Diffusion in Reduced Dimensions. Clemens Bechinger 2. Physikalisches Institut, Universität Stuttgart

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1 Diffusion in Reduced Dimensions Clemens Bechinger 2. Physikalisches Institut, Universität Stuttgart

2 Diffusion in narrow Channels t = 0 t = t 1 3D, 2D: mixing 1D: sequence unchanged 1D diffusion entirely different

3 Realization of SF conditions molecular sieves (zeolites) carbon nanotubes ionic transport through membranes reptation in polymer melts microfluid devices nm

4 Single-File Diffusion Δ 2 lim x(t) = 2F t t Levitt, Phys. Rev. A 8, 3050 (1973) Fedders, Phys. Rev. B 17, 40 (1978) van Beijeren, Kehr, Kutner, Phys. Rev. B, 28, 5711 (1983) Kärger, J. Phys. Rev. A 45, 4173 (1992) σ P = 0.383nm ( CF 4 ) Hahn, Kärger, J. Phys. Chem. B, 102, 5766 (1998)

5 Kärger s Derivation of t 1/2 -Law 1D exclusion model Θ 1: motion of particles and vacancies highly correlated consider vacancy motion vacancy tracer particle J. Kärger, Phys. Rev. A 45, 4173 (1992) L x(t) 0 m = 0 m = L P(m,m,t) P(m,m,t) 0 s(t=0) vacancy m : 0... m :... 0 m [ ] 2 2 x (t) = L ( 1 Θ ) P(m, m, t) + P(m, m, t) dm dm x normal diffusion of vacancies 1/2 2 L 2 P(m,m,t) = exp (Lm Lm ) /(4Dvt) 4πDvt 1/2 1/ Θ t x(t) = L π Θ τ T

6 SFD in Zeolites CF 4 / AlPO 4 5 Θ SFD T Hahn, Kärger, Kukla, Phys. Rev. Lett. 76, 2762 (1996) However: controversial results for CH 4 / AlPO 4 5: SFD and ND no ideal pore structure? interaction across adjacent pores?

7 SFD in colloidal systems channel structures: particles: μm B 3.6μm M(B) Magnetic Moment [M 0 ] Magnetic Field [mt] 1 M = M0 coth( αb) αb M 0 =6 10 μ α = kt B Am -13 2

8 Direct Observation of SFD single particle trajectories MSD B 2 Δx = 2F t Wei, Bechinger, Leiderer, Science 287, 625 (2000)

9 Propagator 1 p(x,t) x=0,t=0 = exp x /4Ft 1/4 4πFt 2 1/2 ( ) (hard rods) : 77s : 385 s : 770 s : 3850 s

10 Channels Made by Optical Tweezers ft ft Single moving trap intermediate regime quasi static toroidal trap Lutz, Reichert, Stark, Bechinger, EPL 74, 719 (2006)

11 Scanning Optical Fields phase locked ν P = ft ν P f T phase slip 2 1 V foc = 2 (2 πr) f T w0 6πηa 2 diffusive Faucheux, Stolovitzky, Libchaber, Phys. Rev. E 51, 5239 (1995)

12 Channels Made by Optical Tweezers ft 300 Hz 2.9 μm PS particles * 2 exp(κσ) βu(r) = (Z ) λ B 1+κσ 2 exp( κr) r 20 μm Advantages In situ control of channel geometry and particle number density Higher particle mobility due to absence of sticking boundary walls

13 Crossover: Normal Diffusion to SFD 100 SFD ρ [1/μm] : <Δx 2 > [µm] 10 1 normal diffusion t cross over ND SFD t 1/2 0,099 0,103 0,119 0,168 0,185 0, t [sec] Lutz, Kollmann, Bechinger, PRL 93, (2004)

14 Propagator 1 p(x, t) x=0,t=0 = exp x /4Ft 1/4 4πFt 2 1/2 ( ) s 501 s 901 s p(x,t) p(x,t) t 1/ x [μm] x/t 1/4 [µm/s 0.25 ] Lutz, Kollmann, C. Bechinger, J. Phys. Cond. Mat. 16, S4075 (2004)

15 SF Mobility 1.0 F [µm 2 /(sec 0.5 )] ρ [µm -1 ] Lutz, Kollmann, Bechinger, PRL 93, (2004)

16 F from Intrinsic System Properties 2 S(q,t = 0) D (q) eff lim 2 Δx (t) = t t ρ π Valid for any pair interaction HI treated pairwise additive infinite system long wavelength limit 1 (q a ) 2F Kollmann PRL 90, (2003) Why MSD is related to collective diffusion coefficient D eff (q)? a 2π λ = q 1D: Decay of density mode trajectory of every single particle

17 Dynamic Structure Factor S(q,t) = exp iq[x (t + τ) x(τ)] N i,j ( j i ) τ ρ=0.203 μm 1 ln ( S(q,t) q=4q min 2π 1 q min = = λmax R q<<a 1 : S(q,t) = t [sec] S(q,0) exp q ( 2 ef D f (q)t) Nägele, Phys. Rep. 272, 215 (1996) F can be obtained at short times (t < t c )!!

18 F from Short-Time Behavior <Δx 2 > [µm] 1 t t 1/ t [sec] F= eff S(q,0) D (q) ρ π 2 Δx = 2F t

19 SFD -Mobility 1.4 F [µm 2 /(sec 0.5 )] S(q,t) q=4*q min S(q,t) q=3*q min ρ [µm -1 ] Lutz, Kollmann, Bechinger, PRL 93, (2004)

20 SFD -Mobility 1.4 F [µm 2 /(sec 0.5 )] S(q,t) q=4*q min S(q,t) q=3*q min MSD ρ [µm -1 ] Lutz, Kollmann, Bechinger, PRL 93, (2004)

21 Finite Size Effects? ρ=0.203 μm S(q,0) q max 2π = a [q/q max ] S(q,0) lim 0 q 0 q

22 From Diffusive to Driven Motion ft ft Single moving trap intermediate regime quasi static toroidal trap Lutz, Reichert, Stark, Bechinger, EPL 74, 719 (2006)

23 Phase-Slipe Regime silica particles, σ = 3μm ethanol (3D tweezing) ft = 76 Hz Δv/v) [%] Φ [rad] constant, non conservative force

24 Circling Particles in Toroidal Trap silica particles, σ = 3μm electrostatic interaction, κ 1 300nm ethanol (3D tweezing) ft = 76 Hz mechanism: max. screening escape of the particle pair two front catches up with from fluid flow particles isolated sphere highest mobility particle pair catches up with isolated sphere t Lutz, Reichert, Stark, Bechinger, Europhys. Lett., 74, 719 (2006)

25 Summary Colloids are versatile model systems for statistical physics Colloids are the computer simulator s dream (Daan Frenkel) realization of SF conditions in colloidal systems topographic structures, optical tweezers transition from normal diffusion to SFD dependence of crossover from particle interaction and density F obtained from collective system behavior asymptotic single particle properties derived from short time collective behavior