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Controlling Particle Diffusion Controlling Particle Diffusion on the Nanoscales Nanoscales on the Fabio Marchesoni 1,2 1 Physics Department, University of Camerino, Italy 2 Physics Department, University of Augsburg, Germany University Augsburg
Fabio Marchesoni1,2
1 Physics Department, University of Camerino, Italy 2 Physics Department, University of Augsburg, Germany
KIAS, July 2008 KIAS, July 2008
in collaboration with
University Augsburg
and
RIKEN
Astumian & Hanggi, Physics Today (2002)
Flashing ratchet: substrate switches on and off periodically Role of noise: rectification is assisted by noise
kinesin ATP hydrolysis
= 2 Dα t α spread around mean α = 1 normal diffusion Impacts efficiency: long
> 2 D1 / ‹|v|›2
PRL 99 PRL 01 Bras & Kets PRL 04
Repulsive interaction: feel ratchet potential; don’t.; both species get rectified in opposite direction
Model: single Brownian particle in a tilted washboard potential
ξ(t): thermal noise 〈ξ〉 = 0, 〈ξ(t)ξ(0)〉 = 2γ kTδ(t) V(x)=V(x+L): periodic substrate Overdamped regime (m = 0, γ = 1): most used for applications
threshold: F3
̶ ‹x(t)›2 = 2 D t normal diffusion with D = D (F,T) L=2π μ=v/F v∞ =F/γ
D/D0 F/F3 Numerics: D(F) peaks at F = F3 to compare with free diffusion D0 = kT
[Costantini, FM, EPL 48 (1999) 491]
Renewal process: diffusion due to uncorrelated hops between adjacent wells Cox' formula yields: ‹t n (a → b)› n-th moment of the a→b FPT in good agreement with numerics.
[Reimann et al, PRL 87 010602 (2001)]
kT=0.1 kT=0.01
Underdamped regime: effect further enhanced [Costantini, FM, EPL 48 (1999) 491] Threshold effect: peaks located by approximate formula γ= 0.01 γ = 0.1 kT=1 Consequence: diffusion enhancement occurs at current onset, eg, in ratchet devices; efficiency gets degraded.
Constrained geometries: from 1D to “narrow” channels. Single file (SF): N non-passing particles diffusing over a line
with ρ = N/L = const [Jepsen gas, 1965] Initial vi and xi : randomly distributed Thermal noise: OFF (ballistic SF); ON (stochastic SF)
SFD: the diffusion of a single particle gets suppressed; independently of F ballistic SFD: normal diffusion stochastic SFD: anomalous diffusion with FD = √D0/π
[Harris 1975, Percus 1974]
D
Corrugated walls: ion channels, zeolites, laser traps, etc for a single particle driven on a periodic substrate Anomalous diffusion: ‹Δx2(t)› = 2FD √t/ρ with FD = √D(F) /π
D F=1 F=0
L V(x) F=1
Bona fide 2D, 3D geometry: suspended
point-like Brownian particles in a “narrow” channel. Channel radius: w(x) = a cos(2πx/L)+b Wmax = 2(b+a); Wmin = 2(b-a) ∝ kT
Ventropic (x) = – kT ln A(x) with A(x) = 2w(x) (2D); π w2(x) (3D)
(x) = T/[1+w'(x)2]α A(x) Wmax Approximate reduction to 1D
[Zwanzig 1992, Reguera Rubi 2001]
Veff (x) α = 1/2, 1/3
Replacing Ventropic (x) and Teff (x) into Cox' formula fits closely numerics Deviations for large F/T: asymptotes ≠ 1; particle piles up against bottlenecks [Hanggi
et al, 2006, 2007]
μ
kT = 0.01, 0.1, 0.2, 04 L=2π a/L=1; b/L=1.02
substrate – ATP power strokes,
– by unbiased forcing 2 x0 (t) ) ( ) ( sin
kick
with residence time τ
by x → y = x – x0 (t)
with Fkick (t) = (–1)n x0 δ(t – tn ); either periodic
with residence time τ
π+ π- Resonant diffusion: τ ~ 1: particle hops right/left by steps
and 2(π-x0 ); hence
π+ ,π
–
right/left probabilities Diffusion coefficient τ → ∞: kicking is irrelevant, τ → 0: potential gets renormalized – cos x → – (cos x0 ) cos x
for the relevant potential amplitude
Work done also in collaboration with: Nori and Savel’ev (RIKEN): PRL 91 010601 (2003); 92 160602 (2004); Taloni (now IoP-AS, Taiwan): 96 020601 (2006); 97 106101 (2006); Borromeo (INFN, Perugia): 99 150605 (2007)
Diffusion plays a crucial role in the control of particle transport at the nanoscales. Relevance for applications in nanotechnology. RMP 2009