Pierre-Gilles de Gennes Pierre-Gilles de Gennes and the puzzle of supersolidity and the puzzle of supersolidity

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Pierre-Gilles de Gennes Pierre-Gilles de Gennes

and the puzzle of supersolidity and the puzzle of supersolidity

Sébastien Balibar

Sébastien Balibar , Frédéric Caupin , Frédéric Caupin and Satoshi Sasaki

and Satoshi Sasaki

Laboratoire de Physique Statistique (ENS-Paris) Laboratoire de Physique Statistique (ENS-Paris)

De Gennes Days, 16 May 2008 De Gennes Days, 16 May 2008 for references, articles and movies, see :

for references, articles and movies, see : http://www.lps.ens.fr/~balibar/http://www.lps.ens.fr/~balibar/

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PG de Gennes PG de Gennes

9 / 02 / 1995 9 / 02 / 1995

« « Le Palais de la Découverte Le Palais de la Découverte est l'un des grands centres européens est l'un des grands centres européens d'initiation à la science. Installé avenue Franklin-Roosevelt, depuis d'initiation à la science. Installé avenue Franklin-Roosevelt, depuis sa fondation par Jean Perrin en 1937, il a marqué des générations sa fondation par Jean Perrin en 1937, il a marqué des générations de lycéens. Nous sommes nombreux à y avoir découvert notre

de lycéens. Nous sommes nombreux à y avoir découvert notre

passion pour la recherche grâce à des expériences simples et bien passion pour la recherche grâce à des expériences simples et bien expliquées.

expliquées.

L'expulsion du Palais de son bâtiment serait un désastre. » L'expulsion du Palais de son bâtiment serait un désastre. » le Palais est en danger, à nouveau. Appel sur

le Palais est en danger, à nouveau. Appel sur http://www.sauvonslepalaisdeladecouverte.fr/

http://www.sauvonslepalaisdeladecouverte.fr/

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Trieste, May 2006

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Trieste, May 2006

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Trieste, May 2006

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Trieste, May 2006

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Trieste, May 2006

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Trieste, May 2006

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what was Pierre-Gilles talking about ?

... as a possible ... as a possible explanation for explanation for Kim and Chan’s Kim and Chan’s

astonishing astonishing

proposal proposal

Nature 427, 225 (2004) Nature 427, 225 (2004)

some of his some of his most recent most recent

ideas...

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a « supersolid » is a solid ...

... which is also superfluid !

a solid a solid : :

transverse elasticity transverse elasticity

non zero shear modulus non zero shear modulus

a consequence of atom localization a consequence of atom localization

crystals - glasses crystals - glasses a superfluid

a superfluid : :

a quantum fluid with zero viscosity a quantum fluid with zero viscosity

a consequence of « Bose-Einstein condensation » a consequence of « Bose-Einstein condensation »

atoms are undistinguishable and delocalized atoms are undistinguishable and delocalized a paradoxical idea

a paradoxical idea

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Could a solid flow like a superfluid ?

E. Kim and M. Chan (Penn. State U. 2004):

a torsional oscillator (

a torsional oscillator (~1 kHz)~1 kHz)

a change in the period of oscillation a change in the period of oscillation below

below ~1~100 mK00 mK

1 % of the solid mass decouples from the 1 % of the solid mass decouples from the oscillating walls ?

oscillating walls ?

Be-Cu

Torsion Rod Torsion Bob containing helium

Drive

Detection

K I

o

π

τ = 2

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some early theoretical ideas

Andreev and Lifshitz 1969:

delocalized defects (vacancies) could exist at T=0 delocalized defects (vacancies) could exist at T=0 ( the crystal would be « incommensurate »)

( the crystal would be « incommensurate ») tunneling frequency

tunneling frequency , coordination number z, coordination number z the bottom of the band could be negative the bottom of the band could be negative

BEC => a fraction of the mass is superfluid => superplasticity BEC => a fraction of the mass is superfluid => superplasticity

EE₀₀ ^{2 z h}^{2 z h}^^

Penrose and Onsager 1956:

generalize Bose Einstein Condensation (BEC) to condensed matter systems generalize Bose Einstein Condensation (BEC) to condensed matter systems Off diagonal long range order in the density matrix

Off diagonal long range order in the density matrix BEC is impossible in a solid

BEC is impossible in a solid

(but they used non-symmetrized wave fonctions) (but they used non-symmetrized wave fonctions)

Reatto and Chester 1969: some models of Q-solids show BEC Reatto and Chester 1969: some models of Q-solids show BEC symmetry of the wave function ?

symmetry of the wave function ? overlap is necessary

overlap is necessary (Imry and Schwartz 1975)(Imry and Schwartz 1975) a large class of Q-solids do NOT show BEC a large class of Q-solids do NOT show BEC Leggett 1970:

Leggett 1970:

non-classical rotation inertia (NCRI) if atoms are delocalized (if there are free vacancies ?) non-classical rotation inertia (NCRI) if atoms are delocalized (if there are free vacancies ?)

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some recent theoretical ideas

Prokofev , Svistunov, Boninsegni, et al. 2005-6 (MC calc.):

no BEC in crystals without free vacancies ;

no BEC in crystals without free vacancies ; ⁴⁴He crystals are commensurate (EHe crystals are commensurate (E_vac_vac = 13K) = 13K) BEC is possible in a

BEC is possible in a ⁴⁴He glass (Boninsegni et al. PRL 2006)He glass (Boninsegni et al. PRL 2006)

controversies with Galli and Reatto 2006 , Clark and Ceperley, Cazorla and controversies with Galli and Reatto 2006 , Clark and Ceperley, Cazorla and Boronat, etc.

Boronat, etc.

possible analogies with the Hubbard model and cold atoms in optical lattices possible analogies with the Hubbard model and cold atoms in optical lattices see I. Bloch, J. Dalibard and W. Zwerger, Rev. Mod. Phys. 2008

see I. Bloch, J. Dalibard and W. Zwerger, Rev. Mod. Phys. 2008

Anderson Brinkman and Huse (Science 2005): 4He crystals are incommensurate ! Anderson Brinkman and Huse (Science 2005): 4He crystals are incommensurate ! a new analysis of the lattice parameter

a new analysis of the lattice parameter a/a (T) and specific heat Ca/a (T) and specific heat C_v_v(T)(T)

not confirmed by new neutron scattering measurements (Blackburn et al. PRB 2007) not confirmed by new neutron scattering measurements (Blackburn et al. PRB 2007) criticized by H.J. Maris and S. Balibar (J. Low Temp. Phys. 147, 539, 2007)

criticized by H.J. Maris and S. Balibar (J. Low Temp. Phys. 147, 539, 2007) Anderson has now switched to a « vortex liquid » model (Nature Physics 2007) Anderson has now switched to a « vortex liquid » model (Nature Physics 2007)

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new experiments:

evidence for the importance of disorder

Rittner and Reppy (Cornell, 2006-7):

annealing destroys supersolid behavior annealing destroys supersolid behavior

quenched cooled crystals : very large « superfluid fraction »

quenched cooled crystals : very large « superfluid fraction » _s_s up to 20 % up to 20 % more experiments (Aoki and Kojima, Clark and Chan, Shirahama...):

more experiments (Aoki and Kojima, Clark and Chan, Shirahama...):

_s_s varies from 0.03 to 20% depending on sample preparation varies from 0.03 to 20% depending on sample preparation

Sasaki, Ishiguro, Caupin, Maris et Balibar (ENS, Science 2006) : Sasaki, Ishiguro, Caupin, Maris et Balibar (ENS, Science 2006) : two communicating vessels filled with solid helium

two communicating vessels filled with solid helium superfluid mass transport

superfluid mass transport onlyonly in the presence of grain boundaries in the presence of grain boundaries

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PGG (CR Physique 7, 561, 2006):

quantum plasticity in a quantum crystal quantum plasticity in a quantum crystal

an edge dislocation an edge dislocation

a weakly bound a weakly bound kink - antikink pair kink - antikink pair

a model at T = 0 : large quantum tunneling of kinks a model at T = 0 : large quantum tunneling of kinks

 weak pairing of kink and antikinksweak pairing of kink and antikinks

 large mobility of dislocationslarge mobility of dislocations

 large mobility of grain boundaries ( « dislocation ladders » )large mobility of grain boundaries ( « dislocation ladders » )

 quantum plasticity at T = 0quantum plasticity at T = 0

 a change in mechanical properties at some finite T ??a change in mechanical properties at some finite T ??

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Pollet et al. PRL 98, 135301, 2007

Grain boundaries are Grain boundaries are ~ ~ 3 atoms thick

3 atoms thick

superfluid for large superfluid for large misorientation

misorientation

not superfluid for small not superfluid for small misorientation

misorientation

edge dislocations are edge dislocations are not superfluid

not superfluid

TT_c_c ~ 0.2 to 1 K~ 0.2 to 1 K depending on depending on orientation orientation

critical velocity ? critical velocity ?

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Boninsegni et al. PRL 99, 035301, 2007 : superfluidity of screw dislocation cores

phase coherence along the core of screw phase coherence along the core of screw dislocations on a distance

dislocations on a distance ~ 1/T~ 1/T a true 1D- supersolid

a true 1D- supersolid

a network of dislocations with density n a network of dislocations with density n_s_s 3D- superfluidity below T

3D- superfluidity below T_c_c ~ 1/l ~ n~ 1/l ~ n_s_s^1/2^1/2

in their model, a very large dislocation density in their model, a very large dislocation density would be needed to build a superfluid density would be needed to build a superfluid density of order 0.1%

of order 0.1%

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in blue: local superfluid density in blue: local superfluid density

a Monte Carlo calculation a Monte Carlo calculation

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G. Biroli and J.P. Bouchaud preprint 2007 G. Biroli and J.P. Bouchaud preprint 2007

the free energy for the creation of kink-antikink pairs should vanish at T the free energy for the creation of kink-antikink pairs should vanish at T_k_k proliferation of kinks

proliferation of kinks

very large transverse fluctuations of dislocations trigger atom exchange in the very large transverse fluctuations of dislocations trigger atom exchange in the bulk solid

bulk solid

supersolidity below some temperature T

supersolidity below some temperature T_c_c < T < T_k_k

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S. Sasaki et al. @ENS Science 313, 1098, 2006

a glass tube (1 cm a glass tube (1 cm ) )

grow a crystal from the superfluid grow a crystal from the superfluid at 1.3 K

at 1.3 K

lower T down to 50 mK lower T down to 50 mK melt the outside => height melt the outside => height difference

difference

follow the level inside

follow the level inside solid heliumsolid helium liquid helium liquid helium

window window

any change in the level inside any change in the level inside requires a mass flow through the requires a mass flow through the solid since

solid since _C_C = 1.1 = 1.1 _L_L

if critical velocity v

if critical velocity v_c_c = 10 = 10 m/s m/s and superfluid density

and superfluid density _s_s = 10 = 10^-2^-2_C_C

=> melting velocity V = 3 mm/h

1 cm1 cm

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filling the tube with solid

⁴⁴

He makes defects He makes defects

solid solid liquid

liquid solidsolid liquidliquid liquid

liquid

the liquid inside the liquid inside crystallizes if a crystallizes if a

substantial stress is substantial stress is applied.

applied.

 grain boundaries grain boundaries

 grooves at the liquid- grooves at the liquid- solid interface

solid interface

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cusps and grain boundaries (GBs)

mechanical equilibrium mechanical equilibrium

of surface tensions of surface tensions

at the liquid-solid interface:

 

_GB_GB

=2 =2

_LS_LS

cos cos  

each cusp signals the existence each cusp signals the existence of an emerging grain boundary of an emerging grain boundary

crystal 1 crystal 1 crystal 2

crystal 2

grain boundary grain boundary liquid phase

liquid phase

 

_GB_GB

22

 

_LS_LS

 

_LS_LS

at Pat P_m_m, most cusps move away in , most cusps move away in a few hours a few hours pinning +

pinning + very fast dynamics of grain boundariesvery fast dynamics of grain boundaries some GBs stay pinned on walls

some GBs stay pinned on walls

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without grain boundaries, no flow

If supersolidity was due to a 1% superfluid If supersolidity was due to a 1% superfluid density in the bulk

density in the bulk

with a critical velocity v

with a critical velocity v_c_c = 10 = 10 m/s m/s the interface should relax at V = 1

the interface should relax at V = 1 m/s, m/s, that is 3.6 mm in 1 hour

that is 3.6 mm in 1 hour

Instead, we see no flow within 50

Instead, we see no flow within 50 m in 4 m in 4 hours, meaning at least 300 times less

hours, meaning at least 300 times less

 supersolidity is not due to the superfluidity of a 1% (even 0.03%) supersolidity is not due to the superfluidity of a 1% (even 0.03%) equilibrium density of vacancies moving at 10

equilibrium density of vacancies moving at 10   m/s. m/s.

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mass flow in crystals with enough grain boudaries

for 3 crystals with some cusps inside the tube we observed a mass flow for 3 crystals with some cusps inside the tube we observed a mass flow crystal 1 : when the cusp disappears, the mass flow stops

crystal 1 : when the cusp disappears, the mass flow stops

superflow of mass through solid

⁴⁴

He is associated He is associated with the existence of grain boundaries with the existence of grain boundaries

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crystal 1 relaxed 1 mm down and stopped

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crystal 2 : many defects

Many grain boundaries Many grain boundaries more in the lower part more in the lower part faster flow down to faster flow down to equilibrium at h = 0 equilibrium at h = 0

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crystal 2 relaxed down to eq. (h = 0)

time

time x x 250 250

5 s = 20 min

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crystal 2:

relaxation relaxation

at 50 mK at 50 mK

0.0 2.0 4.0 6.0 8.0

0 500 1000 1500 2000

crystal #2

time t (seconds)

relaxation is not exponential but linear

relaxation is not exponential but linear with two successive regimes, with two successive regimes, constant velocity : 6

constant velocity : 6 m/s for 0 < t < 500 sm/s for 0 < t < 500 s 11 11 m/s for 500 < t < 1000 sm/s for 500 < t < 1000 s more defects in the lower part of crystal 2

more defects in the lower part of crystal 2

typical of superfluid flow at its critical velocity typical of superfluid flow at its critical velocity

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crystal 1 : a single grain boundary

the relaxation at the relaxation at V = 0.6

V = 0.6 m/sm/s stops stops when the cusp

when the cusp disappears disappears

(the grain boundary (the grain boundary moves away,

moves away,

unpinning from the unpinning from the wall somewhere)

wall somewhere)

7.6 7.8 8.0 8.2 8.4

0 500 1000 1500 2000

height h(t) (mm) crystal #1

time (seconds)

Assume 1 grain boundary (thickness e

Assume 1 grain boundary (thickness e ~ a = 0.3 nm~ a = 0.3 nm , width w ~ D = 1cm , width w ~ D = 1cm) ) the critical velocity inside is

the critical velocity inside is

vv_c_c^GB^GB= (= (DD²²/4ew/4ew_s_s)()(_C_C--_L_L)V = )V = 1.51.5 (a/e)(D/w)((a/e)(D/w)(_C_C / /_s_s)) m/sm/s comparable to 2

comparable to 2 m/sm/s measured by Telschow et al. (1974) on free liquid films of measured by Telschow et al. (1974) on free liquid films of atomic thickness

atomic thickness

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1% superfluid density is large !

(Rittner and Reppy 2007: 20% in thin quenched cooled samples !)

1% matter with grain boundaries 1% matter with grain boundaries

~ ~ 1 atomic layer of superfluid matter1 atomic layer of superfluid matter

 grain size ~ 100grain size ~ 100 nm nm 3 3 m for 0.03%m for 0.03%

Is this possible ? may be Is this possible ? may be

We used to grow single crystals at We used to grow single crystals at constant P from the superfluid, constant P from the superfluid, butbut

crystals grown at constant V from the crystals grown at constant V from the normal liquid are usually polycrystals normal liquid are usually polycrystals with much more disorder

with much more disorder

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a high pressure cell a high pressure cell to grow He crystals at to grow He crystals at

constant V constant V

two cubic cells : 11

two cubic cells : 11 x x 11 11 x 10 mmx 10 mm³³

or 11 or 11 x 11 x 11 x 3 mmx 3 mm³³

thermal contact via 10 mm thick copper walls thermal contact via 10 mm thick copper walls 2 glass windows (4 mm thick)

2 glass windows (4 mm thick) indium rings

indium rings

stands 65 bar at 300K stands 65 bar at 300K

Straty-Adams pressure gauge (0 to 37 bar) Straty-Adams pressure gauge (0 to 37 bar) connection through a 3 cm long CuNi

connection through a 3 cm long CuNi capillary (0.6 mm ID)

capillary (0.6 mm ID)

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at T > 1.8 K from the normal liquid:

growth is dendritic if fast

fast mass injection fast mass injection through the fill line through the fill line in the normal liquid in the normal liquid (here at 1.87 K)

(here at 1.87 K) leads to dendritic leads to dendritic growth

growth

but not slow growth but not slow growth at constant V in a at constant V in a T-gradient

T-gradient

T = 1.87 K T = 1.87 K

11 mm

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more helium

snowflakes ^{T = 2.58 K} ^{T = 2.58 K}

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slow growth at constant volume

slow growth (

slow growth ( ~3 hours) in a ~3 hours) in a temperature gradient

temperature gradient (T (T

_walls_walls

< T < T

_center_center

) )

the solid is the solid is transparent transparent but but

polycrystalline polycrystalline

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hcp solid hcp solid

liquid liquid

liquid liquid 2.56K 2.56K

hcp solid hcp solid

1.95 K 1.95 K

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melting a crystal grown at

constant volume

liquid channels liquid channels appear at the appear at the contact line of contact line of each grain

each grain

boundary with boundary with the windows the windows grain size

grain size ~ ~   m m ripening

ripening

0.04 K 0.04 K

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melting a crystal after fast growth from the

superfluid

a : the fast grown solid a : the fast grown solid

is transparent is transparent but polycrystalline but polycrystalline b to f in 11 seconds b to f in 11 seconds

some bulk some bulk liquid

liquid

appears in f appears in f small size crystal grains small size crystal grains

ripening of the solid foam in ripening of the solid foam in

a few seconds at the a few seconds at the melting pressure melting pressure

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further melting => 2 crystals + 1 grain boundary

the cusp angle

the cusp angle is non-zerois non-zero => the gain boundary energy => the gain boundary energy _GB_GBis strictly < 2 is strictly < 2 _LS_LS

=> partial wetting of the GB by the liquid, the thickness of grain boundaries is

=> partial wetting of the GB by the liquid, the thickness of grain boundaries is microscopic as calculated by Pollet et al. (2007).

microscopic as calculated by Pollet et al. (2007).

complete wetting would imply complete wetting would imply _GB_GB=2=2_LS_LS(2 liq-sol interfaces with bulk liquid in (2 liq-sol interfaces with bulk liquid in between)

between)

a 10 mm thick sample

a 10 mm thick sample a 3 mm thick samplea 3 mm thick sample angle 2

angle 2

the contact of the GB with each window is a liquid channel the contact of the GB with each window is a liquid channel

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angle measurements => the grain boundary energy

here, the GB is parallel to here, the GB is parallel to the optical axis

the optical axis

a fit with Laplace equation a fit with Laplace equation near the cusp leads to

near the cusp leads to   = 14.5 ± 4 ° = 14.5 ± 4 °

  

_GB_GB

  = (1.93 ± 0.04) = (1.93 ± 0.04)  

_LS_LS

other crystals:

  = 11 ± 3 ° = 11 ± 3 °

  = 16 ± 3 ° = 16 ± 3 °

angle 2 angle 2

S.Sasaki et al. PRL 99, 205302 (2007) S.Sasaki et al. PRL 99, 205302 (2007)

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width w ; thickness e width w ; thickness e inversely prop. to the inversely prop. to the depth z (typically depth z (typically 20 20 m at z = 1 cm):m at z = 1 cm):

ll_c_c : capillary length : capillary length

wetting of grain

boundaries near a wall

If If _GB_GBis large enough, is large enough, more precisely

more precisely ifif+ + _c_c< < /2/2

the liquid wets the contact of the GB with the wall.

wetting and premelting of grain boundaries: an wetting and premelting of grain boundaries: an important problem in materials science

important problem in materials science

(see for ex. JG Dash Rep. Prog. Phys. 58, 115, 1995) (see for ex. JG Dash Rep. Prog. Phys. 58, 115, 1995) this effect is responsible for the apparent wetting of this effect is responsible for the apparent wetting of GBs observed with fcc crystals by Franck et al.

GBs observed with fcc crystals by Franck et al.

(Edmonton, 1983-5) (Edmonton, 1983-5)

grain 1 grain 1 grain 2 grain 2 liquid

liquid

wallwall

S. Sasaki, F. Caupin, and S. Balibar, PRL 2007 S. Sasaki, F. Caupin, and S. Balibar, PRL 2007

wallwall

grain 1 grain 1 grain 2

grain 2 ^GB^GB

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hysteresis of the contact angle

advancing angle : 22 ± 6 ° (copper) 26 ± 7 ° (glass) advancing angle : 22 ± 6 ° (copper) 26 ± 7 ° (glass) receding angle : 55 ± 6 ° (copper) 51 ± 5 ° (glass) receding angle : 55 ± 6 ° (copper) 51 ± 5 ° (glass)

more hysteresis on copper rough walls than on smooth glass walls, more hysteresis on copper rough walls than on smooth glass walls,

as expected from E. Rolley and C. Guthmann (ENS-Paris) PRL 98, 166105 (2007) as expected from E. Rolley and C. Guthmann (ENS-Paris) PRL 98, 166105 (2007)

melting

melting growing growing

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the contact line on the window is the contact line on the window is a liquid channel whose width

a liquid channel whose width w w ~ (P-P ~ (P-P

_m_m

) )

^-1^-1

the width w of the triangular the width w of the triangular liquid channel decreases as 1/ z liquid channel decreases as 1/ z

(the inverse of the departure from the equilibrium melting pressure P (the inverse of the departure from the equilibrium melting pressure P_m_m)) consistent with the direct measurement but

consistent with the direct measurement but _c_c is hysteretic is hysteretic the channel should disappear around P

the channel should disappear around P_m_m+ 10 bar (where 2w + 10 bar (where 2w ~1 nm)~1 nm)

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the grain boundary energy

depends on orientation

a stacking fault ? a stacking fault ?

no liquid channel along the no liquid channel along the wall if the GB has a low

wall if the GB has a low

energy (small misorientation)

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two possible interpretations of Sasaki et al. (Science 2006)

the flow could be the flow could be

- either along the GBs (then veither along the GBs (then v_c_c ~ 1 m/s)~ 1 m/s)

- or at the GB-wall contact (then vor at the GB-wall contact (then v_c_c ~ 1 cm/s). ~ 1 cm/s).

This would explain why we saw mass flow up to This would explain why we saw mass flow up to 1.1K at least.

1.1K at least.

- to be checked by changing the shape of the cellto be checked by changing the shape of the cell - or by gluing a piece of graphite on the wallor by gluing a piece of graphite on the wall

liquid liquid liquid

liquid solid solid

solid solid

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next experiment:

measure T

_c

inside a grain boundaries

Pollet et al. (PRL 98, 135301, 2007):

Tc Tc ~ 0.5 K with three layers~ 0.5 K with three layers 3 times less than the bulk T

3 times less than the bulk T__ at the solid density (1.5K) at the solid density (1.5K) a possible measurement:

a possible measurement:

squeeze the liquid channels with an electric field squeeze the liquid channels with an electric field

make a height difference make a height difference

measure the mass flow as a function of T measure the mass flow as a function of T

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Day and Beamish (Nature 2007): a measurement of the shear modulus of crystals grown at constant V

the shear modulus

the shear modulus increases by ~ 15 % below 100 mKincreases by ~ 15 % below 100 mK dislocation pinning by 3He impurity adsorption ? dislocation pinning by 3He impurity adsorption ?

relation to torsional oscillator experiments ? relation to torsional oscillator experiments ?

less inertia for a stiffer crystal ?? a new theoretical challenge less inertia for a stiffer crystal ?? a new theoretical challenge

shear modulus shear modulus

rotational inertia rotational inertia

similar

anomalies

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Clark and Chan :

an increase in stiffness ?

=2

=2 (I/G)(I/G)^1/2^1/2

could the decrease in the could the decrease in the period

period be due to an increase be due to an increase in the shear modulus ,

in the shear modulus ,

consequently the quantity G ? consequently the quantity G ?

c/c c/c ~ ~ 30% for NCRIF = 0.4%30% for NCRIF = 0.4%

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conclusion

- solid helium 4 shows many anomalies at low T but their interpretation is not yet solid helium 4 shows many anomalies at low T but their interpretation is not yet clear

clear

see S. Balibar and F. Caupin, « topical review » see S. Balibar and F. Caupin, « topical review » J. Phys. Cond. Mat. 20, 173201 (2008)

J. Phys. Cond. Mat. 20, 173201 (2008)

- superfluidity of solid helium 4 is not established:superfluidity of solid helium 4 is not established:

check phase coherence check phase coherence

- superfluidity of grain boundaries and dislocation cores is predicted but not yet superfluidity of grain boundaries and dislocation cores is predicted but not yet proved experimentally. New original quantum systems 1D or 2D supersolids.

proved experimentally. New original quantum systems 1D or 2D supersolids.

- dynamics of dislocations and grain boundaries in quantum crystals :dynamics of dislocations and grain boundaries in quantum crystals : solid He is stiffer at low T, apparently not superplastic

solid He is stiffer at low T, apparently not superplastic