S - Dipartimento di Fisica - Università degli Studi di Milano

Nanoparticles in a thermal gradient
R. Cerbino, S. Mazzoni, F. Croccolo, D. Brogioli,
F. Giavazzi, G. Donzelli, A. Zelioli, M. Giglio, A.V.
Dipartimento di Fisica and CNR-INFM,
Università degli Studi di Milano
Nanoparticles
A nanoparticle is a microscopic particle whose size is measured in nanometers.
Nanoparticles are thus larger than angstroms (Å), but smaller than micrometers.
Aggregation and gelation
Phase
transitions
25µm
A. M. Alsayed et al., Science 309, 1207 (2005)
P. N. Pusey, Science 309,1198 (2005)
Applications
D. A. Weitz and M. Oliveria, Phys. Rev. Lett. 52, 1433 (1984)
M. Carpineti and M. Giglio, Phys. Rev Lett. 68, 3327 (1992)
M. Carpineti and M. Giglio, Phys. Rev Lett. 70, 3828 (1993)
Brownian motion and diffusion
Robert Brown
1827
Albert Einstein
1905
D=
kB T
6πηr
Λ
< x2 > ∝ D t
τ diff
Λ
≈
D
Applet courtesy of Michael Fowler, University of Virginia
2
External Fields: gravity
Microscopic scale
h
Mesoscopic scale
g
g
k B T >> (m p − ms ) g h
Archimede
⎧k B T >> mgh random walk
⎨
⎩k B T << mgh sedimentation
τ grav ∝
ν
g Λ2
External fields: thermal gradient and the Soret Effect
Hot
∇T
Joseph Fourier
1807
Charles Soret
1879
Cold
j = − ρ D [∇c + S ∇T ]
Why nanoparticles?
Fick
j =0 ⇒
Soret
∇cSoret = − S ∇T
D S ≈ const
M. Giglio and A. Vendramini, Phys.
Rev. Lett. 34, 561 (1975)
Density profiles
ρ = ρ o (1 + β δc )
∇ρ = ρ0 β ∇c
β=
1 ∂ρ
ρ ∂c
Stable
Unstable
Hot
Cold
∇ρ
g
Cold
∇ρ
Hot
Stable Case: non-equilibrium fluctuations
Mesoscopic scale
Divergence of fluctuations
∇ρ
∇c
g
S ( Λ ) ∝ ∇c Λ
2
Λ
τ≈
D
2
Velocity fluctuations
induce
concentration fluctuations
T. R. Kirkpatrick, E. G. D. Cohen, and J. R. Dorfman,
Phys. Rev A 26, 1812 (1982)
4
Shadowgraphy
Thermal gradient cell
Hot
Cold
Sapphire: κ=0.345 W/(K cm)
Lead:
κ=0.35 W/(K cm)
Copper:
κ=4.0 W/(K cm)
1mm
Non-equilibrium fluctuations
A. Vailati and M. Giglio, Nature 390, 262 (1997)
Critical fluid with correlation length ξ= 8nm
Static and Dynamic shadowgraph
Statics
Dynamics
LUDOX (R)
diff
usi
on
10
ity
av
gravity
gr
diffu
sion
LUDOX (R)
Λ4
6
S(Λ)
τ (s)
10
Λ2
≈
D
τ diff
τ grav ∝
ν
g Λ2
5
10
F. Croccolo, tesi di Dottorato
R. Cerbino
-4
10
Λ (m)
1
-3
10
-4
10
Λ (m)
-3
10
Gravitational Stabilization of Fluctuations
Mesoscopic scale
Time scales
Λ2
=
D
Diffusion
τ diff
Buoyancy
τ grav =
υ
βg∇c Λ2
Rolloff wave length
τ diff = τ grav
Λc = 4
νD
β g∇ c
KAYSER
DTM
Lambda-X
GRADFLEX
Flight scheduled on FOTON M3,
September 2007
Unstable Case: convection
Archimede
F ∝ g Λ3∆ρ = g Λ3 ∇ρ ∆z
Diffusion
Viscosity
Λ2
≈
D
Λ2
τ Diff
τ visc ≈
ν
∇ρ
g
Λ
∆z
Solutal Rayleigh
number
β g ∆c h3
Rs =
νD
Convection threshold:
Rs ≅ 720
Why nanoparticles?
D S ≈ const
small D
large S
∆c = −S ∆T
⇒
large Rs
Boundary layer convection, Ras≈ 107
(heating from above!)
Side view
Ras=4x107 ,h =2mm, z=2cm
A. Zelioli, tesi di Laurea
Ras=1.2x107, 1 frame/ 5 seconds, z=3 cm, heating from above
S. Mazzoni and R. Cerbino
τ =
δ2
D
⇒δ =
Dτ
Heat convection in the Earth’s mantle
∆T ≈ 3000K
h ≈ 3000 km
ν ≈1021 cm2/s
κ ≈10-2 cm2/s
Ra ≈106-108
F. Dubuffet, D. A. Yuen, Geophys. Res. Lett., 27, 17 (2000)
R. Cerbino, A. Vailati and M. Giglio, Phys. Rev. E 66, 055301 (2002)
Scaling in high Rayleigh Number convection
Nu s =
total mass flux
h
=
mass flow by diffusion
δ
τ=
δ2
D
Nus =1 → Diffusion
: j = −ρ D∇c
Goldstein et al., J. Fluid Mech., 213, 111-126 (1990)
Steady state
Nus
R. Cerbino, S. Mazzoni, A. Vailati
and M. Giglio, Phys. Rev. Lett.94,
064501 (2005)
Onset
Nus =
1
δ*
Ras
≈ Ras0.35± 0.03
⇒ δ = Dτ
Competition between stability and instability
G. Donzelli, tesi di Laurea
Density profile
ρ = ρ [1 − α (T − T ) + β (c − c )]
Propagating modes
Rolls
Zip
Grid
Chaotic
Cold
Soret
Heat convection
Soret
Hot
Marzio Giglio
Doriano Brogioli
Roberto Cerbino
Fabio Giavazzi
Stefano Mazzoni
Fabrizio Croccolo
Gea Donzelli
Andrea Zelioli