The metastability of The metastability of liquid helium at high liquid helium at high pressure: pressure: acoustic acoustic crystallization crystallization

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The metastability of The metastability of liquid helium at high liquid helium at high

pressure:

acoustic acoustic

crystallization crystallization

S. Balibar, R. Ishiguro and F. Caupin S. Balibar, R. Ishiguro and F. Caupin

Laboratoire de Physique Statistique Laboratoire de Physique Statistique

Ecole Normale Supérieure (Paris) Ecole Normale Supérieure (Paris) associé au CNRS et aux Universités associé au CNRS et aux Universités

Paris 6 & 7 Paris 6 & 7

ULTI meeting, Lammi 2006 ULTI meeting, Lammi 2006

for references and files :

for references and files : http://www.lps.ens.fr/~balibar/ http://www.lps.ens.fr/~balibar/

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The The

metastability metastability

of liquids of liquids

liquid-gas or liquid-solid:

first order phase transitions first order phase transitions

  metastability is possible metastability is possible energy barriers

energy barriers

against the nucleation of the stable phase against the nucleation of the stable phase

liquid water to - 40 °C or + 200°C at 1 bar, or liquid water to - 40 °C or + 200°C at 1 bar, or - 1400 bar at +35 °C

- 1400 bar at +35 °C

What are the limits of metastability ? What are the limits of metastability ? Acoustics in liquid helium, now in water Acoustics in liquid helium, now in water Cavitation down to -9.5 bar, but

Cavitation down to -9.5 bar, but crystallization ?

crystallization ?

temperature

p r e s s u r e p r e s s u r e

crystallization

solid

solid liquid liquid

gas gas

boiling cavitation

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abstract abstract

1 - search for acoustic crystallization of 1 - search for acoustic crystallization of helium in

helium in

two preliminary experiments two preliminary experiments

2 - experimental evidence for homogeneous 2 - experimental evidence for homogeneous

nucleation of he crystals by acoustic waves nucleation of he crystals by acoustic waves in a recent one

in a recent one 3 - future developments : 3 - future developments :

the vanishing of the roton gap ; superfluidity at high density the vanishing of the roton gap ; superfluidity at high density

a few words about supersolid helium

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acoustic acoustic

crystallization on crystallization on a clean glass plate a clean glass plate

X. Chavanne, S. Balibar and F.

Caupin Caupin

Phys. Rev. Lett. 86, 5506 (2001) Phys. Rev. Lett. 86, 5506 (2001)

acoustic bursts : 6

oscillations, rep. rate ~ 2Hz, P_stat= P_m = 25.3 bar) The crystallization

The crystallization threshold is at :

threshold is at :



 3.1 103.1 10^-3^-3 g/cm g/cm³³ (~2% of (~2% of

_m_m),_),

i.e. i.e. P = 4.3 barP = 4.3 bar

=> heterogeneous nucleation

=> heterogeneous nucleation on 1 defect

on 1 defect

0.170 0.175 0.180 0.185

20 25 30 35 40

11.0 V excitation densité statique 10.4 V excitation

Temps (microsecondes)

0.170 0.172 0.174 0.176 0.178 0.180 0.182 0.184

28.5 29 29.5 30 30.5

densité statique 10.4 Volt 11.0 Volt

temps (microsecondes)

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search for homogeneous search for homogeneous

nucleation of solid helium with nucleation of solid helium with

acoustic waves acoustic waves

F.Werner, G. Beaume, C.Herrmann, A. Hobeika, S.

Nascimbene, Nascimbene,

F. Caupin and S. Balibar (J. Low Temp. Phys. 136, F. Caupin and S. Balibar (J. Low Temp. Phys. 136,

93, 2004) 93, 2004)

remove the glass plate remove the glass plate

increase the amplitude of the acoustic wave increase the amplitude of the acoustic wave

ArAr⁺⁺ laser laser

lenslens

transducer (1 transducer (1

MHz)MHz)

2 cm2 cm

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nucleation nucleation

at high at high pressure:

pressure:

bubbles or bubbles or

crystals ? crystals ?

-50 0 50

0 5 10 15 20 25 30 35

Time (microseconds) cavitation at P

m = 25.3 bar

flight time (22 μs)

18 19 20 21 22 23 24 25 26

540 560 580 600 620 640 660 680

P

stat = - 9.45 + 0.051 ρ_LV_c

cavitationthesholdρ

LV_c(V.kg.m ^-3)

according to previous according to previous measurements (Werner et measurements (Werner et al. 2004):

al. 2004):

 the cavitation threshold the cavitation threshold voltage V

voltage V_c_c (more precisely (more precisely the product

the product _L_LV_V_c_c) ) varies linearly varies linearly

with the pressure in the with the pressure in the cell P

cell P_stat_stat

 extrapolation => extrapolation =>

cavitation occurs at cavitation occurs at

-9.45 bar, in excellent -9.45 bar, in excellent agreement with theory (0.2 agreement with theory (0.2 bar above the spinodal

bar above the spinodal limit at - 9.65 bar) limit at - 9.65 bar)

bubbles,bubbles,

a calibration method for a calibration method for the wave

the wave

no crystallization up to no crystallization up to 160 +/- bar

160 +/- bar

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the extended the extended phase diagram phase diagram

of He4 of He4

the standard theory the standard theory predicts homogeneous predicts homogeneous nucleation of

nucleation of

crystals at 65 bar.

Schneider and Enz Schneider and Enz (1971):

(1971):

an instability when an instability when

rot = 0 ?rot = 0 ?

at 200 bar (Maris)?

300 bar (Vranjes, 300 bar (Vranjes, Boronat) ?

Boronat) ?

The 4 data points (

The 4 data points (  ) assume ) assume linear sound focusing in a

linear sound focusing in a hemispherical geometry ,

hemispherical geometry , but ....

but ....

Superfluidity at high density ? Superfluidity at high density ?

see Nozieres 2004-2006 see Nozieres 2004-2006

Vranjes et al. 2006 Vranjes et al. 2006

Moroni and Boninsegni 2004 Moroni and Boninsegni 2004

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an instability at an instability at

200 bar ? 200 bar ?

0 2 4 6 8 10 12 14

0 5 10 15 20 25

Energy (K)

Wavenumber (nm^-1) 20 bar

svp

phonons

rotons

H.J. Maris H.J. Maris noticed that, noticed that, according to the according to the

density density

functional form functional form of Dalfovo et al.

of Dalfovo et al.

,,

the roton gap the roton gap vanishes around vanishes around 200 bar where the 200 bar where the

density reaches density reaches

0.237 g/cm 0.237 g/cm³³ If true, this If true, this

"soft mode" at

"soft mode" at finite wave finite wave vector could vector could

imply

imply an an instability instability

towards a towards a periodic (i.e.

periodic (i.e.

crystalline ?) crystalline ?)

phase phase

(Schneider and Enz (Schneider and Enz PRL 27, 1186, 1971) PRL 27, 1186, 1971)

Vranjes, Boronat et al. (PRL 2005): the roton gap is Vranjes, Boronat et al. (PRL 2005): the roton gap is

3K at 250 bar 3K at 250 bar

instability at higher P (> 300 bar ?) instability at higher P (> 300 bar ?)

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The condensate fraction vanishes

According to According to

Moroni and Moroni and

Boninsegni (JLTP Boninsegni (JLTP

2004), the 2004), the condesnate condesnate

fraction vanishes fraction vanishes exponentially as exponentially as

the density the density

increases.

Same numerical Same numerical results by Vranjes results by Vranjes

etal.

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a new experiment : spherical geometry a new experiment : spherical geometry

at 140 kHz at 140 kHz

R. Ishiguro, F. Caupin and S. Balibar R. Ishiguro, F. Caupin and S. Balibar

submitted to Europhysics Letters, march 2006 submitted to Europhysics Letters, march 2006

2 2 transduce transduce

rsrs

3 cycles 3 cycles

at 140 at 140

kHzkHz Laser beam

Laser beam

lens outside lens outside (f = 20 cm) (f = 20 cm)

Experimental cell Experimental cell

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2 piezo-electric 2 piezo-electric

transducers

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Echoes in a spherical Echoes in a spherical

geometry geometry

0 20 40 60 80 100 120 140

time t (microseconds)

Accurate measurement of Accurate measurement of

the flight time the flight time

tt_f_f = R/c = R/c

and the radius and the radius R = 9.42 +/- 0.02 mm R = 9.42 +/- 0.02 mm

Excitation : 3 cycles at 1.39 MHz Excitation : 3 cycles at 1.39 MHz

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Non-linear sound Non-linear sound

focusing focusing

focusing with a focusing with a non-linear eq. of non-linear eq. of

state state

leads to sharp leads to sharp positive peaks positive peaks (Appert et al.

(Appert et al.

2003) 2003)

0 10 20 30 40 50 60 70

time t - t

f (microseconds)

22.3 bar

0 bar 2 bar 3.9 bar 10.3 bar excitation voltage V(t)

synchronization synchronization confirms R = 9.42 confirms R = 9.42 mmmm

period : 7.6

period : 7.6 mms s corresponding to corresponding to 132 kHz

132 kHz

QuickTime™ et un décompresseur TIFF (LZW) sont requis pour visionner cette image.

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non-linear non-linear

effects effects

0.154 0.156 0.158 0.160 0.162 0.164 0.166 0.168

32 32.5 33 33.5 34

TIME (microseconds)

At large amplitude, positive pressure peaks At large amplitude, positive pressure peaks

appear, due to the curvature of the appear, due to the curvature of the

equation of state equation of state

C. Appert, C. Tenaud, X. Chavanne, S.

Balibar, F. Caupin and D. d’Humières, Eur.

Phys. J. B35, 531 (2003) Phys. J. B35, 531 (2003)

QuickTime™ et un décompresseur TIFF (LZW) sont requis pour visionner cette image.

A fit with a measurement at 9.8 bar A fit with a measurement at 9.8 bar

in a quasi-spherical geometry

in a quasi-spherical geometryCalculation at larger amplitudeCalculation at larger amplitude

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the the equatio equatio

n of n of state state

of of

liquid liquid helium helium

4 4

at T=0

^-20

0 20 40 60 80

0.1 0.12 0.14 0.16 0.18 0.2

DENSITY (g/cm³)

stable

metastable

metastable P₀ = 0

Pm = 25.324 nucleation

liquid - gas spinodal limit

The equation of state of liquid helium 4 (after Abraham 1970 and Maris 1994)

rather well established cubic law:

rather well established cubic law: P - P P - P

_sp_sp

= a = a ( ( - -

_sp_sp

) ₎

³³

(Maris 1991) (Maris 1991)

Lett. 95, 145302 (2005)

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At which time does it At which time does it

nucleate ? nucleate ?

0 5 10 15 20

time t - t

f (microseconds) crystallization

no crystallization

At the At the

threshold, threshold,

the nucleation the nucleation is random with is random with a probability a probability 0.5. 0.5.

bubbles or bubbles or crystals?

crystals?

For an For an accurate accurate

measurement of measurement of the nucleation the nucleation time, we

time, we substract substract

- the flight - the flight time time

- the upper - the upper signal from signal from the lower one the lower one

Acoustic crystallization at P

Acoustic crystallization at P_m_m=25.3 bar and T = 600 mK=25.3 bar and T = 600 mK

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bubbles or crystals?

nucleation times nucleation times

15 20 25

crystallization at 25.3 bar cavitation at 2 bar

time t - t

f (microseconds)

17.5 21.1

At 25.3 bar, At 25.3 bar, nucleation at nucleation at t -t t -t

_f_f

= 21.1 = 21.1 m m s _s

i.e. 2 + 3/4 i.e. 2 + 3/4 periods, where periods, where

P P

_max_max

is reached is reached 3.6 3.6 m m s , i.e. _{s , i.e.}

half a period half a period

later than later than nucleation at nucleation at low pressure low pressure (cavitation in (cavitation in

the negative the negative

swing) swing)

crystals ! crystals !

tt_f_f = flight time to the acoustic focus = flight time to the acoustic focus

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Pressure Pressure dependence dependence

23.5 24 24.5 25 25.5

18 nov 05 19 nov 05 29 nov 05 30 nov 05 1 dec 05 26 dec 05

0 2 4 6 8 10 12

Pressure (bar)

Pm

close to close to the liquid- the liquid-

solid solid

eq. pressure eq. pressure

P P

_m_m

, ,

the crystals the crystals

grow larger

grow larger crystals ! crystals !

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What is the pressure at which What is the pressure at which

crystals nucleate ? crystals nucleate ?

P > 160 bar if Werner et al. were right but P > 160 bar if Werner et al. were right but their interpretation assumed linear focusing their interpretation assumed linear focusing non-linear effects make the measurement of non-linear effects make the measurement of P difficult

P difficult

Brillouin scattering (in progress):

Measure the local instantaneous pressure Measure the local instantaneous pressure

Calculate P from the sound velocity and the Calculate P from the sound velocity and the

known equation of state P(

known equation of state P(   ) ₎

a possible relation with the predicted instability a possible relation with the predicted instability

where rotons become soft collective modes ? Raman scattering where rotons become soft collective modes ? Raman scattering

superfluidity at high density?

a Brillouin line corresponding to second sound ?

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mass flow through solid mass flow through solid

helium helium

R. Ishiguro , F. Caupin, H.J. Maris*

and S. Balibar and S. Balibar work in progress at work in progress at

Laboratoire de Physique Statistique Laboratoire de Physique Statistique

(ENS-Paris) (ENS-Paris)

* Brown University, Providence (RI,

* Brown University, Providence (RI, USA) USA)

APS march meeting, APS march meeting,

Baltimore 2006 Baltimore 2006

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Motivation and goal

Unlikely but possible critics on 2 previous flow experiments:

. . . . . .

cryst cryst

alal liqui liqui

dd Day, Herman and Beamish (PRL

Day, Herman and Beamish (PRL 2005)

2005)

flow in Vycor glass flow in Vycor glass

the lattice is probably pinned at the lattice is probably pinned at low T,

low T,

mass flow requires motion of the mass flow requires motion of the lattice

lattice

But probably not in the new expt But probably not in the new expt through capillaries (condmat jan through capillaries (condmat jan 06)06)

Bonfait, Godfrin and Castaing (J.

Physique 1989) Physique 1989)

growth inside a thin capacitor growth inside a thin capacitor at T < 20 mK

at T < 20 mK

blocked by a facet at the blocked by a facet at the entrance ?

entrance ?

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experimental setup

Fill a test tube (1 cm Fill a test tube (1 cm

) at 1.3 K_{) at 1.3 K}

lower T down to 100 mK lower T down to 100 mK

melt the outside melt the outside follow the level follow the level

inside inside

P = _{P =}gh = 2.10_{gh = 2.10}^-5^-5 bar bar melting velocity V = 1 melting velocity V = 1

cm/hcm/h

if critical velocity if critical velocity 30 30 mmm/s and superfluid m/s and superfluid

density

density _s_s / _/_c_c = 10_{= 10}^-2^-2

V V

liquid liquid

solid solid

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solid

⁴⁴

He does He does not flow at not flow at

100 mK 100 mK

with a mass flow with a mass flow at the critical at the critical velocity

velocity v v_c_c ~ 30 ~ 30 mmm/s m/s , and , and _s_s / _/

_c_c = 10_{= 10}^-2^-2 the the

interface should interface should move by 1 cm in 1 move by 1 cm in 1 hourhour

=> (

=> (_s_s / _/_c_c)₎ vv_c_c < <

0.3 nm/s 0.3 nm/s

liquid liquid

solid solid Inside a test

Inside a test tube

tube (1 cm

(1 cm ) : no _{) : no} measurable flow measurable flow over 3 hours at over 3 hours at 100 mK

100 mK

R. Ishiguro and S. Balibar, ENS-Paris, 7 march 2006 R. Ishiguro and S. Balibar, ENS-Paris, 7 march 2006 ::

one needs to apply a heat pulse to push the crystal one needs to apply a heat pulse to push the crystal

inside the tube at 1.3 K => a few defects inside the tube at 1.3 K => a few defects

No flow at the glass / He interface either No flow at the glass / He interface either

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No growth either

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No growt

h

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a bad quality crystal grown from the normal liquid phase at high T (

2 K)

Ryosuke Ishiguro, ENS-Paris, 13 march 2006 Ryosuke Ishiguro, ENS-Paris, 13 march 2006

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a bad quality He4 crystal

Ryosuke Ishiguro, ENS-Paris, 13 march 2006 Ryosuke Ishiguro, ENS-Paris, 13 march 2006

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Pressure dependence (Kim and Chan)

• As a function of pressure the supersolid fraction shows a

maximum near 55bars. The supersolid fraction extrapolates

to zero near 170 bars.

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The pressure dependence of the

« supersolidity »:

a possible interpretation

In the range 25 to 55 bar, the In the range 25 to 55 bar, the

number of defects (gain number of defects (gain

boundaries) increases due to boundaries) increases due to

the crystal preparation method the crystal preparation method

(constant volume) (constant volume)

The grain boundaries could be The grain boundaries could be

liquid and have a superlfuid liquid and have a superlfuid

transition in the range 50 to transition in the range 50 to

200 mK 200 mK

Superfluidity in the liquid Superfluidity in the liquid grain boundaries disappears grain boundaries disappears

around 200 bar

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mass flow inside

³

He crystals near 0.32K where E

_vac

< 1K

the latent heat L is negligible, T is highly homogeneous the latent heat L is negligible, T is highly homogeneous

local growth and melting according to gravity, surface tension, and curvature, no facets local growth and melting according to gravity, surface tension, and curvature, no facets

the crystal seems to flow down in less than 1 minute but the lattice is immobilethe crystal seems to flow down in less than 1 minute but the lattice is immobile there must be an inverse flow of vacancies

there must be an inverse flow of vacancies

dripping ( c, g ...) + coalescence ( f, j ...) dripping ( c, g ...) + coalescence ( f, j ...) of single crystals with identical orientation of single crystals with identical orientation except for the last drop (k,l)

except for the last drop (k,l)

Graner et al.

Graner et al. J. Low Temp. Phys. 75, 69 (1989)J. Low Temp. Phys. 75, 69 (1989) Ishiguro et al. PRL 93, 235301 (2004) Ishiguro et al. PRL 93, 235301 (2004)

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Motion of fluid drops inside

⁴

He single crystals

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décompresseur miroMotion JPEG A sont requis pour visionner cette image.

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heterogeneous nucleation with an

electric field

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nucleation of solid nucleation of solid

helium helium

heterogeneous nucleation heterogeneous nucleation occurs

occurs

~ 3 to 10 mbar above P

~ 3 to 10 mbar above P_m_m

(Balibar 1980, Ruutu 1996, (Balibar 1980, Ruutu 1996, Sasaki 1998)

Sasaki 1998)

Balibar, Mizusaki and Sasaki Balibar, Mizusaki and Sasaki

(J. Low Temp. Phys. 120, 293, 2000) (J. Low Temp. Phys. 120, 293, 2000): : it cannot be homogeneous

it cannot be homogeneous nucleation

nucleation,,

since E = 16/3

since E = 16/3 aa³³/_/P_P²² ≈ 10 ≈ 10¹⁰¹⁰ K K

!!

heterogeneous nucleation on heterogeneous nucleation on

favorable sites (graphite dust favorable sites (graphite dust particles ?)

particles ?)

J.P. Ruutu et al., Helsinki, 1996 J.P. Ruutu et al., Helsinki, 1996 consistent with other measurements by consistent with other measurements by

Balibar (1980), Sasaki (1998) Balibar (1980), Sasaki (1998)

pressurizing liquid helium in an ordinary cell:

  acoustic crystallization : eliminate heterogeneous nucleation ? acoustic crystallization : eliminate heterogeneous nucleation ?

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standard theory

(Landau and Lifshitz, Stat. Phys. (Landau and Lifshitz, Stat. Phys.

p553) p553)

: :

the barrier against nucleation is due to the barrier against nucleation is due to

the surface energy the surface energy

a spherical nucleus a spherical nucleus with

with

radius R

radius R and and surface surface energy

energy gg (the _(the

macroscopic surface macroscopic surface tension on the eq.

tension on the eq.

line) line)

F(R) = 4

F(R) = 4   R _R

²²

g g - 4/3 _{- 4/3}

  R _R

³³

  P _P

P : difference in P : difference in free energy per unit free energy per unit volume between the 2 volume between the 2 phases

phases

Critical radius : R

Critical radius : R_c_c = = 2 2 ggP_P

Activation energy : Activation energy : E = (16

E = (16 g g

³³

)(3 )(3 P _P

²²

) )

nucleation rate per unit time and volume :



_

exp(-E/T)

_ : attempt frequency . density of independent sites

-100 0 100 200

0 0.5 1 1.5 2

Bubble radius R (nanometers)

P_l = - 10 bar

P_l= - 6 bar

R_c=2γP

R₁3γP E16πγ³3P²

P_l P_v R

ex : cavitation in liquid helium 4

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superfluidity at high superfluidity at high

density ? density ?

The density 0.237 g/cm

The density 0.237 g/cm³³ is 35 % more than is 35 % more than the maximum density of stable liquid helium the maximum density of stable liquid helium (0.175 g/cm

(0.175 g/cm³³) )

24% more than the density of solid He4 at 25 24% more than the density of solid He4 at 25 bar (0.191 g/cm

bar (0.191 g/cm³³))

Exchange becomes more difficult as the Exchange becomes more difficult as the density increases

density increases

The condensate fraction vanishes, according The condensate fraction vanishes, according to both Vranjes et al. and Moroni et al.

to both Vranjes et al. and Moroni et al.

Does superfluidity disappear when the roton Does superfluidity disappear when the roton gap vanishes ?

gap vanishes ?

an open question: see an open question: see P. Nozieres,

P. Nozieres,

J. Low Temp. Phys. 137, 45, (2004) and 142, J. Low Temp. Phys. 137, 45, (2004) and 142, 91 (2006)

91 (2006)