Structure and phase behaviour of colloidal dispersions. Remco Tuinier

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1 Structure and phase behaviour of colloidal dispersions Remco Tuinier

Yesterday: Phase behaviour of fluids and colloidal

the base understanding fluids Van der Waals theory:

fluid-solid transition Colloidal dispersions similar

2 Yesterday: Phase behaviour of fluids and colloidal dispersions Colloids are everywhere Hard sphere fluid at the base understanding fluids Van der Waals theory: gas-liquid condensation Larger particle concentrations: fluid-solid transition Colloidal dispersions similar behaviour as molecular fluids Colloidal interactions can be tailored

3 Structure and phase behaviour of colloidal dispersions Three parts: Remco Tuinier Phase behaviour of fluids/colloidal dispersions Hard and adhesive colloidal dispersions Colloid-polymer mixtures; depletion

4 central theme: Van t Hoff Laboratory synthesizing colloids physical experimentation (scattering, microscopy) theoretical interpretation

5 Van t Hoff Laboratory Utrecht University, The Netherlands Where is Utrecht?

6 Van t Hoff Laboratory Utrecht University, The Netherlands Utrecht

7 Hard and adhesive colloidal dispersions Introduction; pair interactions Hard sphere gas Hard sphere dispersion Adhesive spheres Adhesive sphere phase behaviour

8 Hard and adhesive colloidal dispersions Introduction; pair interactions Hard sphere gas Hard sphere phase behaviour Adhesive spheres Adhesive sphere phase behaviour O

9 potential O O O O distance Hard spheres (good solvent) Sticky spheres (bad solvent) Soft repulsive Charged spheres

10 Pair interactions in colloidal dispersions Colloids in pure solvent: ω ω ω ( ) ( ) ( ) r = 0< r < 2a r = repulsive/or attractive 2a< r < 2a+ d r = 0 r > 2a+ d

11 Pair interactions in colloidal dispersions Colloids in pure solvent: ω ω ω ( ) ( ) ( ) r = 0< r < 2a r = repulsive/or attractive 2a< r < 2a+ d r = 0 r > 2a+ d Several cases, d<<2a if stable: ω 11 ( r) for 0 < r < σ = 0for r > σ Hard Sphere pair interaction 1 Pair-wise additive? 2 ( a ) ( b )

12 Hard and adhesive colloidal dispersions Introduction; pair interactions Hard sphere gas Hard sphere phase behaviour Adhesive spheres Adhesive sphere phase behaviour

13 The Hard Sphere model prelude Virial expansion of the hard sphere gas: p kt = ρ+ B ρ + ρ B Low density: p Van t Hoff s law = ρ kt

14 J.H. Van t Hoff

The Hard Sphere model Virial expansion of the hard sphere gas: p kt = ρ+ B ρ + ρ + 2 3 B.

15 The Hard Sphere model Virial expansion of the hard sphere gas: p kt = ρ+ B ρ + ρ B Second osmotic virial coefficient B 2 : ω ( r ) HS 2 B = 2π r 1 d = 2 e r 0 Excluded volume: four times the volume of the HS 2πσ 3 HS 3 kt

16 The Hard Sphere model Virial expansion of the hard sphere gas: p kt = ρ+ B ρ + ρ B Higher order virial coefficients: calculation becomes involved for B 5 and higher...

17 The Hard Sphere model Virial expansion of the hard sphere gas: p kt = ρ+ B ρ + ρ B J.E. Mayer 1930s: recognized B n+1 =(n 2 +3n)[V HS ] n Carnahan-Starling (1969): p ρkt 2 n = 1 + (n + 3n)ϕ n=1 p ρkt = ϕ+ ϕ ϕ ( 1 ϕ ) 3

18 Structure of a hard sphere fluid : Integral theory Low concentrations: gr () = exp( ω direct ()/ r kt)

19 Structure of a hard sphere fluid : Integral theory Total correlation function: hr () = gr () 1

20 Structure of a hard sphere fluid : Integral theory Propagation of indirect interactions: h( r r ) = h ( r r ) + d r ρ( r ) h ( r r ) h ( r r )

21 Structure of a hard sphere fluid : Integral theory Propagation of indirect interactions: h( r r ) = h ( r r ) + d r ρ( r ) h ( r r ) h ( r r ) Higher order in concentration connections: h( r r ) = h ( r r ) d r ρ( r ) h ( r r ) h( r r ) h( r r ) = c( r r ) d r ρ( r ) c( r r ) h( r r ) Ornstein-Zernike equation

OZE h( r r ) = c( r r ) + 1 2 1 2 d r ρ( r ) c( r r ) h( r r ) 3 3 1 3 3 2 Integral Theory: h(r)=c(r)+ indirect interactions Ornstein-Zernike

22 OZE h( r r ) = c( r r ) d r ρ( r ) c( r r ) h( r r ) Integral Theory: h(r)=c(r)+ indirect interactions Ornstein-Zernike equation requires closure relations F. Zernike Very useful closure relation for short-ranged attractions by Percus and Yevick (PY): c(r)=g(r)(1 exp[ω(r)/kt])

23 The Hard Sphere model; pressure of a hard sphere gas Percus-Yevick (compr): 2 p 1+ ϕ+ ϕ = 3 acc up to ~0.4 ρkt ( 1 ϕ ) Percus-Yevick (virial): 2 p 1+ 2ϕ+ 3ϕ = 2 acc up to ~0.25 ρkt ( 1 ϕ ) Carnahan-Starling: 2 3 p 1+ ϕ+ ϕ ϕ = acc up to ~ ρkt 1 ϕ ( )

24 Hard Sphere Gas Percus-Yevick versus Carnahan -Starling On the level of the pressure 12 p ρkt 1 10 CS PY c PY v ϕ

25 The Hard Sphere model; pressure of a hard sphere gas 2 p 1+ ϕ+ ϕ Percus-Yevick (compr): = 3 acc up to ~0.4 ρkt 1 ϕ ( ) 2 3 Carnahan-Starling: p 1+ ϕ+ ϕ ϕ = acc up to ~ ρkt 1 ϕ ( )

26 The Hard Sphere model; pressure of a hard sphere gas 2 Π 1+ ϕ+ ϕ Percus-Yevick (compr): = 3 acc up to ~0.4 ρkt 1 ϕ ( ) 2 3 Carnahan-Starling: Π 1+ ϕ+ ϕ ϕ = acc up to ~ ρkt 1 ϕ ( ) Pressure hard sphere gas = Osmotic pressure hard sphere dispersion

27 Osmotic pressure Osmotic equilibrium: allows to fix µ s p 1, 2, 3 2, 3 p R V Colloidal dispersion: solvent 3, added component 2 + colloid 1 Π = p p R

28 Hard and adhesive colloidal dispersions Introduction; pair interactions Hard sphere gas Hard sphere dispersion Adhesive sphere gas Adhesive sphere phase behaviour

29 On the osmotic pressure Low density: p = Π = ρ kt Van t Hoff s law Pressure is hard to measure ρ = N/ V is small However, osmotic compressibility is of order kt Π = kt ρ

30 Colloidal Dispersions Osmotic compressibility is measurable 1 kt Π ρ Compressibility theorem: ( ) 1 2 µ ' = 1+ ρ 4 πrhrdr ( ) 0 hr () = gr () 1 Structure factor: 2 sin qr Sq ( ) = 1+ ρ 4 πrhr ( ) dr 0 qr Structure factor at q=0 (θ 0 or λ ): Π = = Sq= kt ( 0) 2 1+ ρ 4 πrhrdr ( ) ρ µ ' 0 follows from fluctuation theory

31 Hard Sphere Dispersion Osmotic compressibility is measurable Structure factor at q=0: Π = = Sq ( = 0) 2 1 ρ 4 π ( ) ρ + rhrdr kt µ ' Scattering (light, X-rays or neutrons) of a dispersion gives I(q)= ρ f 2 S(q) P(q) Long wavelength limit qσ 0: I(q=0)= ρ f 2 S(q=0)= ρ f 2 kt ρ / Π 0

32 Experiments with Hard Sphere-like Dispersions 1970s, 1980s A. Vrij, C.G. de Kruif and co-workers, Utrecht, The Netherlands: Silica spheres anchored with short C 18 -tail in CHX R. H. Ottewill and co-workers, Bristol, UK: Polystyrene latex spheres The School of Chemistry

33 Experiments with Hard Sphere Dispersions A. Vrij, C.G. de Kruif and co-workers, Utrecht, 1970s/1980s: Silica spheres anchored with short C 18 -tail in CHX pseudo hard-sphere repulsion A. Vrij

34 Experiments with pseudo Hard Sphere Dispersions Silica spheres with a=22 nm ρ 15 Van Helden & Vrij, 1980 Full curve: Percus-Yevick result Π /kt 10 2 Π 1+ ϕ+ ϕ = 3 ρkt ( 1 ϕ ) ( 1+ 2ϕ ) 4 ( 1 ϕ ) Π = kt ρ 2 5 Data points: osmotic compressibility at two wavelengths ϕ 0.4

35 ϕ (1 ϕ) 4 (1+2 ϕ) Hard Sphere Dispersion Percus-Yevick results Scattered intensity: I(q)= ρ f 2 S(q) P(q) Long wavelength limit qσ 0: I(q=0)= ρ f 2 S(q=0)= ρ f 2 kt ρ / Π ~ ϕ(1 ϕ) 4 /(1+2ϕ) 2 I(q) goes through a maximum as a function of ϕ ϕ 0.4

36 Experiments with pseudo Hard Sphere Dispersions Micro-emulsion droplets σ=7.5 nm, Agterof & Vrij, 1976 Full curve: Carnahan-Starling e.o.s. 2 3 Π 1+ ϕ+ ϕ ϕ = 3 ρkt ( 1 ϕ ) Data points: I(q) as a function of droplet concentration I ( θ = 90 o ) c/g cm 3

37 Light Scattering Maximum in a Hard Sphere Dispersion

38 Hard Sphere Dispersions On the Structure Percus-Yevick results g(r) g(r) at ϕ= g (r) Full curve: PY 4 3 Data points: Monte Carlo simulation results r/ σ

39 g(r) is measurable using Confocal Scanning Laser Microscopy Measurement on colloidal crystal of PMMA-PHS spheres with fluorescent silica core at 60 vol% hr () = gr () 1 by Kegel & Van Blaaderen Science 287 (2000) 290.

40 Hard Sphere Dispersion Percus-Yevick results Advantage Percus-Yevick: yields analytical expression for c(r) Important quantity of colloidal dispersions: osmotic compressibility/ pressure Measurement: using scattering Ornstein-Zernike: S(q) 1 =1 ρ c(q) S (q ) S(q=0) 1 = Π /kt ρ 10, 20 and 30% qσ

41 Polydisperse Hard Sphere Dispersions Silica spheres; usually somewhat polydisperse S (q ) Percus-Yevick: can be extended for polydisperse HS 0.5 ( a ) Full curve: PY polydisperse q σ 1.5 Dashed: PY mono S (q ) 1 Data points: MC simulations Frenkel, Vos, de Kruif & Vrij ϕ = 0.30 (a) s = 0.1 (b) s = ( b ) q σ

42 Experiments with Silica Dispersions Concentrated silica dispersions: how to circumvent multiple scattering? Small-angle neutron scattering SANS, Grenoble, France North South

Experiments with Silica Dispersions Silica spheres with a=22.5 + 1.8 nm Data points: SANS measurements Grenoble, France C.G. de Kruif, A. Vrij et al.

43 Experiments with Silica Dispersions Silica spheres with a= nm Data points: SANS measurements Grenoble, France C.G. de Kruif, A. Vrij et al., 1980s S (q ) ϕ = 0.52 S (q ) Curves: polydisperse PY q /nm q /nm 1

44 Let us suppose a cow is a sphere...

45 Casein micelles in milk: hard spheres! Milk: Colloidal dispersion containing: Casein micelles (radius: 100 nm) ~13 vol% including: various types of caseins + CaP caseins: tend to form micelles schematic picture casein micelle

46 Casein micelle: electron microscopy McMahon & McManus, 1998

47 Casein micelle

48 Casein micelles as hard spheres Using dynamic light scattering D= one can probe the diffusion of particles 0 kt 6πηa dilute limit small q : collective diffusion D =1+1.45φ D 0 large q : self-diffusion D =1-1.83φ D 0

49 Casein micelles as hard spheres Dynamic light scattering results on casein micelle dispersions Data points: results for measured diffusion casein micelles Lines: low φ-predictions for a collection of hard spheres D/D collective diffusion D =1+1.45φ D self-diffusion D =1-1.83φ D φ 0 C.G. de Kruif, Langmuir, 1992

50 Can we theoretically predict the fluid-solid transition of hard spheres? Computer simulations: Alder & Wainwright (1957) Wood & Jacobson (1957) Theoretical insight: Kirkwood (1957) Fluid F + C Crystal volume fraction

51 p and µ of an FCC Hard Sphere Crystal Hard sphere pressure p HS ρkt = 3 1 ϕ/ ϕ cp Hall, JCP (1972) From pressure to chemical potential; thermodynamic integration of p: p F HS HS Yields chemical potential: µ HS F HS = N VT, µ HS 3 const + 3ln ϕ = + kt 1 ϕ/ ϕ 1 ϕ/ ϕ cp Computer simulation: const=2.13 (Frenkel & Ladd, 1984) cp

p and µ of a hard Sphere Fluid Hard sphere pressure CS p HS ρkt = 2 3 1+

3 kt ( 1 ϕ ) Solving p f = p c and µ f = µ c : from analytical

52 p and µ of a hard Sphere Fluid Hard sphere pressure CS p HS ρkt = ϕ+ ϕ ϕ ( 1 ϕ ) 3 Yields chemical potential: 2 µ ( 8 9ϕ+ 3ϕ HS ) = lnϕ+ ϕ 3 kt ( 1 ϕ ) Solving p f = p c and µ f = µ c : from analytical expressions: ϕ f =0.492 and ϕ c =0.542 simulation results: ϕ f =0.494 and ϕ c =0.545

53 Hard and adhesive colloidal dispersions Introduction; pair interactions Hard sphere gas Hard sphere phase behaviour Adhesive spheres Adhesive sphere phase behaviour

54 Pair interactions in colloidal dispersions Colloids in pure solvent: ω ( ) 11 0< r < 2a r = repulsive/or attractive 2a< r < 2a+ d 0 r > 2a+ d Short-ranged attractions? Still d<<2a ( r sw ) 0 < r < σ ω = ε σ < r σ + 0 r > σ + Square Well Adhesive Hard Sphere pair interaction Still pair-wise additive...

55 Adhesive Hard Spheres 0 < r < σ ωsw ( r) = ε σ < r σ + 0 r > σ + (r)/ kt ω Square Well Adhesive Hard Sphere pair interaction 0 Sticky spheres r/ σ

56 Adhesive Spheres Virial expansion: p kt = ρ+ B ρ + ρ B Second osmotic virial coefficient B 2 : 0 < r < σ ωsw ( r) = ε σ < r σ + 0 r > σ + ( r) ω 2 B = 2π r 1 d 2 e kt r 0 2 B 3 e kt ε sw 3 2 = πσ 12 2 πσ

57 Adhesive Spheres For small densities the structure factor S(q) can be computed; here ϕ=0.05; /σ=0.1 Some trends: S (q) 1.0 ε/kt = 3 Recall: S(q=0) 1 = Π /kt ρ q σ

Baxter proposed: If <<σ: 0 Adhesive Spheres Baxter s model ω AHS kt ( r) Second osmotic virial coefficient B 2 : parameter τ fully describes the properties of the

58 Baxter proposed: If <<σ: 0 Adhesive Spheres Baxter s model ω AHS kt ( r) Second osmotic virial coefficient B 2 : parameter τ fully describes the properties of the AHS interaction + closed analytical expressions in PY approximation! 0 < r < σ 12τ = ln σ r σ + σ + 0 r > σ + V B 3 ( π ) σ = /6 HS 1 = V 4 τ AHS 2 HS R.J. Baxter

59 PY solution of Baxter s AHS model for ϕ = 0.32 & τ = 0.2 full curve PY/Baxter Adhesive Spheres Baxter s model S (q) MC Computer simulations: Kranendonk & Frenkel, q σ Daan Frenkel

60 Experimental Adhesive Sphere Dispersions A. Vrij & C.G. de Kruif: Silica spheres with C 18 brushes in benzene or dodecane : sticky spheres stickiness τ depends on T Why? Solvency of chains changes with T 0 < r < σ ωsw ( r) = ε σ < r σ + 0 r > σ + (r)/ kt ω r/ σ

61 Experimental Adhesive Sphere Dispersions A. Vrij & C.G. de Kruif: Silica spheres with C 18 brushes in benzene or dodecane : sticky spheres stickiness τ depends on T Why? Solvency of chains changes with T Θ L 1 Θ ε kt T = T 0 T > Θ (r)/ kt ω 0 following Flory-Krigbaum, 1950 P.J. Flory r/ σ

62 Experiments with Adhesive Sphere Dispersions Light scattering at low volume fractions; yields S(q) At small q: S(q=0) 1 = Π /kt ρ =1+2B 2 ρ +... B 2 /V HS Triangles: dodecane Squares: benzene T B =T where B 2 = T/T B

63 Experiments with Adhesive Sphere Dispersions S (q) 1 S (q) 1 Higher concentrations: SANS by M.H.G. Duits et al. 0.5 ( a ) 0.5 ( b ) q /nm 1 q /nm 1 S (q) ( c ) q /nm 1 Curves: Baxter/PY solution Data points: SANS results ϕ = 0.14, 0.19 & 0.28 T = 51.6 (a), 40.8 (b) and (c) 35ºC

64 Experimental Adhesive Sphere Dispersions Protein dispersion: Hen egg white lysozyme, measured using light scattering ρ 1 Π / kt ( a ) ( b ) 0.4 τ ϕ T/ o K R. Piazza et al., 1998

65 Hard and adhesive colloidal dispersions Introduction; pair interactions Hard sphere gas Hard sphere phase behaviour Adhesive sphere gas Adhesive sphere phase behaviour

66 The Dutch School

Phase behaviour of adhesive colloids; gelation in sticky dispersions * When becomes too small, the dispersion becomes unstable * Verduin & Dhont, 1995: Silica

68 Phase behaviour of adhesive colloids; gelation in sticky dispersions * When becomes too small, the dispersion becomes unstable * Verduin & Dhont, 1995: Silica spheres in benzene 19 T/ o C stable cp * Above cp: gelation occurs Dynamically arrested (meta-) stable x x unstable gel x Jan Dhont ϕ

69 Phase behaviour of adhesive colloids; gelation in sticky dispersions Below a certain T: gelation occurs Percolation

70 Phase behaviour of adhesive colloids; gelation in sticky dispersions Theory and computer simulation (from Miller and Frenkel, 2003) 0.14 τ D x xxx x x xx x x x A xxxxxxxxxxx B C ρσ A: PY-c G/L, B: simulation G/L, C: PY-v G/L, D: percolation PY, circles: percolation from simulation

71 Phase behaviour of colloidal dispersions What happens when the range of attraction changes?

72 Phase behaviour of colloidal dispersions Lennard-Jones Lennard-Jones-like pair interaction 2 σ σ () 4ε n V r = r r n=6, 12 and 18 n n affects range of interaction (N, V, T) simulations of many-body LJ particles by Vliegenthart, Lodge & Lekkerkerker, 1999

73 Calculation of binodal points Stable equilibrium; equal µ plus equal p G µ = µ G L and p = G p L L f f µ =, p= f ϕ ϕ ϕ

74 Phase behaviour of colloidal dispersions free energy functions f f µ =, p= f ϕ ϕ ϕ f ( a ) f ( b ) α β ϕ α β ϕ f ( c ) f ( d ) α βγ δ α β ϕ ϕ

75 Phase behaviour LJ fluid n=6 6

76 Phase behaviour LJ fluid n=12 12

77 Phase behaviour LJ fluid n=18 18

78 Phase behaviour of LJ dispersions kt/ ε V/ ε 2.0 ( a ) 2.0 ( b ) ε 0.6 G+S r/σ ρσ intermediate range ( c ) 1.0 ( d ) F F+S S kt/ kt/ε G G+L F F long-range L F+S L+S F+S T t S S short-range ρσ 3 ρσ 3

79 Lennard-Jones pair interaction Various pair interactions in Attractive Yukawa pair interaction DLVO attraction part colloidal dispersions 2 σ σ () 4ε n n V r = r r ; r < σ V() r = σ ε exp [ κσ(1 r/ σ) ]; r σ r V() r = A 2a 2a r 4a + + ln r a r r Phase diagram topology depends only on range of attraction Shape V(r) is hardly relevant Vliegenthart & Lekkerkerker, J. Chem. Phys. 112 (2000) 5364

80 colloidal gas-liquid demixing Phase behaviour of colloidal dispersions

81 Structure and phase behaviour of colloidal dispersions Three parts: Phase behaviour of fluids/colloidal dispersions Hard and adhesive colloidal dispersions Colloid-polymer mixtures; depletion

82 Hard and adhesive colloidal dispersions Summary: Hard sphere fluid: can be treated exact up to the first orders on the virial level Higher concentrations: closure relation: Percus-Yevick semi-empirical e.o.s.: Carnahan-Starling Thermodynamic HS quantities measurable Fluid-Crystal transition at ~50 vol% Simplest way describing attractions: sticky spheres Also using Percus-Yevick Gelation in concentrated sticky sphere dispersions Range of attraction is essential

84 Why R&D?

Important practical questions:

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?