ABSTRACT
Title of thesis: A SCALED PARAMETRIC EQUATION OF STATE
FOR THE LIQUID-LIQUID CRITICAL POINT
IN SUPERCOOLED WATER
Daphne Fuentevilla, Master of Science, 2007
Thesis directed by: Professor Mikhail A. Anisimov
Institute for Physical Science and Technology and
Department of Chemical and Biomolecular Engineering
The second-critical-point scenario is one of the most popular explanations
for the anomalous behavior of supercooled liquid water. According to this scenario,
liquid water at ambient conditions is a "supercritical" ?uid that separates into two
types of liquid water in the supercooled region. However, experimental con?rmation
ischallenging. Inthisworkwedevelopedascaledparametricequationofstate, based
on the principle of critical-point universality, to examine the second-critical-point
scenario from a new direction. The equation of state, built on the growing evidence
for liquid-liquid water separation, is universal in terms of theoretical scaling ?elds
and belongs to the Ising-model universality class. The theoretical scaling ?elds
are postulated to be analytical combinations of the physical ?elds, pressure and
temperature. Theequationof stateenablesustoaccuratelylocatethe"Widomline"
(locus of stability minima) and determine that the critical pressure is considerably
lower than predicted by computer simulations.
A SCALED PARAMETRIC EQUATION OF STATE FOR THE
LIQUID-LIQUID CRITICAL POINT IN SUPERCOOLED WATER
by
Daphne Fuentevilla
Thesis submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial ful?llment
of the requirements for the degree of
Master of Science
2007
Advisory Committee:
Professor Mikhail A. Anisimov, Chair/Advisor
Professor Jan V. Sengers
Professor F. Joseph Schork
Professor Sheryl Ehrman
 
 
 
 
 
 
 
 
 
 
 
 
 
 
?Copyright by 
 
Daphne Anne Fuentevilla 
 
2007 
ii
ACKNOWLEDGMENTS
I want to thank all the members of our research group for their help and
support. I particularly want to thank my advisor, Professor Mikhail A. Anisimov
for his technical guidance, his mentorship, and his tolerance of my work with the
Navy. Without his guidance, this work could not have been done. I also wish to
thank Jintao Wang and Heather St. Pierre for showing me the way and helping me
?nd balance in my work.
I wish to acknowledge Pablo Debenedettik, Osamu Mishima, Jan V. Sengers,
H. Eugene Stanley, and Benjamin Widom for their valuable discussions.
I wish to thank the sta? of the Department of Chemical and Biomolecular
Engineering and of the Institute for Physical Science and Technology for their kind
assistance.
I also want to thank my husband, Christopher Dewey, my parents, William
and Carol Fuentevilla, and my grandparents, William and Jean Bryant and Manny
and May Fuentevilla. Their love and support has been unending.
Finally, IwishtoacknowledgesupportfromtheNavalSurfaceWarfareCenter
Carderock Division.
iii
TABLE OF CONTENTS
LIST OF TABLES............................................................ v
LIST OF FIGURES .......................................................... vi
NOMENCLATURE........................................................... vii
CHAPTER 1 INTRODUCTION .......................................... 1
CHAPTER 2 SUPERCOOLED LIQUID WATER:
WHAT IS KNOWN AND WHAT IS THEORIZED ......... 6
2.1 Supercooled Liquids in General . . . . . . . . . . . . . . . . . . . . . 6
2.2 Supercooled Liquid Water . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Three Scenarios for Supercooled Liquid Water . . . . . . . . . . . . . 13
CHAPTER 3 CRITICAL PHENOMENA ................................. 20
3.1 Phase Transitions and the Critical Point . . . . . . . . . . . . . . . . 20
3.2 Universal Critical Phenomena and Scaling Theory . . . . . . . . . . . 24
3.3 Classical versus Nonclassical (Scaling) Theory of Phase Transitions . 26
3.4 Translating Theoretical Variables into Physical Variables with "Com-
plete Scaling" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
CHAPTER 4 APPLYING CRITICAL POINT UNIVERSALITY TO SU-
PERCOOLED LIQUID WATER ........................... 40
4.1 Relations Between Theoretical and Physical Fields . . . . . . . . . . . 40
4.2 Scaled Parametric Equation of State . . . . . . . . . . . . . . . . . . 46
CHAPTER 5 PREDICTING THERMODYNAMIC PROPERTIES OF SU-
PERCOOLED LIQUID WATER ........................... 52
CHAPTER 6 SUMMARY ................................................ 57
APPENDIX I: TABLES...................................................... 59
APPENDIX II: FIGURES.................................................... 63
iv
APPENDIX III: DERIVATIONS ............................................. 108
REFERENCES CITED.......................................................109
 v
List of Tables 
 
Table 1.  Classical and Non-classical Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . 59 
 
Table 2.  Universal Theoretical Variables for Scaling Theory . . . . . . . . . . . . . . . . . . . 60 
 
Table 3.  Cartoon Analogy between Liquid-Vapor and Liquid-Liquid Critical Points . . 61 
 
Table 4.  Noncritical Background for Ordinary and Heavy Water Properties . . . . . . . 62 
 
 
 vi
List of Figures 
 
Figure 1.  Phase Diagram of Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 
 
Figure 2.  Thermodynamic and Kinetic Limits of Liquid Water . . . . . . . . . . . . . . . . . . . 64 
 
Figure 3.  Isobaric Heat Capacity of Mildly Supercooled Liquid Water . . . . . . . . . . . . . 65 
 
Figure 4.  Isobaric Heat Capacity of Deeply Supercooled Liquid Water . . . . . . . . . . . . 66 
 
Figure 5.  ?No Man?s Land? Schematic of Liquid Water . . . . . . . . . . . . . . . . . . . . . . . . 67 
 
Figure 6.  Structure of LDA and HDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 
 
Figure 7.  LDA to HDA Transition . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 
 
Figure 8.  Stability Limit Conjecture Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 
 
Figure 9.  Corrected Stability Limit Conjecture Illustration . . . . . . . . . . . . . . . . . . . . . . 71 
 
Figure 10. Density of Liquid Water at Atmospheric Pressure . . . . . . . . . . . . . . . . . . . . . 72 
 
Figure 11.  Second-Critical-Point Scenario Illustration. . . . . . . . . . . . . . . . . . . . . . . . . . 73 
 
Figure 12. Phase Diagram for Non-Crystaline Forms of Water . . . . . . . . . . . . . . . . . . . 74 
 
Figure 13. Phase Diagram with Second-Critical-Point . . . . . . . . . . . . . . . . . . . . . . . . . . 75 
 
Figure 14. Temperature ? Density Phase Diagram with Second-Critical-Point . . . . . . . 76 
  
Figure 15. Two Dimensional Ising Model Representation . . . . . . . . . . . . . . . . . . . . . . . 77 
 
Figure 16. Parametric Representation of Scaling Coordinates . . . . . . . . . . . . . . . . . . . . 78 
 
Figure 17. IAPWS-95 Formulation for Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 
 
Figure 18. Coexistence Curve and Widom Line for Ordinary and Heavy Water . . . . . . 80 
 
Figure 19. Isobaric Heat Capacity H
2
O, Experimental and Predicted . . . . . . . . . . . . . . . 81 
 
Figure 20. Graphical Representation of Scaled Parametric Equation of State . . . . . . . . 82 
 
Figure 21. Isothermal Compressibility H
2
O, Experimental and Predicted . . . . . . . . . . . 83 
 
Figure 22. Thermal Expansivity H
2
O, Experimental and Predicted . . . . . . . . . . . . . . . . 84 
 
 vii
Figure 23. Isobaric Heat Capacity D
2
O, Experimental and Predicted . . . . . . . . . . . . . . . 85 
 
Figure 24. Isothermal Compressibility D
2
O, Experimental and Predicted . . . . . . . . . . . 86 
 
Figure 25. Thermal Expansivity D
2
O, Experimental and Predicted . . . . . . . . . . . . . . 87 
 
Figure 26. Heavy Water Melting Curves for Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 
 
 viii
Nomenclature 
 
 
U    Energy 
S     Entropy 
V     Volume 
N     Number of molecules 
X     Canonical independent variable for the energy 
  =  U /  X
j
()
X
1
,X
2
,...,X
j 1
,...,X
n+2
 Conjugate intensive variables (temperature, minus 
    pressure, chemical potential) 
T     Temperature 
C
V
    Isochoric heat capacity 
 
T
    Isothermal compressibility 
C
P
    Isobaric heat capacity 
LDA     Low density amorphous water 
HDA     High density amorphous water 
LDL     Low density liquid water  
HDL    High density liquid water 
T
S
    Spinodal (singular) temperature 
X
b
    Regular background 
 
P
    Thermal expansivity 
TMD     Temperatures of maximum density 
      Density 
?    Chemical potential (Gibbs energy per molecule) 
A     Helmholtz energy per molecule 
      Isothermal susceptibility 
X
c
    Subscript c indicates critical-point value of the property X 
    e.g T
c
, P
c
,  
c
 critical temperature, pressure, and density 
?
X     Circumflex accent above indicates dimensionless property 
    
?
T =
T
T
c
,  
?
  =
 
 
c
, 
?
P =
P
 
c
k
B
T
c
, 
?
? =
?
k
B
T
c
, 
?
A =
A
k
B
T
c
, 
    
?
S =
S
k
B
, 
?
C
P
=
C
P
k
B
, 
?
C
V
=
C
V
k
B
, 
?
 
T
=  
c
k
B
T
c
 
T
, 
?
 
P
=T
c
 
P
, 
    
?
  =
 
 
c
k
B
T
c
 
 ix
      Order parameter 
h
3
= fh
1
,h
2
()=  
1
dh
1
+ 
2
dh
2
 Critical part of field-dependent ?theoretical?  
    thermodynamic potential 
h
1
    ?Ordering? field 
h
2
    ?Thermal? field 
 
1
    Ordering parameter ? conjugate, strongly fluctuating 
    theoretical scaling density 
 
2
     Conjugate, weakly fluctuating theoretical scaling density 
f
?
    Scaling function where ? refers to h
2
> 0 and h
2
< 0 , 
    respectively 
     Asymptotically equal 
 
      Approximately equal 
     Identically equal 
  ,   ,   ,   ,   ,    Universal critical exponents 
?
A
0
, 
?
B
0
 , 
?
 
0
?
   Ising critical amplitudes 
 
1
    ?Strong? scaling susceptibility 
 
2
    ?Weak? scaling susceptibility 
 
12
    ?Cross? scaling susceptibility 
      Critical exponent 
d     Spatial dimensionality in renormalization group theory 
n     Number of components of the order parameter (order 
    parameter dimensionality) 
IAPWS   International Association for the Properties of Water and 
    Steam 
 
 
 
 
 
CHAPTER 1
INTRODUCTION
The simple chemical formula and ubiquity of water belie the challenges in-
volved in understanding its behavior. Although water plays a critical role in our
understanding of such phenomena as weather, sound propagation for communi-
cation, and geo- and bio-evolution, current science does not fully explain water?s
observable behavior.1 One of the most signi?cant challenges is describing the be-
havior of supercooled liquid water. Upon supercooling, water exhibits anomalous
behavior with sharply increasing heat capacity, isothermal compressibility, and the
magnitude of negative thermal expansivity.1 However, exhaustive experimental ex-
ploration of the behavior of supercooled liquid water is hampered by thermodynamic
and kinetic limits of stability.2 While experimental data for liquid water behavior
at stable temperatures and pressures are readily available,3 the limited experimental
data on supercooled water highlight the need for a thermodynamically consistent
equation of state to describe the behavior of water into the supercooled region. Fig-
ure 1 shows the phase diagram for water as well as the limits for empirical equations
of state3, 4 and experimental measurements in the supercooled region.
Several competing theories, consistent with the limited data available, ex-
plain the anomalies in supercooled liquid water.1 One of the leading theories, the
second-critical-point scenario, proposes a liquid-liquid critical point in supercooled
water.5 This scenario suggests the existence of two liquid waters, characterized by
high density and low density, and proposes that their coexistence curve terminates
1
in a critical point. According to the second-critical-point scenario, liquid water
at ambient temperature and pressure is a supercritical ?uid, as the di?erence be-
tween two ?uid phases disappears beyond the critical point and the ?uid becomes
homogenous.
The thermodynamic behavior of a ?uid in the vicinity of a critical point is
described by a ?eld of physics called critical phenomena. The modern theory of
critical phenomena describes the behavior near the critical point with a universal
equation of state based on theoretical variables. These theoretical variables can
be mapped into real physical variables through so-called "complete scaling," an
approach that allows for accurate application of the theory of critical phenomena
to real ?uids.6 We are going to apply this theory to the theorized second-critical-
point and use this new perspective on the observed behavior of supercooled water
to draw new conclusions about the validity of the scenario and the possible location
of this second critical point. Although this scenario has been explored with some
limitedexperiments andwithsimulations of water-like models, athermodynamically
consistent equation of state based on critical phenomena has not yet been developed.
Unlike other approaches, an equation of state developed with the application of
complete scaling would depend on only three adjustable parameters, the critical
pressure, the system dependent amplitude k=a, and the non-critical background,
and provide a way of predicting behavior in the supercooled region.
Thisthesisexaminestheproblemofthermodynamicconsistencyofthesecond-
critical-point scenario using the physics of critical phenomena. We present a scaled
parametric equation of state based on the principle of critical-point universality in
2
order to describe and predict thermodynamic properties of supercooled water. In
the course of developing the equation of state we also answer the following questions:
1. If the critical point in supercooled liquid water exists, where is it located?
2. What is the nature of the criticality at the critical point, mean-?eld or
?uctuation dominated?
3. If the critical point is ?uctuation dominated, what is its universality
class?
Our equation of state is built on the assumption that the critical point of
liquid-liquid separation in supercooled water does exist, and that the critical anom-
alies are associated with the critical ?uctuations of the appropriate order parameter
and exist wherever the correlation length exceeds the range of molecular interac-
tions. This equation of state is universal in terms of theoretical variables and
belongs to the three-dimensional Ising-model class of universality. The theoretical
scaling ?elds are postulated to be analytical combinations of physical ?elds (pressure
andtemperature). The equationof state enables us to accuratelylocate the "Widom
line"7 (the locus of stability minima and maxima of order-parameter ?uctuations)
and the position of the critical point, as well as to predict thermodynamic properties
in the regions that may not be accessible to experiments. The liquid-liquid critical
point is also predicted for heavy water. In particular, we conclude that the theory
of critical phenomena does not invalidate the possibility of a second critical point
in supercooled liquid water and that the critical pressure for the second critical
point must be considerably lower than the critical pressure obtained from computer
simulations. In the future, other alternative scenarios, such as the singularity-free
3
scenario and the possibility of multiple liquid-liquid critical points in supercooled
water will also be analyzed and compared with experimental data. Moreover, pos-
sible e?ects of the existence of the second critical point in supercooled water on the
anomalous behavior observed in some aqueous solutions at ambient conditions as
well as liquid-phase polymorphism in other substances, will be considered in future
research.
This thesis is organized into six chapters, beginning with the introduction.
Chapter II covers supercooled liquid water as it is currently understood, including
the challenges in experimentally exploring the supercooled region, existing experi-
mental data and its accuracy, and the leading theories for the anomalous behavior.
Chapter III introduces the thermodynamics of critical phenomena through discus-
siononthenatureofthecriticalpoint, theuniversalityofcriticalphenomenaandthe
scaled equation of state, classes of universality, and the translation of the theoreti-
cal equation of state to physical variables. Chapter IV describes the development
of the parametric scaled equation of state for supercooled liquid water, based on
the choice of the three-dimensional Ising-model class of universality, the particular
form of the analytic relationship between physical and scaling ?elds, and the sim-
pli?cation of the equation of state through a particular choice of critical entropy.
Chapter V presents our results, describes the limitations of the mean-?eld approx-
imation, and shows comparisons between the thermodynamic properties predicted
by the scaled equation of state and existing experimental data. Implications of
the scaled parametric equation of state for the second-critical-point scenario are
discussed. Chapter VI summarizes the conclusions and suggests future work.
4
The principle results of this study have been announced in a Physical Review
Letter.8
5
CHAPTER 2
SUPERCOOLED LIQUID WATER:
WHAT IS KNOWN AND WHAT IS THEORIZED
2.1 Supercooled Liquids in General
Supercooled liquids are metastable with respect to a solid phase and remain
liquid only because they lack the activation energy to transition to the lower global
free energy of the solid phase. Kinetic e?ects such as vibrations, suspended impuri-
ties, or even contact with the surface of a container can provide the activation energy
required for the new phase to appear before the thermodynamic stability limit is
approached. These experiment-dependent properties a?ect the limit of kinetic sta-
bility. The deeper the supercooling, the larger the energy di?erence between the
two states, and the more likely the energy barrier prohibiting phase change will be
overcome either through spontaneous thermal ?uctuations or with the help of an
activator. The point at which inherent thermal ?uctuations of the material over-
come the energy barrier to a phase change represents the kinetic limit of stability
for a supercooled liquid.
In any experimental observation of supercooled liquid water, both kinetic and
thermodynamic e?ects in?uence the degree of supercooling which can be achieved.
A schematic of the thermodynamic and kinetic limits on supercooling is shown in
Fig. 2.
Kinetic Limits on Supercooling. The characteristic rates and mechanisms de?ne the
kinetic limits of stability for particular systems. In order to study the supercooled
6
?uid, its lifetime must be longer than the observation time. There must be an
experimentally accessible time scale intermediate to the systems lifetime and any
measured properties internal relaxation time in order to characterize the ?uid. In
other words, the system must be in equilibrium for the duration of the observation
time. When these conditions are met, the system can be examined thermodynam-
ically.2
Suspended impurities, vibrations and imperfect boundaries provide higher
energy surfaces for the formation of a new phase. A phase change in the presence
of these imperfections is known as heterogeneous nucleation. If no imperfections
are present, the new phase can still form within the bulk of the metastable ?uid in
a process known as homogeneous nucleation. In homogeneous nucleation, a new
phase is formed when thermal ?uctuations overcome the free energy barrier and
form a crystal of a critical size.2
With a short enough time scale for cooling, even the homogeneous nucleation
limit for supercooling can be avoided. Liquids that have been cooled on a time
scale shorter than the time required for phase change are kinetically arrested in a
metastable amorphous structure and become "glassy." The temperature upon cool-
ing at which they become glassy is referred to as the glass transition temperature.
Upon warming, the temperature when the glass has enough energy to complete the
phase change to a stable crystal structure does not typically occur at the same tem-
perature as the glass transition and is referred to as the spontaneous crystallization
temperature.
7
Thermodynamic Limits. The thermodynamics of supercooled liquid water include
both phenomenological or traditional thermodynamics and mesoscopic thermody-
namics. Traditional thermodynamics determines bulk properties from a macro-
scopic perspective. Stability criteria, given by thermodynamic inequalities and equi-
librium conditions, imply a sharp, absolute boundary between phases. Mesoscopic
thermodynamics recognizes that thermal ?uctuations within the bulk substance im-
ply that stability criteria be treated statistically.9
For the equilibrium state to be stable, the energy of the equilibrium state, U,
mustbeataminimumforallvariationssubjecttoconstantentropy(S), volume(V),
and number of molecules (N). Phenomenologically, the general stability criteria of
a pure ?uid are:
@2U
@
2
@X2

1X3
 0 (2.1)
whereX representsthecanonical independentvariablefortheenergy(S;V;N)and
represents the conjugate intensive variables (temperature, minus pressure, chemical
potential). The conjugate intensive variables are obtained by partial di?erentiation:
j = (@U=@Xj)X1; X2;:::;Xj 1; Xj+1;:::;Xn+2.
Expressing the criteria in measurable quantities, we ?nd that positive val-
ues of the isochoric heat capacity, CV , or the isothermal compressibility, T, are
necessary and su? cient for stability:
@T
@S

V;N
= TC
V
 0; (2.2)

 @P@V

T;N
= 1V
T
 0: (2.3)
8
In other words, the temperature of stable ?uids, T, must increase with the addition
ofheatorthepressuremustincreaseifthebodyiscompressed. Thethermodynamic
limit of stability is not accessible experimentally, since the kinetic limit will always
be reached ?rst.
The stability equations describe the stability of a ?uid with respect to an-
other amorphous phase. They arise from considering ?uctuations that appear to
exhibit di?erent intensive properties than the bulk ?uid but are locally in equilib-
rium. However, supercooled liquids are metastable with respect to an ordered,
translationally modulated or solid phase. The limits of stability of a ?uid phase
with respect to a solid phase have not yet been developed.2
Although the location of the thermodynamic limit on supercooling is more
uniformly applicable to various experimental setups then the kinetic limits, which
depend on experimental conditions, the thermodynamic limit still requires an accu-
rate equation of state. Until a thermodynamically consistent equation of state for
liquid water exists, the location of the thermodynamic limit on supercooling remains
uncertain.
2.2 Supercooled Liquid Water
Most liquids show no signi?cant change in properties when driven into a
metastable state and show no evidence of approaching a condition of impending
loss of stability. In particular, the heat capacity and isothermal compressibility of
most liquids do not increase anomalously below freezing temperatures. In fact, the
9
response functions of most liquids decrease upon supercooling until freezing or vitri-
?cation occurs. The anomalous behavior of supercooled liquid water, therefore, can
not be fully explained in terms of typical liquid behavior and our current depiction
of the phase diagram (Fig. 1).
In 1971, Alexander Voronel speculated that the liquid phase could be de?ned
as a state between two singular points.10 Investigating this hypothesis, Anisimov
et al. used an adiabatic calorimeter to measure the isobaric heat capacity, CP, of
water in the supercooled region (Fig. 3). Although the degree of supercooling
was modest (reaching -8oC), measurements showed a noticeable increase in the
isobaric heat capacity of metastable water upon supercooling.11 In that work, the
authors interpreted the observed anomaly as associated with a singularity in the
supercooled state and even estimated the temperature at which CP would diverge
(255 K). It was not known at that time that the observed anomaly is associated
with the unique phase behavior of water in the supercooled state, a major scienti?c
challenge in the subsequent three decades.1 Two years later, Angell, Shuppert,
and Tucker performed accurate heat-capacity measurements of 1 m droplets of
supercooled water emulsi?ed in n-heptane12 using a procedure ?rst developed by
Rasmussen and MacKenzie.13 Reaching temperatures as low as -39oC, Angell and
co-workers observed a sharp increase in the isobaric heat capacity that resembled a
critical-point-like singularity (Fig. 4).
In addition to anomalies in heat capacity, subsequent experiments found that
the isothermal compressibility and the thermal-expansion coe? cient also exhibit
critical-like anomalies with the isothermal compressibility14 increasing and the ther-
10
mal expansivity15 sharply decreasing upon supercooling. Anomalies also appear in
most transport properties and dynamic properties.1
Since the anomalies in the thermodynamic properties of supercooled liquid
water increase with supercooling, accurate, reliable experimental data for deeply
supercooled water are important. There are two possible approaches to exper-
imentally examining deeply supercooled liquid water and avoiding heterogeneous
nucleation: carefully cool very pure samples of water below the freezing point or
slowly heat glassy water above the glass transition temperature.
The experimental data collected to date suggest that the kinetic limits on
supercooling liquid water exist around -42oC (231 K).16, 17, 18 In order to obtain
data at these low temperatures and avoid crystallization, much of the experimental
evidence for the anomalous behavior of supercooled liquid water was obtained from
small volumes of water in capillary tubes or micro emulsions. Although small
samples are easier to supercool than bulk water, the e?ects of surface energy would
have a magnifying e?ect on perceived anomalies. Several studies have looked at
the relevance of this data to bulk water behavior.19, 15, 20 Although the e?ect of
surface energy cannot be completely discounted, these studies have shown that the
same anomalous behavior exhibited in the small samples also exists in bulk water.
Roughly calculated, 0.6% of the molecules in a 1 m droplet are present in the outer
10 ? of the droplet. Tombari et al.20 and Angell and co-workers, using adiabatic
calorimetry, con?rmed the emulsion data for bulk water (10 cm3) that remained
uncrystallized to -30oC.
There are also kinetic limits imposed when approaching the metastable liquid
11
region by slowly heating glassy water. Below the homogeneous nucleation limit of
around -42oC (231 K) and above the kinetic limit of spontaneous crystallization of
glass, around -123oC (150 K),21, 22 there is a region of "no man?s land," in which the
time scale of the system stability precludes experimental measurements (Fig. 5).
Although the region directly above the spontaneous crystallization to ice Ic,
is unavailable to experiments, the form of glassy water, or amorphous ice, directly
below this limit o?ers insight to the possible behavior of supercooled liquid water in
noman?sland. Glassywaterisformedbycoolingliquidwaterbelowthestablerange
faster than it crystallizes, kinetically arresting its structure. Water, unlike most
substances, forms at least two distinct forms of glass: low density amorphous water
(LDA)23 and high density amorphous water (HDA).24, 25, 26 Although LDA and
HDA amorphous ice are very solid (Fig. 6), their structure is disordered and they
transform into highly viscous liquids above the glass transition temperature of 136
K.21, 27, 28, 29 Experimental measurements of the transition between LDA and HDA
support an apparent ?rst-order transition (Fig. 7).30 However the experimental
di? culties in working close to the kinetic limits leave some room for debate as to
whether the observed di?erences are due experimental methods or distinct phases of
amorphous water.31 Any theory explaining the anomalous behavior of supercooled
liquid water must also explain glassy water.
2.3 Three Scenarios for Supercooled Liquid Water
The existence of these two types of amorphous ice and the anomalous be-
havior of liquid water do not yet ?t into any widely accepted thermodynamically
12
consistent theory. The di? culty in exploring the phase behavior of liquid water
at low temperatures has limited the availability of experimental data and left an
incomplete picture of physical cause behind these anomalies. There are currently
three competing interpretations: the stability limit conjecture, the singularity-free
scenario, and the critical point scenario.
TheStabilityLimitConjecture. Thestabilitylimitconjecture, introducedbySpeedy
in 1982,32 proposes that the liquid-vapor spinodal turns toward positive pressures
in the supercooled region of liquid water, connecting superheated and supercooled
states, at the intersection with the locus of density maxima (Fig. 8). The presence
of the spinodal, a locus of diverging density and entropy ?uctuations,2 predicts
power-law behavior of properties,
X = A
T
Ts  1
 x
+Xb, (2.4)
where X is a property that diverges at the spinodal, A is a constant, TS is the
spinodal (singular) temperature, x is an empirical exponent, and Xb is the regu-
lar background. This behavior empirically accounts for the observed increase in
water?s compressibility,33, 14 isobaric heat capacity,34 and other properties.33, 35, 36
The water-like lattice model of Sastry et al.37 and the lattice-gas implementation
of a random graph model of water?s hydrogen-bonded network of Sasai,38 provide
microscopic validation with possible physical mechanisms accounting for a retracing
of the spinodal toward positive pressures. Although the presence of the spinodal
13
curve, representing instability with respect to the ?uid phase, o?ers a thermody-
namically consistent explanation for the increase in response functions, the theory
did not originally account for metastability with respect to an anisotropic crystalline
phase. The microscopic models or computer simulations account for the instability
of supercooled water with respect to the solid phase, predicting that the response
functions increase sharply but do not diverge upon supercooling.2
Although the properties of supercooled water can be empirically ?t to the
power law predicted by the stability limit conjecture, there is a major conceptual
problem with this scenario. In order for the liquid-vapor spinodal to maintain
a negative slope in the (P;T) plane in the measurable regions of the liquid-vapor
coexistence region before curving up toward positive pressures, the spinodal must
cross the extension of the liquid-vapor equilibrium curve into the metastable region.
Therefore, in order for the liquid-vapor spinodal to curve back to positive pressures,
the metastable extension of the liquid-vapor equilibrium curve must terminate in a
lower liquid-vapor critical point (Fig. 9). Although there are microscopic models
showing a spinodal curve toward positive pressures when crossing the line of tem-
peratures of maximum density (TMD) for any given pressure,37, 38, 39 there is not
currently any evidence supporting the predicted lower liquid-vapor critical point.
The Singularity-Free Scenario. The behavior of the density is probably the most
widely known anomaly of liquid water. At atmospheric pressure, the density of
liquid water reaches a maximum around 4oC and then decreases upon cooling (Fig.
10). This behavior can be attributed to the cohesive nature of the hydrogen bond
14
network and the entropy associated with the expanded and collapsed network struc-
tures.40, 41 According to the singularity-free scenario, the experimentally observed
increases in the response functions of liquid water upon supercooling can be ac-
counted for through the density anomalies and, ultimately, the hydrogen bond net-
work which provides water its unique density properties.42, 43, 44 Isobaric heat ca-
pacity, CP, isothermal compressibility, T, and thermal expansivity, P, all depend
on density, , through the relevant thermodynamic relations,
@C
P
@P

T
= T2
@2
@T2

P
, (2.5)
T = 1
@
@P

T
, (2.6)
P =  1
@
@T

P
, (2.7)
while the TMD is negatively sloped in the (P;T) plane. Therefore, with the den-
sity anomaly, the isothermal compressibility of liquid water increases upon isobaric
cooling, the thermal expansion coe? cient increases upon isothermal compression
and becomes negative upon isothermal decompression, and the isobaric heat ca-
pacity decreases upon isothermal compression.1 The anomalous negatively sloped
TMD locus therefore predicts increases in the response functions, which may remain
?nite,45 without suggesting any singularities.44
The singularity-free scenario is supported by several models which show
anomalousincreasesinresponsefunctionwithoutacorrespondingsingularity. Stan-
ley et al.46, 47, 48 proposed a microscopic model based on bond formation between
neighboring molecules, which predicted density ?uctuations and ?nite anomalies in
15
thermodynamic properties. Sastry et al.42, 44, 49 introduced a lattice model with
nearest-neighbor attraction and directional attractions that also predicted ?nite in-
creases in the response functions. The lattice model of Borick et al.39 shows a re-
tracing density maxima locus but no evidence of singularities or a retracing spinodal.
In addition to these models, small angle X-ray scattering experiments conducted by
Xie et al. show no anomalous growth the correlation length to suggest the presence
of a singularity.45
However, other models are more ambiguous. A ?eld-theory model of water?s
hydrogen bond network by Sasai does not predict any singularities above the liquid-
ice spinodal, but it does shown the response functions diverging along the liquid-ice
spinodal.50 The model of Truskett et al. generates either a singularity-free scenario
or a liquid-liquid critical point in supercooled water depending on the choice of
parameters.51 EvenX-rayscatteringexperimentsarenotconclusiveduetotheweak
scattering in supercooled water. The long accumulation times, separation of critical
behavior and non-critical background, and speci?c cell geometry cause signi?cant
experimentalerrors,45 whilesomestudiesshowanincreaseinthecorrelationlength52
that does not support the singularity-free scenario.
The Second Critical Point Scenario. In 1992, Poole et al. proposed that the ob-
servedanomalies insupercooledliquid waterare associated with density and entropy
?uctuations diverging at a critical point of liquid-liquid coexistence.5 This critical
point terminates a line of ?rst order phase transitions between two liquid phases:
low-density liquid (LDL) and high-density liquid (HDL). The experimentally ob-
16
served LDA and HDA are interpreted as structurally arrested glassy forms of these
two liquid phases.53, 54, 55 Linking the anomalies of supercooled liquid water to
the phase transition between LDA and HDA, the "second-critical-point" scenario
presents as thermodynamically consistent view on the global phase behavior of su-
percooled water.
According to this view, the liquid-liquid coexistence curve in supercooled
water has a negative slope in the (P;T) plane, di?ering from the more common
positive slope seen with the liquid-vapor coexistence curve (Fig. 11). As a result,
water at ambient temperatures and pressures is a supercritical ?uid and "above"
the liquid-liquid critical point where the metastable ?uids LDL and HDL become
indistinguishable. Below the critical point, at lower temperatures and higher pres-
sures, LDA and HDA separate (Fig. 12). As both ?uids are metastable with
respect to the solid crystal, transient domains of long-range tetrahedral symmetry
form spontaneously due to thermal ?uctuations.46, 56 The lower surface interaction
energy between like structures favors larger domains of these lower density, ordered
tetrahedral structures. The transition from the lower density, ordered LDL to the
higher density disordered HDL corresponds to a change from an energy driven to
an entropy driven structure of the hydrogen bond network. The negative slope in
the (P;T) plane of the phase transition locus is due to the higher entropy of the
high density phase. The liquid-liquid transition is also di?erent from the liquid-
vapor curve with a large slope in the (P;T) plane at the critical point (about 30
times larger than the liquid-vapor transition), which indicates the signi?cance of the
entropy change with respect to the density change, and, correspondingly, a greater
17
importance of the entropy ?uctuations.
The hypothesis of a second-critical-point scenario is supported by extensive
Monte Carlo and molecular dynamic simulations of various water and water-like
models,37, 57, 58, 59 and by a modi?ed van der Waals model that includes hydrogen-
bond contributions.60, 51 Limited but impressive experimental evidence for the
existence of the liquid-liquid critical point in supercooled water was obtained by
Mishima and Stanley.61, 62 They observed that the melting curve of metastable
ice IV undergoes a sharp change of slope in the region where it would intersect
the expected liquid-liquid transition. Moreover, similar measurements performed
for several forms of D2O ice showed a decrease in the slope change with decreasing
pressure upon approaching the hypothesized critical point.63 Based on their exper-
iments, Mishima and Stanley61 constructed a Gibbs-energy surface for supercooled
water and estimated the location of the critical point at approximately 1 kbar and
220 K. The equation of state obtained by di?erentiation of the constructed Gibbs-
energy surface is very inaccurate. Therefore, it is not surprising that the exact
location of the liquid-liquid critical point, especially the value of the critical pres-
sure, is uncertain. The simulation data of various water-like models yields a variety
of the critical-pressure values, from negative pressures to 3.4 kbar1 (Fig. 13). On
the other hand, according to Fig. 7 by Mishima, the critical pressure in D2O is most
likely located below or around 0.5 kbar.63
In general, the second-critical point scenario appears more plausible than
either the stability limit conjecture or the singularity-free scenario, especially in
18
view of the experimental evidence of a ?rst-order phase transition between two
amorphous-water glasses, LDA and HDA. However, conclusive evidence of the
second-critical-point scenario requires further experimental studies such as quench-
ing on droplets small enough to avoid crystallization and accurate measurements of
the heat capacity in supercooled water at high pressures.1
19
CHAPTER 3
CRITICAL PHENOMENA
3.1 Phase Transitions and the Critical Point
Critical phenomena is the ?eld of physics describing the behavior of sub-
stances near second-order phase transitions, also known as critical points. Critical
points terminate a line of ?rst-order phase transitions, also known as a coexistence
curve. First-orderphasetransitionscanbede?nedbythecontinuouschangeinther-
modynamic potential and the discontinuity in the ?rst derivative of the thermody-
namic potentials such as entropy and volume through the phase change. Rephrased,
intensive properties remain continuous but extensive properties become discontinu-
ous. In ?rst-order phase transitions, the possibility of metastable states exists as
the stability limit is located beyond the equilibrium coexistence curve. Second-
order phase transitions, however, exhibit discontinuity in intensive properties. The
thermodynamic potentials and their ?rst derivatives, such as molar volume, molar
entropy, molar enthalpy, or concentration, are continuous but the second derivative
of the thermodynamic potentials, such as molar heat capacity, isothermal compress-
ibility, and thermal expansivity, are discontinuous.64 The point where the binodal,
or phase coexistence curve, and the spinodal, the limit of stability, coincide is a crit-
ical point since there is no metastable state beyond the phase transition in which
the thermodynamic quantities can exist (Fig. 14).
Critical phenomena apply to many di?erent types of transitions including
phase transitions between some crystal structures, the transitions characterized by
20
the emergence of magnetismor super?uidity, and the phase transitions in all kinds of
?uids. For pure ?uids, the critical point terminates the coexistence curve between
two ?uid phases, usually vapor and liquid. Beyond the critical point the di?er-
ences between ?uid phases disappear and the ?uid becomes homogeneous. For
one-component ?uids, the critical point is speci?ed by the critical temperature Tc,
critical density c; and critical pressure Pc. For convenience, the dimensionless
thermodynamic properties are de?ned di?erently than the typical de?nitions for a
liquid-vapor critical point (see Section 3.3). The thermodynamic properties reduced
by the critical parameters are de?ned as follows:
^T = T
Tc; ^ =

c;
^P = P
ckBTc;
^ = k
BTc
; ^A = Ak
BTc
; ^S = Sk
B
; ^CP = CPk
B
^CV = CV
kB ; ^T = ckBTcT;
^P = TcP; ^ =
 
@2 ^P
@^2
!
^T
=
@^
@^

^T
= ^2^T; (3.1)
where  is the chemical potential (Gibbs energy per molecule), P is the pressure,
S is the entropy per molecule, CV is the isochoric heat capacity per molecule, A
is the Helmholtz energy per molecule, and ^ is the isothermal susceptibility. In
the (P;T) plane the critical point manifests as the point terminating the coexis-
tence curve. In the (T;V) plane the critical point manifests as the point where
the speci?c volumes of the equilibrium phases coincide (Fig. 10). According to
21
the critical-point-scenario for supercooled water, above Pc and below Tc water sep-
arates into two metastable ?uids, LDL and HDL. Water at ambient pressure and
temperature exists beyond the critical point and is a supercritical ?uid where LDL
and HDL become indistinguishable.
The Role of the Order Parameter. Critical points can exist only when the di?erence
between two phases is a matter of degree and a continuous change from one phase to
another may occur. In fact, the particular phase of a ?uid, liquid or gas, in which
coexistence terminates in a critical point, cannot be assigned except in comparison,
when both ?uids exist simultaneously. A liquid and a gas di?er by the degree of
interaction between the molecules. A liquid and a solid, on the other hand, di?er in
their structure, or what Landau and Lifshitz refer to as their ?internal symmetry.?64
When two substances have di?erent internal symmetry, a de?ning element, such as
a unit cell in a crystal, will exist in one phase but not in the other. This element
will not appear gradually during the phase change as it can with substances with
the same internal symmetry. There can be no critical point between the two phases
of di?erent internal symmetry and the coexistence curve either continues to in?nity
or intersects with the coexistence curve of another phase.
In addition to the density change between a liquid and gas, critical points
can occur between phases where the symmetry element is the displacement of atoms
in a crystal resulting in a rearrangement of the crystal lattice or the ordering of the
crystal structure changing the probability of ?nding one type of atom in a particular
lattice site. In addition to these symmetry elements, critical points can occur with
22
a change of symmetry in the elementary magnetic moments of a substance (Curie
points of ferromagnetic or antiferromagnetic substances). Another example is the
transition of a metal to a state of superconductivity or of liquid helium to a state of
super?uidity. In each example, the body changes continuously but acquires a new
property at the transition point.64
In the 1930?s, in order to describe the change in the structure of the body
when it goes through a phase transition, Landau introduced the concept of the order
parameter, , which breaks the symmetry of a system at the transition point.65 The
order parameter is de?ned as a certain property that is larger in the less symmetrical
or disordered phase and smaller or even zero in the disordered phase. If the order
parameterisspeci?edaszerointhedisorderedphase, itbecomes?niteintheordered
phase. If the order parameter changes at the transition point without discontinuity,
thetransitionisofsecondorder. Iftheorderparameterisdiscontinuousandexhibits
a "jump," the transition is of ?rst order. This de?nition for the order parameter
corresponds to a ?rst derivative of an appropriate thermodynamic potential with
respect to the corresponding ordering ?eld, according to the Ehrenfest classi?cation
of phase transitions.64
For liquid water, an appropriate choice of order parameter is necessary to de-
scribecriticalphenomenaintermsofthegeneraltheoryofphasetransitions.66, 67, 68, 69
In principle, any property, like the surface tension, di?erence in speci?c or molar
volumes, the latent heat - all vanishing at the critical point - can be candidates
for the order parameter. However, the correct choice would result in a universal,
simpli?ed picture for critical phenomena in physically di?erent systems.
23
3.2 Universal Critical Phenomena and Scaling Theory
One of the most important results of the study of critical phenomena was
the concept of critical-point universality: the discovery that the thermodynamic
behavior in the vicinity of a critical point does not depend on the microscopic
structure.70, 71, 66, 72 Instead, long-range ?uctuations of the order parameter, the
e?ect of random deviations from average, dominate over the speci?c e?ects of near-
neighbor interactions in the critical region. The correlation length, the length scale
for the critical ?uctuations, diverges at the critical point. Due to this divergence,
the thermodynamic properties, functions of the correlation length, become singular
at the critical point. Mathematically, the asymptotic critical behavior near critical
points is characterized by scaling laws with universal critical exponents and system-
dependent amplitudes that are universally interrelated.
The universality of critical phenomena means that critical, or ?uctuation in-
duced, behavior can be universally described using theoretical variables, such as
the order parameter or the conjugate ordering ?eld, that can be mapped to speci?c
physical variables for di?erent systems. For real ?uids and ?uid mixtures, it is
commonly accepted that the non-analytic critical behavior can be asymptotically
described by scaling theory in terms of two independent, theoretical scaling ?elds,
namely, h1 ("ordering" ?eld) and h2 ("thermal" ?eld) and two conjugate, theoret-
ical scaling densities, namely, the order parameter 1(strongly ?uctuating) and 2
(weakly ?uctuating). A third ?eld, h3 = f (h1;h2), is the critical part of the ?eld-
dependent theoretical thermodynamic potential, which exhibits a minimum with
24
respect to a variation of the of the order parameter such that
dh3 = 1dh1 +2dh2: (3.2)
Scaling theory is based on the assumption that the critical part of the ther-
modynamic potential h3 has the form of a non-analytical homogeneous function of
the theoretical scaling ?elds, h1 and h2. Asymptotically:
h3 jh2j2  f
 
h1
jh2j2  
!
; (3.3)
where f is a scaling function and the superscript  refers to h2 > 0 and h2 < 0,
respectively. The form of the scaling function is universal for any system; however,
it contains two thermodynamically independent (but system-dependent) amplitudes
and critical exponents speci?c to a particular class of systems. All other asymptotic
amplitudes are related to the selected ones by universal relations. A salient feature
of scaling theory is that the asymptotic behavior of the system near the critical
point can be described through a small number of variables, the critical exponents
 and , universal within a class of critical-point universality. All ?uids and ?uid
mixtures belong to the Ising-model class of universality (in which the order parame-
ter is either a scalar or a one-component vector).66 The Ising values for  and ;
namely  ' 0:109 and  ' 0:326, are well established theoretically and con?rmed
experimentally.66, 67, 68, 69, 73, 74 The two Ising amplitudes, ^A0 and ^B0 can be de-
25
termined by the asymptotic power-law behavior of the two scaling densities in zero
ordering ?eld (h1 = 0),
1 =
@h
3
@h1

h2
^B0jh2j ; (3.4)
2 =
@h
3
@h2

h1

^A0
1 h2jh2j
  ; (3.5)
and of the three scaling susceptibilities, "strong" 1, "weak" 2, and "cross" 12 in
zero ordering ?eld ,
1 =
@
1
@h1

h2
 ^ 0 jh2j  ; (3.6)
2 =
@
2
@h2

h1
 ^A0 jh2j  ; (3.7)
12 =
@
1
@h2

h1
  ^B0jh2j

h2 (h2 < 0); (3.8)
where the critical exponent  = 2  2 ' 1:23973, 74 and the other Ising critical
amplitude ^ 0 isrelatedto ^B0 and ^A0 throughuniversalratios, ^ +0 ^A+0 = ^B20 ' 0:0581;
^ +0 =^  0 ' 4:8, and ^A+0 = ^A 0 ' 0:523.74 While the superscript  refers to the states
at h2 > 0 and h2 < 0; the prefactor  in Eq. (3.4) refers to the branches of the
order parameter at h1 > 0 and h1 < 0 sides, respectively.
3.3 Classical versus Nonclassical (Scaling) Theory of Phase Transitions
Mean-Field Theory. Traditional thermodynamics, which considers only the bulk
properties of a homogeneous ?uid, is commonly based on mean-?eld theory. Mean-
26
?eld theory neglects thermal ?uctuations, simplifying the system until each molecule
in the ?uid has the ?uid?s average properties. The complex picture of interactions
between particles is replaced by a mean ?eld (or a mean force) that acts equally on
all particles. This approximation assumes that the Helmholtz energy is everywhere
an analytic function of volume and temperature. The van der Waals theory and
other approximate statistical mechanical models are mean-?eld theories. Mean-?eld
is also known as the simplest solution or zero-order approximation of any statistical
mechanical model when a local order is either neglected or considered as a small
correction (Ornstein-Zernike approximation). Mean-?eld theory is a particular case
of the more universally applicable scaling theory.
A phenomenological representation of all mean-?eld theories and models is
the Landau theory of phase transitions. The major assumption of the Landau
theory is that the thermodynamic potential is an analytical function of an ordering
?eld, h1, and a thermal ?eld, h2, at the transition point. If so, the critical part of
the appropriate thermodynamic potential, h3, can be expanded in a power series of
the order parameter:
h3 = 12a0h221 + 14!u041  h11. (3.9)
Above the critical point there is only one solution,  = 0. Below the critical tem-
perature, there are two solutions with either a positive or negative order parameter.
Mean-Field and Scaling Universal Critical Exponents. Experimentally, it has been
well established that asymptotically close the critical point, all physical properties
obey simple power laws known as "scaling laws." The universal powers in these
27
laws are called critical exponents. Both mean-?eld and scaling theories all result in
the divergence of some thermodynamic quantities at the critical point,
X  Ajh2j k , (3.10)
where k is the critical exponent for a given thermodynamic property X. The
exponents that describe this divergence for all mean-?eld theories are called classical
criticalexponents. Theexponentsthatdescribethisdivergenceandobeythescaling
laws are called non-classical critical exponents.
As the simplest solution, mean-?eld is often used as the ?rst approximation
of near-critical behavior. However, the behavior of most ?uids near a critical point
is dominated by thermal ?uctuations and non-classical or macroscopic behavior. As
a result, the mean-?eld approximation breaks down near the ?uctuation dependent
critical point. Experimentally, real ?uid behavior is much more closely described
by scaling theory with the nonclassical critical exponents. However, the mean-
?eld approximation becomes exact in systems with long-range interactions such as
superconductors. Therefore, when new kinds of critical points are discovered, it is
not clear a priori whether the critical point is mean-?eld or ?uctuation-dominated.
Renormalization group theory75 provides a theoretical method to calculate
the critical exponents. According to the renormalization group theory, which is
the modern theory of critical phenomena,72 the critical exponents of a system de-
pend only on two parameters: the same spatial dimensionality d and the number of
components of the order parameter (order-parameter dimensionality) n. Systems
28
that have the same order-parameter dimensionality belong to the same universal-
ity class and can be described by the same set of critical exponents. The order
parameter, 1, a theoretical variable corresponding to a system-dependent physical
quantity, vanishes at the critical point, is nonzero on one side of the transition, and
whose ?uctuations diverge at the critical point. For pure ?uids, the order parame-
ter is mostly associated with the density minus the critical density,    c. The
order parameter for pure ?uids is zero along the critical isochore, above the critical
temperature, and nonzero below the critical temperature, where saturated densities
di?er from the critical density. Since this order parameter for all ?uids is a scalar,
its dimensionality n is 1. Fluids, anisotropic magnetic or electric materials, ?uid
mixtures, binary alloys, and some other systems all belong to the three dimensional
Ising-model class universality where n = 1, and d = 3, and can all be described by
the same scaled equation of state asymptotically close to the critical point.
As mentioned earlier, real ?uid behavior is inconsistent with the popular
mean-?eld approximation. Therefore, a scaled equation of state, based on the
principle of critical-point universality, should be considered for the second critical
point in water.
Scaling theory associates all critical anomalies with the divergence of the
correlation length. The correlation length plays the role of a "screening length" for
the correlation function: when  is small, the correlation function is exponentially
short-ranged. However, at the critical point, the correlation length diverges and
the correlation function becomes long ranged - it decays proportionally with the
distance. The amplitude of the order-parameter ?uctuations in the correlation
29
volume is the same as the thermodynamic value of the order parameter in the
ordered phase at the same distance to the critical point.
The nonclassical critical values given in Table 1 are the best theoretical es-
timates of the critical exponents of ?uids.76, 77 These nonclassical values represent
treatment of the system through the correlation function, G(r), a function describ-
ing spatial behavior of ?uctuations of the order parameter. The correlation function
measures how the order parameter at one point correlates to its value at another
point at a distance r. If the correlation function decreases quickly with increasing
distance between points, then far away points are uncorrelated and the system is
dominated by microscopic, short range forces.
The Landau theory satis?es the scaling formulation as far as thermodynamic
quantities are concerned and provides us with the nonclassical critical exponents,
shown in Table 1. Understanding that all critical anomalies depend on the diver-
gence of the correlation length, the introduced six scaling critical exponents are not
independent. The universal relations between the critical exponents for thermody-
namic quantities and the correlation length are:
3 =  + 2 (3.11)
 = 2 d (3.12)
 + 2 + = 2 (3.13)
 =  (2 ) (3.14)
where d is a space of dimensionality. The equation  = 2 d does not satisfy the
30
classical Landau (mean-?eld) theory for dimensions less than four. In scaling theory
only two components and two amplitudes are independent; all other exponents and
amplitudes can be calculated through the given two universal relations.
3.4 TranslatingTheoreticalVariablesintoPhysicalVariableswith"CompleteScaling"
Ising/Lattice Gas Model. The Ising model represents a system of an incompressible
lattice where its con?guration space (set of possible positions) is the assignment of
a +1 or -1 to the otherwise identical vertices on the graph (Fig. 15). In greater
than 1 dimension, the Ising model undergoes a phase transition between an ordered
and a disordered phase. First proposed by Ernest Ising to represent ferromag-
netism,78 the Ising model assigned dipoles or "spins" directed either upward (+1)
or downward (-1) to the vertices in order to describe the magnetic moments. At
high temperatures, entropy overcomes the interaction between these dipoles and the
average magnetization (summation of all spin-vectors) is zero. As the temperature
drops, the system reaches a critical point called the "Curie point" where the order
parameter,  = @h3=@h1, (magnetization) emerges, and increases in value from zero.
At low temperatures, the energy drives the interaction between the dipoles resulting
in a spontaneous magnetization even in zero magnetic ?eld. For ferromagnets, the
ordering ?eld, h1, is the magnetic ?eld.
Adapted for ?uids, this model becomes the lattice gas model describing con-
densation. In the lattice gas model, each site is either occupied by a particle (+1)
or empty (-1) and particle density minus critical density, ( c)=c, becomes the
31
order parameter, 1. At the critical point, half of all sites are occupied and half
are empty and c = 1=2. In ?uids, "zero ?eld" corresponds to the appearance of
the di?erence between the densities of liquid and gas, analogous to the appearance
of spontaneous magnetization in the Ising model.
Mathematically, the Ising, or lattice gas, model is a powerful tool to a uni?ed
theory of phase transitions, describing di?erent systems using one set of theoretical
variables. Table 2 shows the universal theoretical variables associated with scaling
theory and their physical meaning for ferromagnets and for liquid-gas systems.
Given the ?elds, the relevant thermodynamic potential for the lattice gas is
the grand thermodynamic potential divided by volume, 
=V =  P, or negative
pressure.
Real Fluids and the Lattice Gas Model. The lattice gas has perfect symmetry
with respect to the sign of the order parameter, whereas real ?uids approach such
symmetry only asymptotically. This symmetry, not be confused with the "internal
symmetry" discussed in Section 3.1, describes the liquid-vapor coexistence curve
in the (;T) plane and the correlation between the arithmetic mean of the liquid
and vapor densities and the critical isochore. To incorporate ?uid asymmetry into
the scaling theory, in 1970?s Mermin and Rehr79 and Patashinskii and Pokrovskii80
introduced the concept of mixing the independent physical ?elds into the theoretical
scaling ?elds (see also refs. Wilding et al.,81 Anisimov et al.,82 and Anisimov et
al.83). According to their approach, which we will call "incomplete scaling," the
independent scaling ?elds in ?uids are linear combinations of chemical potential
32
and temperature:
h1 = a1^+a2^T; h2 = b1^T +b2^; (3.15)
while the dependent ?eld h3 = c1^P + c2^T . Since any two independent crit-
ical amplitudes can be incorporated into the scaling function f, it is convenient
to adopt a1 = 1 and b1 = 1; then c1 = 1 while c2 becomes  

@ ^P=@ ^T

h1=0
taken
at the critical point. Therefore, the critical part of the ?eld-dependent thermody-
namic potential remains the same as in the lattice gas, since in linear approximation
(P  Pcxc)=ckBTc = ^P  

@ ^P=@ ^T

h1=0;c
^T. Furthermore, as shown by Anisi-
mov et al.,82, 83 since in classical thermodynamics the absolute value of entropy is
arbitrary, the critical value of entropy can be chosen upon practical convenience. It
is seen clearly from the basic thermodynamic relation
dP = d+SdT; (3.16)
that, if the critical entropy is adopted as Sc =  1c (@P=@T)h1=0;c, the coe? cient a2
in Eq. (3.15) vanishes and in linear approximation the chemical potential along the
vapor-liquid coexistence does not depend on temperature. However, the curvature
of this dependence, determined by the second derivative, is well de?ned. With this
choice of the critical entropy, the mixing term b2^ in "incomplete scaling" becomes
also well de?ned, being in lowest approximation the sole contribution to the vapor-
liquid asymmetry in real ?uids. In particular, since ^ = 1+b22 = 1+b2(^^S);
33
this term explains the non-analytic deviation from the law of rectilinear diameter,
the

^T


1  singularity in the "diameter" of vapor-liquid coexistence curve given by
^d = (0 +00)=2c +::: = 1+D1

^T


1  +D
0

^T

+:::. Mapping the asymmetric
?uid criticality into the symmetric lattice model is achieved in "incomplete scaling"
by a rede?nition of the order parameter as 1 = ^   b2(^^S): In "incomplete
scaling"thechemicalpotential (uptothethirdderivative69)isananalyticfunction
of temperature along the vapor-liquid coexistence boundary and along the critical
isochore above the critical point (h1 = 0). Like in the lattice gas, the second
derivative (@2=@T2)h1=0 = (d2=dT2)cxc remains ?nite at the critical temperature
Tc, while (@2P=@T2)h1=0 = (d2P=dT2)cxc diverges proportionally to the isochoric
heat capacity ^CV .
At this point we encounter a major conceptual problem with mapping real
?uids into the lattice-gas even at the mean-?eld level. In the mean-?eld approxima-
tion the critical part h3 of the thermodynamic potential, is represented by Landau
expansion (3.9). When h1 = ^; h2 = ^T +b2^; and 1 = ^ b2(^^S); this
expansion generates asymmetric terms _ b2^T(^)3 and _ b2(^)5: However, in
the simplest equation of state that realistically describes ?uid phase behavior, the
van der Waals equation, the term _ ^T(^)3 is absent, while the term _ (^)5
exists. Furthermore, in most classical equations of state, d^2=d^T2 along the liquid-
vapor coexistence exhibits a discontinuity directly related to the existence of the
independent 5th-order term in Landau expansion. The existence of the indepen-
dent 5th-order term makes exact mapping of ?uids into the lattice-gas model by the
conventional mixing of physical ?elds impossible. This problem was recognized a
34
long time ago67 but was not clearly articulated. On the other hand, a theoretical
renormalization-group treatment of the 5th-order term84, 85, 86, 87, 88 resulted in the
emergence of an independent critical exponent 5 ' 1:3.89 The exponent 5 does
not exist in symmetric models and is expected to belong exclusively to ?uids.
More recently, "incomplete scaling" was challenged by Fisher and his cowork-
ers90, 91 who developed a new approach, known as "complete scaling for ?uids."
They proposed that both (@2=@T2)h1=0 and (@2P=@T2)h1=0 diverge at the critical
point like the isochoric heat capacity. A principle possibility of this e?ect has been
known as the "Yang-Yang anomaly"92 and has been a subject of prolong discussions
fordecades.93 The majorconceptual result of "complete scaling" is that asymmetric
?uids can be consistently mapped into the symmetric Ising criticality by appropri-
ate mixing of the physical ?elds into the scaling ?elds. A rede?nition of the order
parameter, suggested by "complete scaling," makes a special renormalization-group
treatment of the 5th-order term in the e?ective Hamiltonian for ?uids irrelevant, at
least, in practice.
Complete Scaling. "Complete scaling" suggests that all three physical ?elds ^;
^T; and ^P are equally mixed into three scaling ?elds h1; h2; and h3. In linear
approximation:
h1 = a1^+a2^T +a3^P; (3.17)
h2 = b1^T +b2^+b3^P; (3.18)
h3 = c1^P +c2^+c3^T: (3.19)
35
The dependent ?eld h3 is a homogeneous function of h1 and h2 as asymptotically
given by Eq. (3.3). Far away from the asymptotic region, or if the phase-coexistence
locus h1 = 0 exhibits a strong curvature in terms of the physical ?elds, the linear
approximation might be insu? cient and appropriate nonlinear terms should be in-
cluded.
Physical density-like properties, the molecular density and entropy per unit
volume, are given by the thermodynamic relations
^ =
 
@ ^P
@^
!
^T
; ^^S =
 
@ ^P
@ ^T
!
^
: (3.20)
Since the coe? cients c1 and c2 can be absorbed by making the thermodynamic
potential h3 dimensional, as given by Eq. (3.1), while the coe? cient c3 = ^Sc; one
can obtain by applying Eq. (3.20) to Eqs. (3.17-3.19),
^ = 1 +a11 +b221 a
31  b32
; (3.21)
^^S =
^Sc +a21 +b12
1 a31  b32 : (3.22)
Onecanseethatwhilethescaling?eldareexpressedaslinearcombinationsof
the physical ?elds, the physical densities are non-linear combinations of the scaling
densities.
Making "Complete Scaling" Simple. Before we apply complete scaling to describe
asymmetry in ?uids, we note that the relations between scaling and physical ?elds
36
can be further simpli?ed. The coe? cients a1 and b1 can be absorbed by the two
amplitudes in the scaling function f; such that a1 = 1 and b1 = 1: The coe? cient
c3 = ^Sc is determined by the choice of the critical value of entropy. By adopting
^Sc = (kBc) 1 (@P=@T)h
1=0;c =

d^P=d^T

cxc;c
, the slope of the saturation-pressure
curve at the critical point, one obtains a2 =  a3

d^P=d^T

cxc;c
. Indeed, along the
path h1 = 0, asymptotically close to the critical point,
@^
@ ^T

h1=0;c
+a2 +a3
 
@ ^P
@ ^T
!
h1=0;c
= 0: (3.23)
On the other hand, it follows from the thermodynamic relation (3.16) that
d^
d^T +
^Sc   @ ^P
@ ^T = 0: (3.24)
Thus, with adopting ^Sc =

@ ^P=@ ^T

h1=0;c
; we obtain

@^=@ ^T

h1=0;c
= 0 and
a2 +a3
 
@ ^P
@ ^T
!
h1=0;c
= 0: (3.25)
Furthermore, with such a choice of ^Sc, along the path h1 = 0
h2 = ^T

1 b3a2a
3

; (3.26)
and the density of entropy becomes proportional to the weakly-?uctuating scaling
density, (^^S)  (1 +b3)2 _jh2j1  : Withexceptionforatrivial renormalization
of the amplitudes in h2 and (^^S); the coe? cient b3 plays no signi?cant role in
37
asymmetry of ?uid criticality. Indeed, as follows from Eqs. (3.21) and (3.22), this
coe? cient can be independently obtained only from the contributions to the density
behavior of order b312 _jh2j1 + : With 1  +  ' 1:417; this contribution is
of higher order than a321 _jh2j2 and b22 _jh2j1  ; and even signi?cantly weaker
than the linear term. Therefore, for the sake of simplicity, we assume b3 = 0:
Hence, there are only two independent coe? cients that in the ?rst approximation
control the asymmetry in ?uid criticality, namely a3 and b2. In this approximation,
the scaling ?elds read
h1 = ^+a3

^P  

d^P=d^T

cxc;c
^T

; (3.27)
h2 = ^T +b2^; (3.28)
h3 = ^P  ^+

d^P=d^T

cxc;c
^T: (3.29)
Furthermore, byexpandingEqs. (3.21)and(3.22)andneglectingalltermsofhigher-
order than linear of ^T; we obtain
^ ' (1 +a3)1 +a3 (1 +a3)21 +b22 (3.30)


^^S

' b22 (3.31)
As a result, while the order parameter in ?uids is, in general, a nonlinear
combination of density and entropy, the weakly ?uctuating scaling density 2 in
?rst approximation is associated with the density of entropy only.
38
The relations between the theoretical ?elds and the physical ?elds given by
Eqs. (3.27-3.29) are to be built into a theoretical equation of state based on the
scaling formulation given by Eq. (3.3).
39
CHAPTER 4
APPLYING CRITICAL POINT UNIVERSALITY TO
SUPERCOOLED LIQUID WATER
The application of the principle of critical point universality to supercooled
liquid water provides an opportunity to explore the second critical point scenario
from a new direction. Since conclusive experimental evidence supporting one of the
three proposed interpretations of existing data is not available, the role of theoretical
models in supporting these theories has grown. Yet Monte Carlo and molecular dy-
namic simulations are still only approximate models of water?s complex molecular
bonding interactions and they depend heavily on the accuracy of many underly-
ing assumptions.1 An equation of state, developed under the assumption of the
existence of the second critical point and based on the principle of critical point
universality, o?ers a thermodynamically consistent approach to explore the behav-
ior of supercooled liquid water. Using only two adjustable critical parameters, a
system-dependent critical amplitude and the critical pressure, and an adjustable
non-critical background, we can locate the second critical point in liquid water and
predict thermodynamic properties. The comparison between existing experimental
data and the predicted behavior of the thermodynamic properties will o?er a new
tool to evaluate the validity of the second critical point scenario.
4.1 Relations Between Theoretical and Physical Fields
The ?rst step in the development of the equation of state is to map the the-
oretical scaling ?elds to physical variables. Representation of scaling ?elds through
40
linear mixing of physical ?elds is used to incorporate asymmetric ?uid criticality
into the symmetric Ising model.94, 95 We begin with the "complete scaling" rela-
tions, Eqs. (3.17-3.19). The degree of freedom of the system requires only two
independent variables. Therefore, all but two coe? cients in the scaling ?elds may
be absorbed in the two system-dependent amplitudes of the scaling function f. In
this model, we adopt the mixing coe? cients, a2 = 1, b2 =  1, c1 =  1, c2 = 1,
c3 = ^Sc, and simplify the equations:
h1 = a1^+ ^T +a3^P; (4.1)
h2 = b1^T  ^+b3^P; (4.2)
h3 =  ^P + ^+ ^Sc^T: (4.3)
Such a choice anticipates the fact that the coe? cients a1, b1, a3, and b3 can be
very small, since the supercooled water is weakly compressible. The negative sign
of b2 =  1 indicates that the liquid-liquid phase separation in supercooled water
occurs with an increase of pressure (Fig. 1), in contrast to the vapor-liquid phase
separation. Moreover, since the compressibility of supercooled water is very small,
we assume a3 and b3 are small enough so that they can be neglected for the purposes
of our model. The value of a1 and b1 can be determined fromthe shape of the liquid-
41
liquid ?rst-order transition curve. Finally,
h1 = a1+ T; (4.4)
h2 = b1T  ; (4.5)
h3 =  P + . (4.6)
Inordertofurthersimplifythe?elds, wemustadoptacriticalvalueof ^Sc. For
thevapor-liquidcriticalpoint, theconvenientchoicefor ^Sc was(kBc) 1 (@P=@T)h1=0;c.
However, theliquid-liquidcoexistencecurveinsupercooledwaterhasaverydi?erent
slope than the liquid-gas coexistence curve (Fig. 12) and in the case of liquid-liquid
coexistence it becomes convenient to rotate the theoretical coordinates h1 and h2
through a di?erent choice for critical entropy. Adopting a critical entropy of ^Sc = 0
with a3  0 and b3  0 , results in
@^
@ ^T

h1=0;c
= 1^
 
@ ^P
@ ^T
!
h1=0;c
=  a2a
1
; (4.7)
@^
@ ^T

h2=0;c
= 1^
 
@ ^P
@ ^T
!
h2=0;c
=  b1b
2
(4.8)
along the paths h1 = 0 or h2 = 0, asymptotically close to the critical point. In this
approximation, the scaling ?elds h1 and h2 read
h1 ' a1^+a2^T = a1^  a1^

@ ^P=@ ^T

h1=0;c
^T; (4.9)
h2 ' b1^T +b2^ = b1^T + b1^

@ ^P=@ ^T

h1=0;c
^: (4.10)
42
Solving these equations for the scaling densities yields
1 =  
@h
3
@h1

h2
=


^^S

+b1^
1 +a1b1 
^B0jh2j ; (4.11)
2 =  
@h
3
@h2

h1
=
^+a1

^^S

1 +a1b1 
^A0
1  jh2j
1  : (4.12)
If b1 is small, the predominant contribution to the order parameter, 1, comes from
the entropy. If a1 is small, the second scaling density becomes mostly molecular
density.
In addition to the di?erence in direction of the slope of the liquid-liquid
transition line, de?ned as h1 = 0, the liquid-liquid transition also curves strongly
above the critical point. To account for this curvature, we added the non-linear
pressure term, a3^2, in the ordering ?eld, h1, and our scaling ?elds become:
h1 = a1^+a2^T +a3^2 = a1^ a1

@ ^P=@ ^T

h1=0;c
^T +a3^2;(4.13)
h2 = b1^T +b2^ = b1^T +b1

@ ^P=@ ^T

h1=0;c
^: (4.14)
Finally, the scaling ?elds, "ordering" and "thermal," for the liquid-liquid
critical point in supercooled water become
h1 ' ^T  

@ ^T=@ ^P

h1=0;c
^ 1^P  2

@2 ^T=@ ^P2

h1=0;c
^ 2

^P
2
;(4.15)
h2 '  ^ 1^P  

@ ^T=@ ^P

h1=0;c
^T: (4.16)
43
The scaling ?densities?become
1 '


^^S

+b1^
1 + (a1)e b1 ; (4.17)
2 '
^+ (a1)e 

^^S

1 + (a1)e b1 : (4.18)
where ^S  (S Sc)=R, with R being the gas constant, and (a1)e =

@h1=@ ^P

^T =
a1 + 2a3^P. The "strong," "weak," and "cross" susceptibilities remain de?ned as
1 =
@
1
@h1

h2
; (4.19)
2 =
@
2
@h2

h1
; (4.20)
12 = 21 =
@
1
@h2

h1
=
@
2
@h1

h2
; (4.21)
As far as the physical ?elds are mixed into the scaling ?elds, the physical
properties, such as the isobaric heat capacity CP, the isothermal compressibility T,
andthethermalexpansivityP,willnotexhibituniversalpowerlawswhenmeasured
along isotherms or isobars; instead, their apparent behavior will be determined by
a thermodynamic path and by the values of the mixing coe? cients in Eqs. (2) and
(3). As follows from Eqs. (4.15) and (4.16), the critical (?uctuation induced) parts
of the dimensionless isobaric heat capacity, isothermal compressibility, and thermal
44
expansivity are expressed through the scaling susceptibilities as
^
CP

cr
= ^T
 
@^S
@ ^T
!
^P
 
^
CP

b
(4.22)
= ^T a221 + 2a2b112 +b212;
(^T)cr =  1^V
 
@^V
@ ^P
!
^T
 (^T)b (4.23)
= 1^V  (a1)2e 1 + 2(a1)e b212 +b222;
(^P)cr = 1^V
 
@^V
@ ^T
!
^P
 (^P)b (4.24)
=  1^V ((a1)e a21 + ((a1)e b1 +a2b2)12 +b1b22).
where ^T = T=Tc, ^P = P=cRTc, and the subscript "b" indicates the property
backgrounds, behavior not attributable to proximity of the critical point. Given
the small slope of the liquid-liquid coexistence curve in the (P;T) plane, the mixing
coe? cients a1 and b1 are small (Fig. 16). We can see that the thermodynamic
properties depend on susceptibilities such that ^CP is strongly divergent, ^T is mostly
weakly divergent, and ^P is mostly modestly divergent. In practical ranges of
temperatures and pressures, roughly,
^CP / 1; (4.25)
^T / 1; (4.26)
^P / 12: (4.27)
The application of scaling theory to the liquid-liquid critical point in super-
45
cooled water is not a unique result, but it does di?er from the liquid-vapor or even
binary system liquid-liquid translations of the theoretical variables to physical vari-
ables. Liquid crystals such as Blue Phase III have a critical point of the transition
between a molecular-?uid phase and a structural phase that translates into scaling
theory in a similar manner to the liquid-liquid critical point in supercooled water.95
For the blue phase liquid crystal the phase BPI, the defect lies (disclinations) of a
helix structure form simple cubic lattice. For the phase BPII, the defects form a
body centered cubic lattice. In BPIII, defects form an amorphous structure with a
translation between a "liquid" of defects (BPIII) and a "gas" of defects ("molecular
?uid"). Like in supercooled water, the transition temperature is almost indepen-
dent of the second-physical ?eld. A cartoon analogy between the liquid-vapor and
liquid-liquid critical points is given in Table 3.
4.2 Scaled Parametric Equation of State
The universal scaling function and renormalization group theory allow cal-
culation of the singular parts of all thermodynamic properties of a ?uid in the as-
ymptotic vicinity of the critical point given the critical parameters, any two system-
dependent scaling critical amplitudes, and the proportional slope of the coexistence
curve at the critical point. However, the expressions for the scaling function are
implicit and inconvenient for practical engineering use. The phenomenological
parametric representations of a scaled equation of state provide more convenient
coordinates using the parametric variables r and , where r measures the distance
from the critical point and  provides a location along a contour of constant r.
46
We use the simplest form of a scaled parametric equation of state, the so-
called "linear model," which represents the scaling ?elds and scaling susceptibilities
as functions of the ?polar" variables r and 95, 69:
h1 = ar+ 1 2; (4.28)
h2 = r 1 b22; (4.29)
1 = kar c1(),
12 = kr 1c12(), (4.30)
2 = akr c2() Bcr (4.31)
where the coe? cient b2 = (  2)=(1 ) ' 1:36 is a universal constant, while
a and k are system-dependent amplitudes, and Bcr is the so-called "critical back-
ground" of order ak.95 The critical background was studied by Bagnuls and Bervil-
lier in a three-dimensional ?eld theory96 and by Anisimov et al. in a crossover
theory based on a renormalized Landau expansion.97 According to the theory, the
critical background can be found through an explicit equation proportional to the
cuto? wave number of the critical ?uctuations. Unfortunately, the available data
are accurate enough to estimate Bcr through this theory for only a few ?uids such
as methane98and ethane99 in the two-phase region. Although critical background is
a di? cult parameter to obtain accurately, it can be estimated to an order of mag-
nitude as Bcr  ak. The analytical functions c1(), c2(), and c12() are calculated
in ref..95 The remarkable feature of the "linear model" is that the singularities in
47
the thermodynamic functions are only related to the variable r, while the properties
are analytical with respect to . The parameter  is chosen to range from -1 to +1
so that  = 0 corresponds with the critical isochore and  = 1 corresponds to the
coexistence curve. Along the critical isotherm, h2 = 0, the value of theta becomes
 = 1=b. Figure 16 shows the representation of the thermal ?eld h2 and the order
parameter 1 through the variables of the parametric linear model.
Input Parameters for the Scaled Parametric Equation of State. The choice of a co-
ordinate system, reference entropy, and morphology of the liquid-liquid coexistence
curve correlated the universal scaling theory to the physical variables speci?c to the
scaled parametric equation of state for a liquid-liquid critical point in supercooled
water. Until this point, the equation of state remains applicable for both ordinary
and heavy water and the translation from theoretical variables to physical variables
did not introduce any sources of error. However, this scaled equation of state de-
pends on input parameters: density of water at various pressures, the path of the
coexistence curve and its extension into the one-phase region, the location of the
critical point along that curve, and the system-dependent amplitudes a and k. Each
of these parameters introduces error into the equation of state.
Density. The International Association for the Properties of Water and Steam
(IAPWS) provides accepted formulations for the properties of water for scienti?c
and industrial applications. In 1995, IAPWS approved a new formulation of the
properties for water for general and scienti?c use based on existing experimental
48
data. This formulation, serving as the international standard for water?s thermo-
dynamic properties, provides the density of water at various pressures and temper-
atures. In the stable liquid region at ambient pressure, IAPWS-95 is accurate to
0:0001% and represents the most accurate data within the experimental uncer-
tainty available. However, the formulation covers a validity range from 251.2K at
209.9 MPa to 1273 K and 1000MPa, while our equation of state suggests a criti-
cal point in supercooled liquid water below 250K and 50MPa. Nevertheless, the
IAPWS formulation represents the most accurate, available data on the density of
liquid water and the scaled parametric equation of state depends on the IAPWS-95
predicted density values at various temperatures and pressures. Fig. 17 shows plots
of the IAPWS density for various pressures and temperatures.
Theformulationusedforordinarywaterispresentedinref.3 Theformulation
used for heavy water is a dimensionless version of the formulation provided by Hill et
al.100, 101 IAPWS released a Revised Release on the IAPWS formulation for heavy
water in July 2005.
Coexistence Curve and ?Widom line." The ?Widom line" in the one-phase region,
h1 = 0, is an analytical continuation of the liquid-liquid transition curve from C0
to lower pressures and higher temperatures. Based on the most recent estimate of
the liquid-liquid phase transition curve given by Mishima,62 we have obtained the
coe? cients a1 = b1 = 0:039 and a3 = 0:062 for ordinary water and a1 = b1 = 0:03
and a3 = 0:05 for heavy water. This work shows the location of the liquid-liquid
transition line close to the homogeneous ice nucleation locus. Fitting a curve to
49
the estimates of Mishima allows us to specify the location of the critical point with
only one variable, Pc, reducing the number of variables, and inherent error, in our
equation. Figure 18 shows a plot of the liquid-liquid coexistence curve based on
Mishima data for ordinary and heavy water.
System-Dependent Amplitudes. In order to reduce the number of adjustable para-
meters, we assume that the ratio k=a = 1, as obtained for the three dimensional
Ising model with short-range interactions.102 Hence, only two adjustable parame-
ters, namely, Pc and a = k, have been used to describe the anomalous parts of the
thermodynamic properties.
Adjustable Parameters a = k, and Pc. Using high-resolution experimental heat-
capacity data12 shown in Fig. 4, we optimized the location of the critical point
and the system dependent amplitudes. The resulting ?t for the experimental heat
capacity data is shown in Figure 19. The non-critical background of the heat
capacity was approximated as a linear function of temperature. We obtained a =
k = 0:47 and Pc = 27 MPa with the critical temperature corresponding to this
pressure Tc = 232 K for ordinary water. For heavy water, we ?xed a = k = 0:7 and
Pc = 20 MPa with the critical temperature corresponding to this pressure Tc = 238
K. The critical point, obtained from our equation of state, is located at a much
lower pressure than previously predicted from computer simulations (see Fig. 13).
Figure 20 shows the parametric equation of state predicted by scaling theory.
50
Non-critical Background. The scaled parametric equation of state predicts the e?ect
of the critical point on the behavior of thermodynamic properties, which dominates
in the immediate vicinity of the critical point. However, the available experimental
data are usually taken far away from the critical point where classical thermody-
namic behavior becomes predominate. There are several approaches to dealing with
the crossover to classical thermodynamic behavior and for this work we use a simple
polynomial, which grows with distance from the critical point. The backgrounds
for the parameters are given in Table 4.
51
CHAPTER 5
PREDICTING THERMODYNAMIC PROPERTIES OF
SUPERCOOLED LIQUID WATER
With the given amplitudes and location of the critical point, obtained from
the heat capacity data as shown in Chapter 4, we predict the behavior of the com-
pressibility and expansivity, shown in Figs. 21 and 22, by adjusting only their non-
critical backgrounds. Similar predictions were performed for heavy water, shown in
Figs. 23 and 24. The predictions appear to have excellent agreement with the ex-
perimental data.14, 15 While it is di? cult to establish the error bars for the obtained
Pc value, the parametric equation of state certainly excludes the critical pressure
above 50 MPa or below 10 MPa for ordinary water and values of above 40MPa and
below 5MPa for heavy water.
A Lower Pressure for the Second Critical Point. The location of the critical point,
far lower than previously predicted, has some additional experimental support.
Mishima melted various heavy water ice and measured the onset of the change
in sample temperature during the decompression-induced melting experiment. In
heavy water ice V, IV and XIII there appeared to be a sharp change in slope that
suggests the possibility of the HDL to LDL transition line. However, in heavy
water ice III, the line curved smoothly. Figure 25 shows experimental results and
a schematic representation with the second-critical point. These data suggest a
critical point for heavy water below 60 MPa and above 10 MPa, consistent with the
predictions in this thesis.
52
Scaling versus Mean-Field Predictions. We also conclude that the mean-?eld sce-
nario is unlikely. The mean-?eld scenario cannot predict the anomalous behavior
of isothermal compressibility within our model. While the major contribution in
the heat-capacity anomaly is strong susceptibility, 1, (b1 is small) diverging both in
mean-?eld and in scaling theory, the major contribution in the isothermal compress-
ibility anomaly is the weak susceptibility 2, (a1 is small) which shows no anomaly
in mean-?eld approximation. The major contribution in the critical part of the
expansivity comes from the cross susceptibility 12 as both a1 and b1 are small.
These features make the second critical point in water essentially di?erent from the
liquid-vapor critical point where CP, T, and P all diverge strongly, as 1, and from
the liquid-liquid critical points in binary ?uids where CP, T, and P all diverge
weakly, as 2.
Limitations to the Scaled Equation of State. There are obvious limitations of our
equation of state. First, the model used in this work is accurate only asymptotically
close to the critical point (r << 1) while all measurements in supercooled water have
been taken far beyond the asymptotic region. The experimental range of r, the
parametric distance to the critical point, at worst may be as large as 0.5. However,
this is the ?rst estimate of the critical parameters for the second critical point in
water based on experimental data, and not on computer simulations of "water like"
models. Including non-asymptotic corrections to the parametric equation of state
wouldchangetheadjustablebackgroundswhilenotsigni?cantlya?ectingthecritical
parameters. To more accurately describe and predict the properties in a broader
53
range of pressures and densities in supercooled water, a ?global" crossover equation
of state,69 basedon a reliable mean-?eld equation of state, such as a modi?ed van der
Waals model,60 is required. Moreover, we did not address an intriguing possibility
of the existence of multiple critical points in supercooled water, as predicted by
some simulated water models.103, 104
In this work, the order parameter is phenomenologically expressed through
molar volume and entropy, with entropy being the major contribution. A clar-
i?cation of the relation between this phenomenology and the microscopic nature
of the order parameter51 would help in better understanding the physics of phase
transitions in supercooled water.
Nature of the Order Parameter. In supercooled liquid water, the nature of the
order parameter is not associated mostly with density, as is the case of the liquid-
vapor critical point. Studies by Poole et al.60 and Truskett et al.51 have looked
at the connection between the theoretical, microscopically de?ned order parameter,
(1)theoretical = (B0)theoreticaljh2j, and the physical ?elds, while the phenomenologi-
cally de?ned order parameter is:
1 =


^^S

+b1^
1 +a1b1  B0jh2j
 : (5.1)
Poole?s and Truskett?s scalar order parameter is related to an optimal orientation of
hydrogen bonds. The microscopically de?ned order parameter belongs to the Ising
universality class and linearly couples with entropy and density, unlike in ?uids,
54
while dynamically, it may belong to a dynamic universality class of non-conserved
order parameter.
Hajime Tanaka also explored the nature of the order parameter, suggesting
the possibility of a two-order-parameter description and introduced a bond order
parameter de?ned as the local fraction of locally favored structures.106
Other Equations of State. Kiselev and Ely107, 108 were the ?rst to apply ?uctuation
theory to supercooled water and calculate the physical properties of ordinary and
heavy water. Their work encompassed predictions of the thermodynamic properties
ofordinarywater, heavywaterandordinaryandheavywatermixtures, aswellasthe
developmentofacrossoverequationofstate. However, inordertopredictproperties
in both the supercooled and the stable regions and provide reasonable values for
heavy water, ordinary water and mixtures, Kiselev and Ely incorporated over a
dozen system-dependent parameters into their equation of state. The resulting
predictions were more empirical than fundamental. Moreover, the predicted phase
diagram looks unrealistic with the (P;T) slope positive instead of negative, as well
established. The scaled parametric equation of state developed in this thesis is
more limited in scope, but relies on only three adjustable parameters, the critical
pressure Pc, the system dependent amplitude k=a, and the non-critical backgrounds.
Multiple Critical Points. The predictions of the scaled, parametric equation of state
support the hypothesis of a second critical point in supercooled water. However,
this result does not preclude the possibility of multiple critical points. Molecu-
lar dynamic models and monte carlo simulations, such as those of Brovchenko et
55
al. show four possible phases of supercooled liquid water, corresponding with ob-
served densities in amorphous ice.109 The debate regarding the number of phases
of amorphous ice experimentally observed and the level of uncertainty built into
the equation of state support further exploration of the multiple critical points in
supercooled liquid water.
56
CHAPTER 6
SUMMARY
The scaled equation of state for supercooled liquid water o?ers a consis-
tent scaling description of the available experimental data in supercooled water.
Although this work is not absolutely de?nitive in its support of the second-critical-
point scenario, it lends weight to a growing body of evidence and provides another
tool for future evaluations as additional experimental data become available.
This work represents one piece of a much larger puzzle in the development
of a thermodynamically consistent equation of state for liquid water in general.
When complete, the implications for this work are widespread. Supercooled liquid
water occurs naturally in the atmosphere, playing a role in weather and global
warming by absorbing solar and terrestrial energy changes. The high pressures and
low temperatures necessary for supercooled liquid water also occur naturally on
some of our neighboring planets. Storms on Mars and photos of Europa, Jupiter?s
moon, suggest the presence of supercooled liquid water. Aqueous solutions also
represent the importance of supercooled liquid water. The phase behavior of pure
liquid water shifts when put in solution, potentially either shifting the phenomena
previously only seen in deep supercooled regions into an area of the phase diagram
more accessible at ambient temperatures and pressures or making the states of
the solution stable at temperatures below 0oC. Aqueous solutions below 0oC have
implications for deep ocean science, underwater communication and navigation as
well as biological systems. In addition, an understanding of the behavior of liquid
57
water might provide insight into polyamorphism in phosphorous,110, 111 and other
substances, such as SiO2, and GeO2, which exhibit similar anomalous behavior upon
cooling.112, 113
58
 59
APPENDIX I: TABLES 
  
 
 
Thermodynamic 
Property 
Formula Path Classical 
Critical 
Exponent 
Non-classical 
Critical 
Exponent 
Isothermal 
Compressibility 
 
 
  
0
?
 
?
T
()
  
 
  =  
c
 1 1.239 ? 0.002  
Isochoric Heat 
Capacity 
C
V
  A
0
?
 
?
T
()
  
 
  =  
c
 0 0.110 ? 0.003
 
Density 
    
c
  ?B
0
  
?
T
()
 
 
Coexistence 
?
1
= ?
2
 
  0.326 ? 0.002  
Pressure 
P  P
c
  
?
 ()
 
 
T = T
c
 3 4.8 ? 0.02  
Correlation 
Function, Gr() 
Gr()  r
  1  ()
 
Critical 
point;  
large r 
0 0.031? 0.004  
Correlation 
Length,    
    
0
?
 
?
T
()
  
 
  =  
c
 1/2 0.063? 0.001
 
 
 
Table 1.  Classical and Non-classical Critical Exponents.  A
0
?
, B
0
?
,  
0
?
,  
0
?
are critical 
amplitudes above and below the critical temperature.
 60
 
Universal Variables Ising (ferromagnets) Lattice Gas (liquid-gas) 
h
1
 Magnetic Field 
 
?
? = ?   ?
c
()/ RT
c
 
h
2
 
 
?
T   
?
T  
 
1
 Magnetization 
 
?
   
 
2
 
 
?
S =  S / k
B
 
 
?
 
?
S
()
=  S    
c
S
c
()/ rc k
B
 
 
 
Table 2.  Scaling theory universal theoretical variables for ferromagnets and liquid-gas 
systems. 
 
 61
 
Symbol Variable Liquid-Vapor Liquid-Liquid 
(dT/dm) 
Liquid-Liquid 
   (water, BPIII) (binary liquid) 
 
1
 Order parameter   
 
 S   x  
h
1
 Ordering field  ?   T   ?
21
 
h
2
 ?Thermal? scaling 
field 
 T   ?   T  
 
2
  
   S() 
  
 
 S  
 
1
 Strong susceptibility  
T
   
P
  C
P
 C
P
  x /  ?
21
 
 
2
 Weak susceptibility C
V
  
T
  
T
   
P
  C
P
 
 
12
 Cross susceptibility 
   /  T()
h
1
=0
 
 
P
 
 x /  T()
h
1
=0
 
 
 
Table 3.  Cartoon analogy between liquid-vapor and liquid-liquid critical points.   
 
 62
 
Thermodynamic 
Property 
Non-critical Background 
C
P
 (0.1 MPa)  
7.8* T + 2.1 
 
T
 (0.1 MPa)  0.000456
 
 
T
 (10 MPa)  0.000456  
 
T
 (50 MPa)  0.00005* T+0.00039  
 
T
 (100 MPa)  -0.00034* T+0.000436  
 
T
 (150 MPa)  -0.000326* T+0.000396
 
 
T
 (190 MPa)  -0.00024* T+0.00034  
 
T
 (0.1 MPa)  0.00125*ln(1.77297* T^1.05498)+0.002-0.0122* T+0.017* T^2
-(0.00085*ln(0.18418* T^0.33789)
 
C
P
 (0.1 MPa)  2.3* T+8.5
 
 
T
 (10 MPa)   0.178   0.00974*T + 5.5E   5*T ^2
 1.364E   7*T ^3+1.257E  10*T ^4
 
 
T
 (50 MPa)  0.00035* T+0.00033
 
 
T
 (100 MPa)  -0.0006* T+0.00053 
 
T
 (150 MPa)  -0.00042* T+0.00047  
 
T
 (190 MPa)  -0.00024* T+0.00039  
 
T
 (0.1 MPa)  0.000000000000014* T^4-0.0000043* T+0.0023
 
 
Table 4.  Non-critical background polynomials for ordinary water (aqua blue) and heavy 
water (sky blue). 
 63
APPENDIX II: FIGURES 
 
 
 
 
 
Figure 1.  Phase diagram of water showing stable regions of vapor, liquid, and various 
forms of ice.  The region inside the purple curve shows the range of validity for the 
IAPWS-95 empirical equation of state. IAPWS-95 is recommended by the International 
Association for the Properties of Water and Steam (IAPWS) for general and scientific 
use.  Developed to accurately reproduce thermodynamic properties in the stable range, 
the IAPWS-95 formulation for liquid water is valid for temperatures ranging from 240K 
in the metastable region to 1273 K, and for positive pressures up to 1000MPa.
3,4
  The 
IAPWS-95 region of validity is shown as a shaded box,  .  Kanno et al. 
16
 cooled liquid 
water down to -38
o
C at 0.1 MPa and down to -92
o
C at 200 MPa.  The limits of 
experiments from Kanno et al. are shown as green circles,  .  Figure adapted from 
London South Bank University webpage http://www.lsbu.ac.uk/water/phase.html, and  
reprinted with permission from Dr. Martin Chaplin.
114
 
 
 
 64
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 2.  Thermodynamic and kinetic limits for liquid water at atmospheric pressure.  
The thermodynamic limits for water, the binodal or melting temperature (T
M
), and the 
spinodal or limit of stability (T
S
), are system dependent and do not depend on 
experimental conditions.  Although the melting temperature is well known to be 273.15 
K, the spinodal is much less well defined as experiments probing the metastable region 
will first encounter kinetic limits to supercooling.  Determining the actual spinodal 
temperature requires a valid equation of state.  Kinetic limits for heterogeneous 
nucleation (T
h
) and homogeneous nucleation (T
H
   231 K) 
16,17,18
 depend on experiment 
parameters such as sample size, purity, emulsion fluid, etc.  Other kinetic limits apply 
when rapidly cooling liquid water and kinetically arresting its structure.  The resulting 
fluid, glassy water, is defined by the glass transition temperature (T
g
   136 K) 
21,27,28,29
, 
and the spontaneous crystallization line T
X
.
 
21,22
  
Stable Water 
Metastable Water 
273.15 
T/K T/
 
C 
Unstable Water 
T
M
 
T
h
 
T
H
 
T
S
 
0 
St
a
b
l
e
 Ic
e
 
T
X
 
T
g
 
241 - 42 
 65
 
 
                        
 
Figure 3. Temperature dependence of the heat capacity of mildly supercooled water at 
ambient pressure (reproduced from Anisimov, 1972).
11
 Dashed blue line separates stable 
and supercooled liquid water regions.  Solid circles are the isobaric molar heat capacities 
C
P
 at the saturation-vapor pressure; open circles are the isochoric molar heat capacities 
C
V
. In the range 273-285 K the isochoric heat capacities of water have almost the same 
values as the isobaric heat capacities because the thermal expansion coefficient is close to 
zero. 
 
 
 
 
 66
                     
 
 
Figure 4. Temperature dependence of the isobaric heat  capacity of  supercooled ordinary 
water at ambient  pressure (reproduced from Debenedetti, 2003).
1
  Solid circles are the 
measurements of Anismov et al. in 1972.
11
  Open  circles  are  the 1973 measurements of 
Angell et al.
12
   
 
 
 
Stable Liquid 
Supercooled 
Liquid 
 67
 
 
 
 
                     
 
Figure 5.  Schematic illustration indicating the various phases of liquid water found at 
atmospheric pressure, including the region of ?No man?s land,? inaccessible to 
experimentation.  Figure courtesy of Dr. O. Mishima. 
 
 68
 
 
 
 
 
 
 
 
LDA 
HDA 
(a) (b) 
 69
Figure 6.  Structure of LDA and HDA.  (a) Snapshots of the molecular dynamic 
configuration of LDA and HDA.  (b) Pictures of LDA and HDA samples.
 
Figures courtesy of O. Mishima.
115
  
  
Figure 7. Reversible LDA - HDA phase transition (reproduced from Mishima, 1994).
30
  
The data shows the compression of LDA to HDA (a), the decompression (b) from HDA 
to LDA and recompression (c) during warming from 130 to 140 K.  Curve (d) shows the 
compression of ice Ic at 145 K. 
 
 70
 
Figure 8.  Representation of the stability limit conjecture hypothesis and the retracing 
liquid spinodal adapted from Figure 2.16, Metastable Liquids, P.G. Debenedetti, 
Princeton, (1996)
2
 and from Angell, (1988).
116
  Dotted line is 0.1 MPa (1 atm).  Dashed 
line is liquid thermodynamic spinodal.  Dash-dotted line is the liquid-solid 
thermodynamic spinodal.  TRP is triple point. 
 
273 373 473 573 173 
0 
-100 
200 
100 
-200 
300 
 71
 
Figure 9. Stability limit conjecture with the thermodynamically consistent lower critical 
point adapted from Figure 2.16, Metastable Liquids, P.G. Debenedetti, Princeton, (1996)
2
 
and from Angell, (1988).
116
  Dotted line is 0.1 MPa (1 atm).  Dashed line is liquid 
thermodynamic spinodal.  Dash-dotted line is the liquid-solid thermodynamic spinodal.  
TRP is triple point. 
273 373 473 573 173 
0 
-100 
200 
100 
-200 
300 
 72
 
 
 
 
 
Figure 10.  Density of ordinary liquid water at atmospheric pressure (adapted from 
Zheleznyi, 1969). 
117
 
 
 73
 
 
 
Figure 11.  Generalizing Fig. 5 to incorporate pressure and illustrate the second-critical-
point scenario.  Courtesy of Dr. O. Mishima. 
 
 74
 
Figure 12.  Phase diagram for non-crystalline forms of water, illustrating the second-
critical point interpretation (reproduced from Mishima, 1998).
61
  C and C' are the vapor-
liquid and liquid-liquid critical points respectively.  F is the low- and high-density liquid 
(HDL and LDL) coexistence line and H and L are their limits of stability. At low 
temperatures, LDL and HDL transition to their respective kinetically arrested amorphous 
or glassy phases, LDA and HDA. 
300 
400 
C 
 75
 
Figure 13.  Second-critical-point scenario with computer simulation (CS) values, Kiselev 
and Ely EOS 
108
 and our estimate for the second critical point.   
 76
 
 
Figure 14.  Temperature-density phase diagram illustrating the second-critical-point 
scenario with spinodal and binodal, generated from an extended van der Waals model 
(reproduced from Poole et al., 1994). 
60
  C and C' are the vapor-liquid and liquid-liquid 
critical points respectively.  The solid curves are coexistence curves, the dashed and dot-
dashed lines curves are their corresponding spinodals.  The thin dotted line is the TMD 
locus.   
 
 77
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 15.  Two dimensional representation of Ising model.  The spin direction, or (+) 
and (-) are influenced by the nearest neighbors, as illustrated by the red vertice and the 
orange positions that influence it.   
 78
 
 
Figure 16.  Parametric representation of scaling coordinates h
1
=0 and h
2
=0, where r is 
the distance from the critical point and theta is the angle along a curve of constant r.   
 
 79
 
 
 
 
Figure 17.  The IAPWS-95 formulation for density at various pressures and 
temperatures.  Figure reproduced from Wagner and Pru , 2002.
3
 
 
 
 
 80
 
 
 
 
 
 
 
Figure 18.  Coexistence curve and ?Widom line? for ordinary and heavy water.  (a) 
shows the liquid-liquid coexistence curve courtesy of Mishima 
62
 and the fitted curves for 
the parametric EOS.  Figure (b) shows the liquid-liquid coexistence for ordinary water 
and the predicted curve in the (P,T) plane.  
0 50 100 150 200 
240 
200 
160 
120 
Pressure (MPa) 
Tem
p
erat
u
re (
K
)
 
(b) 
(a) 
 81
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 19.  Isobaric heat capacity for ordinary water at atmospheric pressure.  
Experimental data, from Angell, 1982
118
 is shown in open circles.  Solid green line is the 
scaled, parametric equation of state, fitted to the data adjusting the location of the critical 
point P
C
 = 27 MPa, the system-dependent parameter k/a = 0.45, and the non-critical 
background (dashed blue line).  Solid vertical lines denote melting temperature and 
critical temperature. 
 
Temperature (K) 
C
P
/R
 
C
r
i
t
i
c
al
 Te
m
per
atu
r
e 
Mel
t
i
ng T
e
mpe
r
atu
r
e 
Angel et al. JPC (1982) 
                              Calculation 
 
                                Fit of our Theory 
 
    Background   
 
C
P
0
 
C
V
 
C
P
 82
 
 
Figure 20.  Scaling fields and parametric variables for the scaled parametric equation of 
state for supercooled liquid water. 
 
Temperature (K) 
Pre
s
s
u
re
 (M
Pa
) 
 83
 
 
Figure 21.  Isothermal compressibility, experimental
14,15
 and predicted for various 
temperature and pressures.  Non-critical background values can be found in Table 4.  
 
 84
 
 
Figure 22.  Thermal expansivity at atmospheric pressure, experimental and predicted.
15
 
 
P
/
K
-1
 
1986 
 85
 
 
 
Figure 23.  Isobaric heat capacity for heavy water at atmospheric pressure.  Experimental 
data, from Angell, 1982
119
 is shown in open circles.  Solid green line is the scaled, 
parametric equation of state, fitted to the data adjusting the location of the critical point 
P
C
 = 20 MPa, the system-dependent parameter k/a = 0.7, and the non-critical background 
(dashed blue line).  Dotted vertical line separates stable and metastable liquid water. 
230 240 250 260 270 280 
8 
16 
10 
14 
12 
Temperature (K) 
C
P
/
R
 
M
e
lting T
e
mper
atur
e,
 276.
9
 K D
2
O 
1 atm 
 86
 
 
 
Figure 24.  Isothermal compressibility for heavy water at various temperatures and 
pressures.  Experimental data, from Kanno et al., 1982
14
 is shown in open data points. 
Solid lines are predicted from the equation of state.  Non-critical background values can 
be found in Table 4.  Vertical dotted line is the melting temperature for heavy water. 
 
0.0008 
0.0006 
0.0004 
0.0002 
0.0010 
240 250 260 280 290 270 
Temperature (K) 
 
T
/
MP
a
-1
 
10  MPa 
50  MPa 
150 MPa 
100 MPa 
190 MPa 
D
2
O 
 87
 
 
 
 
 
Figure 25.  Thermal expansivity for heavy water at atmospheric pressure.  Experimental 
data, from Hare et al., 1986
15
 is shown in open circles.  Solid lines,  
P
 without non-
critical background in black and  
P
 with background in red, are predicted from the 
equation of state.  Non-critical background values can be found in Table 4.  Vertical 
dotted line is the melting temperature for heavy water. 
D
2
O, 1 atm 
0.0015 
-0.0015 
-0.0005 
240 
0.0005 
-0.0025 
260 320 
280 
300 340 
Temperature (K) 
 
P
/
K
-1
 
 88
 
 
 
 
 
 
 
 
 
Figure 26.  Melting curves of D
2
O ices (III, V, IV, XIII) (reproduced from Mishima, 
2000).
63
  (a) Experimental results (b) Schematic showing hypothesized liquid-liquid 
transition between high-and low-density liquid (HDL, LDL).  C.P. denotes the 
hypothesized critical point.  
 89
APPENDIX III: DERIVATIONS 
 
I.   Calculating mixing coefficients from liquid-liquid coexistence curve and the 
?Widom line? 
 
 
h
1
=
a
1
 P
 
c
RT
c
+
a
2
 T
T
c
+
a
3
 P
2
 
c
RT
c
()
2
 
 
h
2
=
b
1
 T
T
c
+
b
2
 P
 
c
RT
c
 
where 
 
 P = P   P
c
 and 
 
 T = T   T
c
 
 
 
 
 
 
  
 
 
 
 
 
 
 
Figure 1. Phase Diagram for non-crystalline forms of water showing second-critical 
point interpretation.  C and C? are vapour-liquid and liquid-liquid critical points, 
respectively.  C?, the critical point for high and low density liquid water (HDL and LDL) 
occurs at 100MPa and 220K. 
 
The dashed perpendicular lines in the phase diagram above represent the case of h
1
 = 0 
and h
2
 = 0.  The almost vertical axis of h
1
 = 0 corresponds with the phase transition 
between HDL and LDL supercooled liquid water.   
 
 
  P
~
 T
~
 
 
 
 
 
 
 
 
h
1
=0
=  
a
2
a
1
, 
 
  P
~
 T
~
 
 
 
 
 
 
 
 
h
2
=0
=  
b
1
b
2
  
 
 
Data points representing a best guess for the homogeneous nucleation limit, believed by 
O. Mishima to be very close to the liquid-liquid coexistence curve (h
1
=0), were originally 
obtained from Mishima et al.
1
  and then through personal communications between M.A. 
                                                
1
 O. Mishima and H.E. Stanley, Nature 396, 329 (1998). 
Temperature 
h
2
=0 HDL 
LDL C? 
C 
P
r
e
ssu
r
e
 
h
1
=0 
 90
Anisimov and O. Mishima in 2005 and 2007.  The points were fitted with a parabolic 
polynomial, see Fig. 2, to obtain an equation for the ?Widom line.? The fit with O. 
Mishima?s data for the liquid-liquid coexistence curve in ordinary water produced the 
result of: 
 
T = A+ BP+CP
2
= 234.12983428512246   0.06175940215384P  0.000952783743374272P
2
 
 
 
Figure 2.  Estimation for the homogeneous nucleation temperature, T
H
, at various 
pressures in supercooled liquid water.  Red circles represent data points extrapolated 
from a figure in the 1998 Nature publication of Mishima et al.
1
  The sensitivity of the 
equation of state to the location of the coexistence curve and its extension beyond the 
critical point, the ?Widom line?, resulted in further communication with O. Mishima.  
The black squares represent a more accurate estimate of T
H
 at various pressures, provided 
by O. Mishima through personal communication with M.A. Anisimov.  
 
 
Adjusting the equation to fit 
 
h
1
= 0 = a
1
 
 
P + a
2
 
 
T + a
3
 
 
P
2
: 
T  T
c
()+T
c
= A+ BP  P
c
()+ BP
c
+CP  P
c
()
2
+ 2CPP
c
 CP
c
2
 
where  
A = 233.129834 ,  
H
2
O 
 91
B =  0.0167594 ,  
C =  0.0012528 
 T = A T
c
+ BP
c
 CP
c
2
+ 2CP
c
2
 
 
 
 
+ B+ 2CP
c
[]
 P +C P
2
 
 
Rearranging, we find: 
0 = A T
c
+ BP
c
 CP
c
2
+ 2CP
c
2
 
 
 
 
+ B+ 2CP
c
[]
 P   T +C P
2
 
The variables can be made dimensionless by dividing the equation by  
c
RT
c
()
2
.   
Multiplying by   
c
2
R
2
T
c
 gives our  T term a coefficient of 1. 
 
0 =  
A T
c
+ BP
c
 CP
c
2
+ 2CP
c
2
 
 
 
 
T
c
   
c
RB+ 2CP
c
[]
 
 
P +  
 
T    
c
2
R
2
T
c
C 
 
P
2
 
Comparing to h
1
= 0 = a
1
 P
~
+ a
2
 T
~
+ a
3
 P
~
2
and collecting terms with like powers of 
pressure and temperature yields values for the coefficients: 
 
0 =  
A T
c
+ BP
c
 CP
c
2
+ 2CP
c
2
T
c
  
a
1
=   
c
R 2CP
c
+ B
[]
 
 
 
a
3
=   
c
2
R
2
T
c
C  
 
We obtain values for our coefficients by using literature estimates of the critical pressure.  
The actual position of the critical point along the ?Widom line? will be estimated later 
through trial and error comparisons between experimental data and calculated values for 
heat capacity. 
Using R = 8.314Jmol
-1
K
-1
, P
c
 = 100MPa, and  
c
=
1gcm
 3
18gmol
 1
100
3
cm
3
m
3
= 55555.6molm
 3
 , 
we find T
c 
= 218K, 
and our equation 
h
1
= a
1
 P
~
+ a
2
 T
~
+ a
3
 P
~
2
 
 
Where a
1
 = 0.039, a
2
 = 1, and a
3
 = 0.062. 
 
This analysis also shows b
1
 = 0.039 and b
2
 = -1, given that the respective slopes are 
perpendicular to each other, 
 
 
a
2
a
1
=  
1
 b
1
/ b
2
=
b
2
b
1
. 
 
 
a
2
=1
 92
A similar calculation for heavy water was performed.  O. Mishima provided a best guess 
for the location of the liquid-liquid line.  A polynomial fit for the data resulted in: 
 
A = 239.62382,  
B =  0.045217 ,  
C =  0.0008772 
 
Solving the algebra results in mixing coefficients of 
h
1
= a
1
 P
~
+ a
2
 T
~
+ a
3
 P
~
2
 
 
Where a
1
 = 0.042, a
2
 = 1, and a
3
 = 0.042. 
 
 
 
II. Derivations for Thermodynamic Properties based on Universal Equations and 
Theoretical Variables  
 
 
 
h
1
= a
1
 
 
P + a
2
 
 
T + a
3
 
 
P
2
 
 
h
2
= b
1
 
 
T + b
2
 
 
P  
where 
 
  P
~
= P   P
c
()
 
c
RT
c
()
 and 
 
 T
~
= T   T
c
()
T
c
 
 
Inverting the equations for h
1
 and h
2
, we can express  T and  P in terms of h
1
 and h
2
. 
 
 T
~
=
a
1
a
1
b
1
  a
2
b
2
h
2
 
b
2
a
1
b
1
  a
2
b
2
h
1
**    
 P
~
=
b
1
a
1
b
1
  a
2
b
2
h
1
 
a
2
a
1
b
1
  a
2
b
2
h
2
** 
** Only if discounting a
3
 and b
3
 or defining the first term to include the third term. 
 
The field-dependent dimensionless density of a relevant thermodynamic potential relates 
to scaling ?densities? conjugate to h
1
 and h
2
, and strongly and weakly divergent 
susceptibilities associated with the densities.   
 
 
 G
~
(h
1
,h
2
) =
 G
V 
c
RT
c
 
 93
 
 
1
=  
   G
~
 h
1
 
 
 
 
 
 

 	
 	
h
2
, 
 
 
2
=  
   G
~
 h
2
 
 
 
 
 
 

 	
 	
h
1
, 
 
 
1
=
  
1
 h
1
 
 
 
 
 	
 
h
2
, 
 
 
2
=
  
2
 h
2
 
 
 
 
 	
 
h
1
, 
 
 
12
=  
21
=
  
1
 h
2
 
 
 
 
 	
 
h
1
=
  
2
 h
1
 
 
 
 
 	
 
h2
 
 
 
Using Jacobians to change variables: 
 
 
 S
 T
 
 
 
 
 
 
V
=
  S,V
()
  T ,V
()
=
  S,V
()
/   T , P
()
  T ,V
()
/   T , P
()
=
 S
 T
()
P
 V
 P
()
T
 
 V
 T
()
P
 S
 P
()
T
 T
 T
()
P
 V
 P
()
T
 
 V
 T
()
P
 T
 P
()
T
 
 
 S
 T
 
 
 
 
 
 
V
=
 S
 T
()
P
 V
 P
()
T
 
 V
 T
()
P
 S
 P
()
T
 V
 P
()
T
 
 
 
 
 
1
=  
   G
~
 h
1
 
 
 
 
 
 

 	
 	
h
2
=  
  G
~
,h
2
 
 
 
 
 

 	
/   T
~
,  P
~
 
 
 
 
 

 	
  h
1
,h
2
()
/   T
~
,  P
~
 
 
 
 
 

 	
=  
   G
~
  T
~
 
 
 
 
 
 

 	
 	
  P
~
 h
2
   P
~
 
 
 
 
 
 

 	
 	
 T
~
 
 h
2
  T
~
 
 
 
 
 
 

 	
 	
  P
~
   G
~
   P
~
 
 
 
 
 
 

 	
 	
 T
~
 h
1
  T
~
 
 
 
 
 
 

 	
 	
  P
~
 h
2
   P
~
 
 
 
 
 
 

 	
 	
 T
~
 
 h
2
  T
~
 
 
 
 
 
 

 	
 	
  P
~
 h
1
   P
~
 
 
 
 
 
 

 	
 	
 T
~
 
 
 
2
=  
   G
~
 h
2
 
 
 
 
 
 

 	
 	
h
1
=  
  G
~
,h
1
 
 
 
 
 

 	
/   T
~
,  P
~
 
 
 
 
 

 	
  h
2
,h
1
()
/   T
~
,  P
~
 
 
 
 
 

 	
=  
   G
~
  T
~
 
 
 
 
 
 

 	
 	
  P
~
 h
1
   P
~
 
 
 
 
 
 

 	
 	
 T
~
 
 h
1
  T
~
 
 
 
 
 
 

 	
 	
  P
~
   G
~
   P
~
 
 
 
 
 
 

 	
 	
 T
~
 h
2
  T
~
 
 
 
 
 
 

 	
 	
  P
~
 h
1
   P
~
 
 
 
 
 
 

 	
 	
 T
~
 
 h
1
  T
~
 
 
 
 
 
 

 	
 	
  P
~
 h
2
   P
~
 
 
 
 
 
 

 	
 	
 T
~
 
a
1eff
= a
1
+ 2a
3
 P , b
1eff
= b
1
+ 2b
3
 T  
1 0 
 94
 
 
 h
1
   P
~
 
 
 
 
 
 
 
 
 T
~
= a
1eff
, 
 
 h
1
  T
~
 
 
 
 
 
 
 
 
 P
~
= a
2
,  
 
 h
2
  T
~
 
 
 
 
 
 
 
 
 P
~
= b
1eff
, 
 
 h
2
  P
~
 
 
 
 
 
 
 
 
 T
~
= b
2
 
Using Jacobians,  
   P
~
 h
1
 
 
 
 
 
 	
 
 
h
2
=
b
1eff
a
1eff
b
1eff
  a
2
b
2
, 
   P
~
 h
2
 
 
 
 
 
 	
 
 
h
1
=  
a
2
a
1eff
b
1eff
  a
2
b
2
 
  T
~
 h
1
 
 
 
 
 
 	
 
 
h
2
=  
b
2
a
1eff
b
1eff
  a
2
b
2
, 
  T
~
 h
2
 
 
 
 
 
 	
 
 
h
1
=
a
1eff
a
1eff
b
1eff
  a
2
b
2
 
 
 
 
1
R
  G
 T
 
 
 
 
 	
 
P
=   S
~
, from 
 
dG = VdP   SdT  
 
P
c
RT
c
  G
 P
 
 
 
 
 
 
T
=  V
~
, from 
 
dG = VdP   SdT  
 
 
1
=  
 b
2
 S
~
  b
1eff
 V
~
a
2
b
2
  a
1eff
b
1eff
 
 
 
 
 
 
 	
 
 
 
=
 b
1eff
 V
~
  b
2
 S
~
a
1eff
b
1eff
  a
2
b
2
 
 
 
 
 
 
 	
 
 
 
 
 
 
2
=  
 a
1eff
 S
~
  a
2
 V
~
a
1eff
b
1eff
  a
2
b
2
 
 
 
 
 
 
 	
 
 
 
=
a
1eff
 S
~
+ a
2
 V
~
a
1eff
b
1eff
  a
2
b
2
 
 
 
 
 
 
 	
 
 
 
 
 
These densities conjugate to the scaling fields are linear combinations of the physical 
densities 
 
 V = V  V
c
 and
 
 S = S   S
c
. 
 
Rearranging, we can solve for  V and  S in terms of  
 1
 and  
 2
. 
 
 V
~
=
 
1
a
1eff
b
1eff
  a
2
b
2
()
+ b
2
 S
~
 b
1eff
, 
 
 S
~
=
 
1
a
1eff
b
1eff
  a
2
b
2
()
+ b
1eff
 V
~
 b
2
 
 
 V
~
=
 
2
a
1eff
b
1eff
  a
2
b
2
()
  a
1eff
 S
~
a
2
, 
 
 S
~
=
 
2
a
1eff
b
1eff
  a
2
b
2
()
  a
2
 V
~
a
1eff
 
 
 
 95
 
 V
~
=
   G
~
   P
~
 
 
 
 
 
 

 	
 	
 T
~
=   a
1eff
 
1
+ b
2
 
2
()
 
 
 
  S
~
=  
   G
~
  T
~
 
 
 
 
 
 

 	
 	
  P
~
= a
2
 
1
+ b
1eff
 
2
()
 
 
 
 
using Maxwell Relations, 
 
 S
 P
 
 
 
 
 
 
T
=  V
 V
 T
 
 
 
 
 
 
P
, and 
 
 S
 T
 
 
 
 
 
 
P
=
Cp
T
 
 
 
   S
~
  T
~
 
 
 
 
 
 	
 
 
  P
~
= a
2
  
1
  T
~
 
 
 
 
 
 	
 
 
  P
~
+ b
1
  
2
  T
~
 
 
 
 
 
 	
 
 
  P
~
 
 
 
 
 
 	
 
 
 
 
 
  
1
  T
~
 
 
 
 
 
 

 	
 	
  P
~
=
   
1
,  P
~
 
 
 
 
 

 	
/   h
1
,h
2
()
  T
~
,  P
~
 
 
 
 
 

 	
/   h
1
,h
2
()
=
  
1
 h
1
 
 
 
 
 

 	
h
2
   P
~
 h
2
 
 
 
 
 
 

 	
 	
h
1
 
   P
~
 h
1
 
 
 
 
 
 

 	
 	
h
2
  
1
 h
2
 
 
 
 
 

 	
h
1
  T
~
 h
1
 
 
 
 
 
 

 	
 	
h
2
   P
~
 h
2
 
 
 
 
 
 

 	
 	
h
1
 
   P
~
 h
1
 
 
 
 
 
 

 	
 	
h
2
  T
~
 h
2
 
 
 
 
 
 

 	
 	
h
1
 
 
  
1
  T
~
 
 
 
 
 
 

 	
 	
  P
~
= a
2
 
1
+ b
1eff
 
12
()
 
 
  
2
  T
~
 
 
 
 
 
 

 	
 	
  P
~
=
   
2
,  P
~
 
 
 
 
 

 	
/   h
1
,h
2
()
  T
~
,  P
~
 
 
 
 
 

 	
/   h
1
,h
2
()
=
  
2
 h
1
 
 
 
 
 

 	
h
2
   P
~
 h
2
 
 
 
 
 
 

 	
 	
h
1
 
   P
~
 h
1
 
 
 
 
 
 

 	
 	
h
2
  
2
 h
2
 
 
 
 
 

 	
h
1
  T
~
 h
1
 
 
 
 
 
 

 	
 	
h
2
   P
~
 h
2
 
 
 
 
 
 

 	
 	
h
1
 
   P
~
 h
1
 
 
 
 
 
 

 	
 	
h
2
  T
~
 h
2
 
 
 
 
 
 

 	
 	
h
1
 
 
  
2
  T
~
 
 
 
 
 
 

 	
 	
  P
~
= a
2
 
12
+ b
1eff
 
2
 
 
and 
 
 96
 
   S
~
  T
~
 
 
 
 
 
 

 	
 	
  P
~
= a
2
2
 
1
+ 2a
2
b
1eff
 
12
+ b
1eff
2
 
2
()
=
C
p
  C
p
r
T
 
 
The universal form of the heat capacity is: 
 
 
C
p
= Ta
2
2
 
1
+ 2a
2
b
1eff
 
12
+ b
1eff
2
 
2
()
+ C
p
r
 
where r designates the residual, or background, heat capacity. 
 
Heat capacity at constant volume can also be calculated.   
C
v
T
=
 S
 T
 
 
 
 
 
 
V
=
  S,V()/   T,P()
  T,V()/   T,P()
=
 S
 T
()
P
 V
 P
()
T
 
 S
 P
()
T
 V
 T
()
P
 T
 T
()
P
 V
 P
()
T
 
 T
 P
()
T
 V
 T
()
P
=  
 Using Maxwell?s 
 S
 P
 
 
 
 
 
 
T
=  
 V
 T
 
 
 
 
 
 
P
and 
 S
 T
 
 
 
 
 
 
P
=
C
P
T
 
C
V
T
=
C
P
T
 V
 P
 
 
 
 
 
 
T
+
 V
 T
P
 
 
 
 
 
 
2
 
 
 
 
 
 
 V
 P
 
 
 
 
 
 
T
 so that 
C
P
 C
V
=  T
 V
 T
P
 
 
 
 
 

 	
2
 V
 P
 
 
 
 
 

 	
T
=TV 
P
2
 
T
 
 
Isothermal compressibility is defined as 
 
 
 
 
T
  
1
V
 V
 P
 
 
 
 
 

 	
T
 
Compressibility in fluids does not contain any background, and the definition of the terms 
in the compressibility equation is based on  
 
 
 
~
T   
V
c
V
P
c
V
c
 V
 P
 
 
 
 
 

 	
T
=  
1
V
~
 V
~
  P
~
 
 
 
 
 
 

 	
 	
T
~
??? 
 
 V
~
  P
~
 
 
 
 
 
 	
 
 
T
~
=   a
1eff
  
1
  P
~
 
 
 
 
 
 	
 
 
T
~
+ b
2
  
2
  P
~
 
 
 
 
 
 	
 
 
T
~
 
 
 
 
 
 	
 
 
 
 
 97
 
  
1
  P
~
 
 
 
 
 
 	
 
 
T
~
=
   
1
,T
~
 
 
 
 
 	
 
/   h
1
,h
2
()
  P
~
,T
~
 
 
 
 
 	
 
/   h
1
,h
2
()
=
  
1
 h
1
 
 
 
 
 	
 
h
2
 T
~
 h
2
 
 
 
 
 
 	
 
 
h
1
 
 T
~
 h
1
 
 
 
 
 
 	
 
 
h
2
  
1
 h
2
 
 
 
 
 	
 
h
1
  P
~
 h
1
 
 
 
 
 
 	
 
 
h
2
 T
~
 h
2
 
 
 
 
 
 	
 
 
h
1
 
 T
~
 h
1
 
 
 
 
 
 	
 
 
h
2
  P
~
 h
2
 
 
 
 
 
 	
 
 
h
1
 
 
  
1
   P
~
 
 
 
 
 
 

 	
 	
 T
~
= a
1eff
 
1
+ b
2
 
12
 
 
  
2
  P
~
 
 
 
 
 
 	
 
 
T
~
=
   
2
,T
~
 
 
 
 
 	
 
/   h
1
,h
2
()
  P
~
,T
~
 
 
 
 
 	
 
/   h
1
,h
2
()
=
  
2
 h
1
 
 
 
 
 	
 
h
2
 T
~
 h
2
 
 
 
 
 
 	
 
 
h
1
 
 T
~
 h
1
 
 
 
 
 
 	
 
 
h
2
  
2
 h
2
 
 
 
 
 	
 
h
1
  P
~
 h
1
 
 
 
 
 
 	
 
 
h
2
 T
~
 h
2
 
 
 
 
 
 	
 
 
h
1
 
 T
~
 h
1
 
 
 
 
 
 	
 
 
h
2
  P
~
 h
2
 
 
 
 
 
 	
 
 
h
1
 
 
  
2
  P
~
 
 
 
 
 
 	
 
 
T
~
= a
2
 
12
+ b
2
 
2
 
 
and 
 
 
 V
~
  P
~
 
 
 
 
 
 

 	
 	
T
~
=   a
1eff
2
 
1
+ 2a
1eff
b
2
 
12
+ b
2
2
 
2
()
=  V
~
 
~
 
 
 
~
=
1
V
~
a
1eff
2
 
1
+ 2a
1eff
b
2
 
12
+ b
2
2
 
2
()
 
 
The coefficient of thermal expansion is defined as  
 
 
P
 
1
V
 V
 T
 
 
 
 
 	
 
P
 
 
 V
~
=
   G
~
  P
~
 
 
 
 
 
 

 	
 	
 T
~
=   a
1eff
 
1
+ b
2
 
2
()
 
 
 V
~
 T
~
 
 
 
 
 
 	
 
 
P
~
=   a
1eff
  
1
 T
~
 
 
 
 
 
 	
 
 
P
~
+ b
2
  
2
 T
~
 
 
 
 
 
 	
 
 
P
~
 
 
 
 
 
 	
 
 
 
 
From the derivation for entropy: 
 98
 
  
1
  T
~
 
 
 
 
 
 

 	
 	
  P
~
= a
2
 
1
+ b
1eff
 
12
()
 
 
and 
 
 
  
2
  T
~
 
 
 
 
 
 

 	
 	
  P
~
= a
2
 
12
+ b
1eff
 
2
 
 
and 
 
 
 V
~
  T
~
 
 
 
 
 	
 
 

 

 P
~
=   a
1eff
a
2
 
1
+ a
1eff
b
1eff
+ a
2
b
2
()
 
12
+ b
1eff
b
2
 
2
= V
~
 
~
 
 
 
 
~
P
 
1
V
~
  V
~
  T
~
 
 	
 
 
 

 
 
 
 P
~
=  
1
V
~
a
1eff
a
2
 
1
+ a
1eff
b
1eff
+ a
2
b
2
()
 
12
+ b
1
b
2
 
2
 
 
 
III.  Mean-Field or Classical Theory 
 
 
The Landau expansion in powers of the order parameter  
1
 
 
 G
~
h
1
,h
2
()
=
1
2
a
0
h
2
 
1
2
+
1
4
u
0
 
1
4
  h
1
 
1
 
Minimizing the thermodynamic potential with respect to  
1 
yields 
 
u
0
 
1
3
+ a
0
h
2
 
1
  h
1
= 0  
 
 
 
2
=  
   G
~
 h
2
 
 
 
 
 
 

 	
 	
h
1
=  
1
2
a
o
 
1
2
 
 
1
=
 h
1
  
1
 
 
 
 
 

 	
h
2
 1
=
  u
o
 
1
3
+ a
o
h
2
 
1
()
  
1
 
 
 
 
 

 	
h
2
 1
= 3u
o
 
1
2
+ a
o
h
2
()
 1
 
 
 
12
=
 h
2
  
1
 
 
 
 
 

 	
h
1
 1
=
  u
o
 
1
2
a
o
  h
1
a
o
 
1
()
  
1
 
 
 
 
 

 	
h
2
 1
=
 2u
o
 
1
3
  h
1
a
o
 
1
2
 
 
 
 
 

 	
 1
=  
a
o
 
1
2
2u
o
 
1
3
+ h
1
 
 
 
 
 

 	
 
substituting for h1,  
 99
 
12
=  
a
o
 
1
2
2u
o
 
1
3
+u
o
 
1
3
+ a
o
h
2
 
1
 
 
 
 
 	
 
=  
a
o
 
1
3u
o
 
1
2
+ a
o
h
2
 
 
 
 
 	
 
=  a
o
 
1
 
1
 
 
 
2
=
 h
2
  
2
 
 
 
 
 

 	
h
1
 1
=
 h
2
  
1
 
 
 
 
 

 	
h
1
 1
  
1
  
2
 
 
 
 
 

 	
h
1
 1
=
 h
2
  
1
 
 
 
 
 

 	
h
1
 1
  
2
  
1
 
 
 
 
 

 	
h
1
=
 h
2
  
1
 
 
 
 
 

 	
h
1
 1   
1
2
a
o
 
1
2
 
 
 
 
 

 	
  
1
 
 
 
 
 
 
 

 	
 	
 	
h
1
=  a
o
 
1
 h
2
  
1
 
 
 
 
 

 	
h
1
 1
 
 
 
 
 
2
=
 h
2
  
2
 
 
 
 
 

 	
h
2
 1
=  a
o
 
1
 h
2
  
1
 
 
 
 
 

 	
h
2
 1
=  a
o
 
1
 
12
= a
o
2
 
1
2
 
1
 
 
The critical part of the entropy, 
 
 S
~
=   a
2
 
1
+ b
1eff
 
2
()
=
b
1eff
a
o
 
1
2
2
  a
2
 
1
 
 
 
 
 	
 
 
The heat capacity in classical theory is  
 
 
   S
~
  T
~
 
 
 
 
 
 

 	
 	
  P
~
= b
1eff
a
o
 
1
  
1
  T
~
 
 
 
 
 
 

 	
 	
  P
~
  a
2
  
1
  T
~
 
 
 
 
 
 

 	
 	
  P
~
 
 
 
 
 
 

 	
 	
=
C
p
~
  C
p
r
~
T
~
 
 
Since 
 
 
 
  
1
  T
~
 
 
 
 
 	
 
 

 

  P
~
= a
2
 
1
+ b
1eff
 
12
()
=
a
2
  a
o
b
1eff
 
1
a
o
h
2
+ 3u
o
 
1
2
 
 
 
 	
 
 

  
 
we find 
 
 
 C
p
~
=
T
~
a
2
  a
o
b
1eff
 
1
()
2
a
o
h
2
+ 3u
o
 
1
2
()
 
 
  
2
=   
a
o
 
1
2
2
 
 
 
 
 	
 
=  a
o
 
1
  
1
 100 
 
 
 
~
=  
1
V
~
 V
~
  P
~
 
 
 
 
 	
 
 

 

T
~
=
1
V
~
a
1eff
2
 
1
+ 2a
1eff
b
2
 
12
+ b
2
2
 
2
()
=
1
V
~
a
1
2
  2a
1eff
b
2
a
o
 
1
+ b
2
2
a
o
2
 
1
2
3u
o
 
1
2
+ a
o
h
2
 
 
 
 	
 
 

 
 
 
 
 
 
 
 
~
P
 
1
V
~
  V
~
  T
~
 
 

 	
 	
 
 
 
 
 P
~
=  
1
V
~
a
1eff
a
2
 
1
+ a
1eff
b
1eff
+ a
2
b
2
()
 
12
+ b
1eff
b
2
 
2
 P
~
=  
1
V
~
a
1eff
a
2
3u
o
 
1
2
+ a
o
h
2
()
+ a
1eff
b
1eff
+ a
1eff
b
2
()
 
a
o
 
1
2
2u
o
 
1
3
+ h
1
 
 

 	
 
 
 
+
b
2
2
a
o
2
 
1
2
3u
o
 
1
2
+ a
o
h
2
()
 
 

 	
 	
 
 
 
 
 101 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 102 
Parametric Mean-Field Representation: 
 
Using the simplest form of the parametric equations of state, we have a ?linear model? 
for the ?polar? variables r and  . 
h
1
= ar
 + 
  1  
2
()
 
h
2
= r 1 b
2
q
2
()
 
 
1
= kr
 
   
 
where r represents the distance to the critical point,   is the distance along a contour of 
constant r, b is a universal constant, and a and k are system dependent constants. 
 
In mean field (classical) theory, the magnitudes of the critical exponents are   = 0, 
  =1/2,   =1,   =
 
 
+1= 3, where delta is the universal critical exponent.  The value 
of b
2
=     2 ()/  1  2 () can be found by using the   expansion for critical 
exponents, where   = 4   d .  The mean field results are valid in the limit    0.  In the 
first order of  , the expansion gives 2  =1 
1
3
  ,   =1+
1
6
  ,   =
1
6
  , thus b
2
= 3/2. 
 
For scaling densities and susceptibilities in the parametric form: 
 
 
1
= kr
1/2
   
 
2
=  
1
2
akr 
2
=  
1
2
a
0
 
1
2
 
 
1
= k / a()r
 1
= a
o
h
2
+ 3u
o
 
1
2
 
 
 
 
 1
 
 
12
=  kr
 1/2
  =  a
o
 
1
 
1
 
 
2
= ak 
2
=
a
o
2
2u
o
 
2
= a
o
2
 
1
2
 
1
 
where the coefficients are related to k and a as follows: 
a
o
= a / k  
u
o
= a /2k
3
 
ak = a
0
2
/2u
o
 
 
  
 
 
 
 
 103 
IV.  Scaling Theory and the Parametric Linear Model 
 
Equations of state described through power laws are called ?scaling laws?.  They are 
characterized by universal exponents or critical exponents.  In scaling theory these 
exponents have values: 
 
Critical Exponent Scaling Theory Mean-Field Theory 
  
0.11 +/- 0.01 0 
  
0.325 +/- 0.005 0.5 
  
4.81 3 
  
1.24 +/- 0.01 1 
b
2
 1.36 1.5 
  
1.26 +/- 0.02 1.5 in vdw 
  
0.63 +/- 0.01 Not defined unambiguously 
 
Delta is the universal critical exponent, found from the relation    1()  =   . 
 
The simplest form of the parametric equations of state is the ?linear model?, representing 
h
1
 and h
2
 using the ?polar variables? r and  . 
 
h
1
= ar
 + 
  1  
2
()
 
h
2
= r 1  b
2
 
2
()
 
The singular part of the thermodynamic potential is: 
 
 
 
  r, ()= akr
2  
f  ()  
2
1  
2
()
 
 
 
 	
+ ak /6()r
2
1  b
2
 
2
()
2
 
with the last term added to make the model fully consistent with the results of 
renormalization group (RG) theory
2
.  
 
1
= kr
 
  ,  
2
= akr
1  
s  ()  akr 1 b
2
q
2
()
/3 
 
1
=
k
a
 
 
 
 
 

 	
r
  
c
1
 () 
 
12
= kr
  1
c
12
 () 
 
2
= akr
  
c
2
 ()  ak /3 
where r represents the distance to the critical point, the parameter   is the distance along a 
contour of constant r, the coefficient b
2
=     2 ()/  1  2 () is a universal constant, 
and a and k are system dependent constants.   
 
The universal heat capacity is given as: 
 
C
p
=  Ta
2
2
 
1
+ 2a
2
b
1eff
 
12
+ b
1eff
2
 
2
()
+ C
p
r
 
 
                                                
2
 M.A. Anisimov, V.A. Agayan, P.J. Collings, ?Nature of the Blue-Phase-III-Isotropic?? 
 104 
The polar variables r and q used in the parametric representation of the equation of state 
are shown below. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The functions f( ), s( ), and c
i
( ) are known functions of  : 
f  ()= f
0
+ f
2
 
2
+ f
4
 
4
 
f
0
=  
    3()  b
2
  
2b
4
2   ()1  () 
 
f
2
=
    3()  b
2
  1  2 ()
2b
2
1  () 
 
f
4
=  
1  2 
2 
 
s  ()= s
0
+ s
2
 
2
 
s
0
=   2  ()f
0
 
s
2
=   2   ()b
2
1  2 ()f
0
   f
2
 
c
1
 ()= 1  b
2
 
2
1  2 ()/ c
0
 ()
()
 
c
12
 ()=    1     
2
3  () 
 
 
 
/ c
0
 () 
c
0
 ()= 1  3 
2
()
1  b
2
 
2
()
+ 2  b
2
 
2
1  
2
()
 
 
To solve for thermodynamic properties for water at the critical pressure using the scaling 
theory, we find: 
 
 
h
1
= a
2
 
 
T = ar
 + 
  1  
2
()
 
h
2
= b
1
 T = r 1 b
2
 
2
()
 
h
1
 
h
2
 
  = +1/b 
  = 0 
  = -1/b 
  = +1 
  = -1 
 105 
 
Rearranging and combining terms: 
ar
 + 
  1  
2
()
a
2
=
r 1  b
2
 
2
()
b
1
 
r =
a
2
r 1  b
2
 
2
()
ab
1
  1  
2
()
 
 
 
 
 

 	
1/  +   1()
.  Given theta, r and 
 
 
 
T are well defined. 
To solve for thermodynamic properties for water at any pressure using the scaling theory, 
we find: 
 
 
h
1
= a
1
 
 
P + a
2
 
 
T + a
3
 
 
P
2
= ar
 + 
  1  
2
()
 
 
h
2
= b
1
 
 
T + b
2
 
 
P = r 1 b
2
 
2
()
. 
 
Rearranging and combining terms: 
 
r
 + 
a  1  
2
()
a
2
+ r
b
2
 
2
 1
()
b
1
=  
b
2
b
1
 
 
P +
a
1
a
2
 
 
P +
a
3
a
2
 
 
P
2
 
of the form Ar
 + 
+ Br = C  
 
OR 
 
 
 
a
1
 
 
P
a
2
 
a
3
 
 
P
2
a
2
+
ar
 + 
 
a
2
 
ar
 + 
 
3
a
2
=  
 
T  
 
r
b
1
 
rb
2
b
1
 
2
 
b
2
 
 
P
b
1
=  
 
T  
 
 
ar
 + 
a
2
 
3
+
rb
2
b
1
 
2
+
ar
 + 
a
2
  +
b
2
 
 
P
b
1
 
r
b
1
 
a
1
 
 
P
a
2
 
a
3
 
 
P
2
a
2
= 0  
in the form of Ax
3
+ Bx
2
+Cx+ D = 0  
where 
A =  
ar
 + 
a
2
, B =
rb
2
b
1
, C =
ar
 + 
a
2
, 
 
D =
b
2
 
 
P
b
1
 
r
b
1
 
a
1
 
 
P
a
2
 
a
3
 
 
P
2
a
2
 
Substituting y = x +
B
3A
, we get y
3
+ 3Py+Q = 0  
Where P =
3AC   B
2
9A
2
 or P =  
1
3
+
1
9
ra
2
b
2
b
1
ar
 + 
 
 
 
 
 	
 
2
 
 
 
 
 	
 
 
 106 
and Q =
2B
3
  9ABC + 27A
2
D
27A
3
 or 
 
Q =
2
rb
2
b
1
 
 
 
 
 

 	
3
+ 9
ar
 + 
a
2
 
 
 
 
 

 	
2
rb
2
b
1
 
 
 
 
 

 	
+ 27
ar
 + 
a
2
 
 
 
 
 

 	
2
b
2
 
 
P
b
1
 
r
b
1
 
a
1
 
 
P
a
2
 
a
3
 
 
P
2
a
2
 
 
 
 
 

 	
 27
ar
 + 
a
2
 
 
 
 
 

 	
3
 or 
 
Q =  
2
27
a
2
rb
2
ar
 + 
b
1
 
 
 
 
 

 	
3
 
1
3
a
2
rb
2
ar
 + 
b
1
 
 
 
 
 

 	
 
a
2
ar
 + 
 
 
 
 
 

 	
b
2
 
 
P
b
1
 
r
b
1
 
a
1
 
 
P
a
2
 
a
3
 
 
P
2
a
2
 
 
 
 
 

 	
 
Defining  
  =
 Q+ Q
2
+ 4P
3
2
 and   =
 Q  Q
2
+ 4P
3
2
 
The solution to the equation y
3
+ 3Py+Q = 0  is given by 
 
3
+  
3
, e
2 i
3
 
3
+ e
4 i
3
 
3
, e
4 i
3
 
3
+ e
2 i
3
 
3
. 
 
Since e
ix
= cosx + isin x ,  
e
i4 
3
= cos
4 
3
 
 
 
 
 
 
+ isin
4 
3
 
 
 
 
 
 
=  
1
2
  i
3
2
, e
i2 
3
= cos
2 
3
 
 
 
 
 
 
+ isin
2 
3
 
 
 
 
 
 
=  
1
2
+ i
3
2
 
Using this, the solution  
e
2 i
3
 
3
+ e
4 i
3
 
3
=  
1
2
 
3
+ i
3
2
 
3
 
1
2
 
3
  i
3
2
 
3
=  
1
2
 
3
+  
3
()
+ i
3
2
 
3
   
3
()
 
 
 
 
 
 
 107 
 
V.  Actual EOS Equations: 
 
Solving for theta, h
1
= a
1
()
eff
 P + a
2
 T = ar
 + 
  1  
2
()
, from scaling theory 
We simplify the equation and get ar
 + 
 
3
  ar
 + 
  + a
1
()
eff
 P + a
2
 T = 0  in the form 
of  ax
3
+bx
2
+ cx + d = 0 .  Substituting y = x +
b
3a
, we get the equation 
y
3
+ 3py+ q = 0 , where p =
3ac b
2
9a
2
 and q =
2b
3
  9abc+ 27a
2
d
27a
3
.  The discriminant of 
this polynomial is 4p
3
+ q
2
.  The solutions are given by  
3
+  
3
, e
2 i
3
 
3
+ e
4 i
3
 
3
, 
and e
4 i
3
 
3
+ e
2 i
3
 
3
, where   =
 q+ q
2
+ 4p
3
2
 and   =
 q  q
2
+ 4p
3
2
.  If the 
discriminant is +, then 1 root is real and 2 are complex conjugates.  If the discriminant is 
0, then there are 3 real roots of which at least two are equal.  If the  discriminant is 
negative, then there are 3 unequal real roots. 
 
Column C:   
3
=  
a r
 + 
a
2
 
Column D:  
2
=
r  b
2
b
1
()
eff
 
Column E: has b1eff for Column D equation and ?? 
Column F:  
1
=
a r
 + 
a
2
 
Column G:  
0
=  
a
3
  P
2
a
2
 
a
1
+ 2a
3
 P() P
a
2
+
b
2
 P
b
1
()
eff
 
r
b
1
()
eff
 
Column H: P =
3  
3
  
1
   
0
()
2
9  
3
 
Column I: Q =
2   
2
()
3
  9  
3
  
2
   + 27   
3
()
2
 
0
27   
3
()
3
 
Column J: Q
2
+ 4P
3
 
Column K: IF Q
2
+ 4P
3
= 0, then   =  
Q
2
.  If not, then if Q
2
+ 4P
3
> 0 , 
  =
 Q+ Q
2
+ 4P
3
2
.  If Q
2
+ 4P
3
< 0 , then   =  
Q
2
+
Q
2
+ 4P
3
2
i  
 108 
Column L: If Q
2
+ 4P
3
= 0, then   =  
Q
2
.  If not, then if Q
2
+ 4P
3
> 0 , 
  =
 Q  Q
2
+ 4P
3
2
.  If Q
2
+ 4P
3
< 0 , then   =  
Q
2
 
Q
2
+ 4P
3
2
i 
Column M: Solution1 =  
1
3
+ 
1
3
 
Column N: Solution2 =  
1
3
e
2 i
3
+ 
1
3
e
4 i
3
 
Column O: Solution3 =  
1
3
e
4 i
3
+ 
1
3
e
2 i
3
 
Column P:  
1
=  
1
3
+  
1
3
 
 
2
()
3  
3
()
 
Column Q:  
2
=  
1
3
e
2 i
3
+  
1
3
e
4 i
3
 
 
2
()
3  
3
()
 
Column R:  
3
=  
1
3
e
4 i
3
+  
1
3
e
2 i
3
 
 
2
()
3  
3
()
 
Using theta to solve for thermodynamic properties and the critical point 
Column S:  T =
r 1 b
2
 
2
()
b
1
()
eff
 
 Pb
2
b
1
()
eff
 
Column T: T =
 T  T
c
T
c
 
Column U: h
1
= a
1
 P + a
2
   + a
3
 P
2
 
Column V: h
2
= b
1
 T +b
2
 P +b
3
 T
2
 
 
 
 
 
 
 
 
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