The high frequency dynamics of liquids and supercritical fluids

by Filippo Bencivenga

OUTLINEOUTLINE

• Introduction

• Experimental description

• Data analysis

• Experimental results (Dispersions)

• Experimental results (Relaxations)

• Conclusions

• Outlook

LIQUIDS & SUPERCRITICAL FLUIDS (1)LIQUIDS & SUPERCRITICAL FLUIDS (1)

So

lid

Triple Point

Vapor

Liquid

Critical Point

Supercritical

Temperature

Pre

ssu

re

0 9 18 270

2

4

L (p

s-1

)

Q (nm-1)

c sQ

LIQUIDS & SUPERCRITICAL FLUIDS (2)LIQUIDS & SUPERCRITICAL FLUIDS (2)

Thermodynamicproperties

Microscopic

structure (nm)

Microscopic dynamics (ps)

SC fluids: a few cases

Systematic studies: none

What is missing?

Qm~ 2/r0

What is Known:

AIM OF THE THESIS AIM OF THE THESIS

… what is the role of inter- and intra-molecular interactions ?

… what is the difference between a liquid and a SC fluid ?

From a microscopic point of view …

Microscopic dynamics ( ps - nm)

Liquidphase

Supercriticalphase

0.5 1.0 1.5 2.00.1

1

10

P /

Pc

T / Tc

Supercritical

Liquid

H2O NH3 Ne N2

EXPERIMENTS (1)EXPERIMENTS (1)

Ne N2 NH3 H2O

Pc (bar) 27 34 113 221

Tc (K) 45 126 405 647

/ c

H2O 3.2 1.1

NH3 2.8 1.1

N2 2.7 1.8

Ne 2.6 1.2

in out

in

out

EXPERIMENTS (2)EXPERIMENTS (2)

Ne N2 NH3 H2O

Pc (bar) 27 34 113 221

Tc (K) 45 126 405 647

Large Volume HP Cells• Low pressures ( Kbar)• “Large” samples ( cm3)• Versatility (High-T & Low-T)

Sample

Cell body

Pressureconnector

sample

10 mm

NutCell body

X-raybeam Scattered

beam

Sealingsystem

X-ray beam

Scattered beam

Sample

Q,E

Ni Ein, kin

N o E out,

k out

• Q = |kout – kin| 2 kin sin()

• = E / ħ = (Eout– Ein) / ħ

INELASTIC X-RAY SCATTERING (IXS)INELASTIC X-RAY SCATTERING (IXS)

lengths nm

times ps

Q nm-

1

THz

No / Ni S (Q,)

ħ = 1

IXS SPECTRAIXS SPECTRA T = 87 K

0

700

1400

21005 nm-1

0

1000

2000

30008 nm-1

counts

/ 1

50 s

-16 -8 0 8 160

1000

2000

3000 11 nm-1

(meV)

1.5 meV

N2

Q = 8 nm-1

0

1000

2000

3000107 K

c

ou

nts

/ 1

00

s

0

1500

3000

4500148 K

c

ou

nts

/ 8

5 s

-16 -8 0 8 160

1500

3000

4500

(meV)

191 K

co

un

ts /

55

s

N2

DATA ANALYSISDATA ANALYSIS

S(Q)

S(Q,)=

1 (cTQ)2 m’(Q,)

2 - (cTQ)2 - m’’(Q,)[ ]

2 m’(Q,)[ ] 2+

Free

T = 1/DTQ2 ps

m(Q,t) = 2LQ2(t) + (-1)(cTQ)2exp{-t/T}

S(Q

,)

Hydrodynamics

Low-(Q,) limit

Rayleigh-Brillouin Spectrum

(c∞2

- cT2)Q2exp{-t/} + 2(t)

m (Q,t) = (-1)(cTQ)2exp{-t/T} + 2(t)

+ (c∞2

- cT2) Q2

exp{-t/}

Free

Free Free

Fix Fix

Thermal relaxation

Structuralrelaxation

Instantaneousrelaxation

EoS

L(Q) max [2 S(Q,)]

Q

SOUND DISPERSIONS & RELAXATIONS (1)SOUND DISPERSIONS & RELAXATIONS (1)

Visco-Elasticity:

1) Low-Frequency limit:

c0=cs=1/2cT

2) High-Frequency limit: c∞

“High” and “low” frequency

is with respect to -1

-1

L(Q)

c 0Q

Fully relaxed:viscous

Fully unrelaxed:elastic

c ∞Q

L(Q

)

Positive sound dispersion

Structuralrelaxation

Q

-1

L(Q

)

csQ

SOUND DISPERSIONS & RELAXATIONS (2)SOUND DISPERSIONS & RELAXATIONS (2)

Isothermal transition:

“High” and “low” frequency is

with respect to T-1

1) High-Frequency limit:

c∞=cs=1/2cT

2) Low-Frequency limit: c0=cT

Q

-1

L(Q

)

Relaxed

Unrelaxed

Q

L

(Q)

Unrelaxed:adiabatic

c ∞Q L

(Q) T

Relaxed:isothermal

c 0Q

Negative sound dispersion

c sQ

Structural and thermal relaxations:

competing dispersive effects

Structuralrelaxation

Thermalrelaxation

StructuralrelaxationT

-1 = DTQ2

RESULTS (DISPERSIONS)RESULTS (DISPERSIONS)

Good agreement with S(Q)

measurements

0 4 8 120

3

6

9

(m

eV)

Q (nm-1)

T(Q) = cT(Q)QMS(Q)

kBTQ2

T2(Q) =

s(Q) = √T(Q)

∞(Q) = c∞(Q)Q

L(Q)

N2 @ 400 barT/Tc = 0.69

L =

0 8 16 240.0

0.3

0.6

0.9

S (

Q)

Q (nm-1)

0 4 8 120.05

0.1

0.5

(Q) = e

-AQ

(Q)

(ps)

Q (nm-1)

L(Q) max [2 S(Q,)]

0 5 10 15

0

2

4

6T/T

c=1.51

(m

eV)

Q (nm-1)

T

D TQ

2

DISPERSION RELATIONS (NDISPERSION RELATIONS (N22))

L = DTQ2

0 4 8 120

3

6

9T/T

c=0.85

Q (nm -1)

(m

eV)

L =

0

3

6T/T

c=1.17

(m

eV) L ~ s

0

3

6

9 T/Tc=0.76

(meV

) L = L

∞

s

T

0

4

8

12 0.92

(m

eV)

DISPERSION RELATIONSDISPERSION RELATIONS

0

7

14

21T / T

c = 0.57

(m

eV

)

0

6

12

18 0.85

(m

eV

)

0

5

10

15

(m

eV)

T / Tc = 0.76

0

3

6 T / Tc = 0.72

(m

eV)

0

2

4

(m

eV)

1.37

H2O NH3 Ne

c ∞

L

s

L ~ s

D T

Q2

0 4 8 120

1

2

3

Q (nm-1)

(m

eV)

1.62

0 4 8 120

4

8

12

Q (nm-1)

1.09

(me

V)

L=DTQ2

T

L=DTQ2L=DTQ2

0 4 8 12 160

4

8

12

Q (nm-1)

(m

eV)

0.93

COMMON PHENOMENOLOGY (1)COMMON PHENOMENOLOGY (1)

0.1 1 10

0.00

0.25

0.50

0.75

1.00

M (Q

)

L(Q)(Q)

M(Q)=0

Viscous

M(Q)=1Elastic

QQ

QQQM

s

sL22

22

)(

0.1 1 10

0.00

0.25

0.50

0.75

1.00 H

2O

NH3

N2

Ne

M (

Q)

L(Q)(Q)

QQ

QQQM

s

sL22

22

)(

M(Q)=0Viscous

M(Q)=1Elastic

0.1 1 10

0.00

0.25

0.50

0.75

1.00

MT (Q

)

'L(Q)

T(Q)

MT(Q)=0

Isothermal

MT(Q)=1Adiabatic

COMMON PHENOMENOLOGY (2)COMMON PHENOMENOLOGY (2)

MT(Q)=1

Adiabatic

MT(Q)=0

Isothermal

2/1

22

222'

11

QQ

QQQQQ

L

sLL

Dispersive effect of structural relaxation

s(Q) ∞(Q) s(Q) T(Q)

>>2(Q) = ∞

2(Q) -s2

(Q) T2(Q) = s

2(Q) -T2

(Q)

Structural relaxation Thermal relaxation

Vs.

QQ

QQQM

Ts

TLT 22

22'

)(

QQ

QQQM

Ts

TLT 22

22'

)(

0.1 1 10

0.00

0.25

0.50

0.75

1.00 H

2O

NH3

N2

Ne

MT

(Q)

'L(Q)

T(Q)

CONCLUSIONS (SOUND DISPERSION)CONCLUSIONS (SOUND DISPERSION)

Common evolution with T:

• Evidence of a systematic disappearance of the positive dispersion, related to the structural relaxation, close to Tc

• First experimental observation of an adiabatic to isothermal transition of sound propagation, associated to the thermal relaxation.

STRUCTURAL RELAXATION TIMESTRUCTURAL RELAXATION TIME

0.6 1.0 1.4 1.8 2.2

0.1

1

(

0) (p

s)

T c / T

Supercritical Liquid

• H2O• NH3

• N2

• Ne

H2O 12 +/- 0.8

NH39.3 +/- 1.3

N20.55 +/- 0.16

Ne 0.27+/- 0.12

E (KJ/mol)

(0) exp{E/kBT}(Q)= (0)exp{-AQ} A ≈ 0.2 ÷ 0.05 nm

RESULTS (RELAXATIONS)RESULTS (RELAXATIONS)

cs

c∞=(c )2

sQ∞Q

=[cQ Q]2

COMPLIANCE RELAXATION TIMECOMPLIANCE RELAXATION TIME

0 3 6 90

3

6

9

(m

eV)

Q (nm-1)

∞(Q

) = c ∞

Q

EoS

-4 -2 0 2 4 (meV)

S(Q

,) (a

.u.)

c -1(Q)

Q 00.5 1.0 1.5 2.00.1

1

c (p

s)

T c / T

Supercritical Liquid

• H2O• NH3

• N2

• Ne

0.6 1.0 1.4 1.8 2.2

0.1

1

(

0) (p

s)

T c / T

< >

√(d4/M)*(2T)< >

1

0.6 1.0 1.4 1.8 2.2

10

100

C /

< >

Tc / T

• H2O• NH3

• N2

• Ne

Supercritical Liquid

o P. Giura et al.;

Unpublished (2006)

COMMON PHENOMENOLOGYCOMMON PHENOMENOLOGY

0.4 0.7 1.0 1.3 1.6 1.90

2

4

6

T / Tc

2 (

106

m2 /s

2 )

STRUCTURAL RELAXATION STRENGTHSTRUCTURAL RELAXATION STRENGTH

= c∞

2- cs2 =C

• H2O• NH3

• N2

• Ne

< C > a(Pa m6/mole2)

H2O 1.90 +/- 0.02 0.551

NH3 1.71 +/- 0.02 0.423

N2 0.95 +/- 0.02 0.136

Ne 0.16 +/- 0.03 0.021

Lines density

CONCLUSIONSCONCLUSIONS

Common phenomenology:

• Negative sound dispersion Thermal relaxation

• Positive sound dispersion Structural relaxation

• Activation behavior (≈ bond’s energy) of below Tc

• Collision-like behavior of (c) above Tc

• density(correlation with the parameter “a”?)

Structural relaxation related to intermolecuar interactions

OUTLOOKOUTLOOK

0.6 1.0 1.4 1.8 2.2

10

100

C /

< >

Tc / T

Extend the Tc/T range:• high-T for H2O & NH3

• low-T for Ne & N2

H2O NH3 N2 Ne

Other classes of fluids !

ACKNOWLEDGEMENTSACKNOWLEDGEMENTS

• M. Krisch, F. Sette, G. Monaco and all the ID28-ID16 staff (ESRF)

• A. Cunsolo, L. Melesi (ILL)

• G. Ruocco (Universitá “La Sapienza”, Roma)

• L. Orsingher (Universitá di Trento)

• A. Vispa (Universitá di Perugia)

IXS BEAMLINE (ID-28)IXS BEAMLINE (ID-28)

sample

Undulators

MonochromatorSi (n,n,n)

E/E ≈ 10-2

Pre-Monochromator

Si (1,1,1)

≈

E/E ≈ 10-4

75 m

Detecto

r6.5 m

Analyzer

Si (n,n,n)

h Ei(KeV) E (meV) Flux (p/s)

8 15.82 5.5 9 * 1010

9 17.79 3 3 * 1010

11 21.75 1.5 7 * 109

Q

5 Analyzers

Si (n,n,n)

5 Detecto

rs Toroidal mirror

B

E/E ≈ 10 -8

B T-scan ≈ mK

STATIC STRUCTURE FACTORSSTATIC STRUCTURE FACTORS

0 8 16 240.0

0.2

0.4

0.6

0.8

S (

Q) (

a.u

.)

128 K

Q (nm -1)

0 8 16 240.0

0.2

0.4

0.6

0.8

N2 @ 128.5 K

S(Q

) (a

.u.)

Q (nm-1)

400 bar 130 bar 40 bar

0 8 16 240,0

0,5

1,0

1,5

2,0

S (

Q) (

a.u

.)

447 K293 K

Q (nm -1)

H2O @ 400 bar

0 8 16 240.0

0.2

0.4

0.6

0.8

1.0

Ne @ 200 bar

S(Q

) (a

.u.)

Q (nm-1)

32 K 61 K 82 K

N2 @ 400 bar

M(∞)M(0)=c

cs

c∞=(c )2

sQ∞Q

=[cQ Q]2

0.01 0.1 1 10 100

16

19

22

25

Re[M

()]

M(0)

M(∞)= -1

0.04

0.05

0.06

Re[M

-1()]

COMPLIANCE RELAXATION TIMECOMPLIANCE RELAXATION TIME

0 3 6 90

3

6

9

(m

eV)

Q (nm-1)

∞(Q

) = c ∞

Q

-4 -2 0 2 4 (meV)

S(Q

,) (

a.u.

)

= c-1

M-1(0)

M-1(∞)

EoS

c -1(Q)

INSTANTANEOUS RELAXATIONINSTANTANEOUS RELAXATION

H2O 0 7 meV

NH3 0 5 meV

N2 0 1.5 meV

Ne 0 meV

2(Q)(t)

300 500 7000

3

6

9

290 370 4500

2

4

6

T (K)

H2O

5 nm-1 8 nm-1

11 nm-1

T (K)

NH3

8 nm-1

11 nm-1

(Q

) (m

eV)

Lines density

(Q) (Q)(Q)

Intramoleculardegree of freedom?

(Q)(t)

(Q) << ps

(Q)exp{-t/(Q)}

(Q) = g(Q)

0 4 8 120

2

4

6

0 4 8 120

3

6

9 H2O

Q (nm-1)

NH3

Q (nm-1)

g(Q

) (m

eV*c

m3 /g

)

g(Q) = <(Q)/>

(Q) Q

(Q)

(Q)

(Q) const

VISCOSITY (Q-dependence)VISCOSITY (Q-dependence)

0 4 8 12

50

100

500

0 4 8 1210

100

1000

0 4 8 12

100

1000

0 4 8 120.1

1

10

Ne

32 K 42 K 81 K

L (

Q)

(Pa*s

) L (Q

) (Pa*s

)N2

96 K 148 K 190 K

L (

Q)

(Pa*s

)

NH3

Q (nm-1)

293 K 373 K 414 K

L (Q

) (Pa*s

)H2O

Q (nm-1)

293 K 367 K 549 K

L(Q) = Lexp{-BQ}L(Q) = [(Q) (Q) + (Q) / Q2]

VISCOSITY (T-dependence)VISCOSITY (T-dependence)

30 50 70 900

150

300

450

30 50 70 900

2

4

90 120 150 1800

250

500

750

90 120 150 180

1

3

5

290 340 390 4400

400

800

1200

300 340 380 420

2

4

6

300 400 500 600 7000

2

4

6

300 400 500 600

2

4

6

Ne

(

Pa*s

)

L/

S

T (K)

(P

a*s

)

N2

L/

S

T (K)

(

Pa*s

)

NH3

T (K)

L/

S

T (K)

(P

a*s

)

H2O

T (K)

T (K)

L/

S

L / S constant

OUTLOOKOUTLOOK

Disappearance of the positive dispersion?

Study high T/Tc and P/Pc region of the

SC plane

Supercritical

Liquid

0.1

1

10

100

1000

0.4 31

P /

Pc

T / Tc

Vapor

O2

F. Gorelli et al.;

Unpublished (2005)

H2O NH3 N2 Ne

i +i + …

S(Q)

S(Q,)= i +

i +[

-1Im1 [

m2(Q,)

F(Q,t-t’)m3(Q,t-t’)m2(Q,t-t’)m1(Q,t-t’)m2(Q,t’)m3(Q,t’)m4(Q,t’)m1(Q,t’)dm1(Q,t)dm2(Q,t)dm3(Q,t)dF(Q,t)

THEORETICAL FORMALISMTHEORETICAL FORMALISM

MEMORY FUNCTION

S (Q,)Time FourierTransform

F(Q,t)=<(Q,0)(Q,t)>

(Q,t)=jeiQRj(t)

dt= ∫

0

t

MEMORYEQUATION

dt’

0 5 10 150.2

0.4

0.6

0.8

1.0

2 (Q

) /

[ T

2 (Q)+

2 (Q)]

293 K 373 K 414 K 444 K

Q (nm-1)

0 5 10 150.5

0.6

0.7

0.8

2 (Q

) /

[ T

2 (Q)+

2 (Q)]

32 K 41 K 71 K 81 K

Q (nm-1)

0 5 10 15

0.6

0.8

1.0

87 K 107 K 148 K 190 K

2 (Q)

/ [ T

2 (Q)+

2 (Q)]

Q (nm-1)

0 4 8 12 16

0.2

0.4

0.6

0.8

1.0

293 K 423 K 600 K 660 K 706 K

2 (Q

) /

[ T

2 (Q)+

2 (Q)]

Q (nm-1)

RELATIVE RELAXATION STRENGTHRELATIVE RELAXATION STRENGTH

H2O

N2

NH3

Ne

0 5 10 15

0.4

0.6

0.8

1.0

87 K 107 K 148 K 190 K

Q (nm-1)

2

/ [

T

2 T+

2 ]

0 5 10 15

0.2

0.4

0.6

0.8

1.0

2

/ [

T

2 T+

2 ]

293 K 373 K 414 K 444 K

Q (nm-1)

0 5 10 15

0.6

0.8

1.0 32 K 41 K 71 K 81 K

2

/ [

T

2 T+

2 ]

Q (nm-1)

0 4 8 12 160.0

0.2

0.4

0.6

0.8

1.0

293 K 423 K 600 K 660 K 706 K

2

/ [

T

2 T+

2 ]

Q (nm-1)

RELATIVE RELAXATION AMPLITUDERELATIVE RELAXATION AMPLITUDE

H2O

N2

NH3

Ne

0 5 10 15

-1.5

-1.0

-0.5

0.0

87 K 107 K 148 K 190 K

Q (nm-1)

L(Q

) -

s(Q)

- F

( L )

(m

eV

)

0 5 10 15

-4

-2

0

293 K 373 K 414 K 444 K

L(Q

) -

s(Q

) -

F( L )

(m

eV

)

Q (nm-1)

0 5 10 15

-1.5

-1.0

-0.5

0.0

L(Q

) -

s(Q

) -

F( L )

(m

eV

)

32 K 41 K 71 K 81 K

Q (nm-1)

0 4 8 12 16

-6

-4

-2

0

293 K 494 K 549 K 600 K 660 K 706 K

L(Q

) -

s(Q

) -

F( L )

(m

eV

)

Q (nm-1)

NEGATIVE DEVIATIONSNEGATIVE DEVIATIONS

H2O

N2

NH3

Ne

0 9 18 270

2

4

L (p

s-1

)

Q (nm-1)

Liquid Indium

S(Q)

0 8 16 240.0

0.3

0.6

0.9

128 K

S (Q

)

87 K

Q (nm -1)

Nitrogen

S(0) T

Qm -1/3

<r2>

-1

cQ

LIQUIDS & SUPERCRITICAL FLUIDSLIQUIDS & SUPERCRITICAL FLUIDS

Microscopic structure (nm)

Microscopic dynamics (ps)

Thermodynamics

Qm~2/r0

VISCOELASTICITYVISCOELASTICITY

A (t)

t

P (t)

t

A (t)

t

viscouselastic

relaxation time

RELAXATION TIME (Q-dependence)RELAXATION TIME (Q-dependence)

(Q) = (0)exp{-AQ}

0 4 8 120.1

0.2

0.4

0 4 8 12

0.05

0.1

0.5

0 4 8 120.01

0.1

0 4 8 120.01

0.1

1

Ne

41 K 61 K

(Q

) (p

s)

N2

(Q) (p

s)

96 K 148 K

NH3

(

Q) (p

s)

293 K 373 K

Q (nm-1)

H2O

(Q) (p

s)

293 K 367 K 549 K

Q (nm-1)

RELAXATION TIME (T-dependence)RELAXATION TIME (T-dependence)

11 21 31

0.2

0.3

0.4

0.5

4 8 12

0.3

0.4

0.5

2.3 2.8 3.3

0.05

0.1

0.5

1.7 2.5 3.3

0.1

1

30 50 70 900.03

0.07

0.11

0.15

90 125 160 1950.06

0.11

0.16

0.21

290 340 390 4400.01

0.06

0.11

0.16

320 440 560 6800.01

0.08

0.15

0.22

Ne

(

0) (p

s)

(0) (p

s)

N2

(0) (p

s)

NH3

1000 / T (K-1)

(0) (p

s)

H2O

1000 / T (K-1)

A (nm

)

T (K) T (K)

A (nm

)

T (K)

A (nm

)

T (K)

A (nm

)

(0) = 0exp{Ea/kBT}

Tc

Tc Tc

Tc

30 50 70 900.0

0.2

0.4

0.6

90 120 150 1800.0

0.4

0.8

1.2

290 340 390 4400.0

1.5

3.0

4.5

270 420 570 7200

2

4

6

2 (

m2 /s

2 )

Ne

2 (

m2 /s

2 )

2 (m

2/s2)

N2

2 (m

2/s2)

NH3

T (K)

H

2O

T (K)

30 50 70 900.0

0.2

0.4

0.6

90 120 150 1800.0

0.4

0.8

1.2

290 340 390 4400.0

1.5

3.0

4.5

270 420 570 7200

2

4

6

2 (

m2 /s

2 )

2 (m

2 /s2 )

2 (m

2/s2)

2 (m

2/s2)

T (K) T (K)

RELAXATION STRENGTHSRELAXATION STRENGTHS

= c∞

2- cs2

Tc

Tc Tc

Tc