Thermal Casimir Effect for Colloids at a Fluid Interface

Transcription

1 Thermal Casimir Effect for Colloids at a Fluid Interface Jef Wagner Ehsan Noruzifar Roya Zandi Department of Physics and Astronomy University of California, Riverside

2 Outline

3 : Capillary Wave Hamiltonian H int = σa + U int g

4 : Capillary Wave Hamiltonian H int = σa + U int g A = d 2 x 1+ φ 2

5 : Capillary Wave Hamiltonian H int = σa + U int g A = d 2 x 1+ φ 2

6 : Capillary Wave Hamiltonian H int = σ d 2 x 1+ φ 2 + Ug int Ug int = d 2 x ρg 2 φ2

7 : Capillary Wave Hamiltonian H int = σ d 2 x 1+ φ 2 + Ug int Ug int = d 2 x ρg 2 φ2

8 : Capillary Wave Hamiltonian [ ( ) ] H int [φ] =σ d 2 x 1+ φ λ 2 R 2 2 φ2 σ λ = ρg 3cm

9 Colloid Contribution H col = σ da Ω part 1 + U col g part 2 + σ I A I + σ II A II part 3

10 Colloid Contribution H col = σ da Ω part 1 + U col g part 2 + σ I A I + σ II A II part 3 ( Hpart col 1[φ] = σ d 2 x Ω 1+ φ 2 ) d 2 x Ω eq

11 Colloid Contribution H col = σ da Ω part 1 + U col g part 2 + σ I A I + σ II A II part 3 ( Hpart col 1[φ] = σ d 2 x Ω 1+ φ 2 ) d 2 x Ω eq

12 Colloid Contribution H col = σ da Ω part 1 + U col g part 2 + σ I A I + σ II A II part 3 H col part 1[φ] = σ [ ( ) ( d 2 x 1+ φ d 2 x Ω Ω ) ] d 2 x Ω eq

13 Colloid Contribution H col = σ da Ω part 1 + U col g part 2 + σ I A I + σ II A II part 3 H col part 1[φ] = σ [ ( ) ( d 2 x 1+ φ d 2 x Ω Ω ) ] d 2 x Ω eq

14 Colloid Contribution H col = σ da Ω part 1 + U col g part 2 + σ I A I + σ II A II part 3 H col part 1[φ] = σ [ ( ) ] d 2 x 1+ φ Ω[φ] Ω

15 Colloid Contribution H col = σ da Ω part 1 + U col g part 2 + σ I A I + σ II A II part 3 U col g = d 2 x ρg (ρ I d 3 x + ρ II V I 2 φ2 + mgh d 3 x V II ) g

16 Colloid Contribution H col = σ da Ω part 1 + U col g part 2 + σ I A I + σ II A II part 3 U col g = d 2 x ρg 2 φ2 + mgh

17 Colloid Contribution H col = σ da Ω part 1 + U col g part 2 + σ I A I + σ II A II part 3 H col part 3[φ] =σ I A I + σ II A II

18 Colloid Contribution H col = σ da Ω part 1 + U col g part 2 + σ I A I + σ II A II part 3 H col part 3[φ] =(σ I σ II ) A I

19 Colloid Contribution H col = σ da Ω part 1 + U col g part 2 + σ I A I + σ II A II part 3 Young s relation σ II σ I + σ cos θ =0 H col part 3[φ] =σ cos θ A I

20 Total Hamiltonian H tot [φ] =σ ( d 2 x R 2 / Ω i ( ( ) 1+ φ λ 2 + i 2 φ2 ( Ω i + cos θ A ii + m ) ) igh σ ) Expanding for spherical colloids H tot [φ] σ 2 ( d 2 x φ ( 2 + λ 2) φ + R 2 / Ω i i R s δω i dx φ 2 )

21 Proposed Experiment Silver coated hollow glass mircrospheres (10µm R s 100µm). Air Water interface. Potential decays as r 8. Potential has depth of 1k B T at sep 1µm.

22 Scattering Method for Casimir Physics (MIT) Z = Dφe 1 2 φ, ˆDφ R 2 \Ω1 Ω 2 C

23 Scattering Method for Casimir Physics (MIT) Z = Dφe 1 2 φ, ˆDφ R 2 \Ω1 Ω 2 C Delta functional constraints Dφ = C Dφδ δω1 [φ]δ δω2 [φ]

24 Scattering Method for Casimir Physics (MIT) Z = Dφδ δω1 [φ]δ δω2 [φ]e 1 2 φ, ˆDφ R 2 \Ω1 Ω 2 Fourier representation of the delta functional δ δωi [φ] = ψ i e ı ψ i,φ δωi

25 Scattering Method for Casimir Physics (MIT) Z = Dψ 1 Dψ 2 Dφ e 1 2 φ, ˆDφ R 2 \Ω1 Ω +i 2 i φ,ψ i δωi Gaussian functional integral Df e 1 2 f,af = ( det A ) 1 2

26 Scattering Method for Casimir Physics (MIT) Z = Dψ 1 Dψ 2 Dφ e 1 2 φ, ˆDφ R 2 \Ω1 Ω +i 2 i φ,ψ i δωi Gaussian functional integral Df e 1 2 f,af = ( det A ) 1 2

27 Scattering Method for Casimir Physics (MIT) Z = Dψ 1 Dψ 2 e 1 2 i,j ψ i,g 0 ψ j δωi,δω j Complete basis set ψ i = α Ψ iα φ inc iα Dφ i = α dψ iα

28 Scattering Method for Casimir Physics (MIT) { Z = dψ 1α dψ 2α exp 1 2 α ( Ψ1 Ψ 2 ) ) T (Ψ1 )} (Ĝ11 Ĝ 12 Ĝ 21 Ĝ 22 Ψ 2 Reminder: Free Energy βf = ln Z eff

29 Scattering Method for Casimir Physics (MIT) βf = 1 2 ln det ( 1 Ĝ 11 1 ) 1 Ĝ12Ĝ21 Ĝ22 Scattering Properties Ĝ 1 ii φ inc iα + β = T i T i αβ φsct iβ =0 Ĝ ij = U ij

30 Scattering Method for Casimir Physics (MIT) βf = 1 2 ln det ( 1 T 1 U 12 T 2 U 21) Questions: Why is the inverse of the ˆD operator on the space function space L 2 (R \ Ω 1 Ω 2 ) the free Green s function for R 2? How do I treat fluctuating boundaries?

31 Z = ( ) { Dφ Df i δ δωi [φ f i ] exp β 2 φ, ˆDφ β 2 i } f i, Ĥcol f i i Follow previous steps: Expand the delta functions. Integrate out the φ degree of freedom. Write the ψ i and f i fields in a complete basis set {φ iα }. Use the scattering properties.

32 Z = { DV exp 1 } 2 VT MV Vector and Matrix forms Ψ 1 V = P 1 Ψ 2 M = P 2 (T 1 ) 1 ıi U 12 0 ıi H col 0 0 U 21 0 (T 2 ) 1 ıi 0 0 ıi H col

33 Reminder: Free Energy βf = ln Z eff ( (T) 1 ıi ıi H col ) 1 ( ) U 0 = 0 0 ) ( TU 0 X 0 T = T T ( T + H col) 1 T

34 βf = 1 2 ln det ( 1 T 1 U 12 T2 U 21) The Casimir energy is still written in terms of scattering parameters. The term in the Hamiltonian due to fluctuating boundaries changes the scattering matrix.

35 Non-homogeneous Basis Functions Z = ( ) { Dφ δ Ωi [φ] exp β } φ, ˆDφ 2 i Reminder: Question Why is the inverse operator of ˆD on the function space L 2 (R 2 \ Ω 1 Ω 2 ) the free Green s function for R 2? Define the delta functional over the interior. Proceed in a similar fashion as before.

36 Non-homogeneous Basis Functions { Z = Dψ i exp β } ψ i, G 0 ψ j 2 i i,j New orthonormal basis set Let { φ iα } be a complete basis set, and partition the set into the homogeneous solutions to the operator ˆD and the rest. such that { φ iα } = { φ inc iα, ˆφ ia } ˆD φ inc iα =0 ˆD ˆφ ia 0

37 Non-homogeneous Basis Functions { Z = Dψ i exp β } ψ i, G 0 ψ j 2 i New orthonormal basis set The auxiliary field is expanded in the complete basis set i,j ψ i = α Ψ iα φ inc iα + a Ψ ia ˆφ ia Dψ i = α,a dψ iα d Ψ ia

38 Non-homogeneous Basis Functions Z = { DV exp 1 } 2 VT MV Vector and Matrix forms Ψ 1 G αβ G αb U 12 0 V = Ψ 1 Ψ 2 M = G aβ G ab 0 0 U 21 0 G αβ G αb Ψ G aβ G ab

39 Non-homogeneous Basis Functions Useful definition g(x) = d 2 x G 0 (x, x )ψ(x ) Homogeneous and particular solutions Ω ˆDg(x) = { 0 ψ(x) g(x) =g h (x)+g p (x) g h (x) = h α φ α (x) α g p (x) = g α φ α (x)+ α a ĝ a ˆφ a (x)

40 Non-homogeneous Basis Functions Matrix coefficients ˆDg p(x) =ψ(x) = g a = M 1 aβ Ψ β + M 1 Ψ ab b β b g p(x) 0 = g α = δω= S 1 αb M 1 bγ Ψ γ bγ bc Matrix Definitions S 1 αb M 1 bc M { } { α a }b = φ { α a }, ˆD ˆφ b ( M 1 M 1 ) ( ) M = I M S aβ = ˆφ a,φ β δω Ψ c

41 Non-homogeneous Basis Functions Matrix coefficients ( ) ( Gαβ G αb (T) = 1 S 1 M 1 S 1 M 1 ) G aβ G ab Block-wise Inverse ( A B ) 1 = M 1 M 1 C D X ( (A BD 1 C) 1 X ) X

42 Conclusions The thermal Casimir force appears measurable. Incorporated fluctuating boundaries into the scattering method. Is there a better proof that the non-homogenous basis functions do not contribute?

43 Conclusions The thermal Casimir force appears measurable. Incorporated fluctuating boundaries into the scattering method. Is there a better proof that the non-homogenous basis functions do not contribute?

44 Conclusions The thermal Casimir force appears measurable. Incorporated fluctuating boundaries into the scattering method. Is there a better proof that the non-homogenous basis functions do not contribute?