Molecular structure and dynamics of liquid watersu.diva-portal.org/smash/get/diva2:854642/FULLTEXT03.pdf · Molecular structure and dynamics of liquid water Simulations complementing

Molecular structure and dynamicsof liquid waterSimulations complementing experiments

Daniel Schlesinger

[email protected]

c© Daniel Schlesinger, Stockholm 2015

ISBN: 978-91-7649-264-2

Printed by Holmbergs, Malmö 2015Department of Physics, Stockholm University

Abstract

Water is abundant on earth and in the atmosphere and the most crucial liquidfor life as we know it. It has been subject to rather intense research since morethan a century and still holds secrets about its molecular structure and dynam-ics, particularly in the supercooled state, i. e. the metastable liquid below itsmelting point.

This thesis is concerned with different aspects of water and is written froma theoretical perspective. Simulation techniques are used to study structuresand processes on the molecular level and to interpret experimental results.

The evaporation kinetics of tiny water droplets is investigated in simula-tions with focus on the cooling process associated with evaporation. The tem-perature evolution of nanometer-sized droplets evaporating in vacuum is welldescribed by the Knudsen theory of evaporation. The principle of evapora-tive cooling is used in experiments to rapidly cool water droplets to extremelylow temperatures where water transforms into a highly structured low-densityliquid in a continuous and accelerated fashion.

For water at ambient conditions, a structural standard is established in formof a high precision radial distribution function as a result of x-ray diffrac-tion experiments and simulations. Recent data even reveal intermediate rangemolecular correlations to distances of up to 17 Å in the bulk liquid.

The barium fluoride (111) crystal surface has been suggested to be a tem-plate for ice formation because its surface lattice parameter almost coincideswith that of the basal plane of hexagonal ice. Instead, water at the interfaceshows structural signatures of a high-density liquid at ambient and even at su-percooled conditions.

Inelastic neutron scattering experiments have shown a feature in the vi-brational spectra of supercooled confined and protein hydration water whichis connected to the so-called Boson peak of amorphous materials. We find asimilar feature in simulations of bulk supercooled water and its emergence isassociated with the transformation into a low-density liquid upon cooling.

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List of Papers

The following papers, referred to in the text by their Roman numerals, areincluded in this thesis.

I. Evaporative cooling of microscopic water droplets in vacuo:Molecular dynamics simulations and kinetic gas theoryD. Schlesinger, J. A. Sellberg, A. Nilsson, L. G. M. PetterssonManuscript

II. Ultrafast X-ray probing of water structure below the homogeneousice nucleation temperatureJ. A. Sellberg, C. Huang, T. A. McQueen, N. D. Loh, H. Laksmono, D.Schlesinger, R. G. Sierra, D. Nordlund, C. Y. Hampton, D. Starodub,D. P. DePonte, M. Beye, C. Chen, A. V. Martin, A. Barty, K. T. Wikfeldt,T. M. Weiss, C. Caronna, J. Feldkamp, L. B. Skinner, M. M. Seibert,M. Messerschmidt, G. J. Williams, S. Boutet, L. G. M. Pettersson, M. J.Bogan, A. NilssonNature, 510, 381–384 (2014)

III. Benchmark oxygen-oxygen pair-distribution function of ambient wa-ter from x-ray diffraction measurements with a wide Q-rangeL. B. Skinner, C. Huang, D. Schlesinger, L. G. M. Pettersson, A. Nils-son, C. J. BenmoreJ. Chem. Phys., 138, 074506 (2013)

IV. Intermediate range molecular correlations in liquid waterD. Schlesinger, K. T. Wikfeldt, L. B. Skinner, A. Nilsson, L. G. M. Pet-terssonManuscript

V. Highly Compressed Two-Dimensional Form of Water at AmbientConditionsS. Kaya, D. Schlesinger, S. Yamamoto, J. T. Newberg, H. Bluhm, H.Ogasawara, T. Kendelewicz, G. E. Brown Jr., L. G. M. Pettersson, A.NilssonSci. Rep., 3, 1074 (2013)

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https://doi.org/10.1038/nature13266

https://doi.org/10.1038/nature13266

https://doi.org/10.1063/1.4790861

https://doi.org/10.1063/1.4790861

https://doi.org/10.1038/srep01074


VI. The Boson peak in supercooled waterP. Kumar, K. T. Wikfeldt, D. Schlesinger, L. G. M. Pettersson, H. E.StanleySci. Rep., 3, 1980 (2013)

Reprints were made with permission from the publishers.

The following papers are related to but not included in this thesis and arereferred to as ordinary references.

A. Microscopic probing of the size dependence in hydrophobic solva-tionN. Huang, D. Schlesinger, D. Nordlund, C. Huang, T. Tyliszczak, T. M.Weiss, Y. Acremann, L. G. M. Pettersson, A. NilssonJ. Chem. Phys., 136, 074507 (2012)

B. A different view of structure-making and structure-breaking in al-kali halide aqueous solutions through x-ray absorption spectroscopyI. Waluyo, D. Nordlund, U. Bergmann, D. Schlesinger, L. G. M. Petters-son, A. NilssonJ. Chem. Phys., 140, 244506 (2014)

C. X-ray emission spectroscopy of bulk liquid water in “no-man’s land”J. A. Sellberg, T. A. McQueen, H. Laksmono, S. Schreck, M.Beye, D. P.DePonte, B. Kennedy, D. Nordlund, R. G. Sierra, D. Schlesinger, T.Tokushima, I. Zhovtobriukh, S. Eckert, V. H. Segtnan, H. Ogasawara,K. Kubicek, S. Techert, U. Bergmann, G. L. Dakovski, W. F. Schlotter,Y. Harada, M. J. Bogan, Ph. Wernet, A. Föhlisch, L. G. M. Pettersson,A. NilssonJ. Chem. Phys., 142, 044505 (2015)

D. X-ray spectroscopy, scattering and simulation studies of instanta-neous structures in waterA. Nilsson, D. Schlesinger, L. G. M. PetterssonIn book: Proceedings of the International School of Physics "EnricoFermi" Course 187 "Water: Fundamentals as the Basis for Understand-ing the Environment and Promoting Technology", pp.77-135Eds.: P. G. Debenedetti, M. A. Ricci, F. BruniIOS Amsterdam; SIF Bologna, 2015

E. Anomalous Behavior of the Homogeneous Ice Nucleation Rate in“No-man’s Land”

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H. Laksmono, T. A. McQueen, J. A. Sellberg, N. D. Loh, C. Huang, D.Schlesinger, R. G. Sierra, Ch. Y. Hampton, D. Nordlund, M. Beye, A. V.Martin, A. Barty, M. M. Seibert, M. Messerschmidt, G. J. Williams, S.Boutet, K. Amann-Winkel, T. Loerting, L. G. M. Pettersson, M. J. Bo-gan, A. NilssonJ. Phys. Chem. Lett., 6, 2826–2832, (2015)

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Author’s contribution

Paper I: I performed all Molecular Dynamics (MD) simulations of evap-orating droplets and the associated analysis and wrote the ma-jor part of the program to find solutions to the Hertz–Knudsentheory describing the evaporative cooling process. I was in-volved in planning the project and wrote the paper in collabo-ration with my co-authors.

Paper II: I contributed to the temperature calibration with the programdiscussed in Paper I. I provided simulation data for the SPC/Emodel and part of those for the TIP4P/2005 model and wasinvolved in the associated structural analysis. I took part indiscussions and writing the paper.

Paper III: I performed the Reverse Monte Carlo (RMC) simulations inorder to determine the O-H contribution to the total RDF. Theresults were used for the estimate of uncertainties due to thesubtraction of this contribution. I was involved in writing thepaper.

Paper IV: I re-analyzed the experimental data for intermediate-range cor-relations and compared to existing simulation data. I per-formed the structural analysis and wrote parts of the paper.

Paper V: I performed all MD simulations and their subsequent analyses.I was involved in writing the paper.

Paper VI: I performed the major part of the simulations of the TIP4P/2005model. I contributed with dispersion relations and the systemsize scaling analysis. I was involved in all discussions and inwriting the paper.

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List of Abbreviations

AFF Atomic Form Factor

BP Boson Peak

DoD Drop-on-Demand

DOS Density Of States

EPSR Empirical Potential Structure Refinement

GDVN Gas Dynamic Virtual Nozzle

H-bond Hydrogen bond

HDA High-Density Amorphous

HDL High-Density Liquid

LCLS Linac Coherent Light Source

LDA Low-Density Amorphous

LDL Low-Density Liquid

LLCP Liquid-Liquid Critical Point

LSI Local Structure Index

MAFF Modified Atomic Form Factor

MC Monte Carlo

MCT Mode Coupling Theory

MD Molecular Dynamics

ND Neutron Diffraction

PDF Pair-Distribution Function

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RDF Radial Distribution Function

RMC Reverse Monte Carlo

SAXS Small-Angle X-ray Scattering

SPC/E water model: Single Point Charge, Extended

TIP4P water model: Transferable Interaction Potential, 4-Points

TIP4P/2005 water model: Transferable Interaction Potential, 4-Points(2005 derivative)

TIP5P water model: Transferable Interaction Potential, 5-Points

VDOS Vibrational Density Of States

VSEPR Valence Shell Electron Pair Repulsion

XAS X-ray Absorption Spectroscopy

XRD X-Ray Diffraction

xii

Contents

Abstract iii

List of Papers v

Author’s contribution ix

List of Abbreviations xi

1 General introduction 11.1 On the significance of fundamental water research . . . . . . . 11.2 Theories of water . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Models of the structure of liquid water . . . . . . . . . 31.2.2 Water at supercooled conditions . . . . . . . . . . . . 51.2.3 A working hypothesis . . . . . . . . . . . . . . . . . 7

1.3 Overview over this work . . . . . . . . . . . . . . . . . . . . 9

2 Theory and simulation techniques 112.1 Theory of distribution functions . . . . . . . . . . . . . . . . 11

2.1.1 General theory of distribution functions . . . . . . . . 112.1.2 The pair-distribution function . . . . . . . . . . . . . 122.1.3 Relation to the structure factor . . . . . . . . . . . . . 13

2.2 Reverse Monte Carlo simulations . . . . . . . . . . . . . . . . 132.2.1 The Metropolis Monte Carlo method . . . . . . . . . 142.2.2 The Reverse Monte Carlo method . . . . . . . . . . . 15

2.3 Molecular dynamics simulations . . . . . . . . . . . . . . . . 172.3.1 Force fields . . . . . . . . . . . . . . . . . . . . . . . 17

3 Evaporative supercooling of water droplets 193.1 Kinetic theory of evaporative cooling of water droplets . . . . 19

3.1.1 Derivation of the Hertz–Knudsen evaporation rate . . . 203.1.2 Evaporation coefficient . . . . . . . . . . . . . . . . . 203.1.3 Cooling rate . . . . . . . . . . . . . . . . . . . . . . . 21

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3.1.4 Heat conduction within a droplet . . . . . . . . . . . . 213.1.5 Geometrical effects . . . . . . . . . . . . . . . . . . . 233.1.6 Implementation of the Knudsen theory . . . . . . . . . 24

3.2 MD simulations of TIP4P/2005 water droplets in vacuum . . . 263.2.1 Temperature dependence of relevant thermodynamic

properties of the TIP4P/2005 water model . . . . . . . 263.2.2 Results from simulations and comparison with theory . 28

3.3 Evaporative supercooling of microscopic water droplets . . . . 313.3.1 Experimental perspective . . . . . . . . . . . . . . . . 323.3.2 Droplet temperature in the experiment . . . . . . . . . 333.3.3 The structure of supercooled water . . . . . . . . . . . 35

4 Molecular correlations in liquid water 394.1 The oxygen-oxygen radial distribution function of water . . . 39

4.1.1 Analysis of scattering data . . . . . . . . . . . . . . . 394.1.2 The oxygen-oxygen radial distribution function . . . . 43

4.2 Intermediate range correlations in liquid water . . . . . . . . . 444.2.1 Temperature dependence . . . . . . . . . . . . . . . . 464.2.2 Structural analysis . . . . . . . . . . . . . . . . . . . 46

4.3 The structure of water on a Barium fluoride (111) surface . . . 514.3.1 Experimental results . . . . . . . . . . . . . . . . . . 514.3.2 MD simulations of water on a BaF2(111) surface . . . 524.3.3 Results for the molecular structure of water on BaF2(111)

from MD simulations . . . . . . . . . . . . . . . . . . 54

5 Molecular dynamics in liquid water 615.1 Dynamics in liquid water . . . . . . . . . . . . . . . . . . . . 615.2 The Boson peak . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.1 Low-frequency vibrational dynamics of bulk supercooledwater . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Summary and Outlook 67

Sammanfattning 69

Acknowledgement 71

References 73

xiv

1. General introduction

This thesis deals with water, seemingly the most natural substance in our ev-eryday life. It is certainly the most important liquid for living organisms andthe most common solvent and reactant for chemical reactions taking place onearth. Despite this fact and various scientific efforts, this substance is, in manydifferent aspects, not completely understood on a fundamental level. Waterexhibits many anomalies when compared to simple liquids, e. g. argon, due toits extraordinary ability to form hydrogen bonds [1].

1.1 On the significance of fundamental water research

The importance of water to nature and society can hardly be overestimated. In2010, the United Nations General Assembly explicitly recognized the humanright to water and sanitation [2]. Everyone has the “right to sufficient, safe, ac-ceptable, physically accessible and affordable water for personal and domesticuse” [3]. According to an estimate by the World Health Organization (WHO),however, “half of the world’s population will be living in water-stressed areasby the year 2025” [4] and the World Economic Forum names water crises oneof the most significant long-term risks worldwide in its 2015 Global Risks re-port [5]. This most concerning perspective holds a high potential for conflictsin different parts of the world and challenges technological developments.The problem can only be solved in multidisciplinary efforts involving econ-omy, politics, social and natural sciences. Physics and chemistry can, e. g.,contribute with waste water treatment and filtering or desalination of sea wa-ter. Recent research in this area helps to understand the processes taking placeon a molecular level and utilize them, such as simulations of water desalina-tion using nanoporous graphene [6] or membranes with nanotube channels [7].

On a more fundamental level, water research is important to our under-standing of atmospheric processes and climate. Important topics in meteorol-ogy are the properties of supercooled water which is abundant in the atmo-sphere, ice nucleation, evaporation and condensation dynamics (see, e. g., thereview by Koop in [8]). Water in the supercooled state has extraordinary prop-erties [9–11] that are subject to current research and will be discussed below

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in more detail.One of the most important questions in fundamental water research is that

of the role of water for life as we know it. In cells, water is a solvent forthe genetic material, proteins and salts. There are numerous active researchtopics related to the cell-biological relevance of water, e. g. ion hydration,protein hydration, and hydrophobic solvation. Interestingly, the isothermalcompressibility and the isobaric heat capacity of water exhibit minima in thephysiological temperature range and it can thus be speculated that this is not acoincidence.

1.2 Theories of water

The properties of water are strongly dominated by intermolecular hydrogenbonds (H-bonds). The ability to form H-bonds is caused by the two electronlone-pairs on the oxygen atom inside the water molecule. It can be understoodfrom valence shell electron pair repulsion (VSEPR) theory, that the two lone-pairs on the oxygen atom and the two covalent bonds to the hydrogens repeleach other giving rise to hybridization of the 2s and 2p orbitals and thus thebent shape of the molecule and the tetrahedral geometry [12; 13]. A watermolecule can thus accept one H-bond per lone-pair and donate one H-bondper hydrogen.

Generally, H-bonding has rather complex implications when consideringa dense liquid system consisting of many interacting molecules. In such anetwork-like arrangement the molecules cannot rotate freely, resulting in li-bration modes. The charge re-distribution taking place upon H-bonding alsogives rise to a cooperativity effect, i. e. the strength of an H-bond dependson the surrounding H-bond situation and is thus non-additive [14–16]. Fur-thermore, it has been shown recently that relatively strong H-bonds are furtherstrengthened by nuclear quantum effects while weaker ones are further weak-ened [17].

The structure of water in a tetrahedral, H-bonded configuration is enthalpi-cally favorable, but entropically unfavorable at the same time. The anomalousbehavior of water can be understood considering a delicate interplay betweena local enthalpy-favored tetrahedrally coordinated, and an entropy-favored,less constrained molecular arrangement. The local tetrahedral structure is lessdense than the entropy-favored structure with broken hydrogen-bonds [18].

Thus, H-bonding has fundamental consequences on the thermodynamicproperties of water, such as high melting and boiling temperatures, large en-thalpy of vaporization, strong surface tension, when compared to homologouscompounds H2S, H2Se and H2Te [19].

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1.2.1 Models of the structure of liquid water

There have been numerous hypotheses about the molecular structure of theliquid which we briefly want to review here. This brief historical review canonly fail to be complete, it is rather intended to point out different aspects of theproblem and the different attempts to resolve the most fundamental questionsabout water. More complete reviews concerning this matter can, e. g., be foundin the monograph by Eisenberg and Kauzmann [12] and the chapter of H. S.Frank in [20], an older review of early water models by Chadwell [21] anda more recent review by Ludwig [22]. The first models arose from attemptsto explain the most obvious challenges, namely the density anomaly at 4◦Cand properties of aqueous solutions; more recent models were designed toconnect experimental scattering data with molecular real-space structures andrationalize other experimental observations, such as the behavior of water inmetastable states.

Mixture models have the longest history among the molecular structure mod-els of water. H. Whiting, presenting a theory of cohesion for different liquidsand solids in 1883, was one of the first authors to note [23]:

“Conspicuous amongst all liquids in this respect stands water,which expands greatly on solidifying. In all such liquids, the con-tinual formation of solid particles, be it only for an instant, musttend to increase the volume, and the colder the liquid becomes, thegreater will be the proportion of solid particles at any instant; sothat, other things being equal, the liquid will expand by cooling.”

Whiting also stated some rough ratios of how many molecules would be “solidparticles” up to the boiling point, according to his theory. W. C. Röntgendescribed this model of liquid water in more detail in his seminal work of1892 [24]. Postulating “two kinds of differently constituted molecules”, suchthat he calls “ice-molecules” that, upon heating, reversibly interconvert into“molecules of the second kind” that occupy a smaller volume, he explainedthe temperature and pressure dependence of water observed in experiments.Némethy and Scheraga presented a more detailed mixture model of compactclusters of molecules with intact H-bonds that are in dynamic equilibrium withsurrounding molecules with gradually less intact bonds towards the edge of thecluster regions [25]. These ideas have been taken up and further refined. A re-cent theoretical effort was made by Anisimov and co-workers to establish anequation of state that quantitatively models experimental data and assumes theexistence of a second critical point at supercooled conditions [26–28]. Theysuggest a physical interpretation of the model in terms of water as a mixture

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of two states of the same molecular species that could represent different localhydrogen bond arrangements and have different local densities [27]. Russoand Tanaka describe the behavior of water in simulations with a similar two-state model, introducing an order parameter for the translational order of thesecond coordination shell [29]. A more general picture has been developedfrom interpretations of a wide spectrum of experimental results in combinationwith molecular simulations by Pettersson, Nilsson and co-workers [18; 30; 31]which is described below in more detail.

Continuum models. In their 1933 article, Bernal and Fowler presented adifferent model of water that they derived by comparing x-ray diffraction dataof the liquid with that of theoretical scattering patterns to conclude that there isa continuous transformation of the average structure from more tridymite-ice-like below 4◦C, to quartz-like at ambient, to a close packed, ideal liquid struc-ture at higher temperatures. They emphasized that the liquid is homogeneousbut the average structure would more or less resemble these three structural ar-rangements, depending on temperature [32]. Pople further developed this ideain a detailed statistical mechanics analysis and suggested that the structure ofliquid water could be tetrahedral, similar to the molecular arrangement in ice,but with the hydrogen bonds between the molecules gradually distorted to dif-ferent extents depending on temperature and pressure [33]. The H-bonds inthe liquid would bend, to a good approximation, independently in the liquid,which would also be the essential difference between the liquid and the crystal.

Other models. There have been many more structural models of water sug-gested. The hypothesis of the interstitials can be thought of as the liquid beingan open, ice-like structure with defects and a fraction of the molecules diffus-ing between the cavities. On melting of ice Ih, water molecules would breaktheir H-bonds and move into the cavities that the ice structure offers. Dan-ford and Levy used this model to fit the radial distribution function of waterobtained from x-ray diffraction to very high accuracy [34]. They also provedPauling’s water hydrate model to be quantitatively inconsistent with experi-mental data. The interstitial water model can be useful if we think of a watermolecule, its first coordination shell and a molecule penetrating into this firstshell, e. g. for the liquid under pressure.

A random network model has been put forward by Bernal in his 1964 Bak-erian lecture, where he described a molecular H-bond network of ring struc-tures resulting in a complex response to temperature and pressure changes [35].

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1.2.2 Water at supercooled conditions

Water exhibits many anomalies when compared to simple liquids, which is aconsequence of its intermolecular hydrogen-bond network as mentioned above.Prominent properties are the density and the thermodynamic response func-tions: the isothermal compressibility κT , the isobaric heat capacity Cp and theisobaric thermal expansion coefficient αP. Their temperature dependenciesare shown in fig. 1.1 and have been discussed in detail in, e. g., a review byAngell [36].

Water can be supercooled, i. e. it can be cooled far below its melting pointand exist in a metastable state on rather long time scales, depending on tem-perature and provided that there are no structural or dynamical perturbationscausing nucleation events. Upon supercooling, the thermodynamic responsefunctions, κT , Cp and αP, change dramatically and seem to diverge towardstemperatures of 228K (κT ) [9], and 226K (Cp) [10]. The thermodynamic re-sponse functions are associated with fluctuations in entropy and density [41],indicating that these fluctuations sharply increase in water upon supercool-ing. Three major theories have been developed to explain this behavior andare currently under debate: The stability-limit conjecture, the singularity-freescenario and the liquid-liquid critical point hypothesis [11; 42].

The stability-limit conjecture explains the anomalous increase of responsefunctions through a spinodal line possibly retracing back to positive pressuresin the supercooled liquid regime, thus causing a mechanical instability due todiverging density and entropy fluctuations at the spinodal [43].

The singularity-free scenario suggests the presence of a line described bythe pressure dependence of the temperature of maximum density having a neg-ative slope in the T-P phase diagram and thus giving rise to pressure dependentextrema of κT and αP [44]. The isobaric heat capacity Cp would be pressureindependent and not exhibit any singularity.

The liquid-liquid critical point (LLCP) hypothesis [45] is based on thepossible existence of a second critical point in “no-man’s land”, the temper-ature regime between the homogeneous nucleation and the spontaneous crys-tallization temperatures in the P–T plane. It furthermore hypothesizes theexistence of a coexistence line between two distinct phases of a low-densityliquid (LDL) and a high-density liquid (HDL) extending the coexistence linebetween the vitreous low- and high-density amorphous ices (LDA and HDA,respectively) [46], terminating in a critical point in the metastable region, theLLCP.

The aforementioned models by Anisimov and Pettersson and Nilsson bothaim to unify the structural models of water under ambient and the supercooledconditions and naturally connect to the LLCP hypothesis for the supercooled,

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240 260 280 300 320 340 360Temperature [K]

-1.2e-4

-8e-4

-4e-4

0

4e-4

Ơ P [1/

K]

Kell (1975)

80

90

100C P [

J/m

ol/K

]Angell (1982)Archer and Carter (2000)CRC handbook 90th ed.

4.5e-5

5e-5

5.5e-5

6e-5

g Z [1

/bar

]

Kell (1970)

c

b

a

Figure 1.1: Experimental data for the response functions of water: (a) isobaricheat capacity Cp [10; 37; 38], (b) isothermal compressibility κT [39], (c) isobaricthermal expansion coefficient αP [40]. Note the minima in CP (309K) and κT(319K) and the change of sign for αP at the temperature of maximum density(277K).

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metastable state.

1.2.3 A working hypothesis

All of the models discussed above were developed to explain certain aspects ofwater’s anomalous behavior. They seem to emphasize different, complemen-tary perspectives on water and do not necessarily exclude each other.The working hypothesis adapted in this thesis, as put forward in ref. [48], is

Stable water

Supercooled water

HDLLDL

Low densityamorphous ice

High densityamorphous

ice

LLCPWidom

line

TX

TM

TH

200

Pressure [MPa]

Tem

pera

ture

[K]

300

0

200

100

Figure 1.2: Schematic phase diagram of water with metastable regions. The firstorder transition line separating the LDA and HDA states is shown, terminating inthe hypothetical liquid-liquid critical point. The Widom line emanating from thecritical point divides the supercritical region into LDL- and HDL-like regions.The melting temperature, TM , the temperature of homogeneous ice nucleation,TH , and the temperature of spontaneous crystallization, TX , are indicated. Theregion between TH and TX left white has been termed “no man’s land” due to ex-perimental difficulties to probe it. Reproduced from ref. [47] with modifications.

based on the existence of LDL- and HDL-like instantaneous molecular en-vironments. These are pictured to be structural fluctuations due to the in-

7


terplay between enthalpy-favored LDL-structures and entropy-favored HDL-structures, where LDL-structures would be tetrahedrally H-bonded molecules,while the HDL-structures are pictured as molecules with broken or distortedH-bonds. This hypothesis is appealing not least because it seamlessly con-nects to the LLCP hypothesis that can explain the behavior of water undersupercooled conditions. On supercooling at ambient pressure, structural fluc-tuations would gradually favor local LDL-like structures [49]. In the LLCPscenario, this behavior is explained in terms of a hypothetical second criti-cal point in metastable liquid water. A P–T phase diagram is schematicallydepicted in fig. 1.2 and shows the metastable state regions. The supercooledliquid state and the high- and low-density amorphous (HDA/LDA) states canbe observed experimentally, limited by fast ice nucleation or crystallization.It has been shown that the transition between HDA and LDA is of first or-der [50; 51]. This first order transition line has been hypothesized to extendfurther up to higher temperatures and lower pressures terminating in a liquid-liquid critical point in the metastable region. The amorphous phases wouldhave metastable liquid counterparts, HDL and LDL.

It has been shown that the supercritical fluid phase beyond a liquid-gascritical point is actually divided into a liquid-like and a gas-like region as aremnant of the subcritical domain, for argon [52], and for the gas-liquid criti-cal point of water [53]. The dividing line, emanating from the critical point, hasbeen termed Widom line and marks the locus of maximum correlation lengththat decays with growing distance to the critical point, as described in ref. [53].In the supercooled region, there would be such a Widom line emanating fromthe LLCP and the thermodynamic response functions would be expected toexhibit maxima, coinciding with the maximum correlation length in the closevicinity of the critical point. The hypothetical existence of a LLCP in themetastable region could thus explain the anomalous behavior of water, even upto ambient conditions. The LLCP hypothesis for water is supported by theoret-ical investigations of water-like models [54–56] and by experimental evidenceof liquid-liquid transitions in other tetrahedral liquids, e. g. silicon [57].

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1.3 Overview over this work

1.3 Overview over this work

This thesis summarises six different projects which have been grouped intothree different chapters:

Chapter 3 deals with the evaporation of water, in particular the associatedcooling process. We studied evaporative cooling of microscopic water dropletsin vacuum using molecular dynamics simulations, Paper I. This theoretical in-vestigation has been motivated by an experimental effort to supercool waterdroplets evaporatively to below the temperature of homogeneous ice nucle-ation. The structure of water was determined at these extremely low tempera-tures for the first time in Paper II.

The projects described in chapter 4 address molecular correlations in liq-uid water. Paper III is the result of a project which aimed at the determinationof the radial distribution function (RDF) of water to high precision. A newoxygen-oxygen RDF was established as a structural standard for liquid waterat ambient conditions that can be used as a benchmark in the development andimprovement of interaction potentials for simulations. Recent x-ray scatter-ing data by Skinner et al. [58] reveal intermediate range correlations in liquidwater and were re-analyzed with focus on their temperature dependence. Theintermediate range correlations were shown to be present in simulations of theTIP4P/2005 model of water and compared to the experimental data. The re-sults of this project are summarised in Paper IV. The last section in chapter4 is devoted to the investigation of the structure of water at the interface to abarium fluoride (111) crystal surface, Paper V.

The final chapter is focussed on molecular dynamics in the terahertz re-gion. In the physics of disordered materials, an excess in the vibrational den-sity of states (VDOS) over the Debye model is known as Boson peak (BP). Inthe VDOS calculated from simulations of bulk supercooled TIP4P/2005 waterwe investigate a similar feature. The findings have been presented in Paper VI.

9


10

2. Theory and simulationtechniques

In this chapter, the theoretical background will be briefly discussed which isthe technical basis for simulations and analysis in projects described below.The first section deals with distribution functions and particularly the pair-distribution function and its relation to the experimentally observable structurefactor. We then discuss the basics of simulation techniques that have been usedin this work, Reverse Monte Carlo (RMC) and Molecular Dynamics (MD)simulations.

2.1 Theory of distribution functions

The structure of liquids, and disordered materials in general, characterizedthrough a lack of long-range order, can be quantified by the probability distri-butions for finding constituents (i. e. atoms or molecules) in a certain config-uration relative to each other. In the present section, a short general overviewover distribution function theory is given following [41; 59]. This serves asthe basis to develop the theory of the pair-distribution function (PDF). ThePDF is then associated with the structure factor, which makes it a powerfullink between the microscopic atomic or molecular structure and the observ-able scattering pattern in x-ray diffraction experiments.

2.1.1 General theory of distribution functions

Let us consider a canonical ensemble of N particles in a volume V at a temper-ature T and assume an interaction potential U(~r1 . . .~rN) between the particles.The probability density of finding n particles in a certain configuration is givenby

P(n)(~r1, . . . ,~rn)d~r1 . . .d~rn =e−βU(~r1...~rN)

ZNd~rn+1 . . .d~rN , (2.1)

where β = 1/kBT , with kB the Boltzmann factor, and the configurational par-tition function is defined as

ZN =

∫e−βU(~r1...~rN)d~r1 . . .d~rN . (2.2)

11

2. Theory and simulation techniques

If we assume that the particles are identical, we have to introduce a combina-torial factor, that takes care of all the possible permutations of the n particles.The n-particle density distribution can thus be written as

ρ(n)(~r1, . . . ,~rn) =

N!(N−n)!

1ZN

∫e−βU(~r1...~rN) d~rn+1 . . .d~rN . (2.3)

The two-point density distribution can now be expressed as

ρ(2)(~r1,~r2) = N(N−1)

1ZN

∫e−βU(~r1...~rN) d~r3 . . .d~rN . (2.4)

Consider now a decomposition of the probability distribution into a product oftwo one-particle probability distributions: one that describes the conditionalprobability of finding a particle in ~r1 under the condition that there is a particlein ~r2 and one describing the actual probability of finding a particle in ~r2

P(2)(~r1,~r2)d~r1d~r2 = P(1)(~r1 | ~r2)d~r1×P(1)(~r2)d~r2 . (2.5)

From this, it follows directly that

ρ(2)(~r1,~r2) = N(N−1)P(2)(~r1,~r2)

= (N−1)P(1)(~r1 | ~r2)×NP(1)(~r2)

= ρ(1)(~r1 | ~r2) ρ

(1)(~r2)

≡ g(2)(~r1,~r2) ρ(1)(~r1) ρ

(1)(~r2) (2.6)

where we have used the fact that for large distances | ~r1−~r2 |→ ∞, the effectof the local density at ~r2 becomes negligible in ~r1. In the last line of eq. (2.6),we have thus defined the pair-distribution function

g(~r1,~r2)≡ρ(1)(~r1 | ~r2)

ρ(1)(~r1). (2.7)

2.1.2 The pair-distribution function

For homogeneous, uniform and isotropic liquids, the pair-distribution functionin eq. (2.7) becomes a radial distribution function due to spherical symmetryand depends only on the radial distance. With the definition of the density ineq. (2.3) we get ∫

Vρ(1)(~r)d~r = N ⇒ ρ

(1) =NV≡ ρ , (2.8)

ρ being the average density in volume V . The relation between the pair-distribution function and the density pair-distribution defined in eq. 2.6 be-comes

ρ(2)(~r1,~r2) = ρ

2 g(2)(~r1,~r2) . (2.9)

12

2.2 Reverse Monte Carlo simulations

We can further define~r ≡ ~r2−~r1, integrate over all possible ~r2 and, assumingspherical symmetry, arrive at a relation between the local density and the radialdistribution function [41]

ρ(r) = ρ g(r) . (2.10)

The actual calculation of the radial distribution function defined above is usu-ally done by measuring the distances between the atoms, sorting the distancesinto discrete bins and normalizing appropriately [60]. Throughout the thesis,the term radial distribution function (RDF) is used in situations of sphericalsymmetry, while it is called a pair-distribution function (PDF) otherwise.

2.1.3 Relation to the structure factor

The x-ray scattering amplitude of a molecular structure can be calculated asthe Fourier transform of the scattering potential given through its electron den-sity. This can be shown within the first Born approximation to the Lippmann-Schwinger equation [61; 62].

Expressing the density-density correlations within the density distributionin terms of the two-point correlation function h(2)(r)≡ g(r)−1, the sphericallysymmetric Fourier transform becomes

S(q) = 1+4π ρ

∫∞

0(g(r)−1)r2 sin(qr)

qrdr , (2.11)

which is called the structure factor. It is closely related to the coherent dif-ferential x-ray scattering cross section as will be explained in section 4.1.1 inmore detail where the radial distribution function is extracted from scatteringdata. The inverse transform

g(r) = 1+1

2π ρ

∫∞

0(S(q)−1)q2 sin(qr)

qrdq . (2.12)

is employed there to calculate the radial distribution function from the scatter-ing data. In praxis, the integration in (2.12) always has a finite upper boundsince the q-values have experimental limits. This introduces significant arti-facts that are usually minimized by the application of a window function todampen oscillations stemming from the finite q-range as demonstrated in Pa-per III.


The radial distribution function introduced above can be calculated from molec-ular simulations. Here we describe the Monte Carlo and the molecular dynam-ics simulation methods. A certain variant of the Monte Carlo method, called

13


Reverse Monte Carlo, is described below and has been applied to obtain resultsincluded in Paper III.

Generally, Monte Carlo (MC) techniques are used to integrate high-dimen-sional integrals as those encountered in, e. g., statistical physics by randomsampling of the integrand. Integrating the high-dimensional equations of mo-tion is one of its applications. The random sampling of phase space is one ofthe most efficient ways for high-dimensional integrations as the calculation ofexpectation values etc.

2.2.1 The Metropolis Monte Carlo method

In the canonical ensemble, the expectation value of a thermodynamic quantityA is defined as

〈A〉=∫

A({~r})e−β E({~r})d3N~r∫e−β E({~r})d3N~r

, (2.13)

where β = 1kB T , E({~r}) denotes the energy of the configuration {~r} of N par-

ticles and the integrals run over all configurational space.In order to integrate these 3N-dimensional integrals, the configurations are

generated randomly and the more configurations are sampled, the more accu-rate the values of the integrals become. The MC algorithm generally generatesa Markov chain, i. e. a sequence of configurations, where a particular configu-ration depends on the previous configuration only through a random step.

The MC method depends on a configurational energy that is calculated foreach random step. This energy is, e. g., calculated from pair-potentials betweenthe constituents of the system, like Coulomb and Lennard-Jones interactionsbetween molecules. Through a comparison of the energy of the previous andthe new configuration it is decided whether or at what probability the new con-figuration is accepted. The Metropolis-Hastings algorithm generally iteratesthe following steps:

1) set initial configuration {~r}0,

2) calculate the energy E({~r}0) of {~r}0,

3) generate a new configuration through a random propagation {~r}i+1 ={~r}i +∆{~r},

4) calculate the energy difference between the former and the new config-uration∆Ei ≡ E({~r}i+1−E({~r}i)),

5) if the energy is lowered, i. e. ∆Ei < 0, the new configuration is accepted.Otherwise, calculate the probability p(∆Ei) for acceptance, and employ

14


a random number a ∈ [0,1] to decide whether the new configuration isaccepted, for example via the Boltzmann factor to generate a canonicalensemble

p(∆Ei) = e−β∆Ei

{< a ⇒ reject new configuration≥ a ⇒ accept new configuration ,

where steps 3–5 are iterated. There are many details which are not consideredhere, as e. g., biased sampling techniques in order to make the algorithm moreefficient.

The configurations generated by this algorithm will mirror the molecularstructures of the real liquid depending on how well its energy landscape is rep-resented by the potential function defined in the simulation rules (force field).

2.2.2 The Reverse Monte Carlo method

The Reverse Monte Carlo method has been developed by McGreevy and Pusz-tai [63] in order to solve the so-called inverse problem in scattering theory.

The Inverse Scattering Problem. Generally, the term inverse problem refersto a whole class of problems in different contexts of natural sciences and so-ciology, where information about the cause is extracted from the effect that isobserved, i. e. it is the inverse of a model predicting an observation. There canbe severe uniqueness problems, for different causes can lead to the same ef-fect. Measurements, furthermore, often project out certain degrees of freedom,thus the information obtained is not complete and cannot uniquely determinea cause – model relationship.

In a x-ray scattering experiment, the incoming photons are scattered off astructured electron density causing a certain scattering pattern on the detector.The scattering pattern is connected to the structure factor as discussed in sec-tion 4.1.1. In order to extract the real-space structure from the measured data,solutions to the inverse problem have to be approximated. The RMC methodis a powerful tool to accomplish this.

The RMC algorithm. The algorithm described in ref. [63] is used to findmolecular structures that are compatible with experimental scattering data.Here, in contrast to the usual Monte Carlo (MC) simulations, it is not thechange of configurational energy that is used in the acceptance criterion fora propagation step but rather the “distance” between the calculated and the ex-perimental quantity that is fitted to, quantified using an error function χ2. Ifan experimental structure factor Sexp(q) is given, the structure factor Scalc(q)

15


is calculated from the theoretical radial distribution function (RDF) through aFourier transform (eq. 2.11).

The theoretical structure factor Scalc(q) is then compared to the experimen-tal one, Sexp(q) through the error function

χ2i ≡ χ

2({q}i) =N∑

j=1

(Sexp(q j)−Scalc

i (q j)

σ(q j)

)2

, (2.14)

where N is the number of discrete steps in the momentum-space coordinateq and σ denotes the standard deviation of experimental data at a certain qand quantifies how tight the simulation is fitted to the data. Formally, theMetropolis-Hastings algorithm is employed, but the acceptance criterion ischanged from energy-based to a χ2-based criterion (i. e. ∆χ2

i is substitutedfor ∆Ei in the above Metropolis-Hastings scheme) in order to reproduce theexperimental data within their uncertainty. From the mathematical aspect, thisdoesn’t make a difference. It has, however, a completely different meaningwhen looked at from the physical point of view: The energy criterion requiresa potential function to be evaluated for each configuration. This potential isnot known exactly and commonly based on empirical force fields. The fit ofobservables to experimental data is a much more direct determination of thepossible structures causing the diffraction pattern and can be seen as an itera-tive fitting procedure of the model structure to the real scattering structure.

There are, however, problems concerning the uniqueness of the fits. Inthe case of water there are three different relevant correlations, i. e. oxygen-oxygen, oxygen-hydrogen and hydrogen-hydrogen, and it is thus necessary tofit to at least three different data sets with complementary information, as x-ray (XRD) and neutron diffraction (ND) data. In neutron diffraction, isotopesubstitution at different concentrations is often used to disentangle these cor-relations.

It is necessary to impose further constraints on the model structures toavoid unphysical situations, e. g. keep the molecules intact by enforcing certainbond length and angle distributions as described in the supplementary materialof [64]. Even then, it has been shown that a large variety of local structures arecompatible with experimental data [64; 65].

It should be noted that another method, Empirical Potential Structure Re-finement (EPSR), is used extensively as an alternative way of solving the in-verse problem. This method employs empirical force field simulations. Thedifference between simulated and experimental data is used to modify forcefield (or “potential”) parameters to drive the simulated structure factor to re-produce the experimental one as close as possible [66].

16

2.3 Molecular dynamics simulations

2.3 Molecular dynamics simulations

The classical molecular dynamics (MD) method is based on Newtonian me-chanics and is widely used to describe many-particle structures and dynamicson a molecular level using effective descriptions of the interactions betweenatoms. For every particle i, with position~ri and mass mi, the equation of mo-tion reads

d2~ri

dt2 =− 1mi

~∇i

N∑j 6=i

U(~ri−~r j) , (2.15)

assuming a system of N particles that interact through a pair-potential U . Theset of equations of motion for all particles are integrated to obtain a trajectory{~r}(t) of the time evolution of the system. The simulation is propagated intime through an integration scheme which is iterated at small time steps of∆t ∼ 0.1−2.0fs. Different ensembles can be generated using barostat and / orthermostat algorithms to control the pressure and the temperature, respectively.Detailed introductions to MD simulations can be found in refs. [67; 68].

2.3.1 Force fields

The results presented in this thesis were obtained mostly from classical MDsimulations with effective inter-atomic potentials from which the forces arecalculated (eq. 2.15). The functional form of the effective potential togetherwith parameters for different atom types is called the force field. The effec-tive potential typically includes terms describing the non-bonded and bondedinteractions. The non-bonded interactions, on their part, include Coulomb andLennard-Jones-type dispersion interactions. The bonded interactions describebonds, angles and dihedral potentials, often represented by polynomial expan-sions and associated force constants.

For simulations of bulk liquid water, we generically use the TIP4P/2005 model,a re-parametrization of the four-point transferable interaction potential [69]. Itis currently one of the best effective descriptions of water-water interactionsin that it represents the phase diagram qualitatively. The density maximum forthis model is in near-quantitative agreement with the experimental value, whilemelting and boiling points are shifted by about 20K towards lower tempera-ture and 30K towards higher temperatures, respectively. The model underes-timates the rise of the isothermal compressibility towards lower temperatureswhile overestimating it towards high temperatures, which has been explainedby an imbalance between structural and thermal fluctuations [18].

17


18

3. Evaporative supercooling ofwater droplets

This chapter is devoted to the fundamental process of evaporative cooling.Evaporative cooling is an efficient way of energy conversion. It is importantin atmospherical processes and widely utilized in nature by humans, animalsand plants for cooling through transpiration and in technological applicationssuch as cooling of power plants and evaporative coolers for air conditioning. Inatomic physics, evaporative cooling has been applied successfully to achieveBose-Einstein condensation at extremely low temperatures [70].

The present investigation was motivated by an experimental effort to super-cool microscopic water droplets to below the homogeneous nucleation temper-ature of water and probe the structure of the bulk, supercooled liquid. Exten-sive molecular dynamics simulations have been performed in order to validatethe Knudsen theory of evaporation that was used to calibrate the droplet tem-perature.

The theory is introduced first; MD simulations and results are discussedbelow and compared to the theoretical predictions for the TIP4P/2005 modelof water. The experiment and its results for the structure of water below thehomogeneous nucleation temperature are discussed in the end of this chapter.Parts of this chapter have been presented in the licentiate thesis Aspects ofthe local structure and dynamics of water, Daniel Schlesinger, Department ofPhysics, Stockholm University (2013).

3.1 Kinetic theory of evaporative cooling of water droplets

Kinetic theory provides the framework for the description of evaporation andthus we briefly want to review the basics generically applied in literature [71–75].

Starting with a general expression for the evaporation rate, the focus willbe on evaporative cooling of spherical droplets and we aim for a quantitativedescription of the time dependent temperature of the droplet upon evaporationin vacuum.

19

3. Evaporative supercooling of water droplets

3.1.1 Derivation of the Hertz–Knudsen evaporation rate

A mass flux is proportional to the (average) velocity and the mass density ofthe particles

Jm(T ) = v(T )ρ(T ) , (3.1)

both of which depend on the temperature T . The molecules are assumed tofollow a Maxwellian velocity distribution

f (~v) =(

m2πkBT

)3/2

e−m|~v|22kBT , (3.2)

where m is the mass and~v the velocity of the particles. The expectation valueof the velocity in a certain direction, say along the positive surface normal,evaluates to

〈v⊥〉=√

kBT2πm

. (3.3)

The gas phase mass density can be approximated using the equation of statefor an ideal gas ρ(T ) = P0(T )m/kBT , where P0 denotes the vapor pressure.Thus, substituting the terms for the average velocity (3.3) and the density intoequation (3.1), we obtain the mass flux

Jm(T ) = P0(T )√

m2πkBT

. (3.4)

Dividing the mass flux by the particle mass m and multiplying with the (gen-erally time-dependent) surface area A(t) from which the molecules evaporategives the Hertz–Knudsen evaporation rate

Γ(T ) =P0(T )√2πmkBT

A(t) . (3.5)

The evaporation rate derived here is a theoretical maximum rate under idealconditions and does not take into account possible barriers to evaporation.

3.1.2 Evaporation coefficient

The ratio between the experimentally measured evaporation rate and the the-oretical maximum rate is a material constant called evaporation coefficient α

and accounts for barriers to evaporation. For a perfect molecular description,this factor should equal unity. The literature value of the evaporation coeffi-cient for water varies over a wide range [72; 75; 76].

The strong variation of the value for α is not only a problem in exper-iments: In different molecular simulation studies different values have been

20


reported. Recently, Musolino and Trout, using the finite temperature stringmethod, found a theoretical value of α = 0.25 [77], while Varilly and Chan-dler report a value of α ≈ 1 from a transition path sampling study and find thatthe evaporation event is well described by a ballistic escape from a deep wellwithout additional barrier [78].

3.1.3 Cooling rate

The evolution of the surface temperature of a droplet can be described using theHertz–Knudsen evaporation rate (3.5). A single molecular evaporation eventwill result in a loss of energy and a fast distribution of this energy loss ontoall the accessible degrees of freedom, described by the heat equation. Thus,the time dependent-temperature for the evaporative cooling process can be ex-pressed as

dTdt

= −Γ(T )∆H(T )

MCP(T )

= − P0(T )√2πmkBT

A(t)∆H(T )

MCP(T )

= − P0(T )√2πmkBT

A(t)∆V (t)ρ(T )

∆H(T )CP(T )

, (3.6)

where the ratio of the specific enthalpy of vaporization ∆H(T ) per moleculeand the heat capacity CP(T ) ·M is the temperature change per evaporationevent in the evaporation layer of mass M. The evaporation layer has a volume∆V and its mass is thus M = ∆V ρ . Note, that the heat capacity, the enthalpy ofvaporization, the density and the vapor pressure are temperature dependent andthe surface area of a droplet changes due to loss of matter during evaporation.

3.1.4 Heat conduction within a droplet

The theory above describes the evaporation from the surface of the dropletonly. With increasing droplet size, the volume to surface area ratio increases.Thus, heat conduction within the droplet becomes increasingly important andeventually the limiting factor for cooling and has to be taken into account. Heatconduction is described by the heat equation

∂T (~r, t)∂ t

= κ~∇2 T (~r, t) , (3.7)

with the temperature field T (~r, t) at position ~r and time t, and the thermaldiffusivity κ . This can be solved under the assumption that the solution isseparable. Further, we assume spherical symmetry and apply the boundary

21


condition T (r = r0, t) = Tsurf and the initial condition T (r, t = 0) = T0. Thesolution becomes

T (r, t) = Tsurf +2 (T0−Tsurf)∞∑

n=1

(−1)n+1 j0

(nπ

rr0

)e−κ (nπ/r0)

2t , (3.8)

where j0(·) denotes the spherical Bessel function of first kind. If the heatconduction within the droplet is fast enough (or the radius is very small), wecan make the adiabatic approximation that the droplet is in equilibrium for eachevaporation event, i. e. fast heat conduction guarantees a homogeneous droplettemperature. To approximate the rate of temperature change, consider the timederivative of the solution (3.8). The slowest decay (n = 1 term) takes placeon a characteristic time scale of τ ≡ r2

0π2 κ

. The temperature change rate for theslowest decaying term (n= 1) in the center of a droplet (r = 0) is approximatedto ∣∣∣∣∣ ∂T (r = 0, t)

∂ t

∣∣∣∣t=τ,n=1

∣∣∣∣∣≈ 2π2 (T0−Tsurf)κ

r20 e

. (3.9)

In table 3.1, the temperature change rates from conduction and evaporation fordroplet sizes corresponding to those in the simulations and two different ex-perimental techniques are compared to each other. As a result of this estimate,

Droplet radius T-change rate [K/s](conduction)

T-change rate [K/s](evaporation)

r0 = 3nm 3.4 ·1013 2.0 ·109

r0 = 5 µm 1.2 ·107 1.2 ·106

r0 = 25 µm 4.9 ·105 2.4 ·105

Table 3.1: Comparison of the temperature change rates due to heat transportthrough conduction and evaporation, for different droplet sizes at 298K. Thethermal diffusivity of water κ ≈ 1.43 · 10−7 m2/s was used for heat conduction,the evaporation rate was calculated for the whole droplet as heat reservoir.

a crossover is expected for the heat transport limiting process from evapora-tion limited for small droplets to conduction limited for larger droplets. Thecrossover is approximated to occur at about 75 µm droplet radius. Thus, forsimulations as well as for small droplet realizations in the experiment, heatconduction is much faster than evaporative heat transport. For larger droplets,heat conduction is expected to play a role and take over the limitation of cool-ing with increasing droplet radius.

22


3.1.5 Geometrical effects

Surface curvature: In this project, the focus is on evaporative cooling ofsmall droplets with spherical geometry. This is an approximation, becausethe surface can fluctuate locally, in particular for small droplet sizes as thoseconsidered in the simulations.

Water has a very high surface tension due to its hydrogen bond network atthe interface. The surface tension γ causes an enhanced pressure inside smalldroplets depending on their radius r, which is described by the Young–Laplaceequation

∆P = γ2r, (3.10)

giving around 450bar enhanced pressure inside a droplet of r = 3nm and about0.3bar for r = 5 µm and is thus not expected to be important for experimentaldimensions. The variation of the term ∆H(T )

ρ(T )CP(T )due to the additional Young–

Laplace pressure inside the droplet was found to be very small and maximally2 % in the relevant temperature range and thus considered insignificant.

More importantly, the vapor pressure is enhanced due to surface tension.The actual vapor pressure P′0 can be calculated using the Kelvin equation

P′0(T ) = P0(T )e2γVmrRT , (3.11)

where Vm is the molar volume, P0 the vapor pressure at a plane interface andR the universal gas constant. It is important to note that the surface tensionis a function of the length scale and temperature. The temperature dependentsurface tension used below is a parametrization of the Guggenheim–Katayamarelation [79] and the length scale correction was performed using the Tolmanequation [80]. A lower size-limit for the applicability of the Kelvin equationwith Tolman-corrected surface tension has recently been shown to lie below1nm in MD simulations of the mW model of water [81].

The exponential correction factor in eq. (3.11) for a droplet of r = 1nm asin a typical MD simulation evaluates to about 3.1, while for droplet dimensionsof r = 5 µm it is 1.0002 at T = 298K. The correction thus plays a role on asub-micrometer length scale and becomes significant only below 10nm. Note,that the above discussion is valid for an ideal spherical droplet and is just anapproximation here. In the simulations, the surface curvature varies locallydue to fluctuations around the spherical geometry.

Decrease of the droplet radius with particle evaporation: An additionaleffect occurring during evaporative cooling of a droplet is the loss of mass andtherefore a time-dependent decrease in the radius of the droplet and thus its

23


surface area, which in turn influences the evaporation rate. The time dependentradius can be expressed as

r(t) = 3

√3mN(t)

4πρ, (3.12)

where N(t) is the total number of molecules in the droplet after interaction timet. The particle loss is just the evaporation rate (3.5) integrated over time, bywhich the total number of molecules is decreased and the number of moleculescan therefore be written as

N(t) = N0−∆N(t) = N0−∫ t

0Γ(t ′)dt ′ . (3.13)

The time dependent radius thus becomes

r(t) =(

r30−

3m4π ρ(T (t))

∫ t

0Γ(t ′)dt ′

)1/3

(3.14)

with the initial condition r(t = 0) = r0. Thus, surface area and droplet volumehave to be calculated together with the evaporation rate and heat conduction.

3.1.6 Implementation of the Knudsen theory

As described above, there are several effects that have to be taken into accountfor a quantitative description of the evaporative cooling process of a droplet.We implemented a program to calculate the time-dependent temperature of anevaporating droplet including all the previously discussed influences, brieflydescribed below.

Evaporation. The cooling rate (3.6) is calculated from the Knudsen evapo-ration rate (3.5) derived above. In the numerical calculation, the droplet withinitial radius r0 is divided into N spherical shells of width ∆r≡ r0/N and evap-oration changes the temperature in the outermost spherical shell of the droplet.

Heat conduction. A loss of heat in the outermost shell causes a temperaturegradient ∆T/∆r towards the next inner shell of the droplet. Driven by thisgradient, a heat flux is established. Thus, heat is successively re-distributedfrom the center outwards to the droplet surface. The heat flux is described byFourier’s law which, in the discrete case and for spherical symmetry, takes theform

∆Q∆t

=−4π r2κ

∆T∆r

, (3.15)

24


where ∆Q is the heat change in the time interval ∆t and κ denotes the thermalconductivity. For each shell, the difference between the heat flux into and outof the shell is calculated while the temperature change in the shell is given by

∆T =∆Q

Cp ·∆V ·ρ, (3.16)

with the specific heat capacity Cp, the volume of the shell ∆V and the densityρ . There are two special cases: For the innermost shell, which is in fact asphere, the heat flux has to be calculated at one surface, only. The outermostshell, on the other hand, loses heat only through evaporation while gaining heatfrom the next inner shell through heat conduction.

The estimation for the rate of temperature changes in table 3.1 shows that,for small droplets as in the simulations and the small experimental dropletrealization, temperature equilibration due to heat conduction within the dropletis a much faster process than temperature change via evaporation.

25


3.2 MD simulations of TIP4P/2005 water droplets in vac-uum

We performed MD simulations of droplets of TIP4P/2005 water [69] in vac-uum to investigate the temperature change caused by evaporation and com-pare to the predictions of the kinetic theory. Four different droplet sizes weresimulated, r0 = 1nm, r0 = 2nm, r0 = 3nm, and r0 = 4nm, for 100ns totalsimulation time. The following simulation protocol was applied:

1. Spherical droplet from a simulation box with bulk TIP4P/2005 water,based on distances of oxygens to an arbitrary point;

2. NVT simulation for 50ps to obtain homogeneous temperature distribu-tion throughout the droplet and relax the surface (evaporation starts);

3. Vacuum simulation (NVE) for the desired simulation time, time step∆t = 1fs using the leap frog integrator; Coulomb and Lennard-Jonesforces are calculated between all atom pairs without cutoff.

4. Control energy conservation, measure temperature within the droplet,number of molecules in the droplet for each time step or averages oversmall time intervals.

3.2.1 Temperature dependence of relevant thermodynamic propertiesof the TIP4P/2005 water model

To assess the theoretical description of evaporative cooling, we compare theresults from kinetic theory calculations to those from MD simulations. There-fore, the temperature dependencies of thermodynamic quantities of the TIP4P/-2005 model involved in the kinetic theory description were determined. Thetemperature dependence of the density, the enthalpy of vaporization and theisobaric heat capacity were obtained as fits to bulk TIP4P/2005 simulationdata while the saturation vapor pressure, the surface tension and the thermalconductivity were taken from the literature as indicated. The surface area wasmeasured directly along the simulation trajectories. Vega et al. [82] interpolatethe saturation vapor pressure for TIP4P/2005 and obtain (fig. 3.1a)

Ps(T ) = e12.4612−4476.552/(T−41.4984) bar . (3.17)

For the density of TIP4P/2005 we obtain (fig. 3.1b)

ρ(T )≈ 199154.0 − 4798.08T +49.2902T 2−0.278943T 3

+ 0.000940308T 4−1.88965 ·10−6 T 5 (3.18)

+ 2.09734 ·10−9 T 6−9.92214 ·10−13 T 7 kg/m3

26

3.2 MD simulations of TIP4P/2005 water droplets in vacuum

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7HPSHUDWXUH�>.@

7HPSHUDWXUH�>.@7HPSHUDWXUH�>.@

Figure 3.1: Temperature dependencies of properties of the TIP4P/2005 watermodel: (a) saturation vapor pressure; (b) density; (c) isobaric heat capacity; (d)enthalpy of vaporization. The properties (b), (c) and (d) were obtained from ownsimulations; (a) was taken from ref. [82].

The isobaric heat capacity is obtained as (fig. 3.1c)

Cp(T )≈ − 450457.85+11650.27T

− 130.82T 2 +0.83T 3 (3.19)

− 0.0033T 4 +8.27 ·10−6 T 5

− 1.296 ·10−8 T 6 +1.15 ·10−11 T 7

− 4.43 ·10−15 T 8 J/mol/K .

For the enthalpy of vaporization, the fit to simulation results provides (fig. 3.1d)

∆Hvap(T )≈ − 187.44+4.70T −0.034769T 2

+ 0.00012385T 3−2.1674 ·10−7 T 4 (3.20)

+ 1.4959 ·10−10 T 5 kJ/mol

The surface tension for the TIP4P/2005 water model was parametrized byVega et al. [79] as a modified Guggenheim-Katayama relation

γ(T ) = 0.22786(

1− T641.4

)11/9(1−0.6413

(1− T

641.4

))N/m .

(3.21)

27


The surface area of the simulated droplets was found to vary due to changesof the droplet shape. The evaporation rate is directly proportional to the time-dependent surface area and is hence an important ingredient in the kinetic the-ory. Therefore, it was measured along the simulation trajectories through nu-merical integration of the instantaneous liquid interface [83]. Finally, the samethermal conductivity was used as for real water, since data for the TIP4P/2005model are not available in the supercooled regime yet while it is known athigher temperatures of 300K and higher and was shown to overestimate thethermal conductivity under ambient conditions [84]. Note, however, that heattransport is very fast for the length scales considered and for the simulations,the temperature decrease is evaporation-limited as expected from the earlierestimation in table 3.1.

3.2.2 Results from simulations and comparison with theory

From the MD simulations performed by applying the protocol described above,we obtain the droplet temperature depending on the interaction time with vac-uum, see fig. 3.2.

0 20 40 60 80 100

Interaction time [ns]

240

260

280

300

320

340

360

380

Dro

plet

tem

pera

ture

[K]

simulation r0 = 1nmsimulation r0 = 2nmsimulation r0 = 3nmsimulation r0 = 4nmkinetic theory r0 = 1nmkinetic theory r0 = 2nmkinetic theory r0 = 3nmkinetic theory r0 = 4nm

Figure 3.2: Results for the evaporative cooling of TIP4P/2005 water from non-equilibrium MD simulations (see protocol in the text). Comparison of the re-sults for different droplet sizes as indicated, initial temperature 380K. Initially,there were 141 (r = 1nm), 1120 (r = 2nm), 3767 (r = 3nm), 8985 (r = 4nm)molecules in the droplets.

28

3.2 MD simulations of TIP4P/2005 water droplets in vacuum

The size dependence of the temperature evolution was expected since thesurface area-to-volume ratio of the droplets decreases with increasing dropletradius. While the evaporation rate per unit area does not change, the heat isredistributed onto a larger volume. Furthermore, the temperature fluctuationsdecrease with increasing number of molecules in the droplet.

The MD simulation results are compared to kinetic theory calculations thataccount for evaporation, heat conduction, mass loss and the modification of thevapor pressure due to surface tension. For the smallest droplet (r0 = 1nm), in-

0 20 40 60 80 100Interaction time [ns]

84

86

88

90

92

94

Num

ber of molecules in the droplet

Dro

plet

tem

pera

ture

[K]

220

230

240

250

260

270

280

290

Figure 3.3: For the smallest droplet size of initially 1nm radius, individualevaporation events can be identified to cause temperature jumps.

dividual evaporation events can be identified to cause temperature jumps whilethe statistics is poor due to the small number of molecules (141 initially), cf.fig. 3.3.

Furthermore, in the derivation of the evaporation rate a Maxwellian veloc-ity distribution has been assumed. The velocity distribution calculated fromsimulations inside a droplet is Maxwellian indicating that the droplet is inthermal equilibrium. For the droplet of r0 = 4nm, the Maxwellian distribu-tion averaged over the time window of 99.5− 100ns corresponds to an aver-age temperature of ∼ 278K, see fig. 3.4. The velocity distribution of alreadyevaporated water molecules accumulated in the gas phase from the beginningof evaporation process gradually shifts to lower temperatures and thus extendsover a wider velocity range.

29


0 0.5 1 1.5|v| [nm / ps]

0

0.5

1

1.5Re

lativ

e fre

quen

cydroplet moleculesvapor moleculesMaxwell-Boltzmann distribution, T = 278 K

Figure 3.4: Velocity distribution of a 4nm droplet and the gas phase moleculesaveraged over the time window of 99.5−100ns

It is concluded that the theoretical results calculated using the kinetic the-ory of evaporation and those from direct atomistic simulations are in very goodagreement.

30

3.3 Evaporative supercooling of microscopic water droplets


This work was part of an experimental effort to shed light onto the structureof bulk water at extreme supercooled conditions. As discussed in the intro-duction, it is well known that the thermodynamic response functions of watersuch as isothermal compressibility κT and the isobaric heat capacity CP changedramatically upon supercooling and seem to diverge at a temperature of about228K and 226K, respectively, well below the homogeneous nucleation tem-perature of about 232− 233K [85; 86]. It has been under debate whether theresponse functions really diverge or rather exhibit a maximum, the latter beingconsistent with maximum fluctuations at the Widom line in the LLCP hypoth-esis.

There is an ongoing debate about the theoretical explanation of the en-hanced fluctuations and different models were developed to explain the anoma-lous behavior of water, the most prominent being the stability limit conjecture,the singularity-free scenario and the liquid-liquid critical point hypothesis asdescribed in the introduction. The liquid-liquid critical point scenario and thesingularity-free scenario are both consistent with the experimental observa-tions presented in Paper II.

In this project, the first experimental results were presented for the struc-ture of bulk water in the so-called “no-man’s land”, the temperature regionbelow the homogeneous nucleation temperature and above the temperature ofspontaneous crystallization of about 150K (see, e. g. Fuentes-Landete et al.in [8]) at ambient pressure. In this temperature range, ice nucleation takesplace on short time scales that are too fast for conventional techniques. Sofar, measurements on deeply supercooled water have therefore been performedin nano-confined water [87; 88], where ice nucleation is strongly suppressed.This practice has been criticized due to the confinement potentially altering thethermodynamic properties of water [89]. The experimental results presentedin Paper II have been obtained for bulk water in micrometer-sized droplets toavoid confinement effects.

In this section, we focus on the temperature calibration of evaporativelycooled water droplets. We calculated the time evolution of the droplet tem-perature based on the kinetic theory model describe above. The temperatureevolution for experimental dimensions and time scales was described using thesame procedure, while the properties of the TIP4P/2005 water model were re-placed by those of real water. Results on the structure of supercooled waterand its analysis are discussed at the end of this section.

31


3.3.1 Experimental perspective

The experimental investigation has been carried out in the supercooled tem-perature regime, with the objective to measure the bulk water structure factorat extremely supercooled temperatures in “no-man’s land”. The focus was onthree main aspects:

• deeply supercooled droplet temperatures

• non-confined water sample

• coherent and intense x-ray pulses to obtain the diffraction pattern fromsingle droplets

• ultrafast measurement of the structure before crystallization

The requirements on the experiment were met by the use of droplet injectioninto a vacuum chamber, where the droplets were probed by x-ray laser pulses.The droplet injection was realized in two different ways: the Drop-on-Demand(DoD) method and the Gas Dynamic Virtual Nozzle (GDVN) setup [90], re-sulting in two different droplet diameters of 34 and 37 µm, and 9 and 12 µm,respectively.

The experiment was carried out at the Linac Coherent Light Source (LCLS)/ Stanford Linear Accelerator Center (SLAC), a x-ray free electron laser (XFEL).In XFELs, electrons are accelerated to high energies of several GeV in a lin-ear accelerator (LINAC) and injected into a so-called undulator, the heart of aXFEL. The undulator consists of an array of many magnets poled in alternat-ing direction, perpendicular to the electron beam. Electrons in bunches travel-ing through this setup are forced onto an oscillatory trajectory by the Lorentzforce and therefore emit energy in form of electromagnetic radiation. At highintensities of the electromagnetic field, the electrons can interact with the fieldoriginating from their own motion and become organized into sub-structures ofthe bunch, called micro-bunches. These micro-bunches of electrons emit co-herently, i. e. in a collective fashion, and give rise to extremely high intensityand the laser properties of the emitted x-rays [91].

Fig. 3.5a shows the essential part of the experimental setup. The dropletsfall into vacuum and cool down rapidly through evaporation. Thus, the droplettemperature decreases with increasing distance to the injection point. X-raylaser pulses probe the freely falling droplet at the interaction point and cause adiffraction pattern of individual droplets on the detector.

The diffraction patterns, called “shots”, were classified into two differentkinds: Those that showed the characteristic water ring (“water shots”) and suchthat contained Bragg peaks indicating ice formation (“ice shots”), see fig. 3.5b

32


LETTERdoi:10.1038/nature13266

Ultrafast X-ray probing of water structure below thehomogeneous ice nucleation temperatureJ. A. Sellberg1,2, C. Huang3, T. A. McQueen1,4, N. D. Loh5, H. Laksmono5, D. Schlesinger2, R. G. Sierra5, D. Nordlund3,C. Y. Hampton5, D. Starodub5, D. P. DePonte6,7, M. Beye1,8, C. Chen1,4, A. V. Martin6, A. Barty6, K. T. Wikfeldt2, T. M. Weiss3,C. Caronna7, J. Feldkamp7, L. B. Skinner9, M. M. Seibert7, M. Messerschmidt7, G. J. Williams7, S. Boutet7, L. G. M. Pettersson2,M. J. Bogan5 & A. Nilsson1,2,3

Water has a number of anomalous physical properties, and some ofthese become drastically enhanced on supercooling below the freez-ing point. Particular interest has focused on thermodynamic responsefunctions that can be described using a normal component andan anomalous component that seems to diverge at about 228 kelvin(refs 1–3). This has prompted debate about conflicting theories4–12

that aim to explain many of the anomalous thermodynamic prop-erties of water. One popular theory attributes the divergence to aphase transition between two forms of liquid water occurring in the‘no man’s land’ that lies below the homogeneous ice nucleation temper-ature (TH) at approximately 232 kelvin13 and above about 160 kelvin14,and where rapid ice crystallization has prevented any measurementsof the bulk liquid phase. In fact, the reliable determination of thestructure of liquid water typically requires temperatures above about250 kelvin2,15. Water crystallization has been inhibited by usingnanoconfinement16, nanodroplets17 and association with biomolecules16

to give liquid samples at temperatures below TH, but such measure-ments rely on nanoscopic volumes of water where the interactionwith the confining surfaces makes the relevance to bulk water unclear18.Here we demonstrate that femtosecond X-ray laser pulses can be usedto probe the structure of liquid water in micrometre-sized dropletsthat have been evaporatively cooled19–21 below TH. We find experi-mental evidence for the existence of metastable bulk liquid water downto temperatures of 227z2

{1 kelvin in the previously largely unexploredno man’s land. We observe a continuous and accelerating increasein structural ordering on supercooling to approximately 229 kelvin,where the number of droplets containing ice crystals increases rapidly.But a few droplets remain liquid for about a millisecond even at thistemperature. The hope now is that these observations and our detailedstructural data will help identify those theories that best describeand explain the behaviour of water.

Figure 1a sketches our experimental set-up: in vacuum, a liquid jet19

generates spatially unconfined droplets of supercooled liquid water, thestructure of which is then studied using intense, 50-fs, hard-X-ray laserpulses from the Linac Coherent Light Source (LCLS). A Rayleigh jet20

produces a continuous, single-file train of water droplets with a uniformdiameter of 34 or 37mm, and a gas dynamic virtual nozzle21 gives trainsof smaller droplets with a diameter of 9 or 12 mm. The droplets coolrapidly through evaporation and reach an average temperature thatdepends primarily on droplet size and travel time through the vacuum(Supplementary Information, section B.3), which we varied systema-tically by adjusting the distance between the dispenser nozzle and theX-ray pulse interaction region. Scattering patterns (at least 1,800 perdata point; Supplementary Information, section A.1.3) were recordedfrom individual droplets with a single X-ray pulse over the temperature

ranges of 227–252 K and 233–258 K for the droplets generated by gasdynamic virtual nozzle and Rayleigh jet, respectively. Temperature cali-brations were performed using the Knudsen theory of evaporative cool-ing with a typical absolute uncertainty of 62 K at the lowest temperatures(Supplementary Information, section B.3.5; Supplementary Tables 5 and 6;and Supplementary Fig. 20); the Knudsen theory was verified throughextensive molecular dynamics simulations of droplet cooling (Supplemen-tary Information, section B.2).

Figure 1b shows a typical diffraction pattern of a liquid water dropletwith a maximum momentum transfer of q < 3.5 A21 at the corners ofthe detector. The isotropic water diffraction rings uniquely identify the

1SUNCAT Center for Interface Science and Catalysis, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA. 2Department of Physics, AlbaNova University Center,Stockholm University, S-106 91 Stockholm, Sweden. 3Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, PO Box 20450, Stanford, California 94309, USA. 4Department ofChemistry, Stanford University, Stanford, California 94305, USA. 5PULSE Institute, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA. 6Center for Free-ElectronLaser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany. 7Linac Coherent Light Source, SLAC National Accelerator Laboratory, PO Box 20450, Stanford, California 94309, USA. 8Institute forMethods and Instrumentation in Synchrotron Radiation Research, Helmholtz-Zentrum Berlin fur Materialien und Energie GmbH, Wilhelm-Conrad-Rontgen Campus, Albert-Einstein-Strasse 15, 12489Berlin, Germany. 9Mineral Physics Institute, Stony Brook University, Stony Brook, New York, New York 11794-2100, USA.

Droplet dispenser

LCLS X-ray p

ulses

Interactionpoint

Supercooledliquid droplets

Detector(CSPAD)

a

b c

Figure 1 | Coherent X-ray scattering from individual micrometre-sizeddroplets with a single-shot selection scheme. a, A train of droplets(Supplementary Information, section A.1.1) flows in vacuum perpendicular to,50-fs-long X-ray pulses. A coherent scattering pattern from a water dropletwas recorded when a single droplet was positioned in the interaction regionat the time of arrival of a single X-ray pulse. CSPAD stands for, Cornell-SLACpixel array detector. b, c, Each diffraction pattern is classified (SupplementaryInformation, section A.1.3) either as a water shot exclusively containingpure liquid scattering characterized by a diffuse water ring (b), or as an iceshot characterized by intense and discrete Bragg peaks superposed on thewater scattering ring (c).

1 9 J U N E 2 0 1 4 | V O L 5 1 0 | N A T U R E | 3 8 1

Macmillan Publishers Limited. All rights reserved©2014

Figure 3.5: (a) Schematic picture of the experimental setup with the dropletsfalling into vacuum, the x-ray free electron laser probing the structure mani-fested in the scattering pattern on the detector; (b) typical water shot diffractionpattern; (c) typical ice shot, containing Bragg peaks. Reprinted from Paper IIwith permission.

and 3.5c. They were sorted and the water diffraction pattern was analyzedfurther to shed light on the structure of the supercooled water droplets.

3.3.2 Droplet temperature in the experiment

The evaporation kinetics depend on a number of thermodynamic properties ofthe evaporating material under consideration, as can be seen from the deriva-tion of the evaporation rate, eq. (3.5). A major challenge consisted in obtain-ing these properties for water in the supercooled region. Thus far, experimentshave provided data above or down to the homogeneous nucleation temperature.Therefore, for temperatures below the homogeneous nucleation temperature,extrapolations based on different functional forms were tested, suggested inliterature and supported by simulations, as described in the supplementary in-formation of Paper II. The droplet temperature evolution in the experiment wascalculated using the same implementation of the Knudsen theory as has beendescribed in section 3.1.6 and using the properties for water. An additionaltemperature calibration was provided by the fact that the temperature had been

33


liquid. As droplet temperature decreases, broad and bright Bragg peaksfrom hexagonal ice (Supplementary Information, section A.2.1) are some-times superposed on the water scattering ring (Fig. 1c). We denote as‘water shots’ diffraction patterns where we detected only water scat-tering, and as ‘ice shots’ diffraction patterns containing Bragg peaks.Ice could be detected if the illuminated volume of a droplet contained.0.05% of ice by mass (Supplementary Information, section A.2.2). Forthe 12-mm-diameter droplets, the number of ice shots increased dras-tically as the travel time exceeded ,4 ms (Fig. 2). The probing of indi-vidual water droplets with single, ultrashort X-ray pulses allowed us toseparate water shots from ice shots at each temperature, to give the frac-tion of droplets containing ice. Even among the droplets that interactedwith X-ray pulses approximately 5 ms after exiting the nozzle, with thecoldest liquid temperature of 227z2

{1 K, we found more than 100 watershots that contained water diffraction rings and no ice diffraction peaksout of 3,600 total hits; this is below the onset temperature of ice nuc-leation (Fig. 2) and inside water’s no man’s land below TH < 232 K atambient pressure13.

Structural properties of supercooled water in no-man’s land can beextracted from the X-ray scattering data. The total scattering factor, S(q),for each droplet temperature was obtained by averaging the scatteringpatterns from the respective water shots and removing the independentatomic scattering (Supplementary Information, section A.3.1). The result-ing sequence of temperature-dependent S(q) profiles for liquid water isshown in Fig. 3a, illustrating how the principal maximum of S(q) is splitinto a peak S1, located at q1 < 2 A21, and a peak S2, located at q2 < 3 A21.The amplitudes of, and separation between, S1 and S2 increase withdecreasing droplet temperature, indicative of an increase in structuralordering as water is supercooled ever more deeply towards and intono-man’s land. Figure 3b illustrates the shift in the positions of the twopeaks with temperature, as obtained from water droplets using the LCLSX-ray laser, and also based on synchrotron radiation data from metastable

liquid water at only slightly supercooled temperatures and from stableliquid water at ambient temperatures; for the latter two data sets, thetemperature was measured directly. The continuous changes in the S1and S2 peak positions, without apparent break when moving betweenthe independent X-ray laser and synchrotron radiation data sets, stronglysupports the temperature calibration.

At the lowest temperature accessible in the current study, the S1 andS2 peak positions approach the corresponding peak positions for tetra-hedrally coordinated structures (Fig. 3b), represented by low-densityamorphous (LDA) ice as obtained by neutron scattering22 and modelledfor clusters of hexagonal ice using kinematic scattering (SupplementaryInformation, section A.2.3). The rapidly increasing fraction of ice shotson the millisecond timescale below ,229 K shows that glassy water,which would be associated with slow dynamics, is not formed at thesetemperatures (Fig 2). We note that the continuous change in the posi-tions of the diffraction peaks S1 and S2 with temperature resembles thetrend of similar diffraction peaks when high-density amorphous icetransforms into LDA ice23.

The separation between the S1 and S2 peaks is very sensitive to theamount and character of tetrahedrally coordinated configurations favouredby water’s directional hydrogen bonds. Water exhibits a peak in the oxygen–oxygen pair-correlation function, gOO(r), at 4.5 A, corresponding to thesecond-nearest-neighbour distance in tetrahedral coordination,

ffiffiffiffiffiffiffi8=3

pa,

where a < 2.8 A is the nearest-neighbour oxygen–oxygen distance inliquid water24. The inset of Fig. 4a shows that increasing temperaturereduces the amplitude of the second gOO(r) peak (henceforth denotedg2) here obtained from molecular dynamics simulations using the TIP4P/2005 water model (Methods); the same trend has also been observedexperimentally for an increase in pressure25 or temperature26. We findthat g2 can be exploited as a good structural parameter to describe the

1 2 3 4 5

223

228

233

238

243

248

253

258

263

268

273

Travel time (ms)

Tem

pera

ture

(K)

10.35 20.69 31.04 41.38 51.73

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sample–nozzle distance (mm)

Ice shot fraction

Figure 2 | Time dependence of water crystallization during evaporativecooling. Ice shot fraction (green) and estimated temperature (blue) asfunctions of travel time in vacuum for droplets of diameter 12mm and speed10.35 m s21. From the ice shot fraction, shown as mean 6 s.d. of two to sevenindividual recordings, we find the onset of ice nucleation to lie between 232z2

{0:1and 229z2

{1 K. The dashed blue lines represent maximum and minimumtemperatures from the Knudsen model, which consistently overlap withexperimental data sets from SSRL measured at known absolute temperatures(Supplementary Information, sections A.3.2 and B.3.5).

LCLS, 9 μmLCLS, 12 μm LCLS, 34–37 μm SSRL, staticLDA iceIce Ih cluster

1.5 2 2.5 3−2

−1

0

1

2

3

4

5

6

3

q (Å–1)

q (Å

–1)

S(q

)

T (K)

S2

S1

a

323 K

227 K

S1

S2

b

220 240 260 280 300 320

1.8

2

2.2

2.4

2.6

2.8

Figure 3 | Temperature dependence of water scattering peaks. a, Scatteringstructure factor, S(q), obtained from single-shot diffraction patterns(Supplementary Information, section A.3.1). Water temperature decreasesfrom bottom to top (SSRL: 323, 298, 273, 268, 263, 258, 253, 251 K; LCLS: 251,247, 243, 239, 232, 229, 227 K). The data reveal a split of the principal S(q)maximum into two well-separated peaks, S1 and S2 (dashed lines).b, Temperature dependence of the S1 and S2 peak positions, calculated fromthe maxima of local fifth-order polynomial least-squares fits with errorbars estimated by shifting the derivatives of the polynomial fits by 60.05 A(LCLS) and 60.15 A (SSRL) (Supplementary Information, section A.3.1).Green triangles are LCLS data from 12-mm-diameter droplets; red circles areLCLS data from 34- and 37-mm-diameter droplets; and black squares areSSRL data from a static liquid sample. Purple diamonds are LCLS data from9-mm-diameter droplets measured at a separate LCLS run with separateq-calibration (Supplementary Information, section A.1.2). As the temperaturedecreases in no man’s land, the positions of peaks S1 and S2 approach thecharacteristic values of LDA ice (dash–dot blue lines) as determined fromneutron diffraction22 and clusters of hexagonal ice (ice Ih; dashed red lines;Supplementary Information, section A.2.3).

RESEARCH LETTER

3 8 2 | N A T U R E | V O L 5 1 0 | 1 9 J U N E 2 0 1 4


Figure 3.6: The droplet temperature in experiment with estimated uncertaintyand the ice shot fraction depending on the travel time in vacuum. Reprinted fromPaper II with permission.

measured directly in experiments on static water samples at the Stanford Syn-chrotron Radiation Laboratory (SSRL) down to a temperature of 251K. Thetemperature dependence of the structure factor of the water droplets at mod-erately supercooled conditions have to overlap with and continue the trend ofthe SSRL data, see fig. 3.7. Reprinted from Paper II with permission.

A rapidly increasing number of ice shots was observed with decreasingtemperature. The ice shot fraction is shown in fig. 3.6 together with the tem-perature evolution. The data points taken at the lowest temperatures containedmostly ice shots, while a small fraction of the droplets was still found to be liq-uid at a temperature of 227+2

−1 K, well below the temperature of homogeneousice nucleation TH ≈ 232K.

34


3.3.3 The structure of supercooled water

The structure factor was extracted from the water shot data. Since the accessi-ble q-range was too small (0.5 Å−1 ≤ q ≤ 3.2 Å−1), the structure factor couldnot be Fourier transformed directly to obtain a RDF. To draw conclusions aboutthe real-space structure, MD simulations were employed.

liquid. As droplet temperature decreases, broad and bright Bragg peaksfrom hexagonal ice (Supplementary Information, section A.2.1) are some-times superposed on the water scattering ring (Fig. 1c). We denote as‘water shots’ diffraction patterns where we detected only water scat-tering, and as ‘ice shots’ diffraction patterns containing Bragg peaks.Ice could be detected if the illuminated volume of a droplet contained.0.05% of ice by mass (Supplementary Information, section A.2.2). Forthe 12-mm-diameter droplets, the number of ice shots increased dras-tically as the travel time exceeded ,4 ms (Fig. 2). The probing of indi-vidual water droplets with single, ultrashort X-ray pulses allowed us toseparate water shots from ice shots at each temperature, to give the frac-tion of droplets containing ice. Even among the droplets that interactedwith X-ray pulses approximately 5 ms after exiting the nozzle, with thecoldest liquid temperature of 227z2

{1 K, we found more than 100 watershots that contained water diffraction rings and no ice diffraction peaksout of 3,600 total hits; this is below the onset temperature of ice nuc-leation (Fig. 2) and inside water’s no man’s land below TH < 232 K atambient pressure13.

Structural properties of supercooled water in no-man’s land can beextracted from the X-ray scattering data. The total scattering factor, S(q),for each droplet temperature was obtained by averaging the scatteringpatterns from the respective water shots and removing the independentatomic scattering (Supplementary Information, section A.3.1). The result-ing sequence of temperature-dependent S(q) profiles for liquid water isshown in Fig. 3a, illustrating how the principal maximum of S(q) is splitinto a peak S1, located at q1 < 2 A21, and a peak S2, located at q2 < 3 A21.The amplitudes of, and separation between, S1 and S2 increase withdecreasing droplet temperature, indicative of an increase in structuralordering as water is supercooled ever more deeply towards and intono-man’s land. Figure 3b illustrates the shift in the positions of the twopeaks with temperature, as obtained from water droplets using the LCLSX-ray laser, and also based on synchrotron radiation data from metastable

liquid water at only slightly supercooled temperatures and from stableliquid water at ambient temperatures; for the latter two data sets, thetemperature was measured directly. The continuous changes in the S1and S2 peak positions, without apparent break when moving betweenthe independent X-ray laser and synchrotron radiation data sets, stronglysupports the temperature calibration.

At the lowest temperature accessible in the current study, the S1 andS2 peak positions approach the corresponding peak positions for tetra-hedrally coordinated structures (Fig. 3b), represented by low-densityamorphous (LDA) ice as obtained by neutron scattering22 and modelledfor clusters of hexagonal ice using kinematic scattering (SupplementaryInformation, section A.2.3). The rapidly increasing fraction of ice shotson the millisecond timescale below ,229 K shows that glassy water,which would be associated with slow dynamics, is not formed at thesetemperatures (Fig 2). We note that the continuous change in the posi-tions of the diffraction peaks S1 and S2 with temperature resembles thetrend of similar diffraction peaks when high-density amorphous icetransforms into LDA ice23.

The separation between the S1 and S2 peaks is very sensitive to theamount and character of tetrahedrally coordinated configurations favouredby water’s directional hydrogen bonds. Water exhibits a peak in the oxygen–oxygen pair-correlation function, gOO(r), at 4.5 A, corresponding to thesecond-nearest-neighbour distance in tetrahedral coordination,

ffiffiffiffiffiffiffi8=3

pa,

where a < 2.8 A is the nearest-neighbour oxygen–oxygen distance inliquid water24. The inset of Fig. 4a shows that increasing temperaturereduces the amplitude of the second gOO(r) peak (henceforth denotedg2) here obtained from molecular dynamics simulations using the TIP4P/2005 water model (Methods); the same trend has also been observedexperimentally for an increase in pressure25 or temperature26. We findthat g2 can be exploited as a good structural parameter to describe the

1 2 3 4 5

223

228

233

238

243

248

253

258

263

268

273

Travel time (ms)

Tem

pera

ture

(K)

10.35 20.69 31.04 41.38 51.73

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sample–nozzle distance (mm)

Ice shot fraction

Figure 2 | Time dependence of water crystallization during evaporativecooling. Ice shot fraction (green) and estimated temperature (blue) asfunctions of travel time in vacuum for droplets of diameter 12mm and speed10.35 m s21. From the ice shot fraction, shown as mean 6 s.d. of two to sevenindividual recordings, we find the onset of ice nucleation to lie between 232z2

{0:1and 229z2

{1 K. The dashed blue lines represent maximum and minimumtemperatures from the Knudsen model, which consistently overlap withexperimental data sets from SSRL measured at known absolute temperatures(Supplementary Information, sections A.3.2 and B.3.5).

LCLS, 9 μmLCLS, 12 μm LCLS, 34–37 μm SSRL, staticLDA iceIce Ih cluster

1.5 2 2.5 3−2

−1

0

1

2

3

4

5

6

3

q (Å–1)

q (Å

–1)

S(q

)

T (K)

S2

S1

a

323 K

227 K

S1

S2

b

220 240 260 280 300 320

1.8

2

2.2

2.4

2.6

2.8

Figure 3 | Temperature dependence of water scattering peaks. a, Scatteringstructure factor, S(q), obtained from single-shot diffraction patterns(Supplementary Information, section A.3.1). Water temperature decreasesfrom bottom to top (SSRL: 323, 298, 273, 268, 263, 258, 253, 251 K; LCLS: 251,247, 243, 239, 232, 229, 227 K). The data reveal a split of the principal S(q)maximum into two well-separated peaks, S1 and S2 (dashed lines).b, Temperature dependence of the S1 and S2 peak positions, calculated fromthe maxima of local fifth-order polynomial least-squares fits with errorbars estimated by shifting the derivatives of the polynomial fits by 60.05 A(LCLS) and 60.15 A (SSRL) (Supplementary Information, section A.3.1).Green triangles are LCLS data from 12-mm-diameter droplets; red circles areLCLS data from 34- and 37-mm-diameter droplets; and black squares areSSRL data from a static liquid sample. Purple diamonds are LCLS data from9-mm-diameter droplets measured at a separate LCLS run with separateq-calibration (Supplementary Information, section A.1.2). As the temperaturedecreases in no man’s land, the positions of peaks S1 and S2 approach thecharacteristic values of LDA ice (dash–dot blue lines) as determined fromneutron diffraction22 and clusters of hexagonal ice (ice Ih; dashed red lines;Supplementary Information, section A.2.3).

RESEARCH LETTER

3 8 2 | N A T U R E | V O L 5 1 0 | 1 9 J U N E 2 0 1 4


Figure 3.7: The structure factor of water for a wide temperature range (a);q-positions of peaks S1 and S2 as a function of temperature (b), note the overlapbetween data obtained in static experiments with direct temperature measurementand the droplet data. Reprinted from Paper II with permission.

The height of the second peak, denoted as g2, in the RDF has been foundto correlate with the tetrahedrality of the liquid. Using the TIP4P/2005 watermodel, the dependence of the split between the first diffraction peaks in thestructure factor S1 and S2, ∆q, on g2 was established as shown in fig. 3.8a.This correlation curve was used to translate the peak split ∆q to a g2-value forthe experimental data, fig. 3.8b. The data show a much stronger continuousincrease of g2 with decreasing temperature than the water models, SPC/E andTIP4P/2005 and tend towards the value of the low-density amorphous (LDA)state, which had been determined in neutron diffraction experiments [92]. Thisis illustrated in the comparison of a RDF from simulations with a g2 valuecorresponding to the lowest temperature data and experimental data for theRDF of LDA and liquid water at ambient conditions in fig. 3.8c.

We interpret the findings in terms of a continuous and accelerated transition

35


1

1.2

1.4

1.6

1.8

Δq (Å-1)

g 2

3 4 50

1

2

3

4

5

r (Å)

g OO

(r)

g2

a

0

1

2

3

4

5

6

r (Å)

g OO(r)

Exp. 295 KExp. LDATIP4P/2005, 220 K

c

1

1.2

1.4

1.6

1.8

T (K)

b

0.6 0.8 1.0 1.2 220 260 300

2.5 3 3.5 4 4.5 5

g 2

LCLS 9µmLCLS 12µmLCLS 34-37µmSSRL, staticExp. best fitTIP4P/2005SPC/ELDA

Figure 3.8: (a) The peak split ∆q in the structure factor can be correlated withthe peak height of the second coordination shell, g2, in the RDF using MD sim-ulations. (b) The correlation curve from simulations can be used to translate theexperimentally determined ∆q into a g2-value, which can then be used to (c) finda corresponding gOO(r) from simulations. Reprinted from Paper II with permis-sion and modifications.

of the water structure towards an average LDL-like arrangement. This canbe understood within the LLCP hypothesis, where isobaric supercooling ofa water sample would be pictured to first cool it below the melting line Tm

and then, at the extreme supercooling achieved in this investigation, below the

36


temperature of homogeneous ice nucleation TH , see fig. 1.2. Below TH , themetastable state would approach the Widom line TW from above, that wouldextend from the hypothetical LLCP to lower pressures and higher temperaturesand would cause the accelerated structural transition towards LDL, and thusthe increase of g2 upon cooling.

37


38

4. Molecular correlations inliquid water

The structural arrangement of molecules in disordered materials and liquids ismost commonly described by spatial correlation functions, particularly radialdistribution functions in the case of isotropic liquids. The molecular structureis the basis to understanding the behavior of the liquid in different situations.

4.1 The oxygen-oxygen radial distribution function of wa-ter

In the history of x-ray scattering there have been numerous experiments onwater, the pioneering one conducted by Bernal and Fowler in 1933 [32]. ARDF published by Narten and Levy in 1971 [93] has for a long time beenconsidered a structural standard for liquid water. In the meantime, much efforthas been made to improve the measurement techniques, overcome difficultiesand obtain higher precision data.

The result of the project presented here is a new state-of-the-art RDF ofliquid water at ambient conditions that can be used as a benchmark for simula-tions. It has been published in Paper III. This chapter has to a large extent beenpresented in the licentiate thesis Aspects of the local structure and dynamicsof water, Daniel Schlesinger, Department of Physics, Stockholm University(2013).

4.1.1 Analysis of scattering data

In x-ray scattering experiments the differential scattering cross-section I(q) isobserved, which, in the first Born approximation, is proportional to the squareof the Fourier-transformed scattering potential. In order to extract the RDF asthe structural quantity of interest, some further processing is necessary.

The scattering cross-section can be decomposed into different contribu-tions

I(q) = Isel f (q)+ Idist(q) , (4.1)

39

4. Molecular correlations in liquid water

where the first term is called the self -scattering part and the second one thedistinct scattering part. The distinct scattering term provides information aboutthe interatomic structure which is to be extracted. The distinct scattering termcan again be subdivided into an intramolecular and an intermolecular part

I(q) = Isel f (q)+ Iintra(q)+ Iinter(q) . (4.2)

The self-scattering and the intramolecular part have to be subtracted in order toextract the intermolecular correlations. This can be achieved in two differentways using either an atomic or a molecular scheme.

In the molecular normalization scheme, the molecular form factor is de-fined as

Imol(q) = Isel f (q)+ Iintra(q) (4.3)

and the structure factor can be written as

Smol(q) = 1+I(q)− Imol(q)

ω(q), (4.4)

with the weighting factor ω(q) = (∑

i ci fi(q))2 and ci being the number con-

centration of atom type i and fi(q) its atomic form factor. The molecularform factor for water contains contributions to the self-scattering and the intra-molecular scattering

Imol(q) = f 2O(q)+2 f 2

H(q) + 4 fO(q) fH(q)sin(qrOH)

qrOHe−

12 σ2

OH q2

+ 4 f 2H(q)

sin(qrHH)

qrHHe−

12 σ2

HH q2, (4.5)

where rOH and rHH are the intramolecular distances between the atoms indi-cated by the indices and σOH and σHH the corresponding standard deviations.The molecular form factor of water has been determined in quantum chemicalcalculations by Wang et al. [94] and was employed in Paper III.

In the atomic normalization scheme, an independent atom approximationis adopted. Molecular form factors are often unknown or hard to determinewhile the atomic form factors are listed for numerous elements and can be usedin an independent atom approximation. The structure factor is then obtained bysubtracting the atomic self-scattering contribution Iatom(q) =

∑i ci f 2

i (q) fromthe scattering cross sections and is written as

Satom(q) = 1+I(q)− Iatom(q)

ω(q). (4.6)

40

4.1 The oxygen-oxygen radial distribution function of water

0 5 10 15 20Q [Å-1]

0

20

40

60

0 5 10 15 20Q [Å-1]

0

0.5

1

1.5

S(q)

Scat

terin

g cr

oss s

ectio

n

Figure 4.1: The coherent differential scattering cross section and the corre-sponding oxygen-oxygen structure factor (inset) from experiment. The theo-retical molecular form factor (red dashed curve) by Wang et al. [94] was usedto subtract self- and intramolecular contributions from I(q). Furthermore, theintermolecular O-H contribution obtained from neutron diffraction measure-ments [95] was subtracted to determine the O-O correlations (see Paper III forfurther details). The data analysis is summarized in the text. The data presentedare available in the supplementary material of Paper III.

This scheme was improved by Sorensen et al. [96], who introduced the modi-fied atomic form factor (MAFF)

f modi (q) =

(1− ai

zie−

q2

2δ2

)fi(q) (4.7)

to account for electron redistribution in the molecular environment. The con-stant ai/zi is the ratio between the charge transfer due to the bond and thenumber of electrons on the free atom; δ broadens the charge distribution inreciprocal space. This normalization scheme is used in Reverse Monte Carlo(RMC) simulations to calculate the structure factor from the radial distribu-tion function, while the simulation is performed via single atom moves andconstraints [64]. Since we focus on intermolecular oxygen-oxygen correla-tions, the contributions stemming from intermolecular O-H and H-H correla-tions have to be subtracted from the total intermolecular structure factor. Thecontribution from H-H correlations is shown to be negligible in this context

41


Figure 4.2: The O-H contribution to the RDF and the structure factor from NDexperiments [95], MD and RMC simulations. Reproduced with permission fromPaper III. Copyright 2013, AIP Publishing LLC.

(Paper III) due to the small magnitude of the weighting factor. Thus, to a verygood approximation, the oxygen-oxygen structure factor can be expressed as

SOO(q)∼=Smol(q)−ωOHSOH(q)

ωOO, (4.8)

with the weighting factors ωOO = f 2O(q)/ω(q) and ωOH = 4 fO(q) fH(q)/ω(q).

The O-H contributions obtained from neutron diffraction experiments and RMCand MD simulations are compared in fig. 4.2. There are significant differencesbetween the RMC, MD and ND data, particularly in the first peak in ginter

OH (r)around r≈ 2Å. Using the three datasets, it has been shown that the uncertaintyintroduced to O-O RDF is small and significant only in the region of the firstO-H peak, below 2.5 Å.

42

4.1 The oxygen-oxygen radial distribution function of water

4.1.2 The oxygen-oxygen radial distribution function

With the considerations above and the relationship between radial distribu-tion function and structure factor (2.12), the oxygen-oxygen radial distributionfunction can be calculated via a Fourier transformation

g(r) = 1+1

2π ρ

∫ qmax

0M(q,∆(r))(S(q)−1)q2 sin(qr)

qrdq , (4.9)

where the upper integral bound is identical to the maximum q-value of thedata set under consideration and the modified Lorch-function M(q,∆(r)) =sin(q∆(r))/q∆(r) was used with ∆(r) = (1− e−|r−a|/b)π/qmax (a and b pa-rameters). The result of the transformation is shown in fig. 4.3. Two aspects

4 7 10

r [Å]

0

0.5

1

1.5

2

2.5

gO

O(r

)

2 3 5 6 98

Figure 4.3: Oxygen-oxygen radial distribution function of water at ambientcondition. The extension into the region below about 2.4Å is unphysical. Thedata presented are available in the supplementary material of Paper III.

are particularly important to emphasize: The x-ray diffraction data sets usedin this study were measured to very large q-values (particularly the data setby L. Skinner obtained at the Advanced Photon Source (APS)) and a modifiedwindow function was used to remove truncation artifacts. The length of the q-range clearly affects the nearest neighbor peak, both in position and height. Weshow that the peak coordinates converge for large q-ranges above q∼ 20Å. Inthe Fourier transform, a Lorch function is used that both depends on q and onr. This particular window function is chosen to leave the region of the nearestneighbor peak unchanged to determine its coordinates to a high precision.

43


4.2 Intermediate range correlations in liquid water

In a recent study, Skinner et al. [58] presented high-quality x-ray diffractiondata of water in a wide range of temperatures of roughly 250− 340K. Thistemperature range was chosen as to include the compressibility minimum atabout 319K and to investigate the associated structural changes in the liquid.In this current section we re-analyze the data for the oxygen-oxygen RDFs bySkinner et al. for correlations in the intermediate range, particularly the fea-tures found at distances r > 10 Å. We then compare the experimental RDFsto simulation data obtained from large scale simulations of the TIP4P/2005model with 45000 molecules [97] and investigate structural changes with tem-perature. The results of this analysis are summarized in Paper IV.

The RDFs by Skinner et al. were first smoothened to reduce the noise leveland oscillations introduced by the Fourier transform of the structure factor.We use the Gaussian filter [98], i. e. every data point in the smooth RDF iscalculated as the average over data points in the neighborhood of the originaldata point, weighted by a Gaussian function. The width of the normalizedGaussian weighting function, d, can be adjusted. This procedure dampenssharp features, as fast oscillations, but diminishes the height of all varyingfeatures at the same time. We tested different widths of the Gaussian weightingfunction. The original spacing in the distance coordinate r of the experimentaldata is ∆r = 0.01 Å, while the RDFs from simulations have been calculated atdouble this step size. To obtain comparable data sets, we use a Gausian widthof d ≈ 1.5 Å, i. e. average over 74 data points for the simulation and 150 datapoints for the experimental data, respectively. A smoothening of simulationand experimental data at a width of d = 0.6 Å does not qualitatively changethe results presented below.

The smooth RDFs are presented in fig. 4.4 for the intermediate distancerange of 8−20 Å and the insets show the range between 14−20 Å magnified.The excellent agreement of the simulation data with the experimental resultsrules out that the features investigated here are artifacts of the Fourier trans-form or due to noise. All features found in the experimental data are presentin the simulation data with comparable intensity. Clearly, the temperature de-pendence is shifted when comparing experiment and simulations, which is inagreement with the phase diagram of TIP4P/2005 being shifted in tempera-ture: its melting point has been reported to be at about Tm = 252K [69] andthe boiling point is found at Tb = 401K [82], while the model reproduces thetemperature of maximum density near-quantitatively at Tmax = 278K [69].

We now turn to characterize the intermediate range features. The temperature

44


16 18 20

r [Å]

0.99

0.995

1

1.005

gO

O(r

)

254.2 K

263.2 K

268.2 K

277.1 K

284.5 K

295.2 K

307.0 K

312.0 K

321.0 K

337.4 K

342.7 K

0.9998

1

1.0002

1.0004

1412108

8 10 12 14 16 18 20

r [Å]

0.99

0.995

1

1.005

gO

O(r

)

253 K

270 K

278 K

298 K

320 K

340 K

360 K

0.9995

1

1.0005

a

b

4th

5th

6th

7th

8th

Experiment

Simulation

4th

5th

6th

7th8th

Figure 4.4: Oxygen-oxygen radial distribution functions in the intermediaterange for different temperatures from experiment (a) and simulations of theTIP4P/2005 model (b). The long-range features are shown enlarged in the in-sets.

45


dependence is investigated first, where we choose to measure the height ofthe features as a function of temperature. We then decompose the RDFs insimulations in terms of an order parameter into high-density liquid (HDL) andlow-density liquid (LDL)-like contributions.

4.2.1 Temperature dependence

In the present investigation we analyze three different features, at about r5 ≈11 Å, r6 ≈ 13 Å, and r8 ≈ 17 Å and define their heights as g5 ≡ g(r5), g6 ≡g(r6), and g8 ≡ g(r8), respectively. It must be mentioned that at higher tem-perature and for larger distance the features naturally become harder to identifyand uncertainties increase. We also emphasize that the exact heights dependon how the data have been smoothened. Nevertheless, the features are robustand show a very similar behavior in the experimental and simulation data.

Figure 4.5 shows the temperature dependence of g5, g6 and g8. The fifth shellpeak g5 increases sharply upon cooling and experiment and simulation resultsare in excellent agreement. The rise of g5 in the experimental data can be seento accelerate upon cooling, and more so than in the simulation data. This fea-ture has been investigated to great detail in an earlier study by Huang et al. [97]and was found to be associated with the emergence of LDL-like arrangementsin the liquid at ambient temperatures.

Interestingly, the sixth shell feature, g6, is found to first grow upon coolingto then exhibit a maximum and decrease quickly upon further cooling. A simi-lar behavior is observed for the density of water and we therefore compare theexperimental data of g6 to the mass density in fig. 4.6. There is a slight shiftin temperature between the maxima of about 5K. The underlying molecularstructures causing this behavior can be understood within the two-state pictureof water discussed below in more detail.

We further observe a feature between about 16−17.5 Å denoted as g8. Itshifts to higher distances upon cooling. The temperature dependence of g8 isweak and can mainly be observed in the simulations while the scattering in theexperimental data starts to cover the signal.

4.2.2 Structural analysis

In this section we make an attempt to characterize the underlying molecularstructures contributing to the intermediate range features discussed above. Thefeatures between about 8.5−11 Å show a strong temperature dependence and

46


1.002

1.004

1.006

1.008

1.01

g 5

ExperimentSimulation

1.0006

1.0008

1.001

ExperimentSimulation

g 6

1

1.0001

1.0002

1.0003 ExperimentSimulation

260

Temperature [K]280 300 320 340

g 8

Figure 4.5: Temperature dependence of the intermediate range features. Com-parison of experimental and simulation data for g5(T ), g6(T ) and g8(T ).

have previously been analyzed in terms of a structural order parameter, thelocal structure index (LSI) [97]. The LSI value, which has been introduced byShiratani and Sasai [100], is defined for each molecule i as

I(i) =1

n(i)

n(i)∑j=1

(∆( j, i)− ∆(i)

)2, (4.10)

where n(i) is the number of neighbor molecules within a oxygen-oxygen dis-tance of 3.7Å; the neighbors are ordered according to their distances to theoxygen of the central molecule i, r1 < r2 < .. . < 3.7Å< rn(i)+1, such that∆( j, i) = r j+1− r j with the average ∆(i) can be defined. The LSI takes rela-

47


980

990

1000

260 280 300 320 3401.0004

1.0008

1.0012

Temperature [K]

Mass density [kg/m

3]g 6

Figure 4.6: The feature g6 and the mass density as functions of temperature incomparison. Density data taken from [40; 99]

tively high values for ordered molecular arrangements and low values for moredisordered structures [100].

In the present study we divide the molecules into two groups, with LSIvalues above and below the average LSI, respectively. The RDFs are calcu-lated for each of the groups separately. This decomposition is performed forsimulations at two different temperatures of T = 253K and T = 298K and theresults are shown in fig. 4.7. The average LSI values used for the subdivisionwere found to be I = 0.0550Å2 at T = 253K and I = 0.0381Å2 at T = 298K.It should be noted that, in the present investigation, we calculated the LSI val-ues in the direct, instantaneous structure, not the inherent structure [97; 100].The region between 8− 12 Å has been analyzed in detail in a previous studyby Huang et al. [97]. The analysis in terms of the LSI showed that the featureg5 is mainly due to molecules with high LSI values indicative of a LDL-likemolecular arrangement at higher temperatures.

At the position of g6 we observe that the contributions from both, low andhigh LSI RDFs almost coincide. The change of the density with temperature,found to be similar to the changes in g6, can be understood in terms of differ-ent local favorable structures: Upon cooling, thermal fluctuations decrease andnormal thermal contraction cause the liquid to become denser. Upon furthercooling, structural fluctuations into local LDL-like molecular arrangements in-crease, thus causing the density to exhibit a maximum and to expand below thetemperature of 277K. On the macroscopic level, one would observe the aver-age behavior of HDL- and LDL-like contributions. The LSI analysis in fig. 4.7shows that an inward shift of the high-LSI correlation causes g6 to exhibit a

48


maximum and shrink upon cooling. We further notice a redistribution of in-tensity to a feature emerging at 15Å, which is found in the experimental dataat the lowest temperature. According to the decomposition in fig. 4.7, this lat-ter feature stems from a redistribution from HDL- to LDL-like arrangements,similar to what is observed for g4. For the largest distances investigated wecan state that the shift of g8 in position seems to be due to a redistribution ofLDL-like structures to larger distances upon cooling.

8 10 12 14 16 18 20

r [Å]

0.98

1

1.02

1.04

gO

O(r

| L

SI)

T = 253 K, low LSI

T = 253 K, high LSI

T = 298 K, low LSI

T = 298 K, high LSI

0.999

1

1.001

1.002

Figure 4.7: RDFs calculated for molecules with LSI above and below the aver-age value from simulations at T = 253K and T = 298K.

In summary, we have investigated intermediate range correlations in liquidbulk water. The RDFs extracted from experiment and simulations of the TIP4P/-2005 water model are in excellent agreement. The temperature dependence ofthe intermediate range features have been analyzed and the behavior of g6 wasfound to exhibit similarities with that of the density. The underlying struc-tural changes were further analyzed using a decomposition into high-LSI andlow-LSI contributions to the RDFs from simulations. The structural analysisstrongly suggests that there are rather pronounced structural change with tem-perature, which can even be observed around ambient conditions. These struc-tural changes can be understood in terms of different molecular arrangementswith a redistribution between highly ordered, LDL-like, and more disorderedHDL-like local molecular arrangements. Furthermore, the results obtainedhere are consistent with the conclusion drawn from small-angle x-ray scatter-

49


ing of water at ambient conditions that water is inhomogeneous to a lengthscale of about 1nm [101].

50

4.3 The structure of water on a Barium fluoride (111) surface

4.3 The structure of water on a Barium fluoride (111) sur-face

This project has been a joint effort of experimental work and MD simulations.In a x-ray absorption study of water on a BaF2(111) surface it was found thatthe water adapts an HDL-like structure. We performed MD simulations inorder to shed light on the molecular structure and the results were found to beconsistent with the experimental findings.

The BaF2(111) surface has been suggested to be a template for ice forma-tion due to its lattice constant being close to that of hexagonal ice Ih. This hy-pothesis has been subject of earlier experimental investigations by Sadtchenkoet al. [102]. In the present investigation, the template effect was not con-firmed and the surface was rather found to locally disturb the structure of waterseverely, in agreement with the results by Sadtchenko et al.

The results of this project were published in Paper V. This chapter has to alarge extent been presented in the licentiate thesis Aspects of the local structureand dynamics of water, Daniel Schlesinger, Department of Physics, StockholmUniversity (2013).

4.3.1 Experimental results

The oxygen K-edge x-ray absorption spectrum (XAS) has been established asa probe for the local structure of liquid water [103; 104]. Certain features in theXAS have been interpreted to be related to the local H-bond situation in liquidwater, the pre- and main-edge (∼ 535eV) mainly to weakened, asymmetricH-bonds, the post-edge (∼ 540−542eV) mainly to strong H-bonds in a localtetrahedral structure.

The XAS of liquid thin-film water on BaF2, shown in fig. 4.8, exhibits astrong resemblance to that of the high-density liquid (HDL), where the largemain/post-edge ratio indicates a high density [105]. The XAS of ice Ih, fig. 4.11(a, i), clearly shows a strong post-edge, whereas the pre-edge has almost van-ished, consistent with the interpretation above. Fig. 4.11(b) shows the resem-blance of water on BaF2(111) and an aqueous NaCl solution, with a moredisordered average local structure, a strong pre-edge feature and decreasedpost-edge intensity. The structural changes of water due to NaCl have beencharacterized more recently [107]. The suppression of fluctuations into localLDL-like structures in the vicinity of ions has been interpreted to push theLDL/HDL ratio to favor a more disordered, HDL-like environment on the av-erage.

From these findings, it was concluded that the local structure of water ona BaF2(111) crystal surface is locally disturbed with broken or asymmetrical

51


Figure 4.8: a) Comparison of XAS of (i) bulk ice Ih (dashed), ice VII(solid) [105] with the spectra of (ii) 2 monolayers crystalline ice and (iii) su-percooled water on BaF2(111) at 1.5Torr at 259K (90% relative humidity). (b)Comparison of XAS spectra of (i) supercooled water on BaF2(111) at 1.5Torr at259K (90% relative humidity) with (ii, solid) liquid water, and (ii, dashed) 6MNaCl solution measured at room temperature [106]. The shaded area representsthe difference between bulk ambient liquid and thin-film water. Figure takenfrom Paper V, reprinted with permission.

H-bonds between the water molecules.

4.3.2 MD simulations of water on a BaF2(111) surface

We performed MD simulations to investigate the molecular structure of wa-ter on the BaF2(111) surface which was experimentally found to resemble anHDL-like structure as described above. The water structure was characterizedby calculating the oxygen-oxygen pair-distribution functions from the simula-tions. Different force-fields, charge-pairs for the crystal ions and adsorptionenergies were compared to ensure independence of the findings on these prop-erties.

The MD simulations were performed to model the behavior of H2O on afluorine-terminated BaF2(111) surface with 256 BaF2, and 500 H2O molecules,corresponding to about 2.5 monolayers water on the crystal surface. We used

52


the GROMACS package ver. 4.5.3 [108], the OPLS-AA force field [109], andthe TIP4P/2005 water model [69]. Each simulation was equilibrated for atleast 5ns, generic thermodynamic properties were monitored to ensure equili-bration. The production runs were performed in the NPT ensemble for 20nsat a time step of 1fs using velocity rescaling [110] to set the temperature; theBerendsen barostat [111] was applied for the pressure control. All simula-tion results shown here were obtained at P = 1bar and T = 290K or 260K,respectively.

Pair-distribution functions were calculated in layers oriented parallel tothe surface in order to investigate the structure of water close to the BaF2(111)surface. We use a cylindrical normalization which we want to discuss briefly.

A general definition of the pair-distribution function reads [60]

g(r) =VN2

⟨N∑i

N∑j 6=i

δ (~r−~ri j)

⟩. (4.11)

where r ≡ |~r|, ~ri j is the distance vector between atom i and j, and V and Ndenote the total volume and the total number of molecules included in thecalculation, respectively. In practice, we evaluate eq. 4.11, here for oxygenatoms, by counting the oxygen-oxygen distances in small intervals [r;r+δ r].The histogram obtained has to be normalized by the total number of moleculesN and the average number of oxygen-oxygen distances in each bin is dividedby the number of atoms of an ideal gas at the same density in the volumeassociated with the distance bin.

This procedure is further adapted to the cylindrical symmetry of the prob-lem considered here. The total volume is taken to be a flat, cylindrical diskwith the symmetry axis perpendicular to the plane, and the constant factor thusbecomes

VN2 = π h

R2

N2 , (4.12)

where the thickness of the disk and the maximum radius were set to h = 1Åand R≈ 40Å, respectively. The number of oxygen atoms within this layer andthe distance bin [r;r+δ r] is divided by the associated volume of a cylindricalshell

v(r) = π h((r+δ r)2− r2) , (4.13)

so that the pair-distribution function, with the above definition of the constantV/N2, converges to unity for large radii.

53


4.3.3 Results for the molecular structure of water on BaF2(111) fromMD simulations

In the simulations we observe a pronounced layering of TIP4P/2005 watermolecules along the surface, quantified in the inset of fig. 4.9. The densityprofile along the surface normal shows a sharp first peak indicating a stronglocalization of the contact layer of molecules on the surface. A second and in-complete third layer can be identified that are broader in shape. We investigatethe structure of water within the layers by calculating oxygen-oxygen PDFswith the cylindrical normalization introduced above. The layers were definedas cylindrical volumes extending parallel to the surface with 1Å in height. Theresults are shown in fig. 4.9. The structure of the contact layer is dramaticallychanged when compared to the bulk: the second and the third coordinationshells collapse to about 3.3Å and 6Å while the first peak is significantly de-creased.

The structures found for the contact and the second layer are comparedto the structure of the bulk liquid from simulations, bulk water under pres-sure [112] and of a NaCl solution, where the latter was observed to exhibita similar structure as the compressed liquid [113; 114]. The interaction withalkali halide solutions has been studied in more detail recently using transmis-sion mode XAS [107]. It was concluded that the local H-bond configurationbecomes locked into an HDL-like structure at least in the first hydration shellof the ions, thus shifting the LDL/HDL ratio towards more disordered HDL-like average configurations. Recent simulations of potassium halide solutionsshow similar effects, with particularly the fluoride anion affecting the localstructure [115].

From the simulations, a very high local density of 1.23g/cm3 was foundwithin the first layer, integrated from a distance of 1.3Å from the crystal tothe minimum in the density profile (cf. inset in fig 4.9). For T = 260K, thestructure is slightly better defined than for T = 290K, but does not changequalitatively.

The collapse of the second coordination shell is mainly due to two wa-ter molecules H-bonding to the same fluoride ion on the surface, causing theoxygen atoms to be closer together than they would be in the bulk. This isillustrated in fig. 4.10, with a histogram of oxygen-oxygen distances betweenmolecules simultaneously H-bonding to the same fluorine ion at the surface.

The findings from simulations are in qualitative agreement with the resultsfrom the XAS experiments, where the spectra indicate a HDL-like structure.In the experiment, the conclusions even hold for supercooled water at a tem-perature 259K.

54


0

1

2 3 4 5 6 7 8

gO

-O(r

)

r [Å]

b: HDL

a: First layer

d: Bulk H2O

e: Second layer

c: NaCl solution

6distance(O-surface) [Å]

0

1

2

3

rel.

freq.

of o

ccur

ance 2.55 Å

4.65 Å

6.15 Å

2 4 8 10 12

Figure 4.9: Oxygen-oxygen pair-distribution functions of (a) the first layer H2Oon BaF2(111) from simulations, full line T = 260K, dotted line T = 290K; (b)HDL, taken from reference [112]; (c) NaCl solution, mole ratio 1:10 taken fromreference [113]; (d) bulk H2O at T = 260K from simulations, calculated with thesame anisotropic normalization as (a) and (e) for comparison; (e) second layerH2O on BaF2(111) from simulations, full line T = 260K, dotted line T = 290KInset: Histogram of distances between the oxygen atoms of H2O and surfaceatoms. Reprinted from Paper V with permission.

55


2.5 3 3.5 4 4.5r [Å]

0

0.1

0.2

0.3

0.4Re

lativ

e fre

quen

cy

Figure 4.10: Histogram of oxygen-oxygen distances of water molecules simul-taneously H-bonding to the same fluoride ion, as shown in the inset.

Generally, simulations can give valuable insights into structure and dy-namics on the molecular level and complement experimental results. But it isimportant to keep in mind that simulations are performed using models and theresults might not always represent what happens in reality. In order to test howrobust the simulation results were we repeated the simulations using differentforce fields, varied the effective charges of the ions in empirical force field sim-ulations and compared the adsorption energies obtained with empirical forcefields with results from density functional theory (DFT).

Test of different water models. We performed simulations with four differ-ent water models: the three-point models SPC/E and TIP3P and the four-pointmodels TIP4P and TIP4P/2005. All four tested water models exhibit a similarPDF within the first layer at the interface, see fig. 4.11. Interestingly, the shellstructure is less pronounced for the three–point models TIP3P and SPC/E asalso found in simulations of the bulk. Nevertheless, all models predict high-density structures of the first-layer water on the BaF2(111) surface, with astrongly diminished first peak and the inward collapse of the second and thirdcoordination shell, when compared to the bulk structure.

56


2 3 4 5 6 7

r [Å]

0.5

1

1.5

2

2.5

gO

O(r

)TIP4P-2005

TIP4P/OPLS

TIP3P/OPLS

SPCe/OPLS

Figure 4.11: Comparison of the O-O pair-distribution functions in the first layerof H2O on BaF2(111) obtained with different water models. The crystal wasdescribed with the OPLS force field in all cases.

Effective charges of the crystal ion pairs. Coulomb interaction betweenpartial charges within the water molecule and the ionic charges of the sur-face ions provides the dominant interaction in the system under consideration.Thus, the effective charges of the surface ions were expected to play a promi-nent role for the structure observed. MD simulations were performed withthree different charge-pairs for ions in the BaF2(111) slab to investigate theinfluence on the O-O pair-distribution function (PDF) of water on the surface.The results are shown in fig. 4.12. There is a clear effect of the changes incharge on the PDF. However, comparison to the bulk water O-O PDF calcu-lated with the same method clearly shows that even an unrealistically weakcharge-pair (qF =−0.5e, qBa =+1.0e) still results in an HDL-like PDF.

Adsorption energies from MD and DFT. In order to benchmark the empir-ical force fields used in the MD simulations, we compared the adsorption en-ergies of an isolated water molecule on a (3×3) four-layered BaF2(111) slab.The adsorption energies were calculated from energy-minimized simulationboxes using empirical force fields and from density functional theory (DFT)using different functionals for comparison and shown in tables 4.1 and 4.2.The real-space-grid based GPAW [116] code was used with a grid-spacing of

57


4 7 8

r [Å]

0

1

2

3

4

5

gO

O(r

)q

F=-1.0e, q

Ba=2.0e

qF=-0.87e, q

Ba=1.74e

qF=-0.5e, q

Ba=1.0e

bulk water

2 3 5 6

Figure 4.12: Comparison of first-layer O-O pair-distribution functions for wateron BaF2(111) with different effective ionic charges, as indicated. A bulk water O-O pair-distribution function obtained from a 1Å thick slab through the simulationbox (i.e. the same conditions and normalization as for the surface layer, T =260K) is plotted for comparison.

0.2Å.

Water model (BaF2 de-scribed by OPLS forall cases)

Adsorption energy[kJ/mol], charge-pair(-1.0e, +2.0e)

Adsorption energy[kJ/mol], charge-pair(-0.87e, +1.74e)

TIP4P/2005 −41.2 −34.6TIP4P (OPLS) −37.1 −30.7SPC/E (OPLS) −33.5 −26.9TIP3P (OPLS) −27.0 −21.5

Table 4.1: Adsorption energies obtained with the MD force fields for two differ-ent ion charge-pairs. The structures were energy-minimized, they are the sameas obtained from DFT calculations. Reproduced from Paper V with permission.

It is clear that the different force-fields give values for the adsorption en-ergies in good comparison to those obtained with DFT, in particular for thePBE and BEEF functionals, where the latter includes also non-local correla-tions describing van der Waals interactions. In quantum chemical calculationsof ionic systems one typically finds lower charges than the nominal ones; infact, using the intermediate charge-pair brings the force-field values into even

58


DFT functional Adsorption energy [kJ/mol]BEEF-vdW −33.1PBE −30.7RPBE −14.0

Table 4.2: Adsorption energies obtained with different DFT functionals forcomparison with the MD results in table 4.1. Reproduced from Paper V withpermission.

better agreement with the DFT values. Note, however, that the conclusions ona highly compressed contact layer are independent of force-field and charge-pair.

59


60

5. Molecular dynamics in liquidwater

This chapter is devoted to the molecular dynamics of water. After a generaloverview over the time scales of different dynamical processes, the focus willbe on a particular phenomenon called the Boson peak and the observation of asimilar feature in simulations of supercooled bulk water.

5.1 Dynamics in liquid water

The dynamics in liquid water is a very wide field and can be subdivided into thedynamics on different time scales: The fastest possible processes are electronicexcitations and relaxations that take place on a time scale of up to a few fem-toseconds and are relevant for interactions of photons with water molecules, asin x-ray spectroscopies. Intramolecular vibrations appear just below 10fs andlibrations around 50fs. They are probed using infrared (IR) spectroscopies. InMD simulations at ambient temperature, the averaged velocity autocorrelationfunction of water decays on a time scale of a few hundred femtoseconds, in-dicating that oscillations about an instantaneous position are dampened out onthis time scale. On a time scale of several hundred femtoseconds to picosec-onds, proton transfer reactions take place. It is also in this range that inter-molecular and collective vibrations start playing a role which will be discussedin detail below. On longer time scales, phonon excitation and hydrodynamicmotions become dominant.

5.2 The Boson peak

Generally, the low frequency acoustic phonon spectrum of a harmonic crystalcan be described by the Debye model. It is based on the assumption of a lineardispersion relation, which is justified for acoustic modes at small frequencies.For a linear dispersion relation

ω = ck , (5.1)

61

5. Molecular dynamics in liquid water

with frequency ω , sound velocity c and wave vector k, the Debye approxima-tion for the vibrational density of states (VDOS) evaluates to

gD(ω) =3

2π2ω2

c3 (5.2)

thus predicting an ω2-dependence of the density of phonon states, gD(ω) onthe frequency ω [117; 118]. The Boson peak (BP) is an excess over the densityof states predicted by the Debye approximation and is observed to be a univer-sal phenomenon in disordered materials exhibiting a glass transition and oc-curs at frequencies of about 0.5−2.5THz (corresponding to ca. 16−80cm−1

or 2− 10meV). The BP can be observed experimentally in inelastic neutronscattering (INS) [119] and inelastic x-ray scattering (IXS) [120; 121] for mostglasses and supercooled liquids.

There is no general agreement on the origin of the BP, but there are two majorhypotheses. One explains the BP in terms of collective dynamics and propagat-ing modes while the second one associates it with local or quasi-local molecu-lar motions [122].

An interpretation was given by Sokolov et al. [123] in terms of glass tran-sition dynamics in fragile and strong systems and in a mode-coupling theory(MCT) picture. For fragile systems, i. e. such that follow a non-Arrhenius be-havior, the collective motion would be structural relaxation, corresponding tofast β -relaxation. For strong systems, i. e. such that show an Arrhenius-typetemperature dependence of the relaxation time, on the other hand, the collec-tive motions would be “frozen in” into vibrations. Sokolov et al. also reportedthe observation of an excess density of vibrational states in Raman spectrawith the quasi-localized collective atomic vibrations involving about 30−100atoms [123].

The explanation in terms of a quasi-localized nature of the BP was op-posed by Marruzzo et al. [124] who identified the BP related vibrations tobe mainly associated with spatially fluctuating shear stresses. This interpre-tation was developed further as presented by Chumakov et al. [125], with theBP as a remnant of the transverse acoustic van Hove singularity in the parentcrystal. This correspondence would then suggest that the BP originates fromthe piling up of the acoustic states near the boundary of the pseudo-Brillouin-zone [125]. This latter interpretation, however, has been challenged by theory.Schirmacher showed in his textbook [126] that the BP marks the crossoverfrom wave-type vibrational excitations to disorder-dominated vibrations thatwould be characterized by a random Hamiltonian. He emphasizes explicitly,that the BP would not be connected to a van Hove singularity whatsoever.

It becomes clear that the origin of the BP remains puzzling and that it has

62

5.2 The Boson peak

yet to be explained conclusively.

5.2.1 Low-frequency vibrational dynamics of bulk supercooled water

In INS experiments a BP has been reported for protein hydration water [127]and for supercooled water in confinement [88]. We performed extensive MDsimulations of the TIP4P/2005 water model to investigate the VDOS of thebulk supercooled liquid. We ran a series of simulations at temperatures ofT = 210−260K at ambient pressure. The simulations were first equilibratedin the NPT ensemble using the Nosé-Hoover thermostat and the Parrinello-Rahman barostat. We then switched to the NV T ensemble for further structuralrelaxation. In the supercooled temperature region, the equilibration time startsto rise sharply. The structural relaxation time τα can be found from the relax-ation of the incoherent intermediate scattering function defined below. Sinceτα ≈ 20ns at T = 210K, these equilibrations were run for up to 100ns. We fi-nally switched to the microcanonical ensemble (NV E) in which we performedthe production runs for the calculations of dynamic quantities. The simulationswere run at integration time steps of 0.2−1.0fs in double precision mode.

The velocity autocorrelation function

C(t) =

⟨1N

∑i

~vi(t)~vi(0)

⟩(5.3)

was computed from MD simulations for different temperatures and systemsizes and Fourier transformed

g(ω) =

∫C(t)eiωtdt (5.4)

to obtain the VDOS g(ω), shown in fig. 5.1In the low frequency end of the phonon spectrum, we identified peaks that

are due to finite size effects as has been discussed in the literature [128]. Byscaling the system size, they were shown to extrapolate to zero frequency forinfinite system size as shown in fig. 5.1b. These finite-size peaks are connectedto the transverse current spectrum CT (k,ω) at the smallest wave vectors k pos-sible in the simulation box which is shown in fig. 5.1 together with the VDOS.

The BP is most easily identified by removing the Debye-predicted depen-dence in the reduced VDOS (g(ω)−g(0))/ω2 as shown in fig. 5.2. We find abroad feature around 37cm−1, unaffected by system size scaling. In compar-ison, the BPs reported for protein hydration water and supercooled confinedwater are located at about 30cm−1 [127] and 45cm−1 [88], respectively, in

63


g(t

) [a.

u.]

peak 1peak 2

tpe

ak [c

m-1

]

1/L [Å-1]

0

10

20

30

40

0 0.01 0.02 0.03 0.04 0.05 0.06

(b)

0

0 20 40 60 80 100

T=100K iceT=210KT=230KT=250K

CT �N�ѱ�

(a)

t[cm-1]

Figure 5.1: (a) Vibrational density of states (VDOS) together with the transversecurrent spectrum CT (k,ω) which for low values of k is identified to cause thefinite size peaks in the VDOS. (b) The peak positions of the first sharp peaksfound in the VDOS are shown to extrapolate to ω = 0 for infinite system size L.

good agreement with our result. This is the first time that a similar featurecould be shown to emerge in bulk supercooled water. The BP is often observedin experimental data for the dynamic structure factor. To connect the resultsfound in simulations to this quantity we calculated the incoherent intermediatescattering function

FS(k, t) =

⟨1N

∑i

e−i~k(~ri(t)−~ri(0))

⟩, (5.5)

64

5.2 The Boson peak

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 10 20 30 40 50 60 70 80 90 100

T=210KT=150K, LDAT=100K, LDAT=100K, Ice Ih

>J�Ʒ��J

��@�Ʒ2

t[cm��@

Figure 5.2: Reduced VDOS at low temperatures. The broad feature is present inthe supercooled liquid, and blue-shifted in the amorphous LDA states and the iceIh crystal. The simulations quenched into amorphous LDA states at T = 150Kand T = 100K tend towards the excess in the VDOS of hexagonal ice Ih withdecreasing temperature.

that is then Fourier transformed to give the dynamic structure factor

SS(k,ω) =

∫FS(k, t)eiωtdt . (5.6)

The incoherent intermediate scattering function FS(k, t) and the dynamic struc-ture factor SS(k,ω) calculated from our simulations are shown in fig. 5.3.FS(k, t) shows the emergence of a time scale-separation upon supercooling.The two relaxation processes are called α- and β -relaxation, i. e. slow re-laxation at longer times and a faster relaxation at smaller time scales, respec-tively, leading to the plateau-like structure [41; 129]. The intermediate scatter-ing function has been investigated in detail in simulations of the TIP4P watermodel and compared to predictions from MCT, indicating a fragile-to-strongcrossover of the liquid at the Widom line [130].

SS(k,ω), that is observable in experiments, clearly shows a BP-like featurein the liquid. When the liquid simulation is quenched to 150K and 100K intoLDA states at a cooling rate of 2 ·1010 K/s, this feature persists and is shiftedtowards higher frequency with decreasing temperature and observed to shifttowards the position of a similar feature in simulations of hexagonal ice.

We conclude that we found a BP-like feature in simulations of supercooledbulk water and its onset to coincide with crossing the Widom temperature TW

65


S s(k

,ѱ)

0

1

2

3

4

5

6

0 20 40 60 80 100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.01 0.1 1 10 100

F s(k

,t)

100K ice N=432

210K220K230K240K250K260K

100K ice N=3456

ѱ [cm-1]

t [ps]

(a)

(b)

Figure 5.3: (a) Incoherent intermediate scattering function FS(k, t) and (b) itstemporal Fourier transform, the incoherent dynamic structure factor, SS(k,ω).FS(k, t) clearly shows the emergence of relaxation processes on two separate timescales of α- and β -relaxation. SS(k,ω) can be measured in INS experiments. Itshows the emergence of the BP-like feature upon decreasing temperature be-tween T = 240 and 230K .

of the water model. This feature is thus associated with the structural crossoverto a stiffer LDL-like liquid. A similar feature has been observed in the VDOSof LDA states and ice, where it is shifted to higher frequencies. Our observa-tions thus support an interpretation similar to that by Chumakov et al. [125],that the low-frequency vibrational dynamics in the deeply supercooled liquidstarts to resemble that of the related crystalline ice Ih.

66

Summary and Outlook

This thesis has been focussed on water, its properties and their relation to thestructure and dynamics on the molecular level.

Molecular dynamics simulations of the evaporative cooling process fornanometer sized droplets were compared to calculations using the Knudsentheory of evaporation. The results were found to be in very good agreementand we concluded that the generically used evaporation coefficient is unityprovided that the thermodynamic properties of the evaporating material areknown. The results could thus be of use for other liquids than water. Aninteresting perspective would be the application to cloud-microphysics withevaporation and condensation processes of tiny droplets. In simulations, thedroplets could furthermore be doped with ions to investigate the effect on theirevaporation kinetics.

The Knudsen theory of evaporation was further used to calibrate the tem-perature in an experiment in which the evaporative cooling technique was ap-plied to supercool water droplets to unprecedentedly low temperatures belowthe previously accepted temperature of homogeneous ice nucleation. The liq-uid structure probed with high intensity x-ray pulses from the free electronx-ray laser LCLS was found to change continuously and in an acceleratedfashion towards a low-density amorphous structure and can thus be regardedas low-density liquid-like, in agreement with predictions by the liquid-liquidcritical point hypothesis. This experiment might be the key to probe watereven deeper down into “no-man’s land” from above, possibly also at higherpressures, so as to further test the liquid-liquid critical point hypothesis andalternative explanations of water’s anomalous behavior.

We further investigated the dynamics of the bulk liquid upon deep super-cooling in simulations. The low-frequency vibrational spectrum was foundto exhibit an excess over the intensity predicted by the Debye approximation,similar to the Boson peak, which is a universal feature of amorphous materials.This Boson-like peak was observed to emerge upon crossing the Widom lineof the TIP4P/2005 model where the liquid transforms into a low density liquidthat is tetrahedrally ordered and starts to stiffen.

The structure of water on a molecular level is highly adaptable to varioussituations, a property that makes water an outstanding solvent and versatilemedium. Recent high-quality x-ray diffraction data of bulk water at ambi-

5. Summary and Outlook

ent conditions provided the radial distribution function of water to very highprecision. Simulations were employed to determine the O-H contribution tothe radial distribution function and to estimate uncertainties caused by the sub-traction of this contribution obtained from neutron scattering experiments. Theoxygen-oxygen radial distribution function is key to many properties and thestructural data presented can be considered the current standard.

Even more recent experimental data show delicate molecular correlationsto distances of up to 17Å. These intermediate range correlations were foundto be present in simulations of the TIP4P/2005 model and there is a strikingqualitative agreement between experimental and simulation data. The temper-ature dependence of the intermediate range features is non-trivial and mightlead to a deeper understanding of neat water at ambient conditions on meso-scopic length scales. The structural analysis of the intermediate range featureobserved showed that the liquid can be understood within the two-state modelin terms of low-density liquid and high-density liquid-like molecular arrange-ments with a shift of the fluctuations between these structural motifs upontemperature changes in favor of one or the other.

The structure of water at interfaces is of crucial importance since interfa-cial water is the most common form of water in nature and might be helpfulin future applications. In contrast to the structure at extreme supercooled con-ditions, the molecular arrangement in the liquid at a barium fluoride (111)surface resembles much more that of water under high pressure and was thusconcluded to be high-density liquid-like. Our findings confirm that the bar-ium fluoride (111) surface does not promote ice formation but instead stronglysuppresses structural fluctuations into low-density liquid arrangements at theinterface.

All the different aspects of water’s properties and behavior discussed abovecan be understood in terms of the liquid-liquid critical point hypothesis withcompeting structural and thermal fluctuations between the extremes of low-and high-density liquid-like molecular arrangements.

68

Sammanfattning

Vatten är en fascinerande substans som ännu inte är fullständigt förstådd på enfundamental nivå. Den föreliggande avhandlingen handlar om olika aspekterav vätskan vars egenskaper uppvisar många anomalier jämfört med enkla väts-kor som t. ex. argon. Här undersöker vi egenskaper av rent vatten, speciellt denmolekylära strukturen och dynamiken med hjälp av molekylära simuleringar.

Vattens anomala beteende förstärks i underkylda områden, d. v. s. när väts-kan kyls ner under sin smälttemperatur. Detta kan förklaras med en hypotetiskandra kritisk punkt i det underkylda området där vatten är metastabilt. I ettspektakulärt experiment kyldes vattendroppar genom avdunstning till ungefär−46◦ C, där de förblev flytande i några millisekunder. Med hjälp av frielektronröntgenlasern LCLS i Stanford (USA) kunde det visas att vattens struktur änd-ras kontinuerligt och accelererat till en vätska med låg densitet när tempera-turen sjunker, precis enligt hypotesen. Vi jämförde Knudsens förångningsmo-dell, som vi använde att kalibrera temperaturen i experimentet, med resultatfrån molekylära simuleringar och resultaten visade en mycket bra överenstäm-melse. Beskrivningen av förångningsprocessen har en stor betydelse även förforskning inom atmosfärfysik och klimat.

För vätskan i vanliga temperaturområdet etablerade vi en ny standard försyre-syre parfördelningsfunktionen och undersökte det icke-triviala tempera-turberondet av svaga korrelationer så långt ut som 17 Å. Vi undersökte dess-utom vattens struktur vid ytan av en barium flourid (111) kristall, där det visadesig att vattens struktur liknar den av vatten vid högt tryck.

Vi kunde också visa i molekylära simuleringar av underkylt vatten att låg-frekvens vibrationsspektrumet uppvisar en topp som anknyter till en liknandetopp i spektrumet av amorfa material. Toppen uppstår när vatten kyls ner ochändra sin struktur till en mer ordnad vätska med låg densitet.

5. Sammanfattning

70

Acknowledgement

I would like to express my deep gratitude to my supervisor Lars G. M. Pet-tersson and my co-supervisor Anders Nilsson. Lars, thank you for your con-stant guidance and for always being available for discussions, for your strongsupport and patience. Our research was to a large extent directed by the ex-perimental activities and I’m grateful for this guidance and the greater vision,Anders. It was a privilege to work in this group and I’m grateful for all I’velearned about water, simulation and experimental techniques. I’d like to thankyou in particular for the numerous opportunities to go to schools, conferences,beam-times and collaborations.

I’d like to thank some former Ph. D. students of this group: Thor, thank youfor everything I’ve learned from you and your encouragement. It has alwaysbeen a challenge to follow your footprints. Thank you for taking over the roleas my mentor even officially, I couldn’t possibly have found a more suitableone. Micke, thanks for all your help with RMC and the scientific and non-scientific discussions, not least for your great help with housing in Stockholm– Punk’s not dead. And Henrik, your help with GPAW was crucial, thoughnot to the same extent as has been your open-hearted personality warmingthe office. Jonas, thank you for answering theoretician-questions about theexperiment and for scientific discussions, it has been really inspiring to workwith you.

The work in close collaboration with experimentalists has been a greatexperience, mainly due to Jonas (Nature guy), Fivos (man with blue glassesand a beer in a massage chair), Simon (presentation trainer and after-workconnection), Katrin and Peter (bbq masters), Harshad (room mate in Japan)and Stefano (THz). I also want to thank Lawrie, Sarp, Congcong and Hirohitofor collaborations and all I’ve learned from you about experiments; Jerry, well,for the experiments concerning certain solutions and a lot of insights into life.

I’d further like to thank all fellow Ph. D. students and collegues in thechemical physics division, particularly Jörgen, Jesper, Iurii, Oleksii, Rasmus(KTH), Kess, Ida, Sifiso, Oliver, Astrid, Henrik Öström and Richard. Henrik,I estimated that we spent of the order of 400 lunch breaks together, thanksfor those. Thanks also to the FysikShow crew who made teaching a magicexperience: Tor, David, Tanja, Sémeli and Kess; always remember Voland.

Imagine, how incredibly lucky I was to go to Göteborg to learn statisti-

5. Acknowledgement

cal mechanics of liquids and to get correlated with these “particles”: Joakim,Sasha and Aymeric. It has been a great pleasure to study with you. Sasha andJoakim, I’m not only grateful for all the discussions about MD simulations butalso for an uncountable number of badminton matches. Joakim, your intellectis much appreciated but your moral is concerning, we have to talk. Your prag-matic approach to life, Sasha, is admirable and will be missed, don’t forgetthe spot in Bergianska trädgården. Aleks and Zoltan, thank you for numerousdiscussions about science and free will.

I want to express my gratitude to my friends who are not related to work,particularly Mikko, Nathalie and Katrin, Mirko and Pryanka, Alfredo and Vivi,Sébastien and Taija and the gröna villan gang. Special thanks to Nada, the bestbar in the world-tribute for providing the living room.

I’d like to mention all my friends back in Germany, who still remember meeven though we haven’t met sufficiently often during the last five years. I wantto thank my family, Martin, Dorothea, Cornelia and Tobias, for unconditionallove and encouragement. Iida, your incredible patience and support have beenmost essential to me and so is your love, thank you!

Stockholm, September 2015Daniel Schlesinger

72

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