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XCVI CONGRESSO NAZIONALE SOCIETA’ ITALIANA DI FISICA BOLOGNA 20-24 settembre 2010 “Critical Depletion Force”

Stefano BUZZACCARO Prof. Roberto PIAZZA Politecnico di Milano

Prof. Alberto PAROLA Jader COLOMBO Università dell’Insubria Depletion Force 2

SMALL SPHERES CANNOT ENTER HERE!

Osmotic pressure unbalancing yields an ATTRACTIVE force between

IF the depletant can be regarded as an IDEAL GAS

Asakura-Oosawa potential: Ueff = - ΠVexcl

Bologna, 21/09/2010 Stefano Buzzaccaro Depletion Force 2

SMALL SPHERES Π Π Π CANNOT ENTER HERE! Π Π unbalancing Π Π yields an ATTRACTIVE Π force between colloids

IF the depletant can be regarded as an IDEAL GAS

Asakura-Oosawa potential: Ueff = - ΠVexcl

Bologna, 21/09/2010 Stefano Buzzaccaro Critical Casimir Force 3

CLASSICAL CASIMIR FORCE: Electromagnetic field fluctuactions (Van der Waals)

Bologna, 21/09/2010 Stefano Buzzaccaro Critical Casimir Force 3

CLASSICAL CASIMIR FORCE: Electromagnetic field fluctuactions (Van der Waals)

Bologna, 21/09/2010 Stefano Buzzaccaro Critical Casimir Force 3

CLASSICAL CASIMIR FORCE: Electromagnetic field fluctuactions (Van der Waals)

Bologna, 21/09/2010 Stefano Buzzaccaro Critical Casimir Force 3

CLASSICAL CASIMIR FORCE: Electromagnetic field fluctuactions (Van der Waals)

Bologna, 21/09/2010 Stefano Buzzaccaro Critical Casimir Force 3

CLASSICAL CASIMIR FORCE: CONSOLUTION CURVE Electromagnetic field fluctuactions () (Van der Waals)

Bologna, 21/09/2010 Stefano Buzzaccaro Critical Casimir Force 3

CLASSICAL CASIMIR FORCE: CONSOLUTION CURVE Electromagnetic field fluctuactions (PHASE TRANSITION) (Van der Waals)

Bologna, 21/09/2010 Stefano Buzzaccaro Critical Casimir Force 3

CLASSICAL CASIMIR FORCE: CONSOLUTION CURVE Electromagnetic field fluctuactions (PHASE TRANSITION) (Van der Waals)

CRITICAL CASIMIR FORCE: Force induced by the confinement of concentration fluctuations

Bologna, 21/09/2010 Stefano Buzzaccaro Universal Features 4 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:

2 −V (r /) kBT B2 = 2π ∫ drr (1− e ) instability ( “rapid sedimentation” effects ) corresponds to the same virial coefficient (B2 ≈ - 1.3 HS B2 )

Bologna, 21/09/2010 Stefano Buzzaccaro Universal Features 4 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:

2 −V (r /) kBT B2 = 2π ∫ drr (1− e ) Dispersion instability ( “rapid sedimentation” effects ) corresponds to the same virial coefficient (B2 ≈ - 1.3 HS B2 )

Bologna, 21/09/2010 Stefano Buzzaccaro Experimental system 5

Bologna, 21/09/2010 Stefano Buzzaccaro Experimental system 5

Solvent Water

Bologna, 21/09/2010 Stefano Buzzaccaro Experimental system 5

Solvent Water

Sale 250 mM of NaCl

Bologna, 21/09/2010 Stefano Buzzaccaro Experimental system 5

Solvent Water

Sale 250 mM of NaCl

Particles MFA Rc=90 nm

Bologna, 21/09/2010 Stefano Buzzaccaro Experimental system 5

Depletion Agent C12 E8 Non Ionic ; Solvent Tc=70 °C Water Cc=1.8%w/w

Sale 250 mM of NaCl

Particles MFA Rc=90 nm

Bologna, 21/09/2010 Stefano Buzzaccaro Experimental system 5

Depletion Agent C12 E8 Non Ionic ; Solvent Tc=70 °C Water Cc=1.8%w/w

Sale 250 mM of NaCl

Particles MFA Rc=90 nm

Bologna, 21/09/2010 Stefano Buzzaccaro Experimental system 5

Depletion Agent C12 E8 Non Ionic ; Solvent Tc=70 °C Water Cc=1.8%w/w

T L-L coexistence Sale ≈250 70°C mM of NaCl Globular Particles MFA Rc=90 nm EXPERIMENTAL RANGE r ≈3.4 nm ≈ 2% Aggregation number C E concentration 12 8 N ≈ 100

Bologna, 21/09/2010 Stefano Buzzaccaro Φcritical vs T 6

70 C12E8/WATER COEXISTENCE GAP

60

50

C) 40 °

T ( T 30

20

10

0 2 4 6 8 10

volume fraction C 12E8

Bologna, 21/09/2010 Stefano Buzzaccaro Φcritical vs T 6

70 C12E8/WATER COEXISTENCE GAP

60 PHASE SEPARATED 50

C) 40 °

T ( T 30

20 STABLE

10

0 2 4 6 8 10

volume fraction C 12E8

Bologna, 21/09/2010 Stefano Buzzaccaro Osmotic pressure C12 E8/H2O 7

2.0

1.5 TEMPERATURE

Π 1.0

0.5 25 33.4 39 45.6 53 60.5 0 0 0.02 0.04 0.06 0.08 0.10 c

Bologna, 21/09/2010 Stefano Buzzaccaro Separation vs Osmotic 8 Pressure 10

5

2/3 (%) c - c = a Π c 2 sep c - c - sep c 1

0.5

0.01 0.1 1

Π (T sep, csep)

Bologna, 21/09/2010 Stefano Buzzaccaro Separation vs Osmotic 8 Pressure 10

5

2/3 (%) c - c = a Π c 2 sep c - c - sep c 1

0.5 T - Tc ≈ 4°C: Π reduced by a factor of 200! 0.01 0.1 1

Π (T sep, csep)

Bologna, 21/09/2010 Stefano Buzzaccaro Separation vs correlation 9 length

In the only relevant length is the correlation length ξ!

Bologna, 21/09/2010 Stefano Buzzaccaro Separation vs correlation 9 length 10 In critical phenomena the only relevant length is the correlation length ξ! (%) c 1 - c - sep c −λ csep - ccrit = aξ ; λ ≈ 1.8

0.1 1 2 5 10

ξ(csep, Tsep) [nm]

Bologna, 21/09/2010 Stefano Buzzaccaro Again ΠxVesc ?! 10

c − c ≈ Π 2 / 3 sep c 3 3.0 2.0 Πξ ≈ ξ ≈ (Tc − T ) −1.8 csep − cc ≈ ξ

Bologna, 21/09/2010 Stefano Buzzaccaro Again ΠxVesc ?! 10

0.15 c − c ≈ Π 2 / 3 sep c 3 3.0 2.0 Πξ ≈ ξ ≈ (Tc − T ) −1.8 csep 0.10− cc ≈ ξ T B /k 3 Πξ 0.05

0 0 4 8 12 2 ε x 10

Bologna, 21/09/2010 Stefano Buzzaccaro A SIMPLE VIEW: A fluid of independent “soft” droplets with (average) size ξ (Droplet model of critical fluctuations, Oxtoby 1977)

Bologna, 21/09/2010 Stefano Buzzaccaro 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)] functional Ω[][][]n(r) = A n(r) + dr Φ(r) − µ n(r) Φ(r) ∫ -depletant interactions µµµ Chemical potential of the host fluid

Bologna, 21/09/2010 Stefano Buzzaccaro 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 Ω[][][]n(r) = A n(r) + dr Φ(r) − µ n(r) Φ(r) ∫ Colloid-depletant interactions µµµ 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):

Bologna, 21/09/2010 Stefano Buzzaccaro Main Results (I) 13

Continuity between depletion interactions and critical Casimir effect rigorously assessed.

- Far from the critical point Asakura-Oosawa potential

- Approaching Tc Scaling form for the force:

β kBT /1 ν x ∝ δnε (Dietrich model) F(h) = 3 ϑ(x; y )  h  y ∝ hε ν

Bologna, 21/09/2010 Stefano Buzzaccaro Main Results (I) 13

Continuity between depletion interactions and critical Casimir effect rigorously assessed.

- Far from the critical point Asakura-Oosawa potential

- Approaching Tc Scaling form for the force:

β kBT /1 ν x ∝ δnε (Dietrich model) F(h) = 3 ϑ(x; y )  h  y ∝ hε ν

Bologna, 21/09/2010 Stefano Buzzaccaro Main Results (II) 14

● Note of caution: Whereas in the simplest lattice models (particle- hole symmetry) δn has the usual meaning of reduced density n – nc, 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 ”).

Bologna, 21/09/2010 Stefano Buzzaccaro Main Results (II) 14

● Note of caution: Whereas in the simplest lattice models (particle- hole symmetry) δn has the usual meaning of reduced density n – nc, 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:  ∂µ −1 χ =    ∂n 

Bologna, 21/09/2010 Stefano Buzzaccaro Main Results (II) 14

● Note of caution: Whereas in the simplest lattice models (particle- hole symmetry) δn has the usual meaning of reduced density n – nc, 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: −1  ∂µ   ∂Π −1 χ =   χ ∝ c  ∝ I  ∂n  mixtures  ∂c  s

MAX STRENGTH ALONG LINE OF MAX SCATTERING!

Bologna, 21/09/2010 Stefano Buzzaccaro Main Results (II) 14

● Note of caution: Whereas in the simplest lattice models (particle- 60.5°C hole symmetry) δn has the usual meaning of reduced56.5° Cdensity n – nc, 10 53°C in fluids one has field mixing, and δn is generally a linear50°C combination of reduced density and temperature . (the45.6°C “correct” line 42.3°C of approach to the critical point is not the “ critical isochore39°C ”). 36.3°C 33.4°C 30°C 25°C (a.u.) ● However,s 5 both in lattice models and in fluids, theTRANS. line δ POINTSn = 0 I can be identified with the line of maximum susceptibility: −1  ∂µ   ∂Π −1 χ =   χ ∝ c  ∝ I  ∂n  mixtures  ∂c  s 0 MAX STRENGTH0 0.02 ALONG0.04 LINE 0.06OF MAX0.08 SCATTERING!0.10 c

Bologna, 21/09/2010 Stefano Buzzaccaro 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

3 0.10

∆p ξ T sing B /k ≈ 0.1 3 Πξ kBT 0.05

0 0 4 8 12 2 ε x 10

Bologna, 21/09/2010 Stefano Buzzaccaro Grazie per l’attenzione!!!