Structure and phase behaviour of colloidal dispersions. Remco Tuinier

Transcription

1 Structure and phase behaviour of colloidal dispersions Remco Tuinier

2 Yesterday: Hard and adhesive colloidal dispersions Hard sphere dispersions can be described: Percus-Yevick closure relation semi-empirical e.o.s.: Carnahan-Starling Thermodynamic HS quantities measurable in hard sphere dispersions Attractions: sticky spheres Range of attraction is essential

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

4 Soft Matter Triangle

5 Practical example of colloidal dispersions: Latex Paint

6 Concentration of Latex Dispersions Adding 2g/L plant polysaccharides to latex suspension J. Traube, Gummi Z adding polysaccharides latex/polymer mixture phase separation after ~ 9 hours

7 Colloid-polymer mixtures; depletion Introduction; polymer depletion The FOS + Hard Sphere model Phase behaviour of colloidpolymer mixtures Effect of interacting polymers Biopolymer mixtures δ ξ ξ δ

8 Colloid-polymer mixtures; depletion Introduction; polymer depletion The FOS + Hard Sphere model Phase behaviour of colloidpolymer mixtures Effect of interacting polymers Biopolymer mixtures δ ξ ξ δ

9 Polymers at interfaces For polymer solutions near an interface significant change in the composition near this interface: (Positive) Adsorption = increase of the polymer segment density w.r.t. bulk concentration Depletion (Negative adsorption) = decrease of the polymer segment density w.r.t. bulk concentration

10 Polymer depletion ϕ (z ) ϕ b δ Density profile for polymer depletion at a flat wall. Continuous profile: solid curve Dashed lines: step profile width δ; depletion thickness z

11 Polymer concentration profile near flat plate Ideal polymers near a single hard plate: n(x) = n b f(x/r g ) f(x/r g ) calculated from diffusion equation by Eisenriegler (JCP 79 (1983) 1052) E. Eisenriegler depletion thickness δ: 2R g / π 1.13R g

12 ideal chains between two plates Asakura & Oosawa (JCP 22 (1954) 1255): force method: exact result for W depl (h)=f(h/a g ) h

13 Interaction between two flat plates Asakura & Oosawa, 1954

14 Otto von Guericke ( ) Stated that 8 horses would be unable to tear two vacuum copper Magdeburg hemispheres apart

15 The famous experiment In 1654 before Emperor Ferdinand III

16 How depletion works Osmotic pressure pushes spheres together No overlap of depletion layers Unbalanced osmotic pressure

17 How depletion works Osmotic pressure pushes spheres together No overlap of depletion layers AO 1954 Unbalanced osmotic pressure Von Guericke hemispheres -experiment 1654

18 How depletion works Volume restriction effects attraction through repulsion (A. Vrij)

19 Colloid-polymer mixtures; depletion Introduction; polymer depletion The FOS + Hard Sphere model Phase behaviour of colloidpolymer mixtures Effect of interacting polymers Biopolymer mixtures δ ξ ξ δ

20 Simple model: freely overlapping spheres (FOSs) Mixture of hard spheres (HS) + polymer chains mixture of HS + FOS Pair interactions: u u u HS HS FOS FOS HS FOS for r 2a = 0 for r > 2a = 0 for all r HS HS for r = a + a HS 0 > + HS FOS for r a a A. Vrij FOS

21 If at first the idea is not absurd, then there is no hope for it

22 If at first the idea is not absurd, then there is no hope for it A. Einstein

23 Interaction between two flat plates; FOS versus AO radius FOS (depletion layer thickness): 2R g / π 1.13R g

24 Depletion interaction between HS due to FOSs The presence of the FOSs induces an effective attraction between the hard spheres. Depletion potential W dep = Π V overlap ; the osmotic pressure of the FOSs: 3φ FOS Π = ktρ = kt FOS times the overlap 3 volume V overlap. 4π afos What is the overlap volume?

25 Overlap volume The overlap volume is the hatched region. = twice the slice (thickness H) volume of a sphere with radius R: π V H R H 3 2 = (3 ) shaded h R=a HS +a FOS V overlap H=a FOS h/2; R=a HS + a FOS 2 π 3a HS h h = a + + FOS 3 a 2a 2a FOS FOS FOS 2

26 Depletion interaction between HS due to FOSs The interaction potential, W dep (h)= Π V overlap (h), now follows as: 2 W () h 3 dep ahs h h = φ FOS kt 2a 4a 2a FOS FOS FOS W ( h= 0) dep,min 3a = φfos kt 2a HS FOS so, at φ FOS =0.5 and a HS /a FOS =4, W min = 3kT a FOS R g

27 Polymer density of ideal chains near a single hard sphere 2 2 nx () x R x R x = A + B 2 n + b R x x Rg x Rg where B(x)=flat wall result A( x) B( x) Takashi Taniguchi polymer density calculated from Edwards diffusion equation in spherical geometry Taniguchi, T., Kawakatsu, T., Kawasaki, K. (AIP series, 256 (1992) 503)

28 Effective depletion layer thickness around a single hard sphere δ 6 Rg Rg = R π R R δ R g So, δ=o(r g ) only for R g <<R!

29 Some results for depletion interaction 2 3a h h W () h φ for 0 h 2 a + a = kt 0 for h> 2( a + a ) HS FOS ( ) HS dep FOS HS FOS 2a 4a 2a FOS FOS FOS W ( h) dep u(h)/kt kt q=a FO S /a HS φ FO S =0.5; q=0.1 φ FO S =1; q=0.2 φ FO S =0.5; q=0.2 # Constant size ratio, larger φ FOS : stronger attraction # Constant φ FOS + larger FOSs: attraction longer-ranged h/2a HS

30 Depletion Interaction Entropy driving force attraction Range and strength can be adjusted by changing polymer size polymer concentration Tailor-made attractions between particles

31 Depletion-FOS model: comparison with experiment Data points: optical tweezers measurements : two silica spheres (a=625 nm) in DNA polymer solution (R g =500 nm) Curves are FOS model prediction using a FOS =R g DNA W ( h) dep kt φ φ FOS model accurate for a FOS <<a HS Close to and beyond φ FOS =1; polymer chains are non-ideal

32 Colloid-polymer mixtures; depletion Introduction; polymer depletion The FOS + Hard Sphere model Phase behaviour of colloidpolymer mixtures Effect of interacting polymers Biopolymer mixtures

33 Different types of phases in colloidal dispersions Demixed HEC/PS mixtures Colloidal gas Colloidal liquid Colloidal solid

34 Phase behaviour of colloid-polymer mixtures using the FOS model To predict the phase behaviour of hard sphere / FOS mixtures we use the osmotic equilibrium theory: simple and insightful Assumptions: system in equilibrium with reservoir consisting of FOSs and solvent system consists of three components: solvent, FOSs and HSs solvent considered as (continuous) background reservoir system

35 First proposed in 1990 by Henk Lekkerkerker Free volume theory

36 Free volume theory First proposed in 1990 by Henk Lekkerkerker Full theory by: H.N.W. Lekkerkerker W.C.K. Poon P.N. Pusey P.B. Warren & A. Stroobants 1992

37 Phase behavior ~ osmotic equilibrium theory free energy F= (, ) F N HS ( ) µ = FOS µ FOS F N N dµ 0 HS FOS FOS = free energy pure hard sphere dispersion + FOS (polymer) contribution reservoir system all at fixed T,V

38 Phase behavior use osmotic equilibrium theory to compute N FOS : N =ρ V R FOS FOS free 0 R = ρ αv FOS V free 0 the free volume fraction α can be found from Scaled Particle Theory α=f(φ c,a FOS /a HS ) ( ) F N HS ( ) F N,µ = FOS αvktρ R 0 HS FOS

39 Calculation phase behavior Normalized free energy=f (0) =F (0) v HS /ktv= j 3 =v HS /v FOS f yields pressure and chemical potential: f f φc # µ c =, P = f φ c c φc φr binodals: µ # c A µ # c B p + P ca P c B Crystal: f 0 =f crystal Fluid: f 0 =f fluid φ R p f = f j αφ 3 0 p A Liquid Gas Gas-Liquid: both phases: f 0 =f fluid reservoir system B

40 f ( a ) gas-liquid Phase behaviour from f 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

41 Recall: Fluid-Crystal Phase diagram No FOSs: fluid-crystal coexistence:

42 Predicted phase diagrams Phase diagram in the FOS reservoir φ c representation: j=0.1 j<0.3 j=0.4 j>0.3 Fluid-Crystal dominates Gas-Liquid dominates

43 Compare to atomic/molecular fluids with Lennard-Jones interaction: Lennard-Jones interaction: FOS concentration inversed temperature!

44 Compare to atomic/molecular fluids with Lennard-Jones interaction: Lennard-Jones fluid n=12: colloid-polymer mixture:

45 From reservoir to system: Using free volume fraction α=φ s FOS/φ r FOS FOS density in system, φ s FOS, follows. For larger size ratios phase diagram is richer with even a G+L+S three-phase region Left: j=0.1 Right: j=0.4

46 Osmotic equilibrium theory Lekkerkerker et al., 1992 experimental verification (Ilett, Poon & Pusey, PRE, 1995)

47 j=0.6: colloidal gas-liquid phase separation experiments by Ilett, Poon, Pusey Wilson Poon, Science 7 May 2004

48 Colloid-polymer mixtures; depletion Introduction; polymer depletion The FOS + Hard Sphere model Phase behaviour of colloidpolymer mixtures Effect of interacting polymers Biopolymer mixtures δ ξ ξ δ

49 Why and when does the FOS model fail? in order to describe interactions between colloids due to non-adsorbing polymer chains... for chains with a size ~ colloid size beyond very dilute concentrations: depletion thickness ~ correlation length decreases with c/c* osmotic pressure ideal Van t Hoff limit δ ξ ξ

50 General MF expression for depletion thickness Mean-Field equation + numerical lattice calculations (N=400): δ =δ +ξ 10 with 2 ξ = 3ln(1 ϕ ) + 2χϕ p p δ = 0 2 a π g δ δ χ= χ=0.5 Note: N=400, R g = (N/6) = 8.1, δ 0 = Fleer, Skvortsov, RT, Macromolecules 36 (2003) ϕ p

51 Effects of polymer concentration: Results Renormalization Group theory for corr length ξ and osmotic pressure Π Correlation length: Osmotic pressure: (Flory: c 9/ 4 ) Π ~ c g ξ/r 0.6 (Flory: c -3/4 ) ξ ~ c βπ v p c/c* c/c* see also L. Schäfer, Excluded Volume Effects in Polymer Solutions, Springer, 1999

52 Depletion interaction at small and large polymer concentration Dilute: δ R g and Π small Semi-dilute: δ ξ<r g and Π large ξ ξ Π Π

53 Interaction between two parallel flat plates in a solution with EVI polymers Comparison with Self-Avoiding Walk (SAW) chains MC simulations (Bolhuis et al., JCP 114 (2001) 4296) 0.0 n b 1 Π W(h) = dn b ' Γ(h) Γ( ) n' b n' 0 b theory: RT, Aarts, Wensink, Lekkerkerker, PCCP 2003 [ ] βw(h)/r g h/r g

54 Force between two spheres in polymer solution with fully interacting polymers Symbols: SAW computer simulations (Louis, Bolhuis, Meijer, Hansen, JCP 117 (2002) 1893) f W = h 3 q= q= βf(h)r g theory: RT, Aarts, Wensink, Lekkerkerker, PCCP 2003 φ p =0.58 φ p =1.16 φ p = βf(h)r g φ p =0.58 φ p =1.16 φ p = h/r g h/r g

55 Taking into account polymer excluded volume effect Data points are measurements using optical tweezers: two silica spheres (a=625 nm) in solution with DNA polymer (R g =500 nm) Curves are theoretical prediction using chains in the excluded volume limit W ( h) dep kt u dep (h)/kt h/r g

56 Taking into account polymer excluded volume effect Data points are measurements using optical tweezers: two silica spheres (a=625 nm) in solution with DNA polymer (R g =500 nm) Curves are theoretical prediction using chains in the excluded volume limit φ φ W ( h) dep kt u dep (h)/kt h/r g

57 Phase behavior use osmotic equilibrium theory to compute N FOS : N =ρ V R FOS FOS free 0 = ρ αv R FOS V free 0 the free volume fraction α can be found from Scaled Particle Theory Recall : α=f(φ,ξ/r) F( N, µ ) = HS pol dµ pol =(1/n b )dπ=(1/n b )[ Π/ n b ]dn b ( ) F N µ pol N dµ 0 HS pol pol

58 Calculation phase behavior FOS/HS Normalized free energy=f (0) =F (0) v HS /ktv= j 3 =v HS /v FOS f = f j αφ 3 0 p f yields pressure and chemical potential: # f f µ c =, P = f φ c c φ φ c R c φ p φ R p binodals: µ # c A µ # c B + P ca P c B A Crystal: f 0 =f crystal Fluid: f 0 =f fluid Gas Liquid Gas-Liquid: both phases: f 0 =f fluid B reservoir system

59 Calculation phase behavior IP/HS Normalized free energy=f (0) =F (0) v HS /ktv= j 3 φp =v HS /v FOS 3 βπ ' f = f j α dφ 0 ' 0 φp f yields pressure and chemical potential: # f f µ c =, P = f φ c c φ φ c R c R φ p φ p p binodals: µ # c A µ # c B + P ca P c B A Crystal: f 0 =f crystal Fluid: f 0 =f fluid Gas Liquid Gas-Liquid: both phases: f 0 =f fluid B reservoir system

60 Phase behavior ~ effect of EVI 0.5 Dashed: ideal polymers Full: EVI j=0.1 (j=r g /a) j=0.6 r φ p F F+C φ c r 0.8 φ p G+C V+L F F+C C C j=1.0 Aarts, RT, Lekkerkerker (J. Phys.: Cond. Matt. 14 (2002) 7551) r φ p G+C V+L φ c φ c F F+C C

61 Comparison with experiment: PDMS + full curve: EVI theory Dashed curve: FOS theory Ludox silica spheres Exp. Data: silica spheres: a=13 nm PDMS R g =14 nm; j=1.08 E.H.A. de Hoog, Lekkerkerker J. Phys. Chem. B 103 (1999) 5274

62 Colloid-polymer mixtures; depletion Introduction; polymer depletion The FOS + Hard Sphere model Phase behaviour of colloidpolymer mixtures Effect of interacting polymers Biopolymer mixtures δ ξ ξ δ

63 Relevance in food food: proteins + polysaccharides often demix food colloids jointly present in many food products control over physical stability is essential

64 Protein-polysaccharide mixture: depletion (segregative) interaction Whey-protein / polysaccharide mixture a polysaccharide-rich phase coexists with a protein-rich phase

65 Protein-polysaccharide mixture: segregative interaction Gelatin-dextran mixture 0-45 mins after demixing

66 Casein micelles in milk mixed with polymer? 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

67 Comparison with experiment: casein micelles + bacterial polysaccharide Casein micelles; ~assocation colloidal protein particles a=100 nm in milk mixed with EPS; R g = 86 nm (exocellular polysaccharide) Left: casein micelle dispersion with EPS Right: casein micelle dispersion

68 Comparison with experiment: casein micelles + bacterial polysaccharide Casein micelles; ~assocation colloidal protein particles a=100 nm in milk mixed with EPS; R g = 86 nm RT & de Kruif, J. Chem. Phys φ p experiment FOSs Dashed curve: FOSs j=r g /a HS =0.86 HS

69 φ p Comparison with experiment: casein micelles + bacterial polysaccharide polymer chains with full excluded volume Casein micelles; ~association colloidal protein particles a=100 nm in milk mixed with EPS; R g = 86 nm experiment FOSs full curve: osm. equil theory with full excluded volume chains Dashed curve: FOSs j=r g /a HS =0.86 HS

70 Protein crystallization crystallizing proteins = essential for X-ray analysis of proteins number of published crystal forms of proteins: until 1990 s: crystallization was an art since then: considered as scientific problem

71 Protein crystallization ~ using polymer traditionally: adding salts or organic solvent helped to promote crystallisation + improving quality of crystals after mid 1980s: polymer PEG became very popular

72 Can a solution of proteins be compared to a colloid dispersion? Phase behaviour of solution of lysozyme 3% NaCl: # Fluid-Solid coexistence # Hidden Liquid-Liquid phase separation Muschol & Rosenberger, JCP 1997

73 Can a solution of proteins be compared to a colloid dispersion? Phase behaviour of solution of γ crystallin (eye lens protein): T c =5 C # Fluid-Solid coexistence # Hidden Liquid-Liquid phase separation George Benedek and co-workers, 1990s

74 Protein vs colloid dispersion /molecular fluid Phase behaviour of solution Lennard Jones fluid n=18 of γ crystallin (eye lens protein)

75 How to improve conditions for protein crystallization? Adding non-adsorbing polymer chains! polymers help protein crystallisation Often PEG (polethylene glycol) Mechanism for crystallization remained unclear until 1990s Depletion mechanism..

76 Apoferritin crystallization using PEG Apoferritin: Fe-storage protein, radius = 8 nm Stable solution in aqueous salt With PEG: crystallization Molar mass PEG determines the quality of the crystals.

77 Apoferritin crystallization using PEG Results from S. Tanaka & M. Ataka, J. Chem. Phys a) Crystal b) Liquid domains c) Random aggregates

78 Colloid-Polymer Mixtures Summary: Mixing polymers and colloids may lead to phase separation due to depletion interaction Relevant in paint, food products + protein crystallization For size ratios j < 0.5: FOS model works Main trends for depletion interaction + phase behaviour in polymer-protein mixtures: hard spheres + FOSs model Otherwise more involved theories are required

79 Interested in doing a Ph.D. study? Soft condensed matter / Weiche Materie group of Prof. Jan K.G. Dhont

80 Acknowledgement Henk Lekkerkerker*, Utrecht, NL Agienus Vrij*, Utrecht, NL Dirk Aarts, Utrecht, NL Gerard J. Fleer & Martien Cohen Stuart, Wageningen, NL Nel Zoon, Ede, NL Kees de Kruif*, Ede, NL Takashi Taniguchi, Yamagata, Japan Matthias Schmidt, Duesseldorf, Germany Gerrit Vliegenthart, Juelich, Germany Jan Dhont*, Juelich, Germany

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