Critical Depletion Force

Transcription

1 XCVI CONGRESSO NAZIONALE SOCIETA ITALIANA DI FISICA BOLOGNA settembre 2010 Critical Depletion Force Stefano BUZZACCARO Prof. Roberto PIAZZA Politecnico di Milano Prof. Alberto PAROLA Jader COLOMBO Università dell Insubria

2 Depletion Force 2 SMALL SPHERES CANNOT ENTER HERE! Osmotic pressure unbalancing yields an ATTRACTIVE force between colloids IF the depletant can be regarded as an IDEAL GAS Asakura-Oosawa potential: U eff = - ΠV excl

3 Depletion Force 2 Π Π Π Π Π SMALL SPHERES CANNOT ENTER HERE! Π Π Π Osmotic pressure unbalancing yields an ATTRACTIVE force between colloids IF the depletant can be regarded as an IDEAL GAS Asakura-Oosawa potential: U eff = - ΠV excl

4 Critical Casimir Force 3 CLASSICAL CASIMIR FORCE: Electromagnetic field fluctuactions (Van der Waals)

5 Critical Casimir Force 3 CLASSICAL CASIMIR FORCE: Electromagnetic field fluctuactions (Van der Waals)

6 Critical Casimir Force 3 CLASSICAL CASIMIR FORCE: Electromagnetic field fluctuactions (Van der Waals)

7 Critical Casimir Force 3 CLASSICAL CASIMIR FORCE: Electromagnetic field fluctuactions (Van der Waals)

8 Critical Casimir Force 3 CLASSICAL CASIMIR FORCE: Electromagnetic field fluctuactions (Van der Waals) CONSOLUTION CURVE (PHASE TRANSITION)

9 Critical Casimir Force 3 CLASSICAL CASIMIR FORCE: Electromagnetic field fluctuactions (Van der Waals) CONSOLUTION CURVE (PHASE TRANSITION)

10 Critical Casimir Force 3 CLASSICAL CASIMIR FORCE: Electromagnetic field fluctuactions (Van der Waals) CONSOLUTION CURVE (PHASE TRANSITION) CRITICAL CASIMIR FORCE: Force induced by the confinement of concentration fluctuations

11 Universal Features Of Short-range Potentials NORO-FRENKEL GENERALIZED LAW OF CORRESPONDING STATES: All short-ranged spherically symmetric attractive potentials are characterized by the same thermodynamics properties if compared at the same reduced density and virial coefficient: 4 B = 2 2π drr 1 2 ( V ( r) / k T ) B e Dispersion instability ( rapid sedimentation effects) corresponds to the same virial coefficient (B B 2 HS )

12 Universal Features Of Short-range Potentials NORO-FRENKEL GENERALIZED LAW OF CORRESPONDING STATES: All short-ranged spherically symmetric attractive potentials are characterized by the same thermodynamics properties if compared at the same reduced density and virial coefficient: 4 B = 2 2π drr 1 2 ( V ( r) / k T ) B e Dispersion instability ( rapid sedimentation effects) corresponds to the same virial coefficient (B B 2 HS )

13 Experimental system 5

14 Experimental system 5 Solvent Water

15 Experimental system 5 Solvent Water Sale 250 mm of NaCl

16 Experimental system 5 Solvent Water Sale 250 mm of NaCl Particles MFA Rc=90 nm

17 Experimental system 5 Solvent Water Depletion Agent C 12 E 8 Non Ionic; Tc=70 C Cc=1.8%w/w Sale 250 mm of NaCl Particles MFA Rc=90 nm

18 Experimental system 5 Solvent Water Depletion Agent C 12 E 8 Non Ionic; Tc=70 C Cc=1.8%w/w Sale 250 mm of NaCl Particles MFA Rc=90 nm

19 Experimental system 5 Solvent Water T Sale 70 C 250 mm of NaCl Particles MFA Rc=90 nm L-L coexistence Globular Micelles EXPERIMENTAL RANGE 2% C 12 E 8 concentration Depletion Agent C 12 E 8 Non Ionic; Tc=70 C Cc=1.8%w/w r 3.4 nm Aggregation number N 100

20 Φ critical vs T 6 70 C12E8/WATER COEXISTENCE GAP T ( C) volume fraction C 12 E 8

21 Φ critical vs T C12E8/WATER COEXISTENCE GAP PHASE SEPARATED T ( C) STABLE volume fraction C 12 E 8

22 Osmotic pressure C 12 E 8 /H 2 O Π TEMPERATURE c

23 Separation vs Osmotic Pressure c sep - c c (%) 2 1 c sep - c c = aπ 2/ Π (T sep, c sep )

24 Separation vs Osmotic Pressure c sep - c c (%) 2 1 c sep - c c = aπ 2/3 0.5 T - T c 4 C: Π reduced by a factor of 200! Π (T sep, c sep )

25 Separation vs correlation length 9 In critical phenomena the only relevant length is the correlation length ξ!

26 Separation vs correlation length 10 In critical phenomena the only relevant length is the correlation length ξ! 9 c sep - c c (%) 1 c sep - c crit = aξ λ ; λ ξ(c sep, T sep ) [nm]

27 Again ΠxV esc?! 10 c c sep c c c sep c Π ξ 2 / Π ξ 3 ξ 0.3 ( T T ) 0. 2 c

28 Again ΠxV esc?! c sep c c c c sep 0.10 c Π ξ 2 / Π ξ 3 ξ 0.3 ( T T ) 0. 2 c Πξ 3 /k B T ε x 10 2

29 A SIMPLE VIEW: A fluid of independent soft droplets with (average) size ξ (Droplet model of critical fluctuations, Oxtoby 1977)

30 Density Functional Theory (A. Parola& J. Colombo) 12 STRATEGY: Calculating the effective interaction potential between two colloidal particles by minimizing the grand-potential functional Ω[n(r)] of the host fluid evaluated for the non homogeneous density profile n(r) induced by the colloidal particles. In principle exact, provided that we have an expression for the functional: Α[n(r)] Helmholtz free energy functional [ n( r) ] = A[ n( r) ] + dr[ Φ( r) ] n( r) Ω µ Φ(r) Colloid-depletant interactions µ Chemical potential of the host fluid

31 Density Functional Theory (A. Parola& J. Colombo) 12 STRATEGY: Calculating the effective interaction potential between two colloidal particles by minimizing the grand-potential functional Ω[n(r)] of the host fluid evaluated for the non homogeneous density profile n(r) induced by the colloidal particles. In principle exact, provided that we have an expression for the functional: Α[n(r)] Helmholtz free energy functional [ n( r) ] = A[ n( r) ] + dr[ Φ( r) ] n( r) Ω µ Φ(r) Colloid-depletant interactions µ Chemical potential of the host fluid BASIC ASSUMPTIONS: 1) ξ «R Planar geometry Derjaguin approximation 2) A form for Α[n(r)] appropriate for describing long-wavelength fluctuations: (l.d.a.+ gradient corrections):

32 Main Results (I) 13 Continuity between depletion interactions and critical Casimir effect rigorously assessed. - Far from the critical point Asakura-Oosawa potential - Approaching T c Scaling form for the force: F( h) = k T B 3 h ϑ( x; y 1/ ν ) x δnε ν y hε β (Dietrich model)

33 Main Results (I) 13 Continuity between depletion interactions and critical Casimir effect rigorously assessed. - Far from the critical point Asakura-Oosawa potential - Approaching T c Scaling form for the force: F( h) = k T B 3 h ϑ( x; y 1/ ν ) x δnε ν y hε β (Dietrich model)

34 Main Results (II) 14 Note of caution: Whereas in the simplest lattice models (particlehole symmetry) δn has the usual meaning of reduced density n n c, in fluids one has field mixing, and δn is generally a linear combination of reduced density and temperature. (the correct line of approach to the critical point is not the critical isochore ).

35 Main Results (II) 14 Note of caution: Whereas in the simplest lattice models (particlehole symmetry) δn has the usual meaning of reduced density n n c, in fluids one has field mixing, and δn is generally a linear combination of reduced density and temperature. (the correct line of approach to the critical point is not the critical isochore ). However, both in lattice models and in fluids, the line δn = 0 can be identified with the line of maximum susceptibility: χ = µ n 1

36 Main Results (II) 14 Note of caution: Whereas in the simplest lattice models (particlehole symmetry) δn has the usual meaning of reduced density n n c, in fluids one has field mixing, and δn is generally a linear combination of reduced density and temperature. (the correct line of approach to the critical point is not the critical isochore ). However, both in lattice models and in fluids, the line δn = 0 can be identified with the line of maximum susceptibility: χ = µ n 1 mixtures c Π χ c MAX STRENGTH ALONG LINE OF MAX SCATTERING! 1 I s

37 Main Results (II) 14 Note of caution: Whereas in the simplest lattice models (particlehole symmetry) δn has the usual meaning of reduced 56.5 Cdensity n n c, 60.5 C in fluids one C has field mixing, and δn is generally a linear 50 C combination of reduced density and temperature. (the 45.6 C correct line 42.3 C of approach to the critical point is not the critical isochore ). I s (a.u.) 5 However, both in lattice models and in fluids, the TRANS. line δn POINTS = 0 can be identified with the line of maximum susceptibility: χ = µ n 1 mixtures c Π χ c 1 39 C 36.3 C 33.4 C 30 C 25 C 0 MAX STRENGTH ALONG 0.04 LINE 0.06 OF MAX 0.08 SCATTERING! 0.10 c I s

38 Main Results (III) 15 Along the path δn = 0, the product of the singular contribution to the pressure of the host fluid times the correlation volume is constant: 0.15 p 3 singξ k T B B 0.1 Πξ 3 /k B T ε x 10 2

39 Grazie per l attenzione!!!