balls d apart, first term of the expansion in d R: E vdw = AR 12πd

17 DLVO: colloid in a medium Cross interaction between both media, 17/20 u 12 (r) = C 12 r 6, A 12 = π 2 N 1 N 2 C 12 Particles of 2 in material 1 (0 = particles far away) Approximation (combining rule): A 1/2 = A 11 2A 12 + A 22 Then A 12 A 11 A 22 ( A 1/2 = A11 ) 2 A 22

18 DLVO: case study [plot/dlvo.sh] 18/20 TiO 2 in a of NaCl. A(TiO 2 ) = J, A(H 2 O) = J, A J E / (kt/100 nm 2 ) flat interface formulas energy expressed in k B T per 100 nm 2 figure: φ 0 = 0.15 V, c = 0.1 mol dm x/nm dispersion (London) elst. repulsion total energy k B T = thermal quantum barrier 60 k B T/100 nm 2 e E/k BT = e 60 = cubes of side 10 nm will be stable Rule of the thumb: stable for barrier > 25k B T e 25 = 10 11

19 DLVO: case study instability halftime 19/20 Diffusivity of a spherical particle of diameter 2R = 10 nm (Stokes formula): D i = k BT 6πηR i = J K K 6π m 1 kg s m = m 2 s 1 If the particles are about r = R apart, the typical collision time is τ r2 6D = ( m) m 2 s 1 = s For the probability of barrier crossing of π = e 25 = 10 11, the typical timescale of flocculation is: τ π s hours

20 DLVO: summary 20/20 surfaces close together (at contact): attraction (adhesion) surfaces far away: fast decreasing attraction medium distances (according to the potential): energy barrier The barrier grows = stability increases for: potential (surface charge) increases (in abs. value) salt concentration decreases (longer Debye screening range)