Anisimov/Sengers Research Group - 2012
HOW PURE WATER CAN UNMIX
Mikhail Anisimov
Institute for Physical Science &Technology and Department of Chemical &
Biomolecular Engineering, University of Maryland, College Park
108th StatMech , Rutgers University
December 16, 2012
A JOURNEY FROM HOT TO COLD WATER
ONE SUBSTANCE – TWO DIFFERENT LIQUIDS
Discovery of supercooled water
Supercooled water was first described in 1721 by Fahrenheit.
The air temperature in the thermometer was marked
at fifteen degrees [–9 °C ]. After one hour, I found the
water was still fluid in the ball.
[D. G. Fahrenheit, Phil. Trans. 33, 78 (1724)]
Supercooled water exists in nature
• In clouds, water droplets can be liquid down to about
–38 °C (–36 °F)
• When an airplane flies through a supercooled water
cloud, the droplets will freeze on impact: icing
400
-80
-60
-40
0 °C
-20
300
Pressure (MPa)
Metastable liquid water at –90 °C
stable
liquid
water
supercooled water
(metastable)
200
TH
100
Homogeneous
ice nucleation
TM
no man's land
0
180
200
220
240
Temperature (K)
260
280
Supercooled liquid
Despretz (1837)
1000
Density (kg/m3)
Stable liquid
990
Hare and Sorensen (1987)
980
240
260
Temperature (K)
280
300
WATER: ONE SUBSTANCE – TWO DIFFERENT LIQUIDS
High-temperature water: highly compressible, low dielectric constant, no (or
little) hydrogen bonds, good solvent for organics
Low-temperature water: almost incompressible, very high dielectric constant,
strong hydrogen-bond network, good solvent for electrolytes
Hodge and Angell 1978
100
Dielectric constant of water (IAPWS)
80

60
40
Liquid
20
0
Vapor
0
100
200
Temperature (°C)
300
400
ONE SUBSTANCE – TWO DIFFERENT LIQUIDS:
ISOBARIC HEAT CAPACITY OF LIQUID WATER
10
Supercooled liquid water at 1 bar
Saturated liquid water
Liquid water at 1 bar
Angell et al. 1982
Cp (kJ kg-1 K-1)
8
Red is the prediction by our model
6
4
TH
-50
Tc
TM
0
100
200
Temperature (°C)
300
400
6.0
Heat capacity CP (kJ kg-1 K-1)
Supercooled liquid
5.5
Stable liquid
Angell et al. (1982)
Archer and Carter (2000)
5.0
HEAT CAPACITY OF WATER
UPON SUPERCOOLING
4.5
Anisimov and Voronel (1972)
4.0
240
260
280
300
320
Temperature (K)
340
360
WATER’S POLYAMORPHISM:
Second (liquid-liquid) critical point in water (Poole et al. 1992)
400
-80
-60
Pressure (MPa)
300
-40
0 °C
-20
supercooled
water
200
Brown curve: liquid-liquid
coexistence
Continuation is Widom line,
the line of stability minima
stable
water
dP S

dT V
Fluctuations of volume:
 V  2
100
 T
Fluctuations of entropy:
0
 S  2
C
180
200
220
240
260
 CP
280
Temperature (K)
At the liquid-liquid critical point C the P/T slope is 30 times greater than at
the vapor-liquid critical point of water and negative
C the location of LLCP as recently suggested by Holten and Anisimov, 2012
Mishima’s experiment (2000)
400
-80
-60
-40
0 °C
-20
Ice V
300
Pressure (MPa)
Ice III
stable
water
200
TM
TH
100
Ice I
C
0
180
O. Mishima, PRL 85, 334 (2000)
200
220
240
Temperature (K)
260
280
How can pure liquid unmix?
1. Energy driven: a second minimum or a special
shape of the molecular interaction energy
(vapor-liquid is energy driven: lattice gas, van
der Waals)
2. Entropy driven: a “mixture” of two “states”
with negative entropy of mixing (some
polymer solutions, networks)
3. A combination of both
dP S
H


dT V T V
Clapeyron's equation itself does not
answer whether the liquid-liquid
separation is energy-driven or entropy
driven
[Mishima and Stanley, Nature, 396, 329 (1998)]
TWO-STATE MODEL
A
B,
• Assumption: water is a nonideal “mixture” of two configurations of
hydrogen bonds: high-density/high-entropy state and a low-density/lowentropy state
• The fraction of each state is controlled by thermodynamic equilibrium
• Liquid-liquid phase separation occurs when the non-ideality becomes
strong enough
Suggested equation of state: athermal two-state
model A B
G  G  x(G  G )
A
x
B
A
pure A and B states Gibbs energies
 kT [ x ln x  (1  x )ln(1  x )]
ideal mixing entropy contribution
 kT( P) x(1  x)
non-ideal contribution
molecular fraction of low-density structure B. Equilibrium
fraction is found from
=0
GB  GA
 ln K
kT
K is chemical equilibrium constant of “reaction”
thus x is the extent of the reaction
This liquid-liquid phase separation is driven by non-ideal entropy
Regular-solution unmixing (energy driven) versus
athermal-solution (entropy driven) unmixing
Regular solution (equivalent to lattice gas/Ising model)
Energy-driven phase separation
Interaction parameter
G  GA  x(GB  GA )  kT [ x ln x  (1  x)ln(1  x)]  ( P) x(1  x)
ω determines the critical temperature
The critical pressure is determined
by the reaction equilibrium constant:
GB  GA
 ln K (T , P )  0
kT
A
B,
Athermal solution
Entropy-driven phase separation
G  GA  x(GB  GA )  kT [ x ln x  (1  x)ln(1  x)  ( P) x(1  x)]
ω determines the critical pressure
The critical temperature is determined
by the reaction equilibrium constant:
GB  GA
 ln K (T , P )  0
kT
A
B,
Fraction of low-density structure
x
[1]
mW model simulations: Moore and Molinero, J. Chem. Phys. 130, 244505 (2009).
Liquid-liquid transition is zero ordering field h1. The order parameter is entropy change.
For liquid-gas transition the order parameter is the density change.
Relations between scaling and physical fields for liquid–liquid and liquid–
vapor critical points of water
h1  T  aP
102
Pressure (MPa)
h2
h2  P
h1
100
h1
C2 1  S
h2
C1
2  V  
h1  
h2  T
Liquid
Gas
10–2
1    V
2   (  S )
h1= ln K = 0
h1 and h2 are Ising scaling fields
1 and 2 are scaling densities
1 is the order parameter
10–4
200
400
600
Temperature (K)
800
The scaling field h2 determines whether the transition is energy- or entropy-driven.
If h2 = ΔT, the transition is energy driven. If h2 = -ΔP, the transition is entropy driven.
Heat capacity
100
Angell et al. (1982)
Bertolini et al. (1985)
Tombari et al. (1999)
Archer and Carter (2000)
Our model
CP (J K-1 mol-1)
95
90
85
80
75
230
240
250
260
270
Temperature (K)
280
290
Compressibility
Kanno and Angell (1979)
Mishima (2010)
our model
Compressibility (10-4 MPa-1)
7
H2O
6
TM(melting temperatures)
5
P/MPa
0.1
10
4
50
100
150
190
3
220
240
260
280
Temperature (K)
300
Density
380
200
100
0.1 MPa
Temp. of max. density
x = 0.12
Density of cold and supercooled water.
Black curves are the predictions of the
crossover two-state model.
TH is the homogeneous nucleation
temperature. The red line is the line of
maximum density, the green line is a
constant LDL fraction of about 0.12.
Best description of all available
experimental data achieved to date!
Conclusions
• We accurately describe all property data on supercooled water with a
two-state model based on an athermal mixing of two states. This
model assumes that the liquid-liquid transition in water is entropy
driven.
• Heavy water (D2O) shows similar anomalies and can be described by
our model equally well.
• A regular-solution model (purely energy-driven liquid-liquid phase
separation) does not work well (the description quality is an order of
magnitude worse).
Current Activity
• Application to atomistic models of water and to supercooled aqueous
solutions.
• Adding a solute to supercooled water may move the critical point into
the experimentally accessible region.