X-Ray Investigations of theLiquid-Liquid Critical Point
Hypothesis in Supercooled Water
Alexander Späh

Doctoral Thesis in Chemical Physics at Stockholm University, Sweden 2020

X-Ray Investigations of the Liquid-Liquid Critical
Point Hypothesis in Supercooled Water
Alexander Späh
Academic dissertation for the Degree of Doctor of Philosophy in Chemical Physics at
Stockholm University to be publicly defended on Friday 12 June 2020 at 13.00 in sal
FB52, AlbaNova universitetscentrum, Roslagstullsbacken 21, digitally via Zoom:
https://stockholmuniversity.zoom.us/s/239996391
Abstract

This thesis presents experimental x-ray scattering studies on supercooled liquid water. A liquid-liquid transition between
two structurally distinct configurations has been found in deeply supercooled water, indicating the existence of a liquidliquid critical point. The experiments were performed at large-scale x-ray facilities, mostly using free electron x-ray lasers
including PAL-XFEL in Korea, SACLA in Japan, LCLS in the USA, SwissFEL in Switzerland and European XFEL in
Germany, as well as using synchrotrons including APS in the USA, PETRA III in Germany and ESRF in France.
Two conceptually different experimental approaches have been used to investigate the metastable phase of supercooled
water. The first approach is based on rapid evaporative cooling of μm-sized water droplets that are injected into a vacuum
chamber. Using this method, supercooled liquid water samples with temperatures down to approximately 227 K have beenobtained, with the lowest temperature limited by homogeneous ice crystallization occurring after just a few milliseconds.
In a second approach, structurally arrested high-pressure and therefore high-density amorphous ice samples are heated byan ultrafast infrared laser pulse. The fast heating melts the ice into a corresponding high-density liquid. At short time delays
between the heating laser pulse and a subsequent x-ray probe pulse, the supercooled liquefied sample still experiences
the high internal pressure of the initial state. At longer pump-probe delay times the supercooled water sample releases
its internal pressure through structural relaxation. Hence, varying the pump-probe delay allows to probe the sample atdifferent pressures.
Together, these two approaches have been used to access a region within the metastable phase diagram of supercooled
water that has previously been inaccessible. Using elastic x-ray scattering measurements as a structural probe of the liquid,
we identified the existence of a liquid-liquid phase transition in deeply supercooled water. The observed phase transition isinterpreted as the transition between a high-density and a low-density liquid phase. At high pressure this phase transition is
discontinuous or first-order like, featuring a characteristic double-peak feature in the observed x-ray scattering intensity of
the first diffraction maxima. At ambient pressure, however, we observe a continuous shift of the first diffraction maxima thatis consistent with a continuous or second-order phase transition between the two liquids. Further evidence of a continuous
phase transition at ambient pressure is seen in the temperature dependent maxima of the measured correlation length,
isothermal compressibility and heat capacity, which indicate the existence of a Widom line.
In summary, the experiments support the existence of a liquid-liquid critical point where the experimentally observed
Widom line and phase coexistence line would both meet. The main result, however, is the first experimental observationof a liquid-liquid transition within a pure liquid.
Keywords: water, supercooled water, x-ray scattering, free electron x-ray laser, liquid-liquid phase transition, liquidliquid critical point, x-ray speckle visibility spectroscopy.
Stockholm 2020
http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-180847
ISBN 978-91-7911-092-5
ISBN 978-91-7911-093-2

Department of Physics
Stockholm University, 106 91 Stockholm

X-RAY INVESTIGATIONS OF THE LIQUID-LIQUID CRITICALPOINT HYPOTHESIS IN SUPERCOOLED WATER

Alexander SpähX-Ray Investigations of theLiquid-Liquid Critical Point
Hypothesis in Supercooled
Water
Alexander Späh©Alexander Späh, Stockholm University 2020
ISBN print 978-91-7911-092-5
ISBN PDF 978-91-7911-093-2
Printed in Sweden by Universitetsservice US-AB, Stockholm 2020

To my uncle Richard

“Es ist nun begreiflich, dass man nach einer Erklärung dieser
auffälligen Erscheinungen sucht, und eine solche dürfte, nachmeiner Ansicht in der Annahme gefunden sein, dass das flüssige
Wasser aus einem Aggregat von zwei Arten verschieden
constituirter Molecüle besteht. [...] Ich habe mich mit
dieser Frage wohl beschäftigt, möchte jedoch auch in diesem
Falle die Antwort schuldig bleiben, weil vorläufig noch zu
wenig Beobachtungen vorliegen, durch welche man eine solche
Vorstellung genügend begründen könnte.”
Wilhelm Conrad Röntgen (1891)†“Water is not a complicated liquid, but two simple liquids
with a complicated relationship.”Lars G. M. Pettersson (2017)‡†

Röntgen, Wilhelm Conrad. “Ueber die constitution des flüssigen wassers” Annalen
der Physik 281.1 (1892): 91-97.
‡
Stockholm University press release “Water Exists as Two Different Liquids” (2017);
Pettersson, Lars G. M. "A Two-State Picture of Water and the Funnel of Life.” International Conference Physics of Liquid Matter: Modern Problems. Springer, Cham, (2018):
3-39.

List of Papers
This thesis is based on the following papers, which will be referred to by
their corresponding Roman numerals throughout this thesis. Reprints weremade with permission from the publishers.
I. Temperature-independent nuclear quantum effects on the structure of water
Kyung Hwan Kim, Harshad Pathak, Alexander Späh, Fivos Perakis, DanielMariedahl, Jonas A. Sellberg, Tetsuo Katayama, Yoshihisa Harada, Hirohito Ogasawara, Lars G. M. Pettersson, and Anders Nilsson.
Physical review letters 119.7 (2017): 075502
II. Maxima in the thermodynamic response and correlation functions of deeply supercooled water
Kyung Hwan Kim∗ , Alexander Späh∗ , Harshad Pathak, Fivos Perakis, DanielMariedahl, Katrin Amann-Winkel, Jonas A. Sellberg, Jae Hyuk Lee, Sangsoo Kim, Jaehyun Park, Ki Hyun Nam, Tetsuo Katayama, and Anders
Nilsson.
∗ These authors contributed equally to this work.
Science 358.6370 (2017): 1589-1593
III. Coherent X-rays reveal the influence of cage effects on ultrafast water dynamics
Fivos Perakis, Gaia Camisasca, Thomas J. Lane, Alexander Späh, Kjartan Thor Wikfeldt, Jonas A. Sellberg, Felix Lehmkühler, Harshad Pathak,
Kyung Hwan Kim, Katrin Amann-Winkel, Simon Schreck, Sanghoon Song,
Takahiro Sato, Marcin Sikorski, Andre Eilert, Trevor McQueen, Hirohito
Ogasawara, Dennis Nordlund, Wojciech Roseker, Jake Koralek, Silke Nelson, Philip Hart, Roberto Alonso-Mori, Yiping Feng, Diling Zhu, Aymeric
Robert, Gerhard Grübel, Lars G. M. Pettersson, and Anders Nilsson.
Nature communications 9.1 (2018): 1917
IV. Apparent power-law behavior of water’s isothermal compressibility and correlation length upon supercooling
Alexander Späh, Harshad Pathak, Kyung Hwan Kim, Fivos Perakis, DanielMariedahl, Katrin Amann-Winkel, Jonas A. Sellberg, Jae Hyuk Lee, Sangsoo Kim, Jaehyun Park, Ki Hyun Nam, Tetsuo Katayama, and Anders
Nilsson.
Physical Chemistry Chemical Physics 21.1 (2019): 26-31

iV. Intermediate range O–O correlations in supercooled water
down to 235 K
Harshad Pathak, Alexander Späh, Kyung Hwan Kim, Ifigeneia Tsironi,
Daniel Mariedahl, Maria Blanco, Simo Huotari, Veijo Honkimäki, and Anders Nilsson.
The Journal of chemical physics 150.22 (2019): 224506
VI. Temperature dependent anomalous fluctuations in water: shiftof ≈1 kbar between experiment and classical force field simulations
Harshad Pathak, Alexander Späh, Katrin Amann-Winkel, Fivos Perakis,
Kyung Hwan Kim, and Anders Nilsson.
Molecular Physics (2019): 1-9
VII. Experimental Observation of the Liquid-Liquid Transition inSupercooled Water
Kyung Hwan Kim∗ , Katrin Amann-Winkel∗ , Alexander Späh, Fivos Perakis, Harshad Pathak, Marjorie Ladd Parada, Cheolhee Yang, Daniel Mariedahl, Tobias Eklund, Thomas J. Lane, Seonju You, Sangmin Jeong, Matthew
Weston, Jae Hyuk Lee, Intae Eom, Minseok Kim, Jaeku Park, Sae Hwan
Chun, and Anders Nilsson.
∗ These authors contributed equally to this work.
submitted
VIII. Enhancement and Maximum in the Isobaric Specific Heat
Capacity Measurements of Deeply Supercooled Water using Ultrafast Calorimetry
Harshad Pathak, Alexander Späh, Niloofar Esmaeildoost, Jonas A. Sellberg, Fivos Perakis, Katrin Amann-Winkel, Marjorie Ladd-Parada, Jaynath Koliyadu, Thomas J. Lane, Kyung Hwan Kim, Cheolhee Yang, Henrik Till Lemke, Roland Alex Oggenfuss, Philip Johnson, Serhane Zerdane,
Roman Mankowsky, Paul Beaud and Anders Nilsson.
manuscript
IX. Anisotropic X-ray Scattering of Transiently Oriented Water
Kyung Hwan Kim, Alexander Späh, Harshad Pathak, Cheolhee Yang, Stefano Bonetti, Katrin Amann-Winkel, Daniel Mariedahl, Daniel Schlesinger,
Jonas A. Sellberg, Derek Mendez, Gijs van der Schot, Harold Young Hwang,
Jesse Clark, Owada Shigeki, Togashi Tadashi, Yoshihisa Harada, Hirohito
Ogasawara, Tetsuo Katayama, Anders Nilsson and Fivos Perakis.
manuscriptii

X. Double pulse X-ray speckle visibility on atomic length scales
Alexander Späh, Thomas J. Lane, Harshad Pathak, Kyung Hwan Kim,
Katrin Amann-Winkel, Jonas A. Sellberg, Derek Mendez, Simon Schreck,
Felix Lehmkühler, Wojciech A. Roseker, Gerhard Grübel, Yanwen Sun,
Philip Hart, Sanghoon Song, Anders Madsen, Diling Zhu, Anders Nilsson
and Fivos Perakis.
manuscriptiii

The following is a list of papers to which I have contributed but that are
not content of this thesis.
Diffusive dynamics during the high-to-low density transition in
amorphous ice
Fivos Perakis, Katrin Amann-Winkel, Felix Lehmkühler, Michael Sprung,
Daniel Mariedahl, Jonas A. Sellberg, Harshad Pathak, Alexander Späh,
Filippo Cavalca, Daniel Schlesinger, Alessandro Ricci, Avni Jain, Bernhard
Massani, Flora Aubree, Chris J. Benmore, Thomas Loerting, Gerhard Grübel, Lars G. M. Pettersson, and Anders Nilsson.
Proceedings of the National Academy of Sciences 114.31 (2017): 8193-8198
Response to Comment on "Maxima in the thermodynamic response and correlation functions of deeply supercooled water"
Kyung Hwan Kim, Alexander Späh, Harshad Pathak, Fivos Perakis, DanielMariedahl, Katrin Amann-Winkel, Jonas A. Sellberg, Jae Hyuk Lee, Sangsoo Kim, Jaehyun Park, Ki Hyun Nam, Tetsuo Katayama, and Anders
Nilsson.
Science 360.6390 (2018): eaat1729
X-ray scattering and O–O pair-distribution functions of amorphous ices
Daniel Mariedahl, Fivos Perakis, Alexander Späh, Harshad Pathak, Kyung
Hwan Kim, Gaia Camisasca, Daniel Schlesinger, Chris Benmore, Lars G.
M. Pettersson, Anders Nilsson, and Katrin Amann-Winkel.
The Journal of Physical Chemistry B 122.30 (2018): 7616-7624
X-ray studies of the transformation from high-to low-density amorphous water
Daniel Mariedahl, Fivos Perakis, Alexander Späh, Harshad Pathak, Kyung
Hwan Kim, Chris Benmore, Anders Nilsson, and Katrin Amann-Winkel.
Philosophical Transactions of the Royal Society A 377.2146 (2019): 20180164
Brine rejection and crystallite formation upon freezing NaCl aqueous solutions
Ifigeneia Tsironi, Daniel Schlesinger, Alexander Späh, Lars Eriksson, Mo
Segad, and Fivos Perakis.
Physical Chemistry Chemical Physics 22.14 (2020): 7625-7632

iv

Author’s Contribution
As the experiments, performed at large-scale x-ray sources, are the effort
and collaboration of a team of people, it is critical to specify my contribution
to the projects.
In all experiments I took a major or leading responsibility for the design, realization, testing and operation of the experimental setup and sample preparation both in preparation of, and during all beamtimes. I contributed significantly to the data analysis is papers II, IV, VI and X. In all
papers I participated in the discussions of the manuscript and supplementary material. In particular, I contributed in writing of the papers II, IV,
VII, VIII, IX and X and corresponding supplementary materials of papers
II, IV, VII, VIII and X.

v

vi

Contents
List of PapersiAuthor’s Contributionv

1 Introduction

1

2 Liquid Water: A Simplified View
2.1 In Brief: The 2D Ising Model . . . . . . . . . . . . . . . . .
2.2 Phases and Phase Transitions . . . . . . . . . . . . . . . . .
2.3 Pair Correlation Function and Correlation Length . . . . . .
2.4 Compressibility and Heat Capacity . . . . . . . . . . . . . .
2.5 Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Supercooled Water and a Liquid-Liquid Critical Point . . .

7
7
13
16
20
21
23

3 Liquid Water Probed by X-Rays
3.1 X-ray Scattering and the Pair Correlation Function . . . . .
3.2 Phase Transitions: A Question of Pressure . . . . . . . . . .
3.3 Peaking Correlation and Response Functions . . . . . . . .
3.4 Location of the 2nd Critical Point: An Educated Guess . . .
3.5 Dynamics of Supercooled Water . . . . . . . . . . . . . . . .

27
27
30
35
38
40

4 Summary

45

Sammanfattning

47

Acknowledgements

49

References

51

vii

viii

1

Introduction

Water is one of the most anomalous liquids [1]. As such, water is an important subject of current research in chemical physics and physical chemistry [2–12]. In spite of its molecular simplicity, consisting only of one
oxygen and two hydrogen atoms, water as a liquid is most peculiar [13, 14].
The complex interplay between the water molecules, makes water behave
very different in almost all of its properties when compared to other liquids [15–21]. Almost needless to say is the relevance of water for nearly all
biological, geological and chemical processes [22–24]. A better understanding of water is therefore of great interest to many scientific disciplines.
To explain water’s anomalous behavior, Wilhelm Conrad Röntgen proposed the provocative idea that water might consist of an aggregation of
two distinct configurations of water molecules [25]∗ . Each of the two configurations itself behaving as a normal liquid. We can explain this idea on aparticular example, using water’s most prominent anomaly, i.e. the density
maximum at 4 ◦C. Figure 1 schematically shows the density of water asa function of temperature. It also shows the density of the two proposed
configurations of water molecules, indicated by the dashed lines. The two
configurations differ in density, with a high-density liquid at high temperatures and a low-density liquid at low temperatures. Each configurationof water molecules for itself behaves as a normal liquid with the density
monotonically increasing upon cooling. According to the idea of Röntgen,the anomalous behavior of water, i.e. the decrease in density upon coolingand the corresponding existence of a density maximum, are explained bya temperature dependent transformation from one configuration into theother. This also means that water at very high temperatures as well as atvery low temperatures behaves like a normal liquid. Below 0 ◦C, however,liquid water is metastable with respect to its solid ice phase [26]. Due to thiscircumstance, experimental investigation of Röntgen’s idea has been complicated by rapid ice crystallization, and already Röntgen himself noted,that experimental observations were missing at that time to support his
hypothesis.
101 years later, Röntgen’s idea was recycled and put into a rigorous
thermodynamic context, proposing that the anomalous behavior of wateris actually related to supercritical fluctuations between two normal liquids[28]. Again the idea is the same, relating water’s anomalous behavior toa phase transition between two simple liquids. The character of the phasetransition, however, is characterized by fluctuations that emanate from a
critical point between the two liquids. Other examples of critical points
and related phase transitions are commonly observed in binary systems∗

An original citation from reference [25] is given in German, together with a condensed version of this quote in English, at the beginning of the thesis.

1

Figure 1: Schematic of water’s density as a function of temperature, indicated as solid line. Dashed lines indicate the temperature dependence of
two normal liquids with monotonically increasing densities upon cooling.
Water’s anomalous density maximum can be explained due to a transition
between the two simple liquids of different density. Below 0 ◦C water ismetastable with respect to ice, complicating experimental investigation of
the liquid. Adopted from reference [27].
such as mixtures and solutions [29–31]. However, the proposed existence ofa critical point within a single component liquid such as liquid water has
led to heavy disputes within the scientific community [32–50].
In a simplified picture this liquid-liquid critical point is thought to be
similar to the well-known liquid-gas critical point of water. In contrast to
the liquid-liquid critical point which is proposed to be hidden deep in themetastable phase of supercooled water, the liquid-gas critical point is easily
observed in experiments, as for example shown in Figure 2. Here, the twocontributing phases are the liquid and the gas phase, i.e. liquid water andwater vapor. In panel a), liquid water and water vapor form a uniform
mixture, with the mixing ratio between the two phases dependent on temperature. Close to the critical point the water sample turns opaque as seenin panel d). This is because the uniform mixture becomes increasingly nonuniform due to growing regions of the pure phases, i.e. pure liquid wateror pure water vapor. Eventually these regions grow in size comparable to
the wavelength of visible light, causing light to be scattered and the sample
to appear cloudy. This phenomenon is known as critical opalescence. At
temperatures below the critical point liquid water and water vapor separate
and coexist as macroscopic phases, as shown in panel i). Now, coming backto the discussion of the liquid-liquid critical point, at ambient conditions
liquid water itself is thought to be a fluctuating mixture between two liquidsof distinct densities, i.e. a high-density liquid and a low-density liquid. Correspondingly, changing temperature and pressure to the proper conditions,liquid water is expected to show a phase separation into a high-density
liquid and a low-density liquid.
2

a)

b)

c)

d)

e)

f)

g)

h)

i)

Figure 2: Back-illuminated microscope images of water close to its liquidgas critical point. Decreasing temperatures from a) the uniform mixture
between liquid water and water vapor, i.e. the so-called single-phase region,
at T > Tc to d) critical opalescence at T ≈ Tc to i) the phase separation
between liquid water and water vapor at T < Tc . Here, T is temperature
and Tc ≈ 374 ◦C is the critical temperature of water’s liquid-gas criticalpoint. We note that water at the liquid-gas critical point is often seen as amilky colored substance when illuminated from the front. Here however, theback-illumination of the sample causes the water sample to appear black,i.e. opaque to visible light. Reproduced from reference [51] with permissionof T. Walstra and F. Kayzel.

3

Figure 3: Droplet setup used at the Linac Coherent Light Source (LCLS).
Experimental observation of these two structurally distinct configurations within liquid water, requires to resolve the intermolecular configuration of the liquid. Ironically, it was again Röntgen, who himself provided
the crucial key to help answer his own idea. His discovery of the x-ray radiation has nowadays become the most powerful tool to investigate processes
and structures on the intermolecular length scale. However, we shouldnote that modern x-ray sources provide x-rays of much higher quality thancompared to conventional x-ray tubes that have been developed based on
Röntgen’s discovery. A typical measure of the quality of x-ray sources is thebrilliance, defined by the number of photons per second in relation to the
angular divergence, focus size and spectral bandwidth of the x-ray beam.
As a comparison, the brightest modern x-ray sources, namely free electron
x-ray lasers, have a brilliance that is approximately 30 orders of magnitude
higher than the first x-ray tubes. We should also note that these modern
x-ray sources are large-scale facilities that can be several kilometers long.
Conventional x-ray tubes instead easily fit on a small laboratory desk.
In this thesis we make use of such modern x-ray sources to probe theintermolecular structure of water. Figure 3 shows one of the experimental
setups at a free electron laser facility. Usually these experiments are performed in a vacuum environment to prevent background scattering fromsurrounding air molecules. The left panel of the figure shows detectors,
microscopes, vacuum pumps and motorized manipulators for sample diagnostics and manipulation. The middle panel takes a direct look into theinteraction zone where the x-ray beam hits the sample. The visible glow4

a)Detector

Dispenserb)

Detector

Cryostat

∆t
X-ray

X-ray
H2O
Droplets

2 µm
Laser

Amorphous
ice sample

Figure 4: a) Droplet setup to produce µm-sized water droplets. The water droplets are injected into a vacuum chamber where they rapidly cooldue to evaporative cooling. High-intensity x-ray pulses are used to obtainsingle-shot scattering images of individual water droplets. Changing theheight of the droplet dispenser controls the time span in which the droplet
is exposed to the vacuum environment and therefore controls the droplet
temperature. b) High-density amorphous ice samples with a thickness ofless than 100 µm are prepared on a copper grid. The gird is attached tothe tip of a liquid nitrogen cooled cryostat in order to keep the samples
at cryogenic temperature and avoid structural transformation. Before thesample is probed by a single x-ray pulse, an infrared laser pulse heats the
sample into the supercooled liquid state.
however is due to the sample illumination that is needed for the microscope.
On the right panel a vertical array of four x-ray detectors is installed tomeasure the scattered x-ray intensity at a certain scattering angle. For
each experiment details of the setup change according to the needs of theexperiment. Hence, the details of each experiment are found in the scientific papers at the end of this thesis. Here instead, we like to introduce
two conceptually different experimental approaches that have been used toobtain supercooled water samples at temperatures and pressures that havenot been accessible previously.
The first approach, schematically shown in Figure 4a, is based on rapid
evaporative cooling of µm-sized water droplets that are injected into a vacuum chamber [52]. Using this method, supercooled liquid water samples
with temperatures down to approximately 227 K have been obtained, withthe lowest temperature limited by homogeneous ice crystallization occurring after just a few milliseconds. The second approach is illustrated in
Figure 4b. Here, structurally arrested high-density amorphous ice samples,
prepared by applying high-pressure, are heated by an ultrafast infrared laser
5

a)

b)

100 µ m

100 µ mFigure 5: a) Microscope image of a H2 O droplet train with droplets ofapproximately 40 µm in diameter. Breakup of the continuous water jet
into droplets at the top of the image. Image was taken using stroboscopic
illumination. b) Microscope image of an ice sample mounted on a copper
grid. Holes of the grid are approximately 800 µm in diameter.
pulse. The fast heating melts the ice into a corresponding high-density liquid. At short time delays on the order of a few nanoseconds between theheating laser pulse and a subsequent x-ray probe pulse, the supercooledliquefied water sample still experiences the high internal pressure of the initial state. At longer pump-probe delay times the supercooled water sample
releases its internal pressure through structural relaxation. Hence, varying the pump-probe delay allows to probe the sample at different pressures. Figure 5 shows microscope images of both sample preparations. Together, these two approaches have been used to access a region within themetastable phase diagram of supercooled water that has previously been
inaccessible.
The content of the thesis is divided into two chapters. The first chapter
gives an intuitive introduction to the thermodynamics of liquid water inproximity of a critical point. For this purpose we briefly explain the twodimensional Ising model and make use of it to introduce the concepts ofphase transitions and a critical point as well as we define properties such as
the correlation length, density-density pair correlation function, isothermal
compressibility and heat capacity. Finally we introduce the liquid-liquidcritical point scenario for water in greater detail. The second chapter is asummary of the experimental observations that are content of this thesis.
Here we only focus on the main results in connection with the observed
liquid-liquid transition. Detailed discussions of the experiments can befound in the corresponding scientific papers of this thesis.

62Liquid Water: A Simplified View

The concept of a liquid-liquid critical point and its implications on thethermodynamic properties of water are the main subject of this chapter.
A rigorous theoretical description of phase transitions and critical phenomena is out of scope of this thesis and is provided by the renormalizationgroup theory which lead to the Nobel prize in physics in 1982 [53]. For theframework of this thesis instead, the two-dimensional square-lattice Isingmodel serves as a simplified model to qualitatively examine phase transitions and critical phenomena. Simulations on the two-dimensional Ising
lattice are used to illustrate and introduce the concepts of the pair correlation function and correlation length. Thermodynamic response functions
such as isothermal compressibility and heat capacity are derived from thedensity and energy fluctuations in the system. Finally, the picture of theliquid-liquid critical point scenario in supercooled water is presented.

2.1

In Brief: The 2D Ising Model

The Ising model consists of a discrete lattice with each lattice point occupying one of two states. The original interpretation of the Ising model isa magnetic system with each lattice point represented by a particle. Eachparticle has a magnetic moment or spin that is found in one of two states,
i.e. ’parallel’ or ’anti-parallel’ with respect to an external magnetic field [54].
The energy or Hamiltonian, Ĥ, of the Ising system is given by:Ĥ = −X
1X
Jij si sj − H
si
2
i,j

(1)

i

Here, H denotes the magnitude of the external magnetic field. The two
possible states of each Ising lattice point are given by the particle spins,
si . These are oriented parallel or anti-parallel to the external field, withspin values si = ±1, respectively. The first term of equation (1) resembles
the energy contribution to the system due to interactions between different spins. A positive interaction energy, Jij , prefers a parallel alignment
between the two spins i and j, as it is the case in a ferromagnetic system.
A negative interaction energy instead, will result in an anti-parallel alignment between the spins comparable to an anti-ferromagnet. Note that thefactor of 1/2 is due to double counting of the pair interactions. The second term of equation (1) is related to the interaction of the spins with an
external magnetic field. Hence, a spin orientation parallel to the external
field is energetically favored whereas an anti-parallel orientation increases
the system’s energy† . For the rest of this thesis we will limit ourselves to†Strictly speaking, the second term of the Hamiltonian in equation (1) should be7

Figure 6: Two-dimensional Ising model on a 5×5 square lattice. Each lattice
point is occupied by one of two states, i.e. either ’blue’ or ’yellow’. Nearest
neighbours of opposite states increase the total energy of the configuration,
which is illustrated by the red boundaries between opposite colored lattice
points. a) Example of an Ising configuration that is favored by energy. b)
Example of an Ising configuration that is favored by entropy.
positive spin-spin interactions, Jij , as in the case of a ferromagnetic system.
Furthermore, we will only consider nearest-neighbour interactions between
spins, i.e. up, down, left and right. The latter is usually applied in the caseof a two-dimensional square-lattice Ising model.
Besides its original magnetic interpretation, the Ising model has been
applied to many other systems [55], as well as to describe the thermodynamics of supercooled water [41,56–58]. For our purpose, the interpretationof the Ising lattice as a liquid with discrete local densities may be most appropriate. We will therefore consider the two possible states of each lattice
point to resemble configurations of either a ’high’ or a ’low’ local density
within the liquid. Furthermore, the external magnetic field, that was usedin the magnetic picture of the Ising model, is equivalent to an external pressure acting on the liquid. As a result, high external pressures favor Ising
configurations with an increased number of ’high’ density states, whereas’low’ density states are preferred by low external pressures.
Examples of a two-dimensional 5 × 5 square-lattice Ising model are
shown in Figure 6. Here, and throughout the rest of the thesis, the two
possible states of a ’high’ or ’low’ local density are represented by ’yellow’ or’blue’ colored lattice points, respectively. Both of the lattice configurations
shown in Figure 6 contain four lattice points that resemble a high local
density within the simulated liquid, i.e. four yellow squares. Hence, both
positive because the the magnetic moment of an electron is anti-parallel to its spin.
However, by convention a negative term is used.

8

configurations have the same mean density. The total energy of the two configurations is easily obtained from equation (1). For now, we will consider
the case of no external field or pressure, i.e. we will neglect the second termin equation (1). For each pair of nearest neighbours that contains different
states, the energy of the system is increased by the amount of the spin-spin
interaction energy, Jij . In Figure 6, this energy increase is illustrated by
the red boundaries between opposite colored lattice points. Hence, for a
configuration that contains exactly four yellow states, the compact clustershown in Figure 6a minimizes the energy of such a configuration. Any deviation from the compact yellow cluster, as for example in Figure 6b, will
increase the boundary between yellow and blue states and hence increase
the total energy of the system.
Now that we know how to calculate the energy of a particular latticeconfiguration, the next step is to ask, which of all the possible configurations is most likely to occur during a simulation of the Ising liquid? The
answer, of course, depends on the temperature of the simulation and the
distribution of lattice configurations has to follow a Boltzmann probability distribution. A method to incorporate the Boltzmann distribution into
the simulation is achieved by the Metropolis algorithm [59], that works asfollows: In a first step, starting from a given lattice configuration, statesof random lattice points are flipped over into opposite states such that a
new lattice configuration is obtained. In a second step, we make a decision
if the new configuration is accepted or rejected. If the energy of the new
configuration is lower then the energy of the initial configuration, the newly
obtained configuration is directly accepted. However, if the energy of the
new configuration is higher then the energy of the previous configuration,
the newly obtained configuration is only accepted with a probability that is
proportional to the Boltzmann factor of the two configurations and rejected
otherwise.
We can illustrate a simulation step of the Metropolis algorithm in caseof the two lattice configurations shown in Figure 6. First, let us consider
lattice configuration shown in Figure 6a as the initial configuration. We
obtain the new configuration that is shown in Figure 6b by flipping thestates of two lattice points; i.e. one flip of states from yellow to blue and one
flip of a blue state to a yellow state. Then, we calculate the energy of the two
configurations using equation (1) and label the energy of the initial state Ea
and the energy of the new configuration Eb . From the previous discussion
we know that Eb > Ea . Therefore we do not accept the new configurationright away but instead make a decision if the new configuration is acceptedor rejected. The probability that we accept the new configuration is given by
the Boltzmann factor between the two configurations which is calculated asexp(−∆E/kB T ). Here, ∆E = Eb − Ea is the energy difference between thetwo configurations, kB is the Boltzmann constant and T is the temperature9

of the simulated liquid.
A closer look at the Boltzmann factor reveals that, the higher the energy of a configuration, the less likely it is to be accepted. Likewise, aconfiguration of lower energy will always be accepted. Hence, we may raise
the question if this implies that the simulation will always tend toward thelowest energy configuration? To answer this question we have to include
another thermodynamic property of the system, namely its entropy
S = kB ln Ω.

(2)Here, kB is the Boltzmann constant. The entropy S, of a system reflects thenumber of individual microstates Ω, that resemble the same macroscopic
configuration that is characterizing the system. As such, the entropy affects how often a configuration is sampled simply because of the number ofmicrostates that resemble the same configuration. Hence, the answer to thequestion above is, that both energy and entropy of a system will determine
the configuration near equilibrium.
Again, we can make use of the two lattice configurations shown in Figure 6 to exemplify the concept of entropy and its influence on the system.
First, we want to assign a measure of the number of possible microstates,
i.e. the entropy, to each of the two configurations. The 5 × 5 square-latticeconfiguration of Figure 6a features a closed cluster made of four lattice
points in the yellow state. Assuming periodic boundary conditions, it ispossible to obtain 5 × 5 = 25 microstates of the same configuration bysimple translation of the yellow cluster along the horizontal and verticaldirections. The same is true for the lattice configuration in Figure 6b. In
addition however, since the yellow cluster of Figure 6b is not symmetric, it
is possible to rotate or mirror the yellow cluster to obtain eight additional
orientations of the cluster and thus eight-times more microstates of thesame configuration. This means that the lattice configuration of Figure 6b
has eight-times more possibilities to be realized than the lattice configuration shown in Figure 6a. As a consequence, the entropy of the configurationin Figure 6b is higher than the entropy of the configuration in Figure 6a.
In order to see how energy and entropy affect the evolution of the Isingsystem, we again recall the Metropolis algorithm and its two steps: First,
obtain a new lattice configuration by random state flips. Second, accept or
reject the new lattice configuration. The first step is obviously influenced
by the entropy of the new configuration, since it is more likely to end up in aconfiguration that has many microstates associated to it, i.e. a configurationthat has a high entropy. The second step is obviously influenced by theenergy of the new configuration, since, according to the Boltzmann factor,
it is more likely to accept a configuration that has a low energy.
In case of the two example lattice configurations in Figure 6, we can10summarize our previous discussions as follows. The configuration of Figure 6a is favored by energy whereas the configuration of Figure 6b is favored
by entropy. Which of the two configurations is more likely to be found during a simulation of the Ising lattice is therefore a question of the energy
and entropy differences between the two configurations as well as the temperature of the simulation. In general this leads to quite complex latticeconfigurations, as can be seen in Figure 7 for Ising simulations of a twodimensional 512 × 512 square lattice at various temperatures.
As discussed previously, every pair of nearest neighbour lattice points
with opposite states, i.e. opposite colors, increases the energy of the configuration. Hence, at low temperatures, where the energy of the system is low,
the system tends toward configurations that are dominated by one of thetwo states, i.e. configurations of either mostly yellow or mostly blue. Thisbehavior can also be seen from the Boltzmann factor, exp(−∆E/kB T ),which leads to very small acceptance rates and thus prefers low energy configurations if the temperature, T , is low. In fact at T = 0, the system
will converge to the lowest energy configuration with all lattice points inthe same state, i.e. completely yellow or completely blue. In contrast, atvery high temperatures the energy of the system is large enough to easily
allow for state flips even if they correspond to an energy increase of theconfiguration. Hence, the Ising lattice tends toward a random mixture of
all possible configurations. Again, this behavior is reflected in the Boltzmann factor which becomes close to one at very high temperatures andtherefore almost any configuration is accepted in the Metropolis algorithm.
Thus, at high temperatures, the simulation is dominated by configurationsof high entropy. These high entropy configurations typically feature about
the same amount of the two states, i.e. roughly equal amounts of yellow
and blue.
For temperatures that are neither very low nor very high, there is acomplex interplay between energy and entropy. Energy minimization on
the one side, favors clustering of lattice points of the same state as wellas configurations that are dominated by one of the two colors. Entropy
maximization on the other side, prefers configurations with an equal amountof both colors and leads to more open and distorted cluster shapes. Whichof the two effects dominates is a function of temperature and typicallyleads to the formation of regions that are locally dominated by one of thetwo states, as it can be seen in Figure 7. These regions are fluctuating inspace and time during a simulation, and, it is worth noting, that Figure 7
shows a set of representative snapshots of configurations that occur during
these simulations. The size of the fluctuating regions is rather small attemperatures that are either very low or very high. However, starting at
a low temperature and then increasing the temperature, or starting froma high temperature and then decreasing the temperature, the fluctuating
11

Figure 7: Two-dimensional 512 × 512 square-lattice Ising configurations at
various reduced temperatures T /Tc .
12

regions increase in size until eventually their size reaches infinity. The
temperature at which this happens, i.e. where the size of the fluctuations
becomes infinite, is called the critical temperature and is denoted by Tc . A
practical simulation, such as the one shown in Figure 7, is always limited byits simulation-box size, e.g. 512 × 512 lattice points in the case of Figure 7.
This means that close to Tc , the maximum size of the fluctuations becomescomparable to the simulation-box size itself rather than infinite. However,for a simulation with periodic boundary conditions this essentially meansthat the fluctuating regions become infinite. Finally we should note, thatsometimes it is common to express the temperature, T , of the simulationby the so-called reduced temperature that is given by T /Tc .

2.2

Phases and Phase Transitions

We will start this section with a brief comparison between the Ising model,
shown in Figure 7, and the behavior of water close to its liquid-gas criticalpoint, depicted in Figure 2. However, before we continue, we should clarify
the notations that are used in the two cases. In the case of the Ising model,
which is related to a more microscopic representation, the wording of a’state’ has been used to describe the condition of a single lattice point. In
the case of water instead, where we usually think of in a more macroscopic
picture, the phrase of a ’phase’ is usually used to describe the conditionof the system. The difference may sound subtle, but, indeed, a phase isdefined as a part of a system which is uniform in its physical state [60].
We can therefore also interpret the Ising model to resemble a phase when
its configuration is uniform. This is usually the case far away from thecritical point. As such, we will say that the Ising lattice is in a yellow or
blue phase, depending on whether the lattice is uniform, and dominated bylattice points being in either a yellow or blue state, respectively.
Of course it is usually the case, that the Ising lattice contains thermal
fluctuations, i.e. the appearance of states of opposite color. Notably, these
thermal fluctuations are spread uniformly throughout the simulation box,
and we are, therefore, still justified to call it a phase. However, the situation changes near the the critical point where non-uniform anomalous
fluctuations dominate. Hence, the assignment of properties to the system
becomes dependent on the specific choice of the macroscopic subset of thesimulation box and the assignment of phases becomes arbitrary.
Now, let us have a look at the Ising model depicted in Figure 7, andthe behavior of water close to its liquid-gas critical point shown in Figure 2. First, we want to draw the analogy between the two phases thatare present in each system. In case of the Ising model, the two obvious
phases are the yellow phase and the blue phase, i.e. uniform configurationsof the Ising lattice that are dominated by yellow or blue states. Within the13interpretation of the Ising model as a liquid, these two phases correspond toa high density and a low density liquid phase, respectively. The analogous
phases in case of water are the liquid phase with a high density, and thelow density gas phase. At temperatures above the critical temperature Tc ,we can observe a mixture of the two phases. For temperatures that are far
above the critical temperature, this mixture of phases becomes uniform,
and it is justified to refer to the condition of the system as a separate,so-called supercritical phase. In both systems, i.e. the Ising lattice in Figure 7 and water in Figure 2, we can see how the systems become uniform
when we approach the supercritical phase at temperatures far above Tc .
At temperatures that are below the critical temperature, however, we cansee a separation of phases. This is seen in Figure 7 where the Ising latticecondenses into one of the two canonical phases, i.e. yellow or blue, as wellas in Figure 2, where the phase separation is observed by the separation
into the liquid and the gaseous phase. At temperatures very close to Tc ,we can observe the phenomena of critical opalescence, which means thatthe system becomes opaque at all wavelengths, including the wavelength ofvisible light. In the back-illuminated microscope images of Figure 2, this isseen as the water sample becomes opaque and turns black. The underlying
principle is illustrated by the Ising model in figure Figure 7, where criticalopalescence corresponds to the infinite growth of fluctuating regions andhence the scattering of light at all length scales.
So far, we have neglected the second term of equation (1), which represents the interaction between lattice states and an external field. As anext step external fields will be considered as well. As discussed previously
in the case of a magnetic system, the external field is a magnetic field H.
For a liquid however, the external field corresponds to an external pressure P , instead. The influence on the system by both temperature andpressure is conveniently depicted in two-dimensional phase diagrams, withtemperature and pressure defining the two axis of the diagram. A part ofwater’s phase diagram and the phase diagram of the Ising model are shownin Figure 8.
A look at the phase diagram of water in Figure 8a shows the stablephases of water under certain conditions in terms of temperature and pressure. The region of the phase diagram at moderate temperatures and towards high pressures is occupied by the liquid phase of water, whereas atmoderate pressures and towards high temperatures water tends to be in
the gas phase. The two regions of the liquid phase and the gas phase areconnected by a phase coexistence line, also referred to as binodal, that terminates at the liquid-gas critical point of water, located at approximately
374 ◦C and 220 bar. Beyond the critical point, water is found in the supercritical phase that resembles a uniform but fluctuating mixture of the liquidand gas phases. The beginning of the supercritical region in the phase di14

Figure 8: Phase diagram of water and the Ising model. Phase coexistencelines are show as red solid lines. Critical points are shown as red points.
Widom lines are shown as red dashed lines. Supercritical regions are indicated by grey dashed lines.
agram is indicated by grey dashed lines. However, the exact boundaries ofthe supercritical region, can be rather difficult to define, especially near thecritical point, where uniformity breaks down and the assignment of phases
becomes arbitrary. A very similar picture is obtained for the Ising model.
The phase diagram of Figure 8b is partitioned in three regions as well,where the respective phases of a yellow phase and a blue phase resemble
the high and low density phases, in addition to the supercritical phase thatstarts beyond the critical temperature, Tc . Again, a phase coexistence line
separates the yellow phase from the blue phase and terminates in a criticalpoint. We note, that in the simple case of the Ising model, the phase coexistence line is actually a straight line. For a real system such as water,
though, the phase coexistence curve is in general more complicated.
Finally, we like to address the process of a phase transition. Or in otherwords, ask the question, how to convert from one phase into another phase.
The obvious answer is, of course, by changing the temperature or pressureof the system. However, there are two fundamentally different approaches
to go from one phase to the other phase. The straight forward procedure
is to cross the phase coexistence line. In this case, there is a rather abruptchange from one phase into the other phase. This kind of phase transitionis called a discontinuous or first-order phase transition. An example of adiscontinuous phase transition is water at its boiling point, i.e. at a pressureof 1 bar and a temperature of 100 ◦C. When boiling water, the temperature
stays at a constant 100 ◦C although heat is added to the system. This latent
heat is used to gradually convert parts of the system from the liquid phase
into the gas phase.
As a side note, it is worth mentioning that the phase coexistence line
15

defines the loci of phase coexistence for a system at thermodynamic equilibrium. However, it is often possible to cross the phase coexistence line
without the phase transition taking place. This phenomenon is referred toas supercooling or superheating, depending if the phase coexistence line iscrossed by decreasing or increasing temperature, respectively. In such acase, the system is in a metastable state, where thermal fluctuations may
spontaneously trigger a phase transition into the thermodynamic stablephase. There is a limit on how far a system can be supercooled or superheated, namely when an infinitesimal small energy fluctuation is alreadylarge enough to overcome the local energy barrier and initiate the phasetransition. This limit is called a spinodal. Depending on the extend of supercooling or superheating the phase separation is described by nucleation
growth or spinodal decomposition [61]. In general, it is possible to supercool or superheat a system more, the further away from the critical point.
At the critical point the spinodal and the phase coexistence line meet.
Another approach to realize a phase transition is to not cross the phasecoexistence line, but instead take a detour through the supercritical phase.
In this case there is no abrupt phase change, and instead the phases interchange continuously within the supercritical region. Such phase transitions are named second-order or continuous phase transitions. Previously,
we have introduced the supercritical phase as a uniform mixture between
the two contributing phases. Starting from one phase, entering the supercritical phase, and finally fully converting into the other phase, the ratio
between the relative amount of water molecules resembling the initial andfinal phases, gradually changes. The loci of the phase diagram with an evenratio between the two interchanging phases is called a Widom line. Hencethe Widom line can be thought of as a kind of continuation of the phasecoexistence line into the supercritical phase. Widom lines in the case of
supercitical water and the Ising model are shown in Figure 8 as dashed redlines.
We note that there can be several Widom lines associated with different
thermodynamic properties. Each of them is defined as the line of maxima
in the phase diagram for the given property. The most prominent Widom
line is defined by the line of maxima of the correlation length, which we’ll
introduce in the next section.

2.3

Pair Correlation Function and Correlation Length

The aim of this section is to make use of the Ising model in order to introduce the density-density pair correlation function and the concept of thecorrelation length which we’ll encounter later in the discussion of supercooled water. For a real system such as water, it can be very challenging toobtain these properties from experiments. Ising model simulations, in con16

trast, serve as a simplistic model that allows to illustrate and calculate theseproperties directly. For this purpose, simulations on a two-dimensionalsquare-lattice have been performed and the results are discussed in the following. It is worth to note, that all simulation results are given without error estimation and shall be considered as an explanatory illustration rather
than precise readings. For our purpose, however, this will be adequate.
For an isotropic system, such as a liquid or the Ising lattice, there isno distinguished spatial orientation and the density-density pair correlation
function, G(r), simplifies to a function of a scalar length scale, r:
G (r) = hn(r)n(r0 )i − n2(3)Here, r and r0 are positions within the liquid or the Ising lattice and r =|r − r0 | is the distance between the two positions. Furthermore, n(r) andn(r0 ) are the densities at corresponding positions. Angular brackets, h...i,denote the ensemble average and n = hn(r)i = hn(r0 )i is the average densityof the system. Note that the average density is independent of the position
within a spatially uniform system. We can interpret the term hn(r)n(r0 )i as
a measure of a conditional probability to find a particle at position r, when
there is also a particle at position r0 . In other words, this term corresponds
to the spatial correlation of local densities that are separated by a distancer = |r − r0 |. At very long distances the two densities become uncorrelatedhn(r)n(r0 )i → hn(r)ihn(r0 )i = n2

[as r → ∞],(4)and hence the density-density correlation function converges to zeroG (r) → 0

[as r → ∞].

(5)With the help of equation (3), it is straightforward to calculate thedensity-density pair correlation function, G(r), from simulations of the twodimensional Ising lattice. Therefore, we assign the two possible states of theIsing model to two distinct local densities. For example, a straightforward
approach would be to assign the yellow Ising state with a local densityof value 1 and interpret this density as the presence of a particle at thegiven lattice point. Analogously, we assign the blue Ising state to a local
density of value 0 and interpret this density as the absence of a particle,
i.e. an empty lattice point. Figure 9a shows calculated density-density paircorrelation functions of the two-dimensional Ising lattice. Here, we use thesame conditions that have been used previously in Figure 7, i.e. no external
field and the same set of reduced temperatures, T /Tc . As expected, G(r)approaches zero at long distances. In fact, the asymptotic behavior of thepair correlation function at large r is close to an exponential decay and thethe density-density pair correlation function for the two-dimensional Isingmodel is given by17G (r) ∝1
r1/4

e−r/ξ .

(6)

The exponential decay constant of G(r) is the so-called correlation length,
ξ, and serves as a characteristic measure of the spatial extent up to which
density correlations occur in the system. It is important to note that density
correlations also exist on length scales that are both shorter and larger than
the given correlation length, but correlations that are larger than ξ become
exceedingly rare. Fits of the calculated pair correlation functions using
equation (6) are shown in Figure 9a. The corresponding values for thecorrelation length are given in the figure legend.
It is possible to relate the correlation length to the size of the clusters
that occur in an Ising simulation, e.g. the clusters shown in Figure 7. This is
typically done in terms of a Guinier analysis [27,62]. Assuming a simplified
spherical shape of the clusters, it can be shown that the diameter of suchclusters is approximately five times the size of the correlation length∗ . Thisis also seen in the Ising simulations of Figure 7, where the largest clusters
have a spatial extend of approximately five times the correlation lengths
that have been calculated in Figure 9a. In general however, clusters thatform during an Ising simulation feature a distorted or fractal-like shapeand the assignment of a cluster size becomes ambiguous. In this case, acommon approach is to take the radius of gyration of those clusters, i.e.
the root mean square distance from the center of mass, as a measure of thecluster size.
To briefly summarize the last few paragraphs, we conclude that the threeobservations, i.e.: the growth of clusters in the Ising model, the densitydensity pair correlation function becoming long ranged, and the increase of
the correlation length, are all interrelated. The dependence of this behavior
on both the temperature and an external field, is shown in Figure 9b bymeans of the correlation length. Of particular interest is the growth of thecorrelation length in proximity of the critical point, i.e. close to T = Tc and
H = 0. Exactly at the critical point, the correlation length diverges andthe density fluctuations in the Ising liquid become infinite. Remember, thatin case of our finite Ising simulation, this corresponds to the fluctuations
approaching the size of the simulation box. The solid lines in Figure 9b are
guides to the eye. We note that the solid lines are slightly misleading in
two ways. First, the decay of the correlation length is not necessarily thesame above and below the critical temperature, as it can clearly be seen∗

The correlation length ξ, typically used in Ornstein-Zernike analysis of critical phenomena, can be approximately related to the radius of gyration√ RG , that is commonlyused in the Guinier analysis of particle solutions, by RG ≈ 3ξ. [27] Furthermore, theradius of gyration p
for a two-dimensionalp disk or a three-dimensional sphere of radius R
is given by, RG = 1/2R or RG = 3/5R, respectively. [63]18Figure 9: a) Pair correlation function, G(r), (grey dots) and fitted exponentials (colored lines) according to equation (6). Reduced temperatures,
T /Tc , and extracted correlation lengths, ξ, are given in the figure legend. b)
Correlation length as a function of reduced temperature for a set of external
fields H. Solid lines as a guide to the eye.
in the figure. Second, the maxima in correlation length at constant external fields are very close to the critical temperature, even though the solidlines indicate a drift towards higher temperatures with increasing external
fields. Indeed, the maxima in correlation length define the Widom line
and hence resemble the shape of the Widom line itself. For the simplistic
Ising model, the Widom line is expected to be a straight line, whereas, ina real liquid, the Widom line usually exhibits a more complex shape in thephase diagram, see Figure 8. The most striking feature, i.e. the sharpnessof the correlation function as a function of the external field, however, isillustrated nicely by the solid lines in the figure. Close to the critical point,
i.e. H = 0, the correlation length increases very rapid and forms a very
sharp and rather narrow peak. Further away from the critical point, whichmeans with increasing external fields, the increase in correlation length issmaller and the temperature dependence broadens, i.e. becomes less sharp.
A similar decrease and broadening, as shown for the correlation length, is acommon feature of all thermodynamic response and correlation functions.
We will discuss this so-called criticality again in section 2.5. In general,
the further away from the critical point in terms of external field, or equivalently external pressure, the smaller and more broad the response andcorrelation functions become [49, 64]. We will encounter this phenomena
19

in the following sections for two more thermodynamic response functions,
namely the isothermal compressibility as well as the heat capacity.

2.4

Compressibility and Heat Capacity

In this section we introduce the isothermal compressibility and heat capacity at constant volume, which serve as a measure of the density and energy
fluctuations of a system, e.g. in our case the fluctuations within the Isingmodel or a liquid.
The isothermal compressibility, κT , is the relative volume change of thesystem as response to a change of the external field or pressure, respectively.
As such, it can be shown that the compressibility depends on the density
variance, i.e. the density square deviation from the mean [65]:κT ∝h(n − hni)2 i
hni

(7)

Here, n is the number density. In the example of the Ising model, it isstraightforward to obtain the compressibility with help of equation (7).
Again, we assign the density values of n = 0 and n = 1 to the blue and theyellow Ising states in order to calculate the ensemble averages and varianceof the density. It seems obvious that the variance of the density increases
with the growth of density fluctuations in the Ising model and hence thecompressibility increases towards the critical point, as shown in Figure 10a.
The heat capacity at constant volume, CV , is given by the temperaturechange of the system as response to the transfer of heat. In other words,
the heat capacity is a measure of the amount of energy needed to raise
the systems temperature by a certain amount. As such, the heat capacity
is related to the energy fluctuations, or more precisely the variance of theenergy in the system [66]:h(E − hEi)2 i
CV ∝
(8)
T2
Here, T is temperature and E is the energy of the system. The energyof an Ising configuration is given by evaluation of equation (1) and hence
allows calculation of the heat capacity via equation (8). The results of suchcalculations on the 512 × 512 Ising lattice are shown in Figure 10b.
Both κT and CV show a strong increase when approaching the critical temperature Tc . Furthermore, the increase in compressibility or heat
capacity is not just a function of temperature but also depends on the external field, H, of the simulation. At the location of the critical point, i.e.
at T = Tc and H = 0 in case of the Ising model, both quantities reach
their maximum value. In principle both response functions are expected
to diverge, but due to the limitation of a finite simulation box their values
20Figure 10: Compressibility, κT , and heat capacity, CV , as a function of
reduced temperature for a set of external fields H. Solid lines as a guide to
the eye. Data calculated from simulations of the 512 × 512 two-dimensional
Ising lattice.
stay finite. The further away from the critical point, in terms of both temperature and external field, the increase in κT and CV becomes smaller. It
is important to note, that not only the values of the two quantities become
smaller, but also their distribution as a function of temperature broadenswith increasing external fields, as it is seen in Figure 10.
We have already seen a similar behavior in case of the density-density
pair correlation function, G(r), in Figure 9b. In general, similar behavior is expected for other thermodynamic response functions as well, withdetails, however, depending on the specific system and the property thatis considered. Again, as it was the case for G(r), also here the growth offluctuations due to the critical point are cause of the anomalous increaseof κT and CV . This is also seen from equation (7) and equation (8) where
density and energy fluctuations are directly related to the compressibility
and heat capacity, respectively.

2.5

Criticality

The divergence of response functions such as compressibility or heat capacity is caused by the increasingly long-ranged correlations within a system.
We have seen that, in the vicinity of a critical point, the correlation length
grows without a limit and hence fluctuations occur on all length scales. This21

implies that systems that are close enough to its critical point become scale
invariant, i.e. unaffected by any change of length scales. A visualization of
the scale invariance is for example given by the Ising configuration at thecritical temperature T = Tc of Figure 7. Any change of length scales, i.e.
zooming in or out of the picture, will lead to a picture that is essentiallythe same.
A striking consequence of the scale invariance is that systems which
may differ dramatically at finite scales will become increasingly similar as
criticality is approached. This phenomena, where dissimilar systems aredescribed through the same behavior, is called universality. Critical phenomena are commonly described by power laws and a corresponding set
of critical exponents. The three-dimensional Ising model as well as liquids
are in the same universality class, which means that they feature the same
critical exponents. Therefore, we take a look again at the pair correlation
function of equation (6), but this time in more dimensions:
G (r) ∝

1
rd−2+η

e−r/ξ

(9)Here, d is the dimensionality, ξ is the previously introduced correlation
length, and η is the critical exponent of the density-density pair correlation
function. We see that, close to the critical point where ξ is large, theexponent in equation (9) vanishes, and G(r) reduces to a power law. In
two dimensions, when d = 2, we retrieve equation (6) and obtain the criticalexponent of η = 1/4 in two dimensions. In three dimensions, however, η
becomes very small. Some of the most important critical exponents for thethree-dimensional Ising universality class are listed in Table 1. Definitions
through the corresponding power laws are also given in the table. Except
for the pair correlation function, which is a function of spatial distance, thelisted quantities are a function of temperature. We have seen previouslythat the asymptotic behavior of the response functions differ depending onwhether the critical point is approached from temperatures above or below
the critical temperature. Here, the tabulated definitions and exponents are
given for temperatures above Tc . We should note that the Ising universality
class in three dimensions has not been exactly solved [67].
To conclude this section we present a small paradox: It is clear thatshort-range interactions, e.g. spin-spin interactions in the Ising model orinter-atomic forces in a liquid, are responsible for the existence of criticalphenomena. On the one hand, these short-range interactions are clearly
dependent on the specific system. On the other hand, the details of theseinteractions cannot play any role, since the critical exponents stay the same
within a universality class that includes a hole range of systems and hence
significantly different short-range interactions.
22

Table 1: Definition of critical exponents for the three-dimensional Isingmodel and fluid systems. Here,  = T /Tc − 1.
Exponent

Definition

Value

η
ν
γ
α

G(r) ∝ r−(1+η)ξ ∝ −νκT ∝ −γCV ∝ −α≈ 0.036...
≈ 0.630...
≈ 1.237...
≈ 0.110...

2.6

Quantity
pair correlation function
correlation length
isothermal compressibility
isochoric heat capacity

Supercooled Water and a Liquid-Liquid Critical Point

Here in this section, we present the picture of the liquid-liquid critical point
scenario in supercooled water [28]. Among other models [35–37, 68] this is
the only scenario which is fully consistent with all existing experimental
results, including the experiments that are content of this thesis. In this
scenario, water’s anomalous behavior is explained due to the existence of asecond critical point that causes fluctuations between two distinct configurations of water molecules. The two configurations differ in density, likewise
the two states of the Ising model, and are denoted as high-density liquid(HDL) and low-density liquid (LDL) respectively. Each of the two liquids
alone behaves as a simple liquid. It is worth noting that both liquids consist of the same H2 O molecules and the difference between HDL and LDL
is related to their distinct structural configurations. LDL exhibits a morestructured and almost ice-like, tetrahedral coordination which is formed
due to directional hydrogen-bonds. Therefore, the LDL configuration isless dense, and favored in terms of energy. In contrast, the structure of
HDL is more disordered because of the enhanced influence of isotropic vander Waals interactions. Hence, the distorted structure of HDL allows for ahigher density, and is favored in terms of entropy.
According to the liquid-liquid critical point scenario, the two liquids,
LDL and HDL, are separated by a phase-coexistence line, which terminates at a critical point. In order to differentiate from the liquid-gas critical point, it is usually referred to as the liquid-liquid critical point (LLCP)or the second critical point. Beyond the LLCP, water is characterized bysuper-critical fluctuations between these HDL and LDL configurations that
directly relate to the anomalous behavior of the liquid at ambient conditions. The locus of maxima in the correlation length of these fluctuations
defines the Widom line in the pressure-temperature phase diagram, which
emanates from the LLCP as an extension of the phase-coexistence line.
A schematic of the phase diagram of non-crystalline water is shown in
Figure 11. In analogy of the two Ising states, the two simple liquids, HDL
23

Figure 11: Schematic phase diagram of noncrystalline water. Two simple
liquids: the low-density liquid (LDL) in blue and the high-density liquid(HDL) in yellow. LDL and HDL are separated by a phase coexistence line
(red solid line) which terminates in a liquid-liquid critical point (red point).
Beyond the critical point the two liquids coexist as a fluctuating mixture.
Along the Widom line (red dashed line), these fluctuations maximize andeventually diverge towards the critical point. Grey shaded, funnel-shapedlines define the anomalous region, indicating how far in temperature thefluctuations extend at various pressures. Note that these grey shaded lines
are guides to the eye, not representing true phase boundaries. The line ofmelting temperature Tm separates the stable form of liquid water from its
metastable, supercooled phase. Th indicates the homogeneous crystallization line. The temperature of the Widom line is denoted by Tw . Vitreous
forms of LDL and HDL, namely low-density amorphous ice (LDA) andhigh-density amorphous ice (HDA), are found below the crystallization line
Tx . Reproduced and modified from references [44, 49].
24

and LDL, are shown in yellow and blue, respectively. Below the melting
line, water is thermodynamically metastable with respect to hexagonal ice,
i.e. its crystalline form. However, it is possible to supercool water to temperatures near the homogeneous crystallization line Th . We should notethat homogeneous nucleation is a kinetic effect, and, as such, strongly temperature dependent. Usually, Th approximates the temperature at whichmetastable water stays liquid for several minutes or hours. For time scales
that are just a few milliseconds, it is possible to supercool liquid water
down to −46 ◦C at ambient pressure. Unfortunately for experiments, the
LLCP is expected to lie at even lower temperature and elevated pressure,approximately at −60 ◦C and 1 kbar. At the critical point, the fluctuations grow to infinity, and the density pair correlation function as well as
corresponding thermodynamic response functions diverge. With increasing
distance from the LLCP, the anomalous enhancement of the response functions broadens and becomes smaller in magnitude. Likewise, the criticalfluctuations decrease in size and the anomalous region around the Widom
line becomes wider, as indicated in the figure. Interestingly the anomalous
region extends all the way into the ambient regime, i.e. at temperaturesand pressures where life happens.
To complete the description of the phase diagram, we note that there
exist two distinct forms of amorphous ice below the crystallization temperature. The two forms, namely low-density amorphous ice (LDA) andhigh-density amorphous ice (HDA) show a density difference of around
20 % [69,70]. Indeed, LDA and HDA are interpreted as the vitreous forms of
LDL and HDL, thus providing a thermodynamically consistent perspectiveon the global phase behavior of metastable water [44]. This interpretation
has been supported by experimental observations of a high-to-low density,
liquid-liquid transition in the ultraviscous regime [71] and the first-order
like phase transition between LDA and HDA [72], that is interpreted as
the structurally arrested manifestation of an equilibrium phase transition
between LDL and HDL. The direct experimental observation and corresponding implications of such a LDL-HDL liquid-liquid transition will be
the topic of the next chapter.

25

26

3

Liquid Water Probed by X-Rays

In this section we discuss the experimental observations on supercooled water which are analyzed in detail in papers I-X. The aim of this section isto summarize the main results of the papers and emphasize the interpretation of the experimental observations in terms of the previously introduced
liquid-liquid critical point scenario.
First, we briefly present the experimental tool of x-ray scattering that is
utilized to gain insight on the inter-molecular structure of water. Second,
we discuss implications of the two phase transitions, i.e a discontinuous
phase transition at high pressure as well as a continuous phase transitionat ambient pressure, that have both been observed in supercooled water.
We then investigate some of the thermodynamic response functions, in particular the correlation length, isothermal compressibility and isobaric heat
capacity. By comparison between experimental data and molecular dynamics simulations we attempt an educated guess on the location of the 2nd
critical point. Finally we discuss the experimental method of x-ray speckle
visibility spectroscopy as possible approach to study molecular dynamics
in supercooled water.

3.1

X-ray Scattering and the Pair Correlation Function

In the previous chapter we have introduced the concept of a second criticalpoint in supercooled water which, if existent, would explain the anomalous
behavior of water due to supercritical fluctuations between a low-density
liquid (LDL) and a high-density liquid (HDL). The density difference of
these two simple liquids is related to a distinct local structure between thewater molecules of the two liquids. Hence, from an experimental point ofview, we would like to probe the inter-molecular structure of supercooled
water. One possible approach is the experimental technique of elastic x-ray
scattering that has been used in the experiments of this thesis.
A schematic of a x-ray scattering experiment is shown in Figure 12a.
An incoming x-ray beam is scattered by the sample, e.g. a micrometer sizedwater droplet, and the scattering intensity is captured on an area detector.
The scattering of x-rays is sensitive to the electron density within the sample. In case of water, the electron density is dominated by the electrons of
the oxygen atoms. Hence, the measured x-ray scattering pattern contains
mostly information about the structural configuration between the oxygen
atoms. For water, where the intra-molecular structure is very simple, x-ray
scattering provides an excellent probe of the inter-molecular structure of
the liquid. An amorphous sample just like liquid water features no directional order and hence the scattering intensity only shows characteristic
modulations along the radial distance with respect to the incoming beam
27

a)

b)

Detector

low density
high density

Scattering
angle

Water sample

X-ray

Figure 12: Schematic of a x-ray scattering experiment. a) X-ray scatteringgeometry and the typically circular scattering pattern at a characteristic
scattering angle of a disordered sample such as liquid water. b) Schematicfor the first diffraction maximum, obtained from the angular integrated
scattering intensity I, for a low-density and a high-density water sample.
The peak position of the scattering intensity, i.e. the scattering angle or
momentum transfer q, depends on the density of the sample.
but not along the azimuthal orientation. Typically, this is seen as circularor elliptical shaped intensity distributions on the area detector, as depicted
in Figure 12a.
In practice, the scattering intensity is compared to the expected scattering of a single water molecule, i.e. the molecular form factor of a watermolecule. The deviation of the normalized scattering intensity from themolecular form factor is the so-called structure factor. As such, the structure factor relates the observed scattering intensity to that produced by asingle molecule, and, therefore, serves as a measure of the structure within
the sample. Again, in case of an amorphous sample, the structure factor
reduces to a function of the radial distance or scattering angle only. Indeed, a more rigorous treatment shows that the structure factor is givenby the Fourier transform of the density-density pair correlation function
introduced in the previous chapter [63]:Z
S (q) = 1 + ρ

G (r) e−iqr dr(10)Here, ρ is the mean density of the sample and q is the magnitude of the
momentum transfer. It is worth noting the reciprocal relation, through theFourier transform, between the distance r in real space and the momentum
transfer q in momentum space. In essence, this means that large distances
28

Figure 13: a) Oxygen-oxygen contribution to the x-ray scattering structure
factor measured on supercooled water droplets. b) Pair correlation function obtained by Fourier transform of the scattering structure factor andmultiplied with r for better visibility of the oscillations at large distances.
Data at various temperatures are offset from each other for clarity. Hightemperatures were measured on droplets with 40 µm diameter and the twolowest temperatures with droplet diameters of 20 µm. Reproduced frompaper V.
in real space correspond to small distances in momentum space and viceversa. Hence, a sample of high density, i.e. short inter-molecular distances r,
scatters the incoming radiation at larger momentum transfer q, as compared
to a sample of low density. This is schematically illustrated in Figure 12b.
In a x-ray scattering experiment, the momentum transfer is related to thescattering angle by q = 4π/λ sin θ, where λ is the x-ray wavelength and2θ is the scattering angle. Hence, S(q) is relatively easy to obtain fromexperiments. For example, Figure 13a shows the scattering structure factor
measured on supercooled, micrometer-sized water droplets. A close up of
Figure 13 as well as detailed discussions about S(q) and G(r) are found in
paper V. Here we just summarize some of the main features.
First of all, both the structure factor and the pair distribution function
show oscillations that reflect the inter-molecular structure in the liquid.
We should note that the Ising model of the previous chapter does not feature such oscillations in the pair correlation function due to the absenceof structure. In other words, there is no characteristic length scale in the
Ising model as compared to a liquid, where next nearest molecules are separated from each other by approximately the same inter-molecular distance.
In liquid water, this nearest-neighbour distance between the molecules isaround 2.8 Å, as seen in the in the first peak of G(r). The next-nearest
29

distance which is the second peak in G(r) at 4.5 Å, reflects the characteristic distance of tetrahedrally arranged water molecules. For even larger
distances, the coordination shells become less and less pronounced. Notethat in Figure 13b the pair correlation function is multiplied by the radial
distance in order to visualize the higher coordination shells.
Next, we are interested in the temperature dependence of the pair correlation function. The most prominent changes in the experimental dataof Figure 13b are in the first and second coordination shell, which both increase in height towards lower temperatures. In connection, the interstitialregion between the first and second peak becomes less populated. In termsof the two-phase model of water, we can relate these changes to a change
in population of the two competing local structures HDL and LDL. In doing so, we note that LDL is characterized by tetrahedral local order withwell-separated first and second shells in G(r). In contrast, HDL exhibits anincrease of interstitial molecules between the first and second coordination
shells leading to perturbed hydrogen-bonding and higher density. Hence,
we can interpret the rise of the second coordination shell upon cooling withan increasing relative amount of LDL within the liquid. This is consistent
with the second critical point scenario where the liquid is dominated by distorted HDL configurations at high temperatures, and transforms towards
a more structured LDL liquid with decreasing temperature [52, 73–76].
Whereas the radial distribution function G(r) directly shows the formation and population of coordination shells within the liquid, the interpretation of the structure factor S(q) is less intuitive. Here, the principal−1−1
maximum of S(q) is split into two peaks located close to 2 Å and 3 Å .
The amplitudes of, and separation between, the two peaks increase withdecreasing temperature, again indicating an increase in structural ordering
as water is supercooled. In particular the shift of the first diffraction peaktowards lower q values can be affiliated with local regions of tetrahedral
structures [52, 77]. In the following section, we will make use of the firstdiffraction peak as a structural measure of the liquid.

3.2

Phase Transitions: A Question of Pressure

Figure 2 in the introduction shows the first critical point or gas-liquid critical point of water. Here, however, we would like to probe the existence ofa second critical point, i.e. the liquid-liquid critical point (LLCP) that was
shown in Figure 11. Unfortunately, direct observation of critical fluctuations and the appearance of critical opalescence might be difficult in caseof the LLCP due to the metastability of the liquid. In the framework ofthis thesis, we proceed with a more indirect approach towards the proposed
LLCP. Here, the idea is to focus on the phase transition lines, i.e. the phasecoexistence line as well as the Widom line, shown in Figure 11. If existent,
30

Figure 14: Schematic of a phase transition observed by x-ray scattering.
Curves show the scattering intensity I, as a function of momentum transfer
q. Blue and yellow curves show the pure low-density and high-density
states. Purple and grey curves show the intermediate of the transitionin case of a) a discontinuous phase transition or b) a continuous phasetransition. In each case, the purple line marks the middle of the phasetransition.
both lines should eventually meet, and, therefore, lead to the liquid-liquidcritical point.
We have already seen in the previous section how x-ray scattering allows
to probe the structure within the liquid. In particular, the peak positionof the first diffraction maxima serves as a structural indicator of the two
liquids, i.e. the high-density liquid (HDL) with a peak at large momentum
transfer, and the low-density liquid (LDL) with a peak position at lower
scattering angle. Hence, the experimental approach is to probe the existenceof a phase transition between HDL and LDL by observation of the x-ray
scattering intensity as a function of pressure and temperature. Dependingon the location of the LLCP there are two possible realizations of a phasetransition. For pressures that are higher than the pressure of the criticalpoint, we expect a discontinuous phase transition when crossing the phasecoexistence line. Instead for pressures below the critical pressure we expect
a continuous phase transition close to the Widom line.
A schematic view on how the expected phase transitions would manifest
in a x-ray scattering experiment is illustrated in Figure 14. The initial andfinal states of the phase transition are indicated by the principal scattering
maximum of a pure low-density liquid and a pure high-density liquid, thatare shown in blue and yellow, respectively. At the beginning of a discontinuous phase transition, the scattering intensity of the initial state starts
31

to decrease, and a new peak, corresponding to the new state, emerges. The
further the phase transition evolves, more and more of the initial scattering intensity is transferred to the new scattering peak. It is important
to note that, during that process, the peak position, i.e. the momentum
transfer q, stays constant for each of the two scattering maxima. Hence,a characteristic double-peak feature arises in the x-ray scattering intensity
which is typical for a discontinuous phase transition. This is consistent
with the picture that the liquid is macroscopically only found in one of thetwo states, i.e. either the low-density or the high-density state. The relative amount between the two macroscopic phases is reflected in the relative
height of the two corresponding scattering maxima. Exactly in the middle of a phase transition there is an equal amount of the two liquids, and,hence, the scattering intensity between the two peaks is even, as shown by
the purple line in Figure 14a.
In the case of a continuous phase transition, the scattering intensity
continuously shifts from the initial to the final state. This is shown in Figure 14b. Hence, there is no double-peak feature of the scattering intensity incase of a continuous phase transition. This behavior is consistent with thepicture of a mixture between the two pure phases, i.e. a mixture between apurely low-density liquid and a purely high-density liquid. In other words,
the two pure liquids exist only locally within the liquid. Macroscopically,
however, the liquid is seen as a mixture of the two pure liquids, and exhibits
average properties, for example the mean density of the two pure liquids.
The average, of course, taking the relative mixing ratio into account. Hence,in the middle of the phase transition where the mixing ratio is even, thescattering intensity is expected to be approximately in the center of theinitial and final states, as seen by the purple line in Figure 14b. Anothermeasure of how close the liquid is to its phase transition is the rate at which
local phase transformations occur. Closer to the phase transition, it is more
likely for local structures within the liquid to transform into configurationsof the more stable phase. This holds regardless of the direction from which
the phase transition is approached. As a consequence, the rate at which
the phase transition takes place experiences a maximum in the middle ofthe phase transition or, in other words, exactly at the phase transition line.
We have now set the stage for the interpretation of the experimental
results presented in paper VII and II, which are summarized in Figure 15
and Figure 16, respectively.
In paper VII high-density amorphous (HDA) ice samples were probedby x-ray scattering after ultrafast heating into the liquid state as shown in
Figure 15. In preparation of the experiment, the HDA samples have been
kept at cryogenic temperatures to prevent the phase transition into the lowdensity state. This is a common procedure that allows to kinetically arrest
the sample in its high-density state even though the sample is physically32scattering intensity, I(q) [arb. units]

3 s
1 s
100 ns33.2 ns-8.4 ns
1.50 1.75 2.00 2.25 2.50

q [Å 1]Figure 15: Scattering intensity I,(black solid lines) of a high-density
amorphous ice sample after ultrafast heating by an infrared laser
pulse. Delay times between the infrared pump pulse and the x-ray
probe pulse are indicated in the figure legend. With increasing pumpprobe delays the high-density peak−1(yellow) around 2.1 Ågradually vanishes and a new, low−1
density peak (blue) around 1.7 Å
emerges. After approximately 3 µs
the transformation is interrupted
by the formation of crystalline ice.
Here, a linear background was subtracted from I for better visibility. Raw scattering intensities areshown in paper VII.

in an environment of ambient pressure. The HDA samples are heated bya femtosecond infrared pump laser pulse and isochorically transform into aliquid state. A few nanoseconds after the heating laser pulse, the molecular structure of the sample had no time to relax and the liquid is still ina high-density state. However, over time the high-density state will relax
its structure due to the pressure gradient between within the sample andits surrounding. We can experimentally follow the structural relaxation of
the sample by measurements of the x-ray scattering intensity as a function of the delay time between the infrared pump and a subsequent x-ray
probe laser pulse. The results of our pump-probe measurement is shown in
Figure 15.
For the un-pumped case, i.e. a negative pump-probe delay, we see thetypical scattering intensity of a HDA sample with a high-density peak at−1
approximately 2.15 Å . At increasing pump-probe delay times we observe−1
the growth of a new peak around 1.7 Å , indicating a discontinuous transformation towards a more tetrahedral structured liquid of lower density.
Indeed, we can fit the total scattering intensity as the contribution of two
individual components at constant q values, i.e. a high-density and a low33

Figure 16: a) Scattering structure factor S of supercooled, µm-sized water
droplets around the first scattering maximum. Temperatures are given in
the figure legend. b) Temperature derivative of the peak position S1 of thefirst scattering maximum as a function of temperature. The actual peak
position S1 is shown in the figure inset. Solid lines are guides to the eye.
Reproduced from paper II.
density component. In Figure 15 the two fitting components are shown inblue and yellow, respectively. Again we note, that the pump-probe delay
time corresponds to a decrease of pressure within the sample. Hence, within
the framework of the liquid-liquid critical point scenario of Figure 11, we
identify the two contributions as the structural manifestation of pure HDL
and pure LDL. In consequence, the observed double-peak feature in the
scattering intensity is explained by a discontinuous phase transition at thephase coexistence line between HDL and LDL. At last we note, that crystallization of the liquefied samples sets in at longer time scales of around
3 µs, interrupting the liquid-liquid transition.
In paper II we attempt to probe the phase diagram of supercooled water at ambient pressure. The experimental approach is based on rapid
evaporative cooling of approximately 14 µm sized water droplets that are
injected into a vacuum chamber. Using this method, supercooled liquidwater samples with temperatures down to approximately 227 K have beenobtained, with the lowest temperature limited by homogeneous ice crystallization occurring after just a few milliseconds. Individual water droplets
were probed using intense x-ray pulses from an x-ray free electron laser
source. For droplet temperatures below 230 K an increasing fraction of
the recorded x-ray scattering patterns showed Bragg peaks that indicate
the growth of crystalline ice. Thus, the recorded data was selected on a
shot-by-shot basis, and only x-ray scattering images of pure liquid water,
34

i.e. scattering patterns without Bragg peaks, were used for further analysis.
The normalized scattering intensity, i.e. the scattering structure factor, ofthe first scattering maxima is shown in Figure 16a for a set of temperatures.
We see a continuous shift of the peak position towards lower momentum
transfer upon supercooling, which is also shown in the inset of Figure 16b.
The continuous shift in the first scattering maximum indicates a continuous increase of tetrahedral coordinated structures within the liquid. Interestingly, the rate at which the peak position changes with respect tothe change in temperature experiences a maximum around 229 K, as seenin Figure 16b. In the picture of the liquid-liquid critical point scenario,
this observation is consistent with a continuous phase transition from thehigh-density liquid, HDL, to the more tetrahedral-like low-density liquid,
LDL. Indeed, as we will see in the following section, other thermodynamic
response functions also exhibit a maximum around 229 K, adding further
evidence to the existence of a continuous liquid-liquid phase transition atambient pressure.

3.3

Peaking Correlation and Response Functions

In this section, we take a closer look at the continuous liquid-liquid transition in supercooled water that was observed at ambient pressure. In theprevious section, we found a maximum in the shift of the first scatteringpeak position with respect to changes in temperature. From our discussionof the Ising model we expect also other thermodynamic response functions
to exhibit a maximum close to the phase transition. This is due to thegrowth of fluctuations in the liquid. The previously discussed maxima inthe change of the x-ray scattering peak position, for example, is related tostructural fluctuations. A measure of the spatial extend of such fluctuations is given by the correlation length ξ. In our introduction to the Isingmodel, ξ was used as the decay constant of the pair-correlation function.
Indeed, taking the inverse Fourier transform of the pair-correlation function shows that an increase of the correlation length relates to an increaseof the scattering structure factor S, at small momentum transfer q. In
practice, S(q) is often described by Ornstein-Zernike analysis where the increase of the structure factor at small momentum transfer is approximatedby a Lorentzian, with half-width at half-maximum related to the correlation
length [65]:S OZ (q) ∝ 1/(ξ −2 + q 2 )(11)

In Figure 17a we show x-ray measurements of the structure factor at
small scattering angles, i.e. at small values of q. At room temperaturewater shows almost no increase at small momentum transfer. However,
35

a)

b)

structure factor

structure factor
q [Å-1]q [Å-1]ousqq [Å-1]Figure 17: a) Normalized structure factor derived from small angle x-ray
scattering on supercooled µm-sized water droplets, measured at PAL-XFEL(left panel) and SACLA (right panel). Depicted structure factors correspond to temperatures ranging from 275 K (dark red) down to 228 K (darkblue). Dashed lines indicate the experimental limit for low q. Grey solidlines show Ornstein-Zernike fits of the structure factor. b) Decomposition of
the structure factor (blue) into a normal component (red) and an anomalous
component (green). Isothermal compressibility κT and correlation length ξ
are retrieved from the fit. Reproduced from paper II.
when water is cooled to lower and lower temperatures we observe a growing increase of the scattering structure factor at small q. In practice we
can fit the small-angle structure factor by linear combination of two competing contributions: One of the contributions resembles the behavior of anormal liquid, where the scattering intensity is expected to just monotonically decrease towards small q. The other contribution is well described by
Ornstein-Zernike analysis featuring a Lorentzian line shape. An example
of such a fit is depicted in Figure 17b. As mentioned above, we can extract
ξ from the half-width at half-maximum of the anomalous component. Furthermore it can be shown that the isothermal compressibility, κT is directly
related to the scattering structure factor at q = 0 by the relation [78]S (0) = kB T nκT .

(12)Here, kB is the Boltzmann constant, T is temperature and n is the molecularnumber density.
The two quantities, ξ and κT , obtained from small-angle x-ray scattering are shown in Figure 18. First of all we note a strong increase ofboth properties upon cooling. Solid lines in the figure show a power-law
fit as guide to the eye. At approximately 229 K, the correlation length aswell as the compressibility suddenly deviate from the power-law behavior,
36

Figure 18: a) Correlation length ξ and b) isothermal compressibility κT ofsupercooled water. Solid lines are powerlaw fits up to the maximum values.
Insets show the existence of maxima in both, ξ and κT at approximately
229 K. Reproduced from paper II and IV.
which otherwise describes the data at higher temperatures reasonably well.
As a result, at approximately 229 K we observe a maximum in both thecorrelation length and compressibility. This is at the same temperatureas the maximum of the temperature derivative of the first scattering peak
position discussed earlier. In the discussion of the Ising model, we showedthat these maxima are related to the crossing of the Widom line where fluctuations between two competing phases reach a maximum. In the pictureof the liquid-liquid critical point scenario, the two competing phases are
the high-density liquid HDL, and the low-density liquid LDL. In practice,
we can probe such fluctuations in different ways. For example, a direct
measure of the spatial extent of the fluctuations is given by the correlation
length. Whereas, in case of the isothermal compressibility the fluctuations
are probed through density fluctuations within the liquid.
Another approach is presented in paper VIII, where we measured theheat capacity of supercooled water, which is directly related to the energy fluctuations of the system. The experiment is briefly explained here.
Again, we made use of evaporatively cooled µm-sized water droplets toachieve temperatures down to approximately 228 K. The droplets were hit
by a femtosecond infrared laser pulse to heat the water samples by roughly
one Kelvin. A subsequent x-ray laser pulse was used to probe the resulting
temperature rise through the temperature dependent shift in the first scattering peak. With the excitation frequency of the optical pump laser far
from any resonance in water’s absorption coefficient, we can also estimate
37the amount of energy that is absorbed during the heating. The ratio of
both, increase in the droplet temperature and the energy that is absorbedby the sample, provides the heat capacity. We should note that the time
between the heating laser pulse and the x-ray probe was chosen to be longenough for the sample structure to relax, i.e. approximately one microsecond. In other words this means that the sample is probed at constantpressure and the measured heat capacity is the isobaric heat capacity, CP .
In the experiment we observe an increase of CP upon cooling by almost afactor of three in comparison to its value at room temperature. Although
the measured data come with a relative large uncertainty, there is again an
indication of a maximum close to 229 K, i.e. the same temperature where
the previous response functions showed their maximum values.
At last we note that the correlation length as well as the isothermal
compressibility in paper II, have also been measured for D2 O, i.e. heavy
water. The results are essentially the same as in case of H2 O, again indicating a continuous phase transition and the existence of a Widom line
at ambient pressure. However, the corresponding maxima are found at ahigher temperature of around 233 K. This is consistent with the study ofpaper I, where we show that the magnitude of nuclear quantum effects on
the structure of water is temperature independent. This is seen as the structure factor of D2 O is similar to that of H2 O if the temperature is shiftedby a constant 5 K. This constant shift in temperature is found to be valid
from ambient temperatures down to the deeply supercooled conditions.

3.4

Location of the 2nd Critical Point: An Educated GuessIn the previous sections we presented experimental evidence for the existence of a liquid-liquid transition between a high-density and a low-density
liquid in supercooled water. Depending on the external pressure, the observed transition manifests in two different types. At high pressure weobserve a discontinuous phase transition, featuring a phase coexistence between the two liquids. At ambient pressure, on the contrary, we find acontinuous phase transition, characterized by peaking response functions
around the Widom line. According to thermodynamics, the two phasetransitions have to eventually meet in a critical point. This is commonlyreferred to as the liquid-liquid critical point scenario, which has been introduced at the end of the previous chapter and is summarized in Figure 11.
In this section we would like to address the question of where such aliquid-liquid critical point might be located. Or, in other words, where in
the phase diagram of supercooled water, i.e. at what temperature and pressure, do the phase coexistence line and the Widom line meet. To answer
that question we make use of the previous discussion about the thermodynamic response functions that have been calculated from simulations using38

the Ising model. There we have seen that for pressures which are further away from the critical point, the response functions become smaller
in magnitude and broaden with respect to temperature. An essentiallyvery similar but much more accurate behavior of water’s thermodynamic
response functions can be obtained from molecular dynamics simulations.
In paper IV and VI we compare classical force field simulations withthe experimental results obtained on supercooled water. Here, we like tofocus on the comparison of the isothermal compressibility, κT , between theexperiment and molecular dynamics simulations. Figure 19a shows suchcomparison for a set of well established force fields. We note that theexperimental data and simulations are both shown at ambient pressure. At
moderate temperatures most of the simulations agree reasonably well withthe experiment. At supercooled temperatures, however, the simulations
dramatically underestimate κT . Furthermore, the broadening around thecompressibility maximum is much more prominent in the simulations. We
can explain this discrepancy between the experiment and simulations due
to a liquid-liquid critical point that is closer to ambient pressure in caseof real water than compared to simulations. Indeed, we show in paper VI
that real water is described best if the classical force field simulations are
shifted by approximately 1 kbar.

a)

b)

500
400
300
200

100

50

Figure 19: Isothermal compressibility κT . a) κT at 1 bar of various molecular dynamics models (dashed lines) in comparison to experimental data
(black dots). b) Maxima of the isothermal compressibility κmax
(black dots)T
as well as power-law exponents γ (red squares) obtained by simulations of
the iAMOEBA water model at various pressures. Dashed lines are guides
to the eye. Solid lines indicate the experimental values of κmaxand γ at amTbient pressure. The iAMOEBA water model features a liquid-liquid criticalpoint at approximately 1750 bar. Adapted from paper IV and VI.
39

One of the more accurate force field models is the iAMOEBA watermodel. It features a liquid-liquid critical point at around 1750 bar [79]. In
Figure 19b we compare the height of the compressibility maximum from theiAMOEBA water model at various pressures with the experimental value
at ambient pressure. We note the logarithmic scale of the figure. The
experimental value of κT at ambient pressure is indicated by the solid linein the figure. It corresponds best to simulations around 800 bar, or in otherwords approximately 1 kbar below the critical pressure of the simulation
model. In paper IV, a similar result is obtained for comparison of the powerlaw exponents γ, when κT is fitted in terms of a power law. The γ exponent
thereby serves as a measure of the broadening around the compressibility
maximum.
In conclusion, the experimental compressibility is best described by simulations that are approximately 1 kbar below the critical pressure. Together
with the discussion in paper VII, where we estimate the critical temperature to be a few Kelvin below the homogeneous nucleation temperature, weestimate that the liquid-liquid critical point of supercooled water might belocated at around 215 K and approximately 1 kbar.

3.5

Dynamics of Supercooled Water

The experimental observations presented in this thesis strongly suggest theexistence of a liquid-liquid critical point deep in the metastable phase diagram of supercooled water. As such, experiments that aim to measure
the critical behavior in proximity of the second critical point have to be
performed on time scales that are short enough to beat ice crystallization.
From crystallization rates found in paper VII we can give a crude estimateof such time scales to be on the order of microseconds or even less. However, this estimate might be off by orders of magnitude. This is because not
only structural properties but also molecular dynamics, i.e. the time scalesof molecular motion, can change dramatically with temperature.
A direct example is the diffusion constant in supercooled water whichchanges by many orders of magnitude upon cooling. Interestingly, the rate
at which the dynamics slow down in supercooled water has also been assigned to the existence of a liquid-liquid transition between two, structurally
and dynamically, distinguished structures [80]. However, more experimental data is needed to tell whether a liquid-liquid critical point really manifests in the metastable phase of supercooled water. Or, in other words,
even if a liquid-liquid critical point exists in supercooled water, it is still an
open question whether the accessible time scales are long enough for fluctuations to grow and reach true criticality before ice nucleation occurs. In
this section, we present two experimental approaches that have been used
to probe the dynamics of supercooled water and hence might open a way
40

to answer the above question in future experiments.
The first approach is based on optical Kerr effect (OKE) spectroscopy
and is discussed in paper IX. Here we briefly summarize the main idea of theexperiment. First, a strong electric field provided by a polarized laser pulse
induces a transient reorientation of the water molecules within the sample.
Then a subsequent x-ray pulse is used to monitor the anisotropic scattering pattern as a function of the pump-probe delay. The time evolutionof the induced anisotropy reflects the relaxation and vibrational responseof the molecules in the sample. We note that, typically, an optical probe
pulse is used in OKE spectroscopy [81]. Here however, utilizing an intense x-ray probe pulse, we can directly relate the dynamics of liquid water
to the structural information such as the pair correlation function. As aresult, we observe a decay of the induced anisotropy after approximately
0.3 ps. Around 1 ps after the probe pulse the transient reorientation of themolecules dissipates into heat. Both the experimental anisotropic scattering pattern and its time trace are well reproduced by molecular dynamics
simulations, which indicate the existence of two distinct local structuresthat show different dynamics.
The second approach is based on x-ray speckle visibility spectroscopy(XSVS) and is discussed in paper III and X. In brief, µm-sized water
droplets are probed by coherent x-rays pulses from a free electron laser.
A speckle pattern arises due to the interference between wavefronts thatoriginate from coherent x-ray scattering by the electron density within thesample. In case of water the electron density is dominated by the oxygen
atoms. Any change in their exact atomic positions will be reflected in acorresponding change of the speckle pattern in reciprocal space. In Figure 20a we schematically show the effect of different exposure times on the
distribution of molecules in real space as well as the corresponding speckle
patterns. As the exposure time ∆t becomes comparable to or longer than
the time scale of molecular motion, the real-space arrangement of atoms isblurred and so is the corresponding speckle pattern in momentum space.
In paper III the exposure time is changed by variation of the x-ray
pulse duration ∆t. Results of the speckle contrast, i.e. a measure of theblurring of a speckle pattern, are shown in Figure 20b as a function of ∆t.
The observed decay of the speckle contrast is attributed to the molecularmotion during the exposure time. Through comparison with molecular
dynamics simulations, we conclude that the slowing down upon cooling is
not only due to simple thermal ballistic-like motion, but that cage effects
due to hydrogen-bonding play an important role on time scales over 25 fs.
We note that in practice the measured photon count on the detector ofa single x-ray pulse is much smaller than suggested by the image in Figure 20a. Indeed, typical photon counts obtained in the study of paper III
are on the order of 0.01 photons/pixel/pulse. This means that a typical x41

a)large ∆tsmall ∆tspeckle contrast [arb. units]b)real space0
momentum space

25

50

75 100 125 150∆t [fs]Figure 20: a) Effect of the exposure time ∆t on the image acquisition.
Schematic illustration of images taken in real space (left column) and momentum space (right column) for a short exposure time (top row) and longexposure time (bottom row). b) Decay of the speckle contrast as a functionof the exposure time. Symbols are the experimental data at two different
temperatures. The lines are predictions by different theoretical models.
Reproduced from paper III and X.
ray scattering image on the detector consists mostly of pixels which do notrecord any scattering intensity or the scattering of a single x-ray photon.
Occasionally however, very few pixels measure the scattering intensity of
two photons. In paper III the speckle contrast is calculated form the ratioof the two-photon and one-photon events. Obtained values of the speckle
contrast vary significantly between single x-ray shots due to the low photon
count. The contrast values and corresponding error bars reported in Figure 20b are therefore averages and corresponding standard deviations fromapproximately 104 shots per data point.
In paper X, we attempt to probe the dynamics of supercooled water atlonger time scales. Hence, instead of varying the exposure time by stretching the pulse duration of the x-rays, the idea is to split a single x-ray pulse
into two and delay one of the pulses. With this approach exposure times,
i.e. delay times between the two x-ray pulses, between 300 fs and 10 ps havebeen investigated. Unfortunately, due to reflections of the x-ray beam thatare necessary to split and delay the pulses, the pulse intensity drops byroughly an order of magnitude and so does the photon count on the detector. As a result, we obtained large uncertainties for the speckle contrast
that prohibit further interpretation on the molecular dynamics.
In conclusion we note that both approaches, x-ray probe OKE and42

XSVS, provide a useful tool to study dynamics in supercooled water. Bothexperiments are challenging due to the measurement of small signals. In
case of OKE this is mainly because of the the almost isotropic molecularpolarizability of water. For XSVS the limitation is due to low photon
counts. However, with the capabilities of new x-ray facilities with increased
repetition rates such as the European XFEL or LCLS II these kind ofexperiments promise to play an important role in the study of molecular
dynamics in liquids.

43

44

4

SummaryThis thesis presents experimental x-ray scattering studies on supercooled
liquid water. A liquid-liquid transition between two structurally distinct
configurations has been found in deeply supercooled water. The observed
liquid-liquid transition strongly indicates the existence of a liquid-liquidcritical point to cause the anomalous behavior of water.
The experiments were performed at large-scale x-ray facilities such as
x-ray free electron lasers and synchrotrons. Two conceptually different experimental approaches have been used to investigate the metastable phaseof supercooled water. The first approach is based on rapid evaporative
cooling of µm-sized water droplets. In a second approach, structurally arrested high-density amorphous ice samples which have been prepared underhigh pressure, are heated by an femtosecond infrared laser pulse into thesupercooled liquid state. Together, these two approaches have been used toaccess a region within the metastable phase diagram of supercooled waterthat has previously been inaccessible.
Using elastic x-ray scattering measurements as a structural probe of
the liquid, we identified the existence of a liquid-liquid phase transition in
deeply supercooled water. The observed phase transition is interpreted as
the transition between a high-density and a low-density liquid phase. At
high pressure this phase transition is discontinuous or first-order like, featuring a characteristic double-peak feature in the observed x-ray scattering
intensity of the first diffraction maxima. At ambient pressure, however, weobserve a continuous shift of the first diffraction maxima that is consistent
with a continuous or second-order phase transition between the two liquids.
Further evidence of a continuous phase transition at ambient pressure isseen in the temperature dependent maxima of the measured correlation
length, isothermal compressibility and heat capacity, which indicate theexistence of a Widom line.
Together, the experiments presented in this thesis strongly support theexistence of a liquid-liquid critical point deep within the metastable phase
diagram of supercooled water. From comparison of our experimental results and predictions from molecular dynamics simulations we estimate thelocation of such a liquid-liquid critical point to be roughly at 215 K and
1 kbar. However, even if a liquid-liquid critical point exists at such deeply
supercooled conditions, it is still an open question whether the accessible
time scales are long enough for fluctuations to grow and reach true criticality before ice nucleation occurs. In order to answer this question in future
experiments, we successfully demonstrate the capability of coherent x-ray
scattering to serve as a probe of molecular dynamics within supercooled
liquid water.
In conclusion, the experiments support the existence of a liquid-liquid45critical point where the experimentally observed Widom line and phasecoexistence line would both meet. The main result, however, is the firstexperimental observation of a liquid-liquid transition within a pure liquid.

46

Populärvetenskaplig Sammanfattning
För att förstå hur det kommer sig att vatten - fysikaliskt sett - är så olikalla andra vätskor har jag och mina kollegor undersökt vattenprover i denunderkylda delen av fasdiagrammet. I underkylt vatten finns två strukturellt olika vätskefaser och vi har hittat en fasövergång mellan dessa vilket
indikerar att en så kallad kritisk punkt existerar. Bortom den kritiska
punkten samexisterar de båda faserna genom snabba fluktuationer och någon åtskillnad mellan faserna kan inte göras. Vi har använt oss av storaröntgenanläggngar för att med hjälp av röntgenspridning undersöka dessa
metastabila faser av underkylt vatten.
Två konceptuellt olika metoder har använts, den första metoden baseras
på en mycket snabb kylning genom spontan avdunstning när vattendroppar på mikrometerskala injiceras i en vakuumkammare. Med denna metod
kunde vi skapa underkylda vattenprover med temperaturer så låga som
227 K (−46 ◦C) innan droppen oundvikligen kristalliseras och blir till isnågra millisekunder därefter.
I vår andra metod utsätter vi isprover för höga tryck, vilket gör att
deras strukturer kollapsar och amorfa isar av mycket hög densitet skapas.
Vid uppvärmning med hjälp av en ultrasnabb infraröd (IR) laserpuls smälter isen till en motsvarande vätskefas med hög densitet. Under några kortaögonblick efter att den första laserpulsen värmt upp provet upplever vätskeprovet samma höga tryck inifrån som i det fasta initialtillståndet. Vi mäter
spridningen från proverna med en andra puls (denna gången av röntgenstrålning) på olika tidsavstånd efter det att IR-pulsen har träffat provet ochdetta ger oss möjligheten att se hur provets struktur förändras som följd
av interntryckets utveckling.
Tillsammans ger dessa två metoder tillgång till ett område i det metastabila fasdiagrammet som tidigare inte kunnat nås det är där vi har identifierat en övergång mellan två vätskefaser i underkylt vatten. Vi tolkar dennaövergång som den mellan två olika sorters underkylt vatten som skiljer sig
genom att ha olika täthet. Det är alltså en högdensitetsfas och en lågdensitetsfas. Vid högre tryck är övergången faserna emellan mycket tvär
– en så kallad första ordningens övergång - vilket medför en karaktäristisk
dubbeltopp i röntgenspridningen vid det första diffraktionsmaximat. Vid
atmosfärstryck däremot ser vi ett kontinuerligt skift av det första diffraktionsmaximat vilket stämmer överens med en övergång av andra ordningen.
Vidare evidens för en kontinuerlig fasövergång vid atmosfärstryck kan
fås om man undersöker temperaturberoendet hos tre observabler hos vatten,
nämligen dess korrelationslängd, isotermala kompressabilitet och värmekapacitet. Vi har påvisat ett maximum hos dessa storheter som indikerar
existensen av en gräns mellan två superkritiska faser– en så kallad Widomlinje.
47

Sammanfattningsvis så stöder dessa experiment hypotesen att en kritisk
punk mellan två vätskefaser existerar i fasdiagrammet hos underkylt vatten.
Den kritiska punkten finns där den experimentellt observerade Widomlinjen slutar och möter den skarpa gränsen mellan de två vätskefaserna
i djupt underkylt vatten. Det huvudsakliga resultatet i denna avhandlingär dock den första experimentella observationen av en fasövergång mellan
två vätskor i ett prov bestående av en enda ren vätska.

48

Acknowledgements
Most of all I like to thank my supervisor professor Anders Nilsson for excellent supervision and guidance throughout my time as PhD student. Anders, thank you for your support, encouragement and reliance. You neverrun out of daring ideas to propose new experiments and push forward.
You always had confidence and good advice whenever difficulties occurred.
You are a brilliant researcher and supervisor but also a fabulous person. I
leave with plenty of good memories including fun travelling, successful xray beamtimes and interesting conferences but also tasty Julboards, joking
discussions, unexpected medical x-ray imaging and Swedish Fika. I promise
to take diving classes as soon as time permits. I want you to know, that I
really enjoyed my time as your PhD student. Thank you so much.
I also like to thank my co-supervisor professor Lars G. M. Pettersson.
Lars, you are a great teacher. You manage to formulate the most complicated things in precise but yet understandable and didactic ways. One of
the many examples is your quote at the beginning of this thesis. Anotherexample is your course on simulation methods in statistical physics, which
was one of the most fruitful classes that I took during my studies. You
really care about the education and progress of your students. Thank you
Lars.
A big thank-you goes to the water people of the group. All of you
contributed with tremendous expertise and exceptional team spirit, making
all these projects possible.
Harshad, you are the secret star of the crew. I had a lot of fun together
with you working in the lab, constantly looking to improve the setup andsample delivery. Your dry sense of humour, calmness and persistence areunmatched. We had countless funny moments like sledding in Switzerland,
kayaking in Holderness, shoeless beamtimes and failing beamstops. Thankyou for your help over all these years, especially when I just started andit was only the two of us working on the setup. I learned a lot from you.
Without you much of this work would not have been possible. Many thanks
Harshad.
Kyung Hwan, it was really great to have you as an office mate. You
are a cheerful guy, always helpful with a sharp mind. You are a man ofconsistency and efficiency but also a good friend. I’m still grateful andimpressed how you handled my hospitalization and the beamtime in Korea,
both at the same time. I’m looking forward to have Korean barbecue
together some time again. Kyung Hwan, thank you so much for all you
did for me.
Fivos, I learned so much from you, scientifically but also personally.
There are lots of good memories, e.g. Q-bar at APS, Hutch 8 at Stanford,
climbing in Stockholm, games, food and running with Akiko, Tollwood in49

Munich and so much more. You are always optimistic and encouraging.
Both Akiko and you became really good friends. Thank you so much for
all your time and advice. Let’s stay in contact.
Katrin, it was nice to have you in the lab, teaming up to take action toward a good lab environment and proper testing of the setup and samples (ifpossible) before an upcoming beamtime. Thank you for your patience when
I was asking you so many questions about water, ice and other things. You
and Peter always ensure a positive atmosphere within the group. Thanks
to both of you for taking care of so many, often unnoticed, but crucial details. A special thanks to you Katrin, for letting me win the chess battle inPohang. I’m sure Felix will become a great researcher one day.
Jonas, you have always been a highly appreciated help, both at the
beginning of my PhD with useful input to the experimental setup but also
at all those beamtimes we did together.
I also thank my office mates for creating such a nice working atmosphere
during all these years. David and Daniel, it was great to start together as
new PhD students and do most of the courses with you guys. Daniel, I
wish you all the best in your job. You are a nice guy and I always enjoyedbeamtimes and lab work with you. David, thanks for your help with theSwedish summary, I wish you all the best, for the rest of your PhD, andyour family. Jonatan you really pushed the social life of the group (in agood way). You are a straight-forward guy with a ’decent’ sense of humor. I always enjoyed going climbing or grabbing a beer with you. You
are a really good friend. Iurii and Chang, chatting with you was always apleasure. TJ, you are filled with skills and energy, always feeling the need
– the need for speed. I really enjoyed working and chatting with you at
beamtimes and during your long-term visit in Stockholm. I’d work withyou any time again. Marjorie, you are a mindful and cooperative person,
immediately fitting in and supporting the team. In addition, you are agreat baker, always good for a sweet surprise. Gaia, you are the female
copy of Jonatan. Thank you for the fun time we had. Daniel S., thanks foryour help at the very beginning of my PhD. Joakim, I’m looking forward tocontinue our world-class football battles at Stanford. Rich, you have been
my perfect mentor, always having time for quick discussions. Many thanks
for your useful comments on my licentiate and final PhD thesis. Jesper,
you made being a PhD student look so temptingly easy. Thanks for not
bragging about being so much better in climbing than me. Finally, thanks
to everybody in the XSoLaS group and people from the fourth floor. It wasalways fun hanging out with you.
MaPa, Dani and Tati you are the best family. I’m sorry for the longtimes we haven’t seen each other. Julia, my love, thank you for taking mymind off from work, initiating and sharing this adventure with me.
50

References[1] Lars Gunnar Moody Pettersson, Richard Humfry Henchman, and Anders Nilsson. Water—the most anomalous liquid. Chemical Reviews,
116(13):7459–7462, 2016. PMID: 27405667.
[2] Paola Gallo, Katrin Amann-Winkel, Charles Austen Angell,
Mikhail Alexeevich Anisimov,
Frédéric Caupin,
Charusita
Chakravarty, Erik Lascaris, Thomas Loerting, Athanassios Zois
Panagiotopoulos, John Russo, et al. Water: A tale of two liquids.
Chemical reviews, 116(13):7463–7500, 2016.
[3] Gerardo Andrés Cisneros, Kjartan Thor Wikfeldt, Lars Ojamäe, Jibao
Lu, Yao Xu, Hedieh Torabifard, Albert P Bartók, Gábor Csányi, Valeria Molinero, and Francesco Paesani. Modeling molecular interactions
in water: From pairwise to many-body potential energy functions.
Chemical reviews, 116(13):7501–7528, 2016.
[4] Michele Ceriotti, Wei Fang, Peter G Kusalik, Ross H McKenzie, Angelos Michaelides, Miguel A Morales, and Thomas E Markland. Nuclear
quantum effects in water and aqueous systems: Experiment, theory,and current challenges. Chemical reviews, 116(13):7529–7550, 2016.
[5] Thomas Fransson, Yoshihisa Harada, Nobuhiro Kosugi, Nicholas A
Besley, Bernd Winter, John J Rehr, Lars GM Pettersson, and Anders
Nilsson. X-ray and electron spectroscopy of water. Chemical reviews,
116(13):7551–7569, 2016.
[6] Katrin Amann-Winkel, Marie-Claire Bellissent-Funel, Livia E Bove,
Thomas Loerting, Anders Nilsson, Alessandro Paciaroni, Daniel
Schlesinger, and Lawrie Skinner. X-ray and neutron scattering of water. Chemical reviews, 116(13):7570–7589, 2016.
[7] Fivos Perakis, Luigi De Marco, Andrey Shalit, Fujie Tang, Zachary R
Kann, Thomas D Kühne, Renato Torre, Mischa Bonn, and Yuki Nagata. Vibrational spectroscopy and dynamics of water. Chemical reviews, 116(13):7590–7607, 2016.
[8] Silvina Cerveny, Francesco Mallamace, Jan Swenson, Michael Vogel,and Limei Xu. Confined water as model of supercooled water. Chemical
reviews, 116(13):7608–7625, 2016.
[9] Nico FA Van Der Vegt, Kristoffer Haldrup, Sylvie Roke, Junrong
Zheng, Mikael Lund, and Huib J Bakker. Water-mediated ion pairing: Occurrence and relevance. Chemical reviews, 116(13):7626–7641,
2016.
51

[10] Noam Agmon, Huib J Bakker, R Kramer Campen, Richard H Henchman, Peter Pohl, Sylvie Roke, Martin Thämer, and Ali Hassanali.
Protons and hydroxide ions in aqueous systems. Chemical reviews,
116(13):7642–7672, 2016.
[11] Marie-Claire Bellissent-Funel, Ali Hassanali, Martina Havenith,Richard Henchman, Peter Pohl, Fabio Sterpone, David van der Spoel,
Yao Xu, and Angel E Garcia. Water determines the structure anddynamics of proteins. Chemical reviews, 116(13):7673–7697, 2016.
[12] Olle Björneholm, Martin H Hansen, Andrew Hodgson, Li-Min Liu,David T Limmer, Angelos Michaelides, Philipp Pedevilla, Jan Rossmeisl, Huaze Shen, Gabriele Tocci, et al. Water at interfaces. Chemical
reviews, 116(13):7698–7726, 2016.
[13] Pablo G Debenedetti. Supercooled and glassy water.
Physics: Condensed Matter, 15(45):R1669, 2003.

Journal of[14] Pablo G Debenedetti and Frank H Stillinger. Supercooled liquids andthe glass transition. Nature, 410(6825):259–267, 2001.
[15] CA Angell, J Shuppert, and JC Tucker. Anomalous properties ofsupercooled water. Heat capacity, expansivity, and proton magnetic
resonance chemical shift from 0 to -38◦ C. The Journal of Physical
Chemistry, 77(26):3092–3099, 1973.
[16] RJ Speedy and CA Angell. Isothermal compressibility of supercooled
water and evidence for a thermodynamic singularity at -45◦ C. The
Journal of Chemical Physics, 65(3):851–858, 1976.
[17] Pradeep Kumar and H Eugene Stanley. Thermal conductivity minimum: A new water anomaly. The Journal of Physical Chemistry B,
115(48):14269–14273, 2011.
[18] CA Angell, WJ Sichina, and M Oguni. Heat capacity of water at
extremes of supercooling and superheating. The Journal of Physical
Chemistry, 86(6):998–1002, 1982.
[19] H Sato, K Watanabe, JMH Levelt Sengers, JS Gallagher, PG Hill,
J Straub, and W Wagner. Sixteen thousand evaluated experimental
thermodynamic property data for water and steam. Journal of physicaland chemical reference data, 20(5):1023–1044, 1991.
[20] O Conde, J Teixeira, and P Papon. Analysis of sound velocity insupercoled H2 O, D2 O, and water-ethanol mixtures. The Journal ofChemical Physics, 76(7):3747–3753, 1982.
52

[21] H Kanno and CA Angell. Water: Anomalous compressibilities to 1.9
kbar and correlation with supercooling limits. The Journal of Chemical
Physics, 70(9):4008–4016, 1979.
[22] Philip Ball. Water as an active constituent in cell biology. Chemical
reviews, 108(1):74–108, 2008.
[23] Marie-Claire Bellissent-Funel.
Hydration processes in biology:
Theoretical and experimental approaches, volume 305. IOS Press, 1999.
[24] G Wilse Robinson, Sheng-Bai Zhu, Surjit Singh, and Myron W Evans.
Water in biology, chemistry, and physics: Experimental overviews andcomputational methodologies, volume 9. World Scientific, 1996.
[25] Wilhelm Conrad Röntgen. Ueber die constitution des flüssigen wassers.
Annalen der Physik, 281(1):91–97, 1892.
[26] Pablo G Debenedetti. Metastable liquids: Concepts and principles.
Princeton University Press, 1996.
[27] A Nilsson, D Schlesinger, and LGM Pettersson. X-ray spectroscopy,
scattering and simulation studies of instantaneous structures in water.
Water: Fundamentals as the Basis for Understanding the Environment
and Promoting Technology, 187:77, 2015.
[28] Peter H Poole, Francesco Sciortino, Ulrich Essmann, and H Eugene
Stanley. Phase behaviour of metastable water. Nature, 360(6402):324–
328, 1992.
[29] Sander Woutersen, Bernd Ensing, Michiel Hilbers, Zuofeng Zhao, andC Austen Angell. A liquid-liquid transition in supercooled aqueous
solution related to the HDA-LDA transition. Science, 359(6380):1127–
1131, 2018.
[30] John S Huang, Walter I Goldburg, and Allan W Bjerkaas. Study ofphase separation in a critical binary liquid mixture: Spinodal decomposition. Physical Review Letters, 32(17):921, 1974.
[31] Kiwing To and Hyoung J Choi. Polymer conformation near the criticalpoint of a binary mixture. Physical review letters, 80(3):536, 1998.
[32] Peter H Poole, Francesco Sciortino, Ulrich Essmann, and H Eugene
Stanley. Spinodal of liquid water. Physical Review E, 48(5):3799, 1993.
[33] Peter H Poole, Ulrich Essmann, Francesco Sciortino, and H Eugene
Stanley. Phase diagram for amorphous solid water. Physical Review
E, 48(6):4605, 1993.
53

[34] Francesco Sciortino, Peter H Poole, Ulrich Essmann, and HE Stanley.
Line of compressibility maxima in the phase diagram of supercooled
water. Physical Review E, 55(1):727, 1997.
[35] Srikanth Sastry, Pablo G Debenedetti, Francesco Sciortino, and H Eugene Stanley. Singularity-free interpretation of the thermodynamics ofsupercooled water. Physical Review E, 53(6):6144, 1996.
[36] C Austen Angell. Insights into phases of liquid water from study of its
unusual glass-forming properties. Science, 319(5863):582–587, 2008.
[37] Robin J Speedy. Stability-limit conjecture. An interpretation of theproperties of water. The Journal of Physical Chemistry, 86(6):982–
991, 1982.
[38] Hajime Tanaka. Simple physical model of liquid water. The Journalof Chemical Physics, 112(2):799–809, 2000.
[39] Hajime Tanaka. Simple view of waterlike anomalies of atomic liquids
with directional bonding. Physical Review B, 66(6):064202, 2002.
[40] Hajime Tanaka. Bond orientational order in liquids: Towards a unified description of water-like anomalies, liquid-liquid transition, glasstransition, and crystallization. The European Physical Journal E,
35(10):113, 2012.
[41] DA Fuentevilla and MA Anisimov. Scaled equation of state for supercooled water near the liquid-liquid critical point. Physical review
letters, 97(19):195702, 2006.
[42] CE Bertrand and MA Anisimov. Peculiar thermodynamics of the second critical point in supercooled water. The Journal of Physical Chemistry B, 115(48):14099–14111, 2011.
[43] V Holten and MA Anisimov. Entropy-driven liquid-liquid separationin supercooled water. Scientific reports, 2(1):1–7, 2012.
[44] Osamu Mishima and H Eugene Stanley. The relationship between
liquid, supercooled and glassy water. Nature, 396(6709):329–335, 1998.
[45] HE Stanley, P Kumar, G Franzese, L Xu, Z Yan, Marco G Mazza,
SV Buldyrev, S-H Chen, and F Mallamace. Liquid polyamorphism:
Possible relation to the anomalous behaviour of water. The European
Physical Journal Special Topics, 161(1):1–17, 2008.
[46] Limei Xu, Pradeep Kumar, Sergey V Buldyrev, S-H Chen, Peter H
Poole, Francesco Sciortino, and H Eugene Stanley. Relation between
54the widom line and the dynamic crossover in systems with a liquidliquid phase transition. Proceedings of the National Academy of Sciences, 102(46):16558–16562, 2005.
[47] David T Limmer and David Chandler. The putative liquid-liquid transition is a liquid-solid transition in atomistic models of water. The
Journal of chemical physics, 135(13):134503, 2011.
[48] David T Limmer and David Chandler. The putative liquid-liquid transition is a liquid-solid transition in atomistic models of water. II. The
Journal of chemical physics, 138(21):214504, 2013.
[49] Anders Nilsson and Lars GM Pettersson. The structural origin of
anomalous properties of liquid water. Nature communications, 6(1):1–
11, 2015.
[50] Lars GM Pettersson. A two-state picture of water and the funnel oflife. In International Conference Physics of Liquid Matter: Modern
Problems, pages 3–39. Springer, 2018.
[51] Taco Walstra and Frank Kayzel. Critical point of water with the declichigh temperature insert, 2007.
[52] Jonas A Sellberg, C Huang, Trevor A McQueen, ND Loh, H Laksmono,
Daniel Schlesinger, RG Sierra, D Nordlund, CY Hampton, Dmitri Starodub, et al. Ultrafast x-ray probing of water structure below the homogeneous ice nucleation temperature. Nature, 510(7505):381–384, 2014.
[53] Press release. NobelPrize.org. Nobel Media AB 2020. Thu. 19
Mar 2020. <https://www.nobelprize.org/prizes/physics/1982/pressrelease/>.
[54] Ernst Ising. Beitrag zur theorie des ferromagnetismus. Zeitschrift für
Physik, 31(1):253–258, 1925.
[55] Dietrich Stauffer. Social applications of two-dimensional ising models.
American Journal of Physics, 76(4):470–473, 2008.
[56] Claudio A Cerdeiriña, Jacobo Troncoso, Diego González-Salgado,
Pablo G Debenedetti, and H Eugene Stanley. Water’s two-criticalpoint scenario in the ising paradigm. The Journal of chemical physics,
150(24):244509, 2019.
[57] Paola Gallo and Francesco Sciortino. Ising universality class for theliquid-liquid critical point of a one component fluid: A finite-size scaling test. Physical review letters, 109(17):177801, 2012.
55

[58] Claudio A Cerdeirina and H Eugene Stanley. Ising-like models withenergy-volume coupling. Physical review letters, 120(12):120603, 2018.
[59] Nicholas Metropolis, Arianna W Rosenbluth, Marshall N Rosenbluth,
Augusta H Teller, and Edward Teller. Equation of state calculations byfast computing machines. The journal of chemical physics, 21(6):1087–
1092, 1953.
[60] JB Clarke, JW Hastie, LHE Kihlborg, R Metselaar, and MM Thackeray. Definitions of terms relating to phase transitions of the solidstate (IUPAC Recommendations 1994). Pure and applied chemistry,
66(3):577–594, 1994.
[61] Kyohei Takae and Hajime Tanaka. Role of hydrodynamics in liquid–
liquid transition of a single-component substance. Proceedings of the
National Academy of Sciences, 117(9):4471–4479, 2020.
[62] André Guinier, Gérard Fournet, and Kenneth L Yudowitch. Smallangle scattering of x-rays. 1955.
[63] Jens Als-Nielsen and Des McMorrow.
physics. John Wiley & Sons, 2011.

Elements of modern X-ray[64] P Gallo, D Corradini, and M Rovere. Widom line and dynamical
crossovers as routes to understand supercritical water. Nature communications, 5(1):1–6, 2014.
[65] HE Stanley. Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, 1987.
[66] James J Binney, Nigel J Dowrick, Andrew J Fisher, and Mark EJ
Newman. The theory of critical phenomena: An introduction to therenormalization group. Oxford University Press, 1992.
[67] David Poland and David Simmons-Duffin. The conformal bootstrap.
Nature Physics, 12(6):535–539, 2016.
[68] Peter H Poole, Francesco Sciortino, Tor Grande, H Eugene Stanley,
and C Austen Angell. Effect of hydrogen bonds on the thermodynamic
behavior of liquid water. Physical review letters, 73(12):1632, 1994.
[69] Thomas Loerting, Katrin Winkel, Markus Seidl, Marion Bauer, Christian Mitterdorfer, Philip H Handle, Christoph G Salzmann, Erwin
Mayer, John L Finney, and Daniel T Bowron. How many amorphousices are there? Physical Chemistry Chemical Physics, 13(19):8783–
8794, 2011.
56

[70] O Mishima, LD Calvert, and E Whalley. ‘melting ice’i at 77 k
and 10 kbar: A new method of making amorphous solids. Nature,
310(5976):393–395, 1984.
[71] Fivos Perakis, Katrin Amann-Winkel, Felix Lehmkühler, Michael
Sprung, Daniel Mariedahl, Jonas A Sellberg, Harshad Pathak, Alexander Späh, Filippo Cavalca, Daniel Schlesinger, et al. Diffusive dynamics
during the high-to-low density transition in amorphous ice. Proceedingsof the National Academy of Sciences, 114(31):8193–8198, 2017.
[72] O Mishima, LD Calvert, and E Whalley. An apparently first-ordertransition between two amorphous phases of ice induced by pressure.
Nature, 314(6006):76–78, 1985.
[73] Harshad Pathak, JC Palmer, Daniel Schlesinger, Kjartan Thor Wikfeldt, Jonas A Sellberg, Lars GM Pettersson, and Anders Nilsson. The
structural validity of various thermodynamical models of supercooled
water. The Journal of chemical physics, 145(13):134507, 2016.
[74] Chris Benmore, Leighanne C Gallington, and Emmanuel Soignard.
Intermediate range order in supercooled water. Molecular Physics,
117(18):2470–2476, 2019.
[75] Congcong Huang, KT Wikfeldt, D Nordlund, U Bergmann, T McQueen, Jonas Sellberg, Lars GM Pettersson, and Anders Nilsson.
Wide-angle x-ray diffraction and molecular dynamics study of mediumrange order in ambient and hot water. Physical Chemistry Chemical
Physics, 13(44):19997–20007, 2011.
[76] Lawrie B Skinner, CJ Benmore, Joerg C Neuefeind, and John B Parise.
The structure of water around the compressibility minimum. The Journal of chemical physics, 141(21):214507, 2014.
[77] Daniel Mariedahl, Fivos Perakis, Alexander Späh, Harshad Pathak,
Kyung Hwan Kim, Gaia Camisasca, Daniel Schlesinger, Chris Benmore, Lars Gunnar Moody Pettersson, Anders Nilsson, et al. X-ray
scattering and O-O pair-distribution functions of amorphous ices. The
Journal of Physical Chemistry B, 122(30):7616–7624, 2018.
[78] RW Hendricks, PG Mardon, and LB Shaffer. X-ray zero-angle scattering cross section of water. The Journal of Chemical Physics, 61(1):319–
322, 1974.
[79] Lee-Ping Wang, Teresa Head-Gordon, Jay W Ponder, Pengyu Ren,
John D Chodera, Peter K Eastman, Todd J Martinez, and Vijay S
Pande. Systematic improvement of a classical molecular model of water. The Journal of Physical Chemistry B, 117(34):9956–9972, 2013.
57

[80] Rui Shi, John Russo, and Hajime Tanaka. Origin of the emergent
fragile-to-strong transition in supercooled water. Proceedings of the
National Academy of Sciences, 115(38):9444–9449, 2018.
[81] A Taschin, P Bartolini, R Eramo, R Righini, and R Torre. Evidenceof two distinct local structures of water from ambient to supercooled
conditions. Nature communications, 4(1):1–8, 2013.

58