Transcript Critical Casimir effect and wetting by helium mixtures

Acoustic crystallization
S. Balibar, F. Werner, G. Beaume, A. Hobeika,
S. Nascimbene, C. Herrmann and F. Caupin
Laboratoire de Physique Statistique
Ecole Normale Supérieure, Paris
for references and files, including video sequences,
go to http://www.lps.ens.fr/~balibar/
Kyoto, nov. 2003
abstract
we study phase transitions with acoustic waves
very high intensity : up to 1000 bar amplitude
liquid-gas : acoustic cavitation
liquid-solid : acoustic crystallization
why acoustic waves ?
eliminate the influence of impurities walls and defects :
homogeneous nucleation
test the intrinsic stability limits of the liquid state of matter
pressure
metastable liquids
liquid-gas or liquid-solid:
first order phase transitions
-> metastability is possible
crystallization
solid
liquid
boiling
gas
cavitation
temperature
liquids can be supercooled or overpressurized
before crystalization occurs, i.e. before crystallites nucleate
they can also be overheated , or underpressurized
before boiling or cavitation occurs (before bubbles nucleate)
ex: water down to - 40 °C, + 200°C or - 1400 bar
the barrier against nucleation
is due to the surface energy
Standard nucleation theory (Landau and Lifshitz, Stat. Phys. p553):
ex : cavitation in liquid helium 4
200
Free energy F(R) (Kelvin)
and surface energy g (the macroscopic
surface tension)
R
Pl Pv
Pl = - 6 bar
F(R) = 4p R2 g - 4/3 p R3 DP
100
DP : difference in free energy per unit
3
E=16pg /3P
volume between the 2 phases
2
Pl = - 10 bar
Critical radius : Rc = 2 g/ DP
Activation energy : E = (16p g3)/(3 DP2)
0
Rc =2g / P
R > Rc  growth
R1 =3g / P
-100
a spherical nucleus with radius R
0
0.5
1
1.5
Bubble radius R (nanometers)
nucleation rate per unit time and volume :
2
The critical nucleus is in unstable
equilibrium
 DP = (1 - rv/rl)(Peq - P)
G = G0 exp(-E/T)
G0 : attempt frequency x density of independent sites
supercooling water:
Taborek ’s experiment
(Phys. Rev. B 32, 5902, 1985)
Avoid heterogeneous nucleation:
- divide the sample into micro-droplets
- minimize surface effects (STS not STO)
Regulate T : the heating power P increases
exponentially with time
The time constant t =1/VJ
The nucleation rate J varies exponentially with T
Compare with standard theory of homogeneous
nucleation :
Taborek used his nucleation experiment to measure the (unknown)
tension of the ice/water interface : it is 28.3 erg/cm2 at 236 K
(see also Seidel and Maris 1986 for H2 crystals)
the surface tension of helium 4 crystals is accurately known
pressure (bar)
the surface of helium crystals
solid
25
normal liquid
gas
superfluid
0
1
2
temperature (K)
a paradoxical situation :
model systems
for very general properties of crystal surfaces
for ex: the roughening transitions
unusual growth dynamics
due to quantum properties
for ex: crystallization waves
for review articles, see:
S. Balibar and P. Nozières, Sol. State Comm. 92, 19
(1994)
S. Balibar, H. Alles and A. Ya. Parshin, to be published
in Rev. Mod. Phys. (2004).
video He crystals
crystallization waves
helium 4 crystals grow from a superfluid (no viscosity, large thermal conductivity)
the latent heat is very small (see phase diagram)
the crystals are very pure wih a high thermal conductivity
-> no bulk resistance to the growth, the growth velocity is limited by surface effects
smooth surfaces : step motion
rough surfaces : collisisions with phonons (cf. electron mobility in metals)
v = k Dm with k ~ T -4 : the growth rate is very large at low T
helium crystals can grow and melt so fast that crystallization waves propagate at their
surfaces as if they were liquids.
2 restoring forces :
-surface tension g
(more precisely the "surface stiffness" g )
- gravity g
inertia : mass flow in the liquid ( rC > rL)
rL
3
 =
2 gq  (rC  r L )gq
(rC  rL )
2
 experimental measurement of the surface stiffness g
superfluid
crystal
video waves
Qui ckTime™ et un décompresseur Animation sont r eq uis pour vi sionner cette imag e.
surface stiffness measurements
the surface tension a is anisotropic
the anisotropy of the surface stiffness
g= a   2a/q2 is even larger.
a mean value for the surface tension is
a = 0.17 erg/cm2
E. Rolley, S. Balibar and C. Guthmann
PRL 72, 872, 1994 and J. Low Temp. Phys. 99, 851, 1995
D.O. Edwards et al. 1991
g1
a
g2
nucleation of solid helium
pressurizing liquid helium in an ordinary cell:
heterogeneous nucleation occurs
~ 3 to 10 mbar above Pm
(Balibar 1980, Ruutu 1996, Sasaki 1998)
Balibar, Mizusaki and Sasaki
(J. Low Temp. Phys. 120, 293, 2000):
it cannot be homogeneous nucleation,
since E = 16/3 p a3/DP2 ≈ 1010 K !
J.P. Ruutu et al., Helsinki, 1996
consistent with other measurements by
Balibar (1980), Sasaki (1998)
heterogeneous nucleation on favorable sites
(graphite dust particles ?)
 acoustic crystallization : eliminate heterogeneous nucleation ?
the optical
refrigerator
at ENS-Paris
superfluid helium cell :
300 cm3
0 to 25 bar ; 0.02 to 1.4 K
sapphire
windows
piezo-électric
transducer (1 MHz)
heat exchangers
acoustic crystallization on a
clean glass plate
11.0 V excitation
densité statique
10.4 V excitation
3
)
0.180
densité (g/cm
X. Chavanne, S. Balibar and F. Caupin
Phys. Rev. Lett. 86, 5506 (2001)
0.185
liquid helium
P = P m = 25.3 bar
Ar + laser
lens
(20 mm)
0.175
0.170
20
piezo-electric
transducer (1 MHz)
25
30
35
40
30
30.5
Temps (microsecondes)
0.184
4% reflexion
densité statique
10.4 Volt
11.0 Volt
0.182
0.180
acoustic bursts (6 oscillations, rep. rate ~ 2Hz)
wave amplitude at the crystallization threshold:
± 3.1 10-3 g/cm3 (~2% of rm),
i.e. ± 4.3 bar according to the eq. of state
densité (g/cm
3
)
solid helium
glass plate
0.178
0.176
0.174
0.172
0.170
28.5
29
29.5
temps (microsecondes)
80
The equation of state of liquid helium 4
(after Abraham 1970 and M aris 1994)
the
equation
of state
of liquid
helium 4
PRESSURE (bar)
60
metastable
40
nucleation
20
liquid - gas
spinodal limit
0
Pm = 25.324
stable
P0 = 0
metastable
-20
0.1
0.12
0.14
0.16
DENSITY (g/cm 3)
a rather well established cubic law (Maris 1991)
P - Psp = a (r  rsp)3
0.18
0.2
nucleation is stochastic
transmission
no nucleation
nucleation
0.178
3
densité (g/cm )
0.176
0.174
reflexion
0.172
0.170
0.168
20
transmission signals
are not averaged,
so that the nucleation
probability is easily
obtained by counting
events
22
24
26
28
temps (microsecondes)
30
32
a selective averaging
is made on reflexion
signals, in order to
measure the wave
amplitude at the
nucleation threshold
on a clean glass plate, nucleation of solid He is still heterogeneous
quantum nucleation ?
Nucleation probability
1.0
classical nucleation
(thermally activated)
0.8
0.6
0.4
0.2
0.0
∂rc/∂T = - 2.6 10-4 g/cm3K
∂E/∂r = -3.84 104 Kcm3/g
-8
-6
-4
-2
0
10 (r  rc)
5
2
4
6
8
3
(g/cm )
the nucleation probability S increases continuously from 0 to 1
in a small density interval, as expected for nucleation due to thermal or quantum
fluctuations. This is the usual "asymmetric S-shape curve":
S = 1 - exp (- G0 Vt exp (-E/T) = 1 - exp {- ln2 exp [ - (1/T)(∂E/∂r) (r - rc)] }
from S (r) and rc(T), we obtain the activation energy E = T . ∂E/∂r . ∂rc(T)/∂T = 6 T
 heterogeneous nucleation on the glass (~ 1 preferential site)
(at Pm + 4 bar the homogeneous nucleation barrier would be ~ 3000 K)
cavitation
in helium 3
same "asymmetric S-shape" law
for the nucleation probability:
S = 1 - exp (- G0 Vt exp (-E/T)
= 1 - exp {- ln2 exp [ - (1/T)(dE/d ) (
F. Caupin and S. Balibar,
Phys. Rev. B 64, 064507 (2001)
-
c)]
}
search for homogeneous nucleation of solid
helium with acoustic waves
transducer (1 MHz)
Ar+ laser
lens
2 cm
remove the glass plate
increase the amplitude of the acoustic wave
calibrate the wave amplitude from the
known cavitation threshold (- 9.4 bar)
m
acoustic cavitation
in liquid 4He
at high pressure
= 25.3 bar
0
-50
Signal (arb. units)
Excitation (Volt)
cavitation at P
50
flight time (22 ms)
0
5
10
15
20
25
30
35
640
660
680
Time (microseconds)
26
25
P
s tat
= - 9.45 + 0.051 rLVc
static pressure P
stat
(bar)
24
23
22
21
20
19
18
540
560
580
600
620
cavitation threshold r Vc (V.kg.m-3)
L
 the cavitation threshold voltage Vc
(more precisely the product rLVc)
varies linearly
with the pressure in the cell Pstat
 agreement with the linear
approximation for the amplitude of the
wave at the focus:
dP ≈ R 2rLV
 in our hemispherical geometry, nonlinear efects must be small.
 extrapolation => cavitation occurs at
-9.45 bar, in excellent agreement with
theory (0.2 bar above the spinodal limit
at - 9.65 bar)
 a calibration method for the wave
increasing the acoustic amplitude
* as one increases the
excitation voltage, cavitation
occurs on earlier and earlier
oscillations. This is due to
the finite Q factor of the
transducer
(we measured Q = 53)
* here, for bursts of 3
oscillations and at 25 bar, 55
mK:
- no cavitation at 119 V
- cavitation on third oscillation
at 120 V
-on second oscillation at 125 V
- on first oscillation at 140 V
PMT signal (arb. units)
140 V
125 V
120 V
119 V
0
5
10
15
20
time (microseconds)
25
30
pressure
principle of an ideal experiment
0
1
2
3
4
5
6
pressure
time (microseconds)
0
1
2
3
4
time (microseconds)
5
6
In liquid helium at 25 bar,
we emit a sound pulse, which starts with a
negative pressure swing
cavitation is observed for a threshold voltage Vc,
when the pressure reaches - 9.45 bar
at the acoustic focus at time tflight + 0.25 ms.
 calibration:
Vc corresponds to a 25 + 9.45 = 34.45 bar amplitude
We reverse the voltage applied to the transducer.
We increase this voltage V as much as possible,
looking for nucleation of crystals
at the same time tflight + 0.25 ms.
A maximum positive pressure
P max = 25 + 34.45(V/Vc) bar
is reached at this time
a real experiment
PMT signal (arb. units)
cavitation
no cavitation
18
20
22
24
26
28
time (microseconds)
when starting with a negative pressure swing
we have found a cavitation threshold for Vc = 340 Volt
PMT signal (arb. units)
liquid helium 4 up to 163 bar
265 V
265 V
340 V
700 V
1000 V
1180 V
1370 V
21.8
22
22.2
22.4
time (microseconds)
22.6
22.8
after reversing the excitation voltage,
no nucleation of crystals up to 1370 Volt.
this sound amplitude corresponds to a maximum pressure
Pmax = 25 + 34.45 (1370/340) = 163 bar
the standard nucleation theory
fails
some comments
liquid-solid spinodal ?
200
the standard theory predicts homogeneous
nucleation at 65 bar.
It assumes a pressure independant surface
tension, but this assumption was criticized
by Maris and Caupin
(J. Low Temp. Phys. 131, 145, 2003)
extended
phase diagram
is liquid helium superfluid at
163 bar ?
Pressure (bar)
150
nucleation line
(standard theory)
100
 line extrapolation ?
50
liquid-solid equilibrium
 line
liquid-gas equilibrium
0
liquid-gas spinodal
0
0.5
1
1.5
T ( K)
2
2.5
It is unlikely that crystals nucleated but
were not detected, since they should grow
even faster at 163 bar than at 29.6 bar,
except if liquid helium is no longer
superfluid (rL ~ 0.227 gcm-3, much more
than rL = 0.172 or rC = 0.191 at 25 bar).
The extrapolation of the  line is not
precisely known, but it should reach T = 0
at 200 bar, where the roton gap vanishes
according to H.J. Maris, and where the
liquid should become unstable (Schneider
and Enz, PRL 27, 1186, 1971).
an instability at 200 bar ?
(Schneider and Enz PRL
27, 1186, 1971)
14
20 bar
12
Energy (K)
Maris noticed that,
according to the density
functional form of
Dalfovo et al. ,
the roton gap vanishes
around 200 bar where the
density reaches
0.237 g/cm3
If true, this "soft mode"
at finite wave vector
could imply an instability
towards a periodic (i.e.
crystalline ?) phase
svp
10
8
6
rotons
4
phonons
2
0
0
5
10
15
Wavenumber (nm -1)
20
25
future experiments:
reach 200 bar or more
0.168
2 transducers
Ar+
laser
3
DENSITY (g/cm )
0.166
0.164
0.162
0.160
0.158
0.156
0.154
32
lens
32.5
33
33.5
TIME (microseconds)
2 cm
use 2 transducers (full spherical geometry)
due to non-linear effects, positive swings are larger than negative swings
easy to reach + 200 bar
difficult to calibrate the amplitude
improve numerical calculations of the sound amplitude
(see C. Appert , C. Tenaud, X. Chavanne, S. Balibar, F. Caupin, and D. d'Humières
Euro. Phys. Journal B 35, 531, 2003)
34