Equation of state of liquid mercury up to 7 GPa and 520 K Simon Ayrinhac, Michel Gauthier, Frédéric Decremps, Livia Bove, Marc Morand, Gilles Le Marchand, Frédéric Bergame, Julien Philippe
EHPRG 51 – London – 1-6 september 2013
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Liquid mercury Simple liquid Eslami J.Nucl.Mat. 336 135 (2005) • well described by hard sphere model • no orientational or internal vibrational degrees of freedom Not simple metal • relativistic effects : 6s-5d orbitals hybridization Singh PRB 49 4954 (1994)
contracted 6s orbitals (weak bond liquid) • gradual metal-nonmetal
Norrby, J.Chem.Edu, 68, 110 (1991) transition (~9 g/cm3) F.Hensel, Metal-to-Nonmetal Transitions (book), pp 23-35 (2010)
accurate pair potential needed for simulations Bomont JCP 124 054504 (2006)
the repulsive part of the effective pair potential play a dominant role in M-NM transition S.Munejiri et al JPCM 10 4963 (1998) 2
Equation of state (EOS) • P-V-T relation (unique for equilibrium chemical phase) derived thermodynamical quantities
P
1 V V T P
P BT V V T
B BT' T P T
...
searching of an accurate analytical form V(P,T)
• EOS can be obtained from the velocity measurements at high P Davis & Gordon J. Chem. Phys. 46 2650 (1967) Decremps et al RSI 80 073902 (2009)
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picosecond acoustic technic H.J. Maris, Brown University, 1986
Generation (pump) pulsed laser 80 MHz =800 nm
Detection (probe)
heated region
light impulsion 100 fs
t (delay line)
R (t ) R
RR
photodiode
surface displacements refractive index variations 0
2
4
6
8
10
12
t (ns)
Lock-in amplifier : measure of small changes in the optical reflectivity, R/R 10-5
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picosecond acoustics @ IMPMC B.Perrin et al, Physica B 263, 571 (1999) F.Decremps et al, PRL 100, 3550 (2008)
Ti:sapphire laser /2
PBS1
A.O.M.
pump
delay line (~13.5 ns)
probe
pol.
A
/2
DAC
pol. /2 PBS2
B PBS3 /4
/4
/4
/4 ref.mirror
surface imaging
Michelson interferometer + detection
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Surface imaging Y. Sugawara, O.B. Wright, O. Matsuda, et al PRL 88, 185504 (2002)
in the liquid, at ambiant
duration : 13.2 ns flaser=80MHz Tlaser =12.55 ns 100 m
parallel and undeformed culets homogeneous sample
100 m 6
Movie analysis r/r
l-Hg, p=1 GPa, T=30°C 100
(a.u.)
90
4
80
Radius (µm)
integrated intensity profile @ t=5.2 ns
70
2.8
60
1.5
50 40
0.25
30 20
-1
10 0 0
5
10
15
20
Time (ns)
Tlaser 12.55 ns
25
30
35
Time (ns)
How to extract sound velocity ? 7
acoustic diffraction • spot : ~ 3 m • ~ 0.6 GHz ac~ 1 m
e(t)
R(t) pump
diam.
e0
l-Hg
probe
diam.
R(t ) e 2 (t ) e02 e0 v(t0 pTlaser ) e(t ) v(t pTlaser ) v : sound velocity t0: time when the perturbation reach the surface R(t=t0)=0 : pump-probe coincidence delay (fixed) p : integer (fixed)
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r/r
l-Hg, P=1 GPa, T=30°C 100
(a.u.)
L T
90
4
Radius (µm)
80 70
2.8
fit R(t) : free parameters v and t0 thickness e0
60
1.5
50 40
- reflexions (calculus) 1 and 2 round-trips
0.25
30 20
-1
10 0 0
5
t0
10
15
20
Time (ns)
25
30
35
- surface skimming bulk waves (SSBW) in the diamond
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Adiabatic sound velocity 2200
movies (slow !) peak shift (faster !)
Davis & Gordon T=23°C T=110°C T=193°C T=240°C
Sound velocity (m/s)
2100 2000 1900
[J. Chem. Phys. 46 2650 (1967)]
1800 1700 1600
l-Hg
1500 1400 0
e0 v t0 pTlaser
1
2
3
4
5
6
Pressure (GPa)
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Equation of state P0
...
P1
P2
P2=P1+ P P=0.01 GPa T=1 K
T
input
P
( P, T ) 1 (T )
output P2
v( P, T ) (smoothed and interpolated)
( P0 , T ) P ( P0 , T ) C P ( P0 , T )
Davis J. Chem. Phys. 46 2650 (1967) Daridon et al, International journal of thermophysics 19 145 (1998)
P
dP T 2 2 (T ) 1 (T ) 2 ( P, T )dP v C P 1 P1 P1 2
1 2 2 T p
(smoothed)
repeated until convergence of 2
( P, T ) 0.15% P ( P, T ) 3.8% C P ( P, T ) 3.4%
2 1 ( P P1 ) 1 P2 P1
( P, T )
T Cp 2 P T
1 2 1 T P
C CP2 P ( P2 P1 ) CP 1 P T 11
Results density densité thermal expansion coeff. P
• in good agreement with : Holman J. Phys. Chem. Ref. Data (1994) Grindley J. Chem. Phys. 54, 3983 (1971) • in good agreement with : Holman J. Phys. Chem. Ref. Data (1994) Davis J. Chem. Phys. 46 2650 (1967) 12
bulk modulus BT P BT V V T
first derivative B'T B BT' T P T
BT(P=0) and BT'(P=0) essential parameters for the analytic EOS 13
Equations of state
Murnaghan Kumari-Dass
B' V V0 1 0 P B0
1/ B0'
(Vcalc-Vexp)/Vexp (%)
2 Vcalc Vexp B0 and B0' known (%) _ B0=18.5 GPa Vexp B0'=10.7 1 comparison between EOS
l-Hg, T=240°C
Kumari-Dass Birch-Murnaghan generalized Rose
0
GMA
-1
V [(1 )e ZP ]1/ V0
Taylor expansion
Vinet Onat-Vaisnys
-2 0
Taylor expansion
Murnaghan
V 1 V ' P B0 B0 ( B0 1) V 2 V
1
2
3
4
5
6
7
2
P (GPa)
1/ 3
Vinet
Birch-Murnaghan
V 1 X (1 X ) 3 P 3B0 e ; X ; ( B0' 1) 2 X 2 V0 3B P 0 2
7 5 2 3 3 3 V 0 V 0 3 V 0 ' 1 (4 B0 ) 1 V V 4 V
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Conclusions • surface imaging technic at nanosecond scale • sound velocity (and elastic constants) measurements at high T and P • equation of state of l-Hg up to 7 GPa • Birch-Murnaghan
Perspectives - alkali metals at high density (complex phase diagram, possible liquid-liquid transitions) - geological fluids (l-Fe, FeSi, etc) - metallic glasses (amorphous-amorphous transitions)
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THANK YOU FOR YOUR ATTENTION
EHPRG 51 – London – 1-6 september 2013
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Liquid Hg phase diagram W.Klement, A. Jayaraman, and G. C. Kennedy, Transformations in mercury at high pressures, Phys. Rev. 131 1 (1963)
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