Download Structure, Dynamics and Thermodynamics of Liquid Water

Structure, Dynamics and Thermodynamics
of Liquid Water
Insights from Molecular Simulations
Kjartan Thor Wikfeldt
Thesis for the Degree of Doctor of Philosophy in Theoretical Physics
Department of Physics
Stockholm University
Stockholm 2011
c Kjartan Thor Wikfeldt
ISBN 978–91–7447–287–5
Printed by Universitetsservice US-AB, Stockholm
It is wrong to think that the task of physics is to find out how nature is.
Physics concerns what we can say about nature...
Niels Bohr
Abstract
Water is a complex liquid with many unusual properties. Our understanding
of its physical, chemical and biological properties is greatly advanced
after a century of dedicated research but there are still many unresolved
questions. If answered, they could have important long-term consequences
for practical applications ranging from drug design to water purification.
This thesis presents results on the structure, dynamics and thermodynamics
of liquid water. The focus is on theoretical simulations applied to
interpret experimental data from mainly x-ray and neutron scattering
and spectroscopy techniques. The structural sensitivity of x-ray and neutron
diffraction is investigated using reverse Monte Carlo simulations and
information on the pair-correlation functions of water is derived. A new
method for structure modeling of computationally demanding data sets
is presented and used to resolve an inconsistency between experimental
extended x-ray absorption fine-structure and diffraction data regarding
oxygen-oxygen pair-correlations. Small-angle x-ray scattering data are
modeled using large-scale classical molecular dynamics simulations, and the
observed enhanced scattering at supercooled temperatures is connected to
the presence of a Widom line emanating from a liquid-liquid critical point in
the deeply supercooled high pressure regime. An investigation of inherent
structures reveals an underlying structural bimodality in the simulations
connected to disordered high-density and ordered low-density molecules,
providing a clearer interpretation of experimental small-angle scattering data.
Dynamical anomalies in supercooled water observed in inelastic neutron
scattering experiments, manifested by low-frequency collective excitations
resembling a boson peak, are investigated and found to be connected to the
thermodynamically defined Widom line. Finally, x-ray absorption spectra
are calculated for simulated water structures using density functional theory.
An approximation of intra-molecular zero-point vibrational effects is found
to significantly improve the relative spectral intensities but a structural
investigation indicates that the classical simulations underestimate the
amount of broken hydrogen bonds.
i
Kjartan Thor Wikfeldt
Stockholm
May 2011
ii
List of Papers
I. Diffraction and IR/Raman Data Do Not Prove Tetrahedral Water
M. Leetmaa, K.T. Wikfeldt, M.P. Ljungberg, M. Odelius, J. Swenson, A.
Nilsson and L.G.M. Pettersson
J. Chem. Phys. 129, 084502 (2008)
II. On the Range of Water Structure Models Compatible with X-ray and
Neutron Diffraction Data
K.T. Wikfeldt, M. Leetmaa, M.P. Ljungberg, A. Nilsson and L.G.M.
Pettersson
J. Phys. Chem. B 113, 6246 (2009)
III. The inhomogeneous structure of water at ambient conditions
C. Huang, K.T. Wikfeldt, T. Tokushima, D. Nordlund, Y. Harada, U.
Bergmann, M. Niebuhr, T.M. Weiss, Y. Horikawa, M. Leetmaa, M.P.
Ljungberg, O. Takahashi, A. Lenz, L. Ojamäe, A.P. Lyubartsev, S. Shin,
L.G.M. Pettersson and A. Nilsson
Proc. Natl. Acad. Sci. (USA) 106, 15214 (2009)
IV. SpecSwap-RMC: a novel reverse Monte Carlo approach using a
discrete set of local configurations and pre-computed properties
M. Leetmaa, K.T. Wikfeldt and L.G.M Pettersson
J. Phys.: Cond. Matter 22, 135001 (2010)
V. Oxygen-oxygen correlations in liquid water: Addressing the
discrepancy between diffraction and extended x-ray absorption
fine-structure using a novel multiple-data set fitting technique
K.T. Wikfeldt, M. Leetmaa, A. Mace, A. Nilsson and L.G.M. Pettersson
J. Chem. Phys. 132, 104513 (2010)
VI. Enhanced small-angle scattering connected to the Widom line in
simulations of supercooled water
K.T. Wikfeldt, C. Huang, A. Nilsson and L.G.M. Pettersson
Submitted to J. Chem. Phys. (2011)
VII. X-ray diffraction study of temperature dependent structure of liquid
water
C. Huang, K.T. Wikfeldt, D. Nordlund, U. Bergmann, T. McQueen, J.
Sellberg, L.G.M. Pettersson and A. Nilsson
Submitted to
VIII. Possible origin of low-frequency excitations in supercooled bulk and
protein-hydration water
iii
P. Kumar, K.T. Wikfeldt, D. Schlesinger, L.G.M. Pettersson and H.E.
Stanley
Manuscript
IX. Bimodal inherent structure in simulated liquid water from 200 to
360 K
K.T. Wikfeldt, A. Nilsson and L.G.M. Pettersson
Submitted to Nature (2011)
Papers not included in this thesis
X. Reply to Soper et al.: Fluctuations in water around a bimodal
distribution of local hydrogen-bonded structural motifs
C. Huang, K.T. Wikfeldt, T. Tokushima, D. Nordlund, Y. Harada, U.
Bergmann, M. Niebuhr, T.M. Weiss, Y. Horikawa, M. Leetmaa, M.P.
Ljungberg, O. Takahashi, A. Lenz, L. Ojamäe, A.P. Lyubartsev, S. Shin,
L.G.M. Pettersson and A. Nilsson
Proc. Natl. Acad. Sci. (USA) 107, E45 (2010)
XI. Complementarity between high-energy photoelectron and L-edge
spectroscopy for probing the electronic structure of 5d transition
metal catalysts
T. Anniyev, H. Ogasawara, M.P. Ljungberg, K.T. Wikfeldt, J.B.
MacNaughton, L.Å. Näslund, U. Bergmann, S. Koh, P. Strasser,
L.G.M. Pettersson and A. Nilsson
Phys. Chem. Chem. Phys. 12, 5694 (2010)
XII. In situ X-ray probing reveals fingerprints of surface platinum oxide
D. Friebel, D.J. Miller, C.P. O’Grady, T. Anniyev, J. Bargar, U. Bergmann,
H. Ogasawara, K.T. Wikfeldt, L.G.M. Pettersson and A. Nilsson
Phys. Chem. Chem. Phys. 13, 262 (2010)
XIII. Increasing correlation length in bulk supercooled H2 O, D2 O, and
NaCl solution determined from small angle x-ray scattering
C. Huang, T.M. Weiss, D. Nordlund, K.T. Wikfeldt, L.G.M. Pettersson
and A. Nilsson
J. Chem. Phys 133, 134504 (2010)
XIV. Ab initio van der Waals interactions in simulations of water alter
structure from mainly tetrahedral to high-density-like
A. Møgelhøj, A. Kelkkanen, K.T. Wikfeldt, J. Schiøtz, J.J. Mortensen,
L.G.M. Pettersson, B.I. Lundqvist, K.W. Jacobsen, A. Nilsson and J.K.
Nørskov
Submitted to J. Phys. Chem. B (2011)
iv
List of Abbreviations
AFF – atomic form factor
AIMD – ab initio molecular dynamics
CPF – critical point free
EPSR – empirical potential structure refinement
EXAFS – extended x-ray absorption fine-structure
FS – fragile-to-strong
DD – double donor
∆KS – delta Kohn-Sham
DFT – density functional theory
DW – Debye-Waller
FMS – full multiple scattering
GGA – generalized gradient approximation
H-bond – hydrogen bond
HDA – high-density amorphous ice
HDL – high-density liquid
HH – hydrogen-hydrogen
INS – inelastic neutron scattering
IS – inherent structure
ISF – intermediate scattering function
IXS – inelastic x-ray scattering
LDA – local density approximation
LDA – low-density amorphous ice
LDL – low-density liquid
LFE – low-frequency excitation
LLCP – liquid-liquid critical point
LSI – local structure index
MAFF – modified atomic form factor
MD – molecular dynamics
MS – multiple scattering
ND – neutron diffraction
OH – oxygen-hydrogen
OO – oxygen-oxygen
OZ – Ornstein-Zernike
v
PCF – pair-correlation function
PES – potential energy surface
QENS – quasi-elastic neutron scattering
RMC – reverse Monte Carlo
SAXS – small-angle x-ray scattering
SD – single donor
SF – singularity-free
SL – stability-limit
SPC/E – extended simple-point charge
TIP4P – transferable interaction potential with 4 points
TMD – temperature of maximum density
TP – transition potential
vdW – van der Waals
VDOS – vibrational density of states
XAS – x-ray absorption spectroscopy
XD – x-ray diffraction
XES – x-ray emission spectroscopy
ZP – zero-point
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Contents
Abstract
i
List of Papers
iii
List of Abbreviations
v
1 Introduction
1
2 Structure of water
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Time-independent correlation functions . . . . . . . . . . .
2.2.1 Pair-correlation functions and structure factors . .
2.2.2 The Ornstein-Zernike equation . . . . . . . . . . .
2.2.3 Experimental methods: elastic scattering . . . . . .
2.3 Structure modeling . . . . . . . . . . . . . . . . . . . . .
2.3.1 Reverse Monte Carlo (RMC) . . . . . . . . . . . .
2.3.2 SpecSwap-RMC . . . . . . . . . . . . . . . . . . .
2.4 Many-body correlation functions . . . . . . . . . . . . . .
2.4.1 Bond angles . . . . . . . . . . . . . . . . . . . . .
2.4.2 Voronoi polyhedra . . . . . . . . . . . . . . . . . .
2.4.3 Local structure index . . . . . . . . . . . . . . . .
2.4.4 Experimental methods: EXAFS, XAS . . . . . . . .
2.5 Pair correlation functions of water . . . . . . . . . . . . .
2.6 Tetrahedrality and local structure index of simulated water
2.7 Long-range correlations . . . . . . . . . . . . . . . . . . .
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3 Dynamics of water
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
3.2 Time-dependent correlation functions . . . . . . . . .
3.2.1 van Hove correlation functions . . . . . . . .
3.2.2 Velocity and current autocorrelation functions
3.2.3 Experimental methods: inelastic scattering . .
3.3 Molecular dynamics . . . . . . . . . . . . . . . . . .
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CONTENTS
3.4
Low-frequency excitations in supercooled water . . . . . . . . . .
3.4.1 Finite-size effects in simulations . . . . . . . . . . . . . .
3.4.2 Dynamic correlation functions . . . . . . . . . . . . . . .
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4 Thermodynamics of water
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Theoretical scenarios . . . . . . . . . . . . . . . . . . . . . .
4.3 Thermodynamic response functions from molecular simulations
4.4 Enhanced small-angle scattering and the Widom line . . . . .
4.5 Bimodality in inherent structures . . . . . . . . . . . . . . . .
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5 Electronic structure of water
5.1 Theoretical background . . . . . . . . . . . . . . . . . . .
5.1.1 Density functional theory . . . . . . . . . . . . . .
5.1.2 Theoretical spectrum calculations . . . . . . . . .
5.2 Extended x-ray absorption fine-structure . . . . . . . . . .
5.2.1 Experimental results . . . . . . . . . . . . . . . . .
5.2.2 Theoretical calculations and structural information
5.3 X-ray absorption spectroscopy . . . . . . . . . . . . . . .
5.3.1 Experimental results . . . . . . . . . . . . . . . . .
5.3.2 Importance of zero-point vibrational effects . . . .
5.3.3 Structural sensitivity of XAS . . . . . . . . . . . .
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6 Summary of main results
6.1 Water, tetrahedral or not - Paper I . . . . . . . . . . . . . . . . .
6.2 Bounds on liquid water structure from diffraction - Paper II . . . .
6.3 Inhomogeneous structure of water - Paper III . . . . . . . . . . .
6.4 SpecSwap-RMC: a new multiple-data set structure modeling
technique - Paper IV . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Inconsistency between diffraction and EXAFS on the first
coordination shell in water - Paper V . . . . . . . . . . . . . . . .
6.6 Enhanced small-angle scattering and the Widom line in supercooled
water - Paper VI . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Improved x-ray diffraction experiments on liquid water - Paper VII
6.8 The boson peak in supercooled water - Paper VIII . . . . . . . . .
6.9 Bimodal inherent structure in simulated ambient water - Paper IX
6.10 A note on my participation in Papers I-IX . . . . . . . . . . . . .
71
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7 Conclusions and Outlook
85
Acknowledgements
89
Sammanfattning
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References
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viii
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1
Introduction
It is no wonder that water has attracted the attention of physicists, chemists,
biologists and scientists in other related fields. Water is the “matrix of life” [1]
and as such, to the best of our knowledge, essential to all forms of life. It
is the environment in which most chemical reactions take place on Earth
and it comprises the major driving force in molding the geology of Earth
as well as controlling its climate. Due to its energetic stability and relative
simplicity, the water molecule is one of the most abundant molecules in
the universe, and the formation of more complex molecules in interstellar
space is believed to be catalyzed by the presence of amorphous solid water
on the surface of dust grains [2]; indeed, it has been suggested that life on
Earth was seeded by the arrival of complex molecules formed in outer space.
That water is important for the scientific understanding in a large range of
disciplines is not the sole reason for investigating its properties. More than a
century’s worth of systematic research has revealed that condensed-phase
water is an unusually complex substance with many properties that can
rightly be termed anomalous since they differ from the properties of other
substances. Historically, the most well known anomaly is the fact that ice
has lower density than liquid water, discussed already by Galileo Galilei in
1612 1 . Compilations of water anomalies can be found on the world wide
web 2 , but keeping such a list up-to-date may be difficult since discoveries of
new anomalies are regularly reported even in recent years [3, 4].
The long history of water research alluded to above has taken many
twists and turns. More so than for many other topics, water research has
undergone bursts of controversies and conflicting experimental and/or
theoretical results. At present, two debates relevant to the topics addressed
in this thesis concern the interpretation of x-ray spectroscopic measurements
on liquid water and the possible existence of two distinct liquid phases in
1 G.
Galilei, Discorso intorno alle cose che stanno in su l’acqua, o che in quella si muovono (1612)
2 http://www.lsbu.ac.uk/water/
1
1 Introduction
the metastable supercooled region of the phase diagram. A third debate,
which is also given ample room in this thesis, but which can perhaps be
considered to be resolved in light of the most recent investigations, concerns
the height and shape of the first peak of the oxygen-oxygen pair-correlation
function (PCF), gOO (r ). Diffraction experiments provide information on the
PCFs in liquid water, but experimental challenges and sometimes conflicting
methods of analysis have rendered an exact determination of their form
elusive. Furthermore, a recent application of the extended x-ray absorption
fine-structure (EXAFS) technique [5] to study the local coordination in liquid
water caused some confusion since the reported gOO (r ) deviated strongly
from diffraction results. Work described in this thesis has helped shed light
on these issues by, on one hand, thoroughly investigating the structural
content of diffraction data, and, on the other, developing and applying a
novel structure-modeling technique to address the discrepancy between
diffraction and EXAFS.
Of a very different nature, the question whether water can exist in two
distinct liquid phases, accompanied by a liquid-liquid critical point (LLCP),
has been actively investigated since the scenario was proposed in 1992 [6].
By many researchers it is considered to be the most likely scenario for real
liquid water. Its basic tenet is the first-order liquid-liquid phase transition
between a high-density liquid (HDL) and a low-density liquid (LDL) at
deeply supercooled temperatures. So far this transition has been impossible
to observe directly in experiments due to the homogeneous nucleation into
ice several degrees above the anticipated phase transition temperature, and
thus remains unproven and somewhat controversial. Research based on
molecular simulations presented in this thesis provides support for the LLCP
scenario by connecting experimentally observed small-angle x-ray scattering
(SAXS) data of supercooled water with thermodynamic behavior pertinent
to the LLCP hypothesis.
Lastly, the x-ray spectroscopy debate revolves around the interpretation
of x-ray absorption and emission spectroscopies (XAS and XES) applied to
water. In 2004, it was proposed [7] that a strong pre-edge peak in the liquid
XAS spectrum and overall large differences between spectra from hexagonal
ice and liquid water originated from a large amount of broken hydrogen (H-)
bonds in the liquid, beyond what is expected from the standard textbook
picture of water structure. In 2009, the XAS results were combined with XES
spectra and SAXS results in a unified interpretation [8] supporting the LLCP
hypothesis, and extending this picture to the ambient regime by suggesting
that pronounced fluctuations between LDL-like and HDL-like local structures
take place in water all the way up to the boiling point. Both the XAS and XES
interpretations are still controversial, but further work by many independent
research groups will hopefully establish a consensus in the coming years.
Broadly speaking, this thesis deals with topics relating to the structure,
dynamics, thermodynamics and electronic structure of liquid water. A list of
2
this sort may give the false impression that a comprehensive account of these
most diverse aspects is to follow. Because of the sheer amount of accumulated
knowledge on the physical nature of water, such a task would be impossible.
Instead, a limited range of topics roughly corresponding to the contents of
Papers I-IX listed above will be discussed. When relevant, extra material not
contained in the papers will be presented.
3
2
Structure of water
This chapter will deal with various aspects of the structure of liquid water.
The theory of static correlation functions will be reviewed and two structural
modeling techniques which have been applied in Papers I, II, IV and V,
i.e. reverse Monte Carlo (RMC) and SpecSwap-RMC, will be described. A
few results on the pair-correlation functions and many-body correlation
functions of water will then be presented, followed by results on long-range
correlations and the large-scale structure of water.
2.1
Introduction
All atoms in a liquid are in constant motion and the microscopic structure
is constantly changing due to diffusion, vibrations around local or semilocal potential energy minima and collective structural relaxation events. The
dynamics and thermodynamics of supercooled liquids is sometimes analyzed
in the inherent structure (IS) formalism [9, 10], which is the configuration of
atoms obtained by quenching the liquid structure to its nearest minimum on
the multidimensional potential energy surface (PES). One can view the IS
as an “underlying” structure of the liquid at any instantaneous moment in
time [9]. The real structure of a liquid, where thermally excited motion can
be seen as being superimposed on the IS, is at non-zero temperatures shifted
above local minima of the PES. In water, fluctuations between different local
structures associated with qualitatively different PES minima can nonetheless
be anticipated. This is due to the anisotropic intermolecular potential and
the competition between energetic stabilization, realized in open local
structures with strong hydrogen (H-) bonds such as in hexagonal ice, and
entropic stabilization obtained by a denser local structure with population
of interstitial sites and perturbed H-bonding geometries. What makes water
unusual is the fact that strong intermolecular bonding correlates with a
below-average local density, a property also seen in other tetrahedral liquids
5
2 Structure of water
such as silica and germanium. Many of the thermodynamic anomalies of
water are indeed believed to originate from the formation of local regions
of strongly H-bonded molecules, the rate of which rapidly increases at
decreasing temperatures and in particular in the supercooled region. The
temperature of maximum density at 4◦ C, below which the liquid density
decreases far into the supercooled regime, directly reflects this formation
of low-density strongly bonded regions. An important question is whether
the incommensurate driving forces for energy- and entropy-stabilization
ultimately lead to the existence of distinct structurally different liquid
phases, separated by a phase transition in the supercooled region. This
topic will be discussed in Chapter 4. Here, the focus will be on structural
properties as quantified by pair- and many-body correlations, but several
connections to the thermodynamic behavior will be established.
2.2
Time-independent correlation functions
2.2.1
Pair-correlation functions and structure factors
The particle density of a system can be defined microscopically as [11, 12]
N
ρ(r) =
∑ δ(r − ri )
(2.1)
i =1
where N is the number of particles and the average density at point r (singleparticle density) is the ensemble average: ρ(1) (r) = hρ(r)i. The mean number
R
density is the average over all space: V1 ρ(1) (r)d3 r = ρ.
Pair-correlation functions (PCFs), sometimes called radial distribution
functions (RDFs), denoted g(r ) can be defined as the ensemble average over
delta functions centered on all interatomic distances present in the system,
normalized by the number of particles:
*
+
N N
1
ρg(r ) =
(2.2)
∑ δ(r − rj + ri ) .
N i∑
=1 j 6 = i
In molecular simulations, the PCF between two species α and β is more
conveniently calculated from distance histograms:
gαβ (r ) =
nαβ (r )
4πρr2 ∆r
(2.3)
where nαβ (r ) is the histogram. Generally, PCFs can be calculated out to one
half of the box length of the simulated system, rmax = L/2. By changing the
6
2.2 Time-independent correlation functions
normalization in eq. (2.3), distances out
√ to the edges of the simulation box
can also be used, obtaining rmax = L/ 2.
Higher-order correlation functions, such as three-body functions, are
calculable on given atomic structures but cannot in most cases be directly
obtained experimentally. Note that in a system sensitive to many-body
correlations, such as liquid water, all higher-order correlations are to a
certain extent reflected in the two-body partial pair-correlation functions.
However, the problem of extracting many-body correlation functions from
the pair-wise ones is not uniquely solvable.
One often needs the microscopic density in k-space. It is obtained from
the Fourier components of the density in eq. (2.1), i.e.
N
ρk =
∑ exp (−ik · ri ) .
(2.4)
i =1
From this representation one obtains the structure factor as
S(k) =
1
hρ ρ i .
N k −k
(2.5)
This equation can be used to calculate the structure factor from a molecular
simulation. For isotropic systems, another approach is to reexpress the
structure factor as the Fourier transform of g(r ) [12] as
S(k)
= 1+ρ
= 1+ρ
Z
g(r ) exp(−ik · r)d3 r
Z
( g(r ) − 1) exp(−ik · r)d3 r + ρ
= 1 + 4πρ
Z ∞
0
r 2 ( g (r ) − 1)
Z
exp(−ik · r)d3 r
sin kr
dr + ρδ(k )
kr
(2.6)
where k = |k|. The delta function at k = 0 has no effect since the forward
scattering direction is not observable experimentally. In multicomponent
systems the generalization of the structure factor is
Sαβ (k ) = 1 + cα c β 4πρ
Z ∞
0
r2 ( gαβ (r ) − 1)
sin kr
dr
kr
(2.7)
where ci is the concentration of the i’th component. This quantity is called the
partial structure factor and the corresponding functions in real space, gαβ (r ),
are the partial pair-correlation functions.
The low-k limit of the structure factor describes long-wavelength density
fluctuations, and in the thermodynamic limit k → 0 there exists a relation to
the isothermal compressibility,
S(0) = ρk B Tκ T
7
(2.8)
2 Structure of water
where ρ is the number density, N/V. This useful relation can be applied
to extrapolate simulated or experimental small-angle structure factors to the
thermodynamic limit.
2.2.2
The Ornstein-Zernike equation
Ornstein-Zernike (OZ) theory is a widely applied theoretical correlation
function framework in the description of critical phenomena [13]. At its
foundation is the OZ integral equation,
h(r12 ) = c(r12 ) + ρ
Z
c(r13 )h(r23 )d3 r3
(2.9)
where rij = rj − ri , h(r ) = g(r ) − 1 is called the total correlation function
and c(r ) has been introduced as the direct correlation function. ρ is the
number density as before. This decomposition of the total correlation
function can be intuitively understood; the total correlation function between
two particles is the sum of their direct correlation and the indirect correlations
propagated via all other particles in the system.
The Ornstein-Zernike equation can be expressed in k-space by
multiplying through with exp(ik · r12 ) and integrating over r1 and r2 .
The translational invariance of the system is necessary to complete the
derivation; the left hand side of eq. (2.9) is for example Fourier transformed
as
Z
Z
V1
V2
h(|r2 − r1 |)eik·(r2 −r1 ) d3 r1 d3 r2
=V
Z
0
V0
h(r 0 )eik·r d3 r0
=
Z
orig
d3 rorig
= V h̃(k)
Z
0
V0
h(r 0 )eik·r d3 r0
(2.10)
where we have translated the origin to r2 and integrated over its position,
and the Fourier transform of h(r ) is denoted h̃(k ). The second term on the
right hand side of eq. (2.9) is calculated in the same way, except the origin is
translated to r3 . Each term gives an integral over all space, yielding a volume
factor V which of course diverges, but this divergence is cancelled. Thus, in
k-space the OZ equation takes the form
h̃(k ) = c̃(k ) + ρ h̃(k )c̃(k ).
(2.11)
Since the Fourier transform of the total correlation function can also be
expressed in terms of the structure factor as (see eq. (2.6))
S(k ) = 1 + ρh̃(k )
8
(2.12)
2.2 Time-independent correlation functions
one can use the OZ equation in k-space to arrive at the relation
1
.
1 − ρc̃(k )
S(k) =
(2.13)
The Ornstein-Zernike approximation [13] consists of expanding the direct
correlation function in powers of k as
∞
c̃(k ) =
∑ al (ρ, T )kl
(2.14)
l =0
in which all terms with odd powers of k can be seen to vanish upon angular
integration of the odd derivatives of the Fourier transform of c(r ), and
finally truncating this expansion at the second power of k. This is a good
approximation for small k. One finally arrives at the expression
S(k) =
C (ρ, T )
ξ −2 + k 2
(2.15)
where C (ρ, T ) is related to the fourth moment of the function c(r ) and
depends on temperature and density, and ξ has been introduced as a
variable that has the dimension of length and is called the correlation length.
The large-r asymptotic behavior of the pair-correlation functions is therefore
g (r ) − 1 ∝
e−r/ξ
.
r
(2.16)
ξ is interpreted as the length-scale of correlations between density
fluctuations. A signature of critical (diverging) fluctuations is that ξ increases
sharply upon approaching the critical region and can be described by a
power law,
ξ = ξ 0 e−ν
(2.17)
where e = T/TS − 1, TS is the critical (or singular) temperature and ν is a
critical exponent [13].
A common way to analyze the critical behavior of both response functions
and the small-angle structure factor is to make a division into “normal”
(regular) and “anomalous” (singular) parts. The OZ expression for S(k ) is
taken to be the anomalous part,
S ( k ) = Sn ( k ) +
C (ρ, T )
ξ −2 + k 2
(2.18)
and ξ is obtained by assuming a functional form of the normal part
Sn (k ) and fitting eq. (2.18) to experiments or simulation data. A constant
background, Sn (k ) = const, is sometimes assumed [14], but a more physical
9
2 Structure of water
approximation [15] consists of using the structure factor of a fluid of
hard-spheres in the Percus-Yevick approximation [16],
# −1
η (3 − η 2 ) − 2 j1 (kσ)
Sn (k ) ∝ 1 − 12η
kσ
(1 − η )4
"
(2.19)
where j1 is the first order Bessel function, σ is the hard sphere diameter and η
is the volume fraction (η = πρσ3 /6). In Papers III, VI and XIII this approach
was successfully applied to fit simulated and experimental S(k ) and extract
correlation lengths, which showed power-law behavior according to eq. (2.17)
at supercooled temperatures.
2.2.3
Experimental methods: elastic scattering
The wide-angle region
The primary experimental probes of structure factors and PCFs in crystalline
and amorphous compounds are elastic scattering techniques, where neutron
and x-ray diffraction (ND and XD) play the most important role. Neutrons
interact predominantly with atomic nuclei and the strength of the interaction
is quantified by the atomic scattering length b, which is a scalar quantity
(since nuclei are to a good approximation point-like particles in this context)
that varies irregularly across the periodic table. The neutron scattering
intensity (or differential cross section) can therefore be written as
I (k)
≡
dσ
=
dΩ
*
N
=
2 +
*
N
∑ bi exp(−ik · ri )
i =1
N
∑ ∑ bi bj exp(−ik · (rj − ri ))
+
(2.20)
i =1 j =1
where the angular brackets denote the ensemble and orientational averages.
To obtain a useful expression, we introduce the following notation for the
scattering lengths [12] :
D E D E
bi2 ≡ b2 ,
bi b j = h bi i b j ≡ h b i 2
D E
2
2
,
b2 − hbi2 ≡ binc
.
(2.21)
hbi2 ≡ bcoh
10
2.2 Time-independent correlation functions
The neutron scattering intensity can then be written as
*
+
D E
N N
2
2
I (k) = N b + hbi
∑ ∑ exp(ik · rij )
i =1 j 6 = i
=
2
Nbinc
+
2
Nbcoh
S(k)
(2.22)
where the connection to the structure factor is established. binc is the
incoherent part of the neutron scattering, which is particularly large for the
hydrogen atom, while bcoh is the coherent part which provides the sensitivity
of neutrons to intermolecular structure. Note that binc and bcoh represent
isotopic averages over the isotopes present in the scattering volume.
An equivalent expression can be obtained for x-ray scattering by summing
over electrons instead of atomic nuclei in eq. (2.20), and viewing the factors
bi as electron scattering lengths. A summation over all electrons of an atom
yields the atomic form factor (AFF)
*
+
Z
f 0 (k) =
∑ bi
(el )
exp (ik · (ri − r))
i =1
(2.23)
QM
where in addition to the angular averaging the quantum mechanical
expectation value is also performed. This is equivalent to taking the Fourier
transform of the atomic electron density, which for a spherically symmetric
distribution simplifies to
f 0 (k ) = 4π
Z ∞
0
r 2 ρ (r )
sin kr
dr
kr
(2.24)
It can be noted that a general expression for the AFF is [17]
f (k, E) = f 0 (k ) + f 0 ( E) + i f 00 ( E)
(2.25)
where it is seen that an energy-dependent correction with real and complex
parts contributes to the total AFF. This dispersive correction only becomes
appreciable when the energy of the incoming photons approaches absorption
edges in the atom. Away from absorption edges the normal coherent form
factor f 0 (k ) dominates and the index 0 will be dropped in the following.
Integrating ρ(r ) over all space gives the total number of electrons Z of
the atom, which means that f (k = 0) = Z; form factors thus increase with
atomic number. This implies that XD measurements on water are dominated
by the scattering from electrons located around the oxygen atoms, and a
Fourier transform of the XD structure factor gives to a good approximation
11
2 Structure of water
the oxygen-oxygen PCF. The final expression for x-ray scattering becomes
*
+
I (k)
=
N f ( k )2 + f ( k )2
N
N
∑ ∑ exp(ik · rij )
i =1 j 6 = i
=
N f (k)
1
1+
N
2
*
N
N
∑ ∑ exp(ik · rij )
+!
= N f (k)2 S(k). (2.26)
i =1 j 6 = i
Using AFFs for oxygen and hydrogen, obtained from Dirac-Fock calculations
and listed in crystallographic tables [18], for the analysis of XD data on
water may introduce uncertainties. This is because of the electronegativity
of the oxygen atom which polarizes the water molecule through a transfer of
electrons from the covalently bonded hydrogen atoms to the oxygen, thus
significantly modifying the electron density of the atoms away from the
gasphase electron densities for which the form factors have been tabulated. A
correction procedure suggested in ref. [19] modifies the form factors to take
the polarization effect into account:
2
2
(2.27)
f i0 (k) = 1 + αi e−k /2δi f i (k )
where α describes the charge transfer and δ can be tuned to realistically
represent the enhanced delocalization of the valence electrons due to covalent
bonding.
Total x-ray and neutron diffraction structure factors of multicomponent
systems are combinations of the partial structure factors, defined in eq. (2.7).
In the convention used in RMC simulations (a structure-modeling technique
to be described below), the neutron and x-ray total structure factors are
written as
S ND (k ) = ∑ ∑ cα c β hbα i bβ Sαβ (k ) − 1
(2.28)
α
SXD (k)
=
β
∑ ∑ cα c β f α (k ) f β (k )
α
β
Sαβ (k) − 1 /
∑ ci f i ( k )
!
2
(2.29)
i
where the relative weights of each partial is determined by the scalar
scattering lengths bi in the case of neutrons and the k-dependent form
factors f i (k ) for x-rays. Table 2.1 shows the weights of the O-O, O-H and
H-H partials to the total neutron structure factors, for different isotopic
mixtures between H2 O and D2 O, and for the x-ray structure factor for the
value k = 2 Å−1 (position of the first peak in SXD (k )) for both the normal
AFF and the modified AFF according to eq. (2.27) with realistic values of α
and δ. As seen, there is a large sensitivity to the O-H (O-D) and H-H (D-D)
12
2.2 Time-independent correlation functions
partials in ND, in contrast to XD where the largest contribution by far is
the O-O partial. An inherent difficulty in ND experiments is however the
large incoherent background from hydrogen atoms, limiting the accuracy in
measurements on isotopic mixtures of water containing a large fraction of
H2 O.
Table 2.1: The contribution of each partial structure factor to the total ND and XD structure
factors. Note that the O-H partial is added in negative for the two most hydrogeneous
samples since hydrogen has a negative scattering length. The last two lines show the partial
contributions to the x-ray structure factor at k = 2 Å−1 , the position of the first maximum
of SXD (k), using the unmodified atomic form factors and the modified form factors with
α H = 0.5, corresponding to a charge transfer of 0.5 electrons from each hydrogen atom to
the oxygen (see eq. (2.27)) which is seen to give the best fit to XD data in RMC simulations.
Data set
100% D2 O
75% D2 O
50% D2 O
25% D2 O
100% H2 O
XD α H = 0.0
XD α H = 0.5
% O-O
9.2
17.3
44.1
51.6
19.1
70.2
79.8
% O-H
42.3
48.6
44.6
40.4
49.2
27.1
19.1
%H-H
48.6
34.1
11.3
8.0
31.7
2.6
1.1
X-ray diffraction experiments reported in Paper VII were performed
at the Stanford synchrotron radiation light source (SSRL). A conventional
approach was used with a monochromatic beam from the synchrotron
and angular-dispersive setup obtaining a k-range of 0.5 to 16.0 Å−1 ,
where k = 4π sin(θ )/λ (λ=0.7 Å is the incident beam wave length and
2θ is the scattering angle). The water sample was a free-flowing water jet
in a helium environment which eliminated the need to subtract sample
holder scattering. Compton scattering was experimentally eliminated by
using a Germanium crystal analyzer with high energy resolution and thus
filtering out scattering intensity outside of the elastic peak. The free-flowing
water jet and the experimental elimination of Compton scattering are two
significant improvements over previous experiments. The experimental setup
is schematically shown in fig. 2.1 and further details are discussed in Paper
VII.
The small-angle region
The small momentum-transfer region of the structure factor contains
information on long-range correlations and density fluctuations, and in
the limit k → 0 the macroscopic value of the compressibility is obtained,
13
2 Structure of water
Ge(111) crystal
entrance slits
2θ
exit slits
beamstop
x-ray beam
water jet
PMT
Figure 2.1: X-ray diffraction experimental setup. 2θ is the scattering angle and PMT is the
photo-multiplier tube which registers scattered x-ray photons.
see eq. (2.8). Small-angle scattering techniques are employed to study
this region, i.e. small-angle neutron scattering (SANS) and small-angle
x-ray scattering (SAXS). Which approach is chosen for the analysis of
experimental small-angle scattering data depends on the system under
study. The study of macromolecules in solution requires analyses in terms
of particle form factors, where information on the shape and size of the
macromolecule can be obtained [20]. Two-component systems can be
analyzed by decomposing experimental data into concentration and number
density fluctuations, i.e. Bhatia-Thornton structure factors [21]. An excess
scattering at k = 0 above the compressibility limit can in this case be
observed from concentration fluctuations. One-component systems, such as
water, need to be analyzed differently, since a description in terms of particle
form factors is inappropriate due to the constantly rearranging H-bond
network and concentration fluctuations can be ruled out since even if there
could exist distinct types of structural environments there is no conservation
of different chemical species. The Ornstein-Zernike relation described above
can however be used to decompose experimental data into “normal” and
“anomalous” parts, where the latter is taken to be the OZ structure factor;
this is the approach described in connection with eq. (2.18) above.
14
2.3 Structure modeling
2.3
Structure modeling
The possibility to build three-dimensional structure models consistent
with experimental data has significantly impacted our understanding of
the structure of disordered matter. Several methods are in use today, two
common approaches being Reverse Monte Carlo (RMC) [22] and Empirical
Potential Structure Refinement (EPSR) [23]. Both are geared towards
structure modeling of neutron and x-ray diffraction data, but both can in
principle also handle EXAFS data, although only in the single-scattering
approximation [24].
2.3.1
Reverse Monte Carlo (RMC)
The RMC method is based on and inspired by the Metropolis Monte Carlo
(MMC) algorithm. While configurations of atoms in MMC are generated
to comply to a Boltzmann distribution of energies (where the energy is
calculated using a model potential), the RMC method instead samples
structures that agree with imposed experimental or geometrical constraints.
Contrary to MMC, RMC does not strictly sample structures from a known
thermodynamic ensemble [25, 26]; one should rather view RMC simply as
a multidimensional fit of structure models that obey the given constraints
sufficiently well. A RMC simulation is based on the following basic steps
(here we choose for illustration the fit of experimental structure factors,
but RMC can be applied to any properties calculable sufficiently fast from
atomistic structures):
1. Start with an initial configuration of atoms; it can either be a random
structure or a structure obtained by other simulations, e.g., a hardsphere Monte Carlo run.
2. Calculate gαβ (r ) between all unique pairs of atoms. Fourier transform
gαβ (r ) to obtain partial structure factors Sαβ (k ) and combine into total
structure factors S(k ).
3. Calculate the error function, which is determined by the difference
between the calculated and fitted quantities, as
χ2o =
Ndata Npoints
∑ ∑
M =1 i =1
C,i
E,i
fM
− fM
i )2
(σM
2
(2.30)
where the sums are over all supplied data sets, e.g. x-ray and neutron
C,i
structure factors, and their corresponding data points. f M
is the value
E,i
of the computed quantity for data set M and data point i and f M
is the
i
value which is being fit. σM is the inverse weight given to each data set
and data point which determines its contribution to the error function.
15
2 Structure of water
4. Move one atom randomly. If two atoms come too close (closer than
a predefined cutoff distance) reject the move. Otherwise update all
computed quantities, e.g., structure factors.
5. Compute a new value of the error function, χ2n . If χ2n < χ2o the move
is accepted and the new configuration becomes the old one. If χ2n >
χ2o the move is accepted with a probability given by min(1.0, exp(-(χ2n χ2o ))) with the new configuration replacing the old, thus allowing some
moves that worsen the agreement with the data.
6. Iterate from step 4 and continue until the simulation oscillates around
a converged χ2o . Collect statistics on correlation functions and structural
properties.
2.3.2
SpecSwap-RMC
A generalization of RMC to other types of experimental data sets on
disordered materials is provided by the SpecSwap-RMC method, introduced
in Paper IV and applied in Paper V. RMC is inherently limited to data sets
that can be computed on-the-fly from atomistic RMC structure models,
but many properties, such as x-ray and vibrational spectroscopies, require
time-consuming quantum chemical calculations. This makes it practically
impossible to model these data in normal RMC since each atomic move
and consecutive updating of calculated properties would take hours or days
on the computer. SpecSwap was developed to overcome this limitation by
discretizing configurational phase space; specifically, SpecSwap replaces the
periodic simulation box of atoms in normal RMC by a finite but large number
of clusters of atoms, the relevant properties of which can be computed prior
to the actual fit to experimental data. It is important that the atomic clusters
thus used as a basis in coordinate space contain structures representative for
the disordered material being modeled.
In a typical SpecSwap-RMC run the basis set of atomic clusters is
extracted from previous normal RMC runs and/or MD simulations and
possibly augmented with other plausible structures obtained from other
types of simulations. Generally the average over the original structural basis
set does not give agreement with the (computationally demanding) data set
being modeled; the whole purpose of SpecSwap is to reweight local structures
found in the basis set to better conform to the modeled data. After a
successful run, a reweighted linear combination of basis elements (i.e. atomic
clusters along with their calculated properties) reproduces the modeled data.
Each basis element has acquired a weight, wi , signifying its importance in the
basis set expansion in reproducing the data, and these weights can be used
to calculate reweighted geometrical properties of the basis set. Comparison
to the unweighted properties of the basis set then reveals the geometrical
information imposed on the weights in the basis set expansion through the
16
2.4 Many-body correlation functions
data modeling. Unweighted and reweighted properties are calculated as
P=
1
N
N
∑ pi
Pw =
i
1
N
N
∑ wi p i
(2.31)
i
where summation is taken over all elements in the basis set, P and Pw are
the averaged unweighted and reweighted properties and pi is the property
of basis element i. Note that any property can in principle be applied, given
that it can be precomputed and numerically stored for a large set of atomic
clusters. Similarly to RMC, geometrical constraints can be used alongside the
data modeling to test structural hypotheses or to investigate the structural
content and sensitivity of the data.
2.4
Many-body correlation functions
Diffraction experiments probe the pair-correlations present in a system.
These may determine higher-order correlations in a crystalline system, but
for disordered materials there is a large range of many-body correlation
functions that are equally consistent with the same set of pair-correlation
functions. The many-body correlation functions g(r1 , r2 , . . . , r N ) are multidimensional and difficult to represent graphically. It is therefore convenient
to investigate different projections of the many-body correlation functions
onto fewer dimensions to quantify structural order. Various many-body
correlation functions which are important measures of local structural order
in liquid water will be presented below.
2.4.1
Bond angles
Liquid water is characterized by strongly orientational interactions due to the
shape of the water molecule, where the intramolecular HOH angle is close to
the tetrahedral angle 109◦ , and the possibility for each molecule to form up
to four strong H-bonds with its nearest neighbors. Two important structural
parameters quantifying this order are, firstly, ensemble distributions of
angles formed between the donating O-H bond of one molecule and the
accepting oxygen atom of another, and secondly the angles formed between
three nearest neighbor oxygen atoms.
A geometric criterion for H-bonds, specifying whether a bond is intact or
broken, was suggested by Wernet et al. [7]. It is based on results from the
comparison between calculated and experimental XAS spectra, where one
broken H-bond on the donating side of a water molecule was required to
reproduce the strong pre-edge feature seen in the experimental spectrum of
17
2 Structure of water
liquid water. The criterion for a H-bond to be intact is written
max
rOO < rOO
− 0.00044(∆α HOO )2
(2.32)
where the distance between nearest neighbor oxygen atoms rOO must lie
max , and the squared
within the cone defined by a maximum O-O distance rOO
deviation of the H-bond angle α HOO from linearity. Figure 2.2 illustrates the
α HOO angle and the rOO distance for a dimer.
θ
roo
α
Figure 2.2: Left: Illustration of the parameters rOO and α used for classifying H-bonds in
eq. (2.32). Right: The angle θ between nearest-neighbor oxygen atom triplets, used in the
definition of the tetrahedrality parameter in eq. (2.33).
A commonly used structural order parameter for tetrahedral substances
is the tetrahedrality parameter Q [27, 28]. It is based on the angles formed
between nearest neighbor oxygen atom triplets and can be written for a
molecule i as
4 3 3
1 2
Q (i ) = 1 − ∑ ∑
cos(θ jik ) +
(2.33)
8 j =1 k = j +1
3
where the sums over j and k run over the four nearest neighbors of molecule
i. The angle θ is illustrated in fig. 2.2. Since the cosine of the tetrahedral
angle 109.5◦ equals -1/3, a perfectly tetrahedral system, such as hexagonal
ice, will have Q(i ) = 1 for all molecules. By integrating the angle θ over
a random distribution, such as for an ideal gas, one obtains the ensemble
average <Q>=0 [29]. Q is thus a convenient structural order parameter for
water since water molecules can be found in environments ranging from
ice-like, giving a Q value near 1, to disordered where Q takes low values.
2.4.2
Voronoi polyhedra
Since liquids lack geometrical symmetries there is no unique Wigner-Seitz
cell, as in the case of crystalline systems, to describe the local coordination.
One can nonetheless define the so called Voronoi cell around any given
18
2.4 Many-body correlation functions
particle in a liquid; it is defined as the polyhedron whose volume is given
by the set of points lying closer to the given particle than to any other
particle in the system. Voronoi polyhedra can be used in various ways to
quantify local order present in water by calculating the ensemble averages
over polyhedra. A useful parameter which quantifies the local order in a
liquid is the asphericity parameter, defined in terms of the volume V and
area A of the Voronoi polyhedron of a particle as [30]
η (i ) =
A3i
36πVi2
.
(2.34)
For water, η takes the value 2.25 for a molecule in a perfect hexagonal ice
environment, while lower values indicate a larger degree of disorder. The
limiting value is η = 1, which is however never realized since it corresponds
to a perfect sphere. In contrast to Q, η takes into account higher coordination
shells since neighboring molecules beyond the 4th nearest neighbor add
surfaces to the Voronoi cell, thus making it more spherical.
2.4.3
Local structure index
A structural order parameter which correlates with the tetrahedrality
parameter Q, but is perhaps more appropriate to describe fluctuations
between ordered and disordered local environments in liquid water, is the
local structure index (LSI) [31, 32]. Its definition is somewhat more involved
than that for Q: for each molecule i, arrange the nearest neighbors j in
descending order of proximity to molecule i as r1 < r2 < · · · < r j < r j+1 <
· · · < rn(i) < 3.7 Å < rn(i)+1 , where n(i ) is the index of the distance closest to
3.7 Å from below. The LSI, I (i ), is then defined as
I (i ) =
1
n (i )
n (i )
∑ [∆( j; i) − ∆¯ (i)]
2
(2.35)
j =1
¯ (i ) is the average of ∆( j; i ) over the neighbors
where ∆( j; i ) = r j+1 − r j and ∆
j of molecule i. A molecule with a well separated first coordination shell
will display a large value of I while a molecule in a disordered environment
with neighboring molecules in interstitial positions around 3.7 Å will have
low values of I. This structural parameter has been shown to exhibit a clear
bimodality at supercooled [33] and ambient temperatures (Paper IX), but only
in the so called inherent structure (IS). The IS is obtained by an instantaneous
energy minimization to the nearest local minimum on the potential energy
surface, thus removing thermally activated vibrations around that minimum.
19
2 Structure of water
2.4.4
Experimental methods: EXAFS, XAS
Information on many-body correlations in liquid water can be obtained
from various experimental techniques. Two techniques which lie outside the
scope of this thesis are nuclear magnetic resonance (NMR) and vibrational
spectroscopies probing the intramolecular O-H stretch band. The former has
been used to derive information on the distribution of H-bonding angles [34]
and to show that supercooled confined water exhibits a sharp change in
structural order [35]. In vibrational spectroscopy, the O-H stretch band has
a featureless, yet complex, shape which has been interpreted to indicate
the presence of subensembles of water molecules [36, 37], supported by the
existence of isosbestic points in vibrational spectra, or to show that H-bonds
are only transiently broken [38], supported by a spectral diffusion from
blue-shifted to red-shifted frequencies when exciting with a pump-pulse in
pump-probe experiments.
The experimental techniques sensitive to many-body correlations which
have been theoretically modeled in this thesis are extended x-ray absorption
fine-structure (EXAFS) and x-ray absorption spectroscopy (XAS). Both
methods, when applied to water, probe the absorption cross section for the
1s oxygen core-electrons (denoted the K-edge) into unoccupied bound or
free-electron states. Structural information on water can be derived from
experimental XAS spectra either by comparison to different experimental
model spectra, such as ice polymorphs, amorphous ices and the surface
of hexagonal ice [39–41], or from theoretical calculations of structure
models [42]. A more thorough discussion of XAS and EXAFS will be
presented in Chapter 5.
2.5
Pair correlation functions of water
Oxygen-oxygen correlations
The most direct measure of structure in a liquid is provided by the paircorrelation functions (PCFs). These can be straight-forwardly obtained from
diffraction experiments for many liquids, but for water it has proved difficult
to reach a consensus on important details of the PCFs, such as peak shapes
and positions. Particularly the height and position of the first peak in the
oxygen-oxygen partial, gOO (r ), have been subject to controversies in the
literature [19, 43–51]. X-ray diffraction experiments on liquid water have
been used for a long time to provide information on gOO (r ) (for two old
studies, see refs. [52, 53]), and another commonly used source of information
are the neutron diffraction experiments and structure modeling work of
Soper [54–56]. Neutrons are however rather insensitive to O-O correlations
since both hydrogen and deuterium scatter neutrons more strongly than
oxygen, making ND however the best source of information on the O-H
20
2.5 Pair correlation functions of water
and H-H correlations. Figure 2.3 shows a comparison between various
experimental determinations of gOO (r ) and simulation models. The ND
results from 2000 [56] show a peak height of 2.75 which is similar to
many classical MD force fields, but recent diffraction results where XD
data have either been directly Fourier transformed or included in joint
diffraction structure modeling show a significantly lower peak height of
around 2.3 [44–47, 57], and a larger distance around 2.8 Å compared to
the ND results. Yet another recent XD data set where supercooled water
has also been measured [48] conforms with a peak height of around 2.3
and a position at 2.8 Å. Deviating results derived from XD data have been
presented by Head-Gordon and coworkers [19, 43, 58], where a peak height
of 2.6-3.0 was reported. This can possibly be ascribed to an inherent bias
in the analysis procedure applied, since gOO (r ) functions from MD models
were used to guide the analysis, and the total scattering intensity, I (k ), was
modeled instead of the structure factor, S(k ), which is more sensitive to the
shape and position of the first O-O peak. A direct Fourier transform of these
data could not be performed as with the other XD data mentioned above
( [47, 48] and Paper VII) due to the limited k-range (k max =10.8 Å−1 ), but both
EPSR [44] and RMC (Papers I and II) structure modeling of the same data
set instead give a peak height of around 2.3.
A striking conclusion from fig. 2.3 is that the recent application of the
x-ray Raman EXAFS experimental technique to liquid water [5] gives a
seriously deviating gOO (r ) compared to results from XD analyses. This
disagreement between XD and EXAFS regarding the first peak in gOO (r )
was investigated in Paper V, which is discussed further in Chapter 5 below.
It was found that the discrepancy could be resolved by taking into account
a strong multiple-scattering focusing effect which arises due to linear Hbonds. EXAFS can thus provide novel structural information on many-body
correlations, but further attempts to measure EXAFS on liquid water would
be desirable to confirm the earlier data and to investigate other effects, such
as the temperature dependence.
One main conclusion from inspecting the MD results in fig. 2.3 is that
common parametrized classical force fields (SPC/E, TIP4P-Pol2, TIP4P/2005)
overestimate the sharpness of the first O-O peak and predict a somewhat
too short average nearest neighbor distance. It appears that good agreement
with thermodynamic properties, such as for the TIP4P/2005 model, does
not guarantee a faithful reproduction of experimental PCFs. The requirement
for realistic thermodynamic behavior of a classical point-charge model may
possibly even be difficult to reconcile with accurate PCFs. This is perhaps
not surprising due to the simplicity of classical force fields; effects from
intramolecular dynamics, polarization, electronic cooperativity and quantum
delocalization are all neglected or very approximately accounted for through
simplified model interactions.
Another interesting observation in fig. 2.3 is the performance of ab
21
2 Structure of water
initio molecular dynamics (AIMD) models of water. Results for two DFT
functionals are displayed, the common PBE functional [59] and the recently
introduced vdW-DF2 functional [60, 61]. PBE is seen to severely overestimate
the height and underestimate the distance to the O-O peak. This can be
understood from the fact that PBE at room temperature and normal density
describes a supercooled liquid state since the melting temperature of ice Ih,
as described by PBE, has been found to be 417 K [62]. It has furthermore
been found that PBE’s equilibrium density at 1 bar is 0.85-0.9 g/cm3 [63, 64],
i.e. 10-15% lower than the experimental value. These discrepancies of an ab
initio model such as PBE are likely related to the lack of a proper description
of dispersion, or van der Waals (vdW), interactions. Dispersion interactions
are more isotropic compared to the highly directional electrostatic H-bond
interaction, and they lead to the emergence of additional potential energy
minima between two water molecules of a dimer model in non-H-bonding
geometries, at so called interstitial distances [63]. An ab initio model including
dispersion interactions can thus be expected to show a more densely packed
and less structured first coordination shell. This is indeed found for the
recently proposed vdW-DF and vdW-DF2 density functionals, which are
the first DFT functionals to explicitly treat non-local dispersion interactions
from first principles [60, 61] (empirical schemes to include vdW forces have
been suggested earlier [65, 66] and also applied to water [64, 67]). As seen
in fig. 2.3, the model however overcorrects the overstructuring seen for the
PBE functional and produces an O-O PCF with a lower first peak centered
around 2.9 Å, larger than found from structure modeling of XD and ND
data, and where in particular the second coordination peak around 4.5 Å is
almost entirely smeared out. This could be due to a too repulsive short-range
interaction in the functional, which is an important topic for future studies,
but it is interesting to note that the O-O PCF from vdW-DF2 simulations
resembles high-density water [68–70]. A surprisingly large effect of vdW
forces is thus seen in ab initio simulations.
Oxygen-hydrogen correlations
It is important to also investigate oxygen-hydrogen (O-H) correlations
since these give information on the H-bonding network in water; the
first O-H peak reflects directly the nearest neighbor H-bonding geometry.
Figure 2.4 shows the O-H PCF (gOH (r )) obtained from RMC [46] and
EPSR [44, 56] fits to experimental diffraction data compared with results
from classical MD simulations and from AIMD simulations. There is
reasonable agreement between RMC and EPSR structure modeling of
diffraction data concerning the peak heights, but the exact position of the
first O-H peak has uncertainties. The systematic difference between EPSR
and RMC is most likely related to the use of a reference MD potential in EPSR
modeling to guide the analysis. This may either bias the resulting structural
22
2.5 Pair correlation functions of water
3.5
3
2.5
gOO(r)
3.5
RMC-FREE ND+XD(PCCP)
Soper 2007 ND+XD(Narten)
Soper 2007 ND+XD(PCCP)
Soper 2000 ND
Fu 2009 XD
Bergmann 2007 EXAFS
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
2
2.5
3
3.5
r [Å]
4
4.5
vdW-DF2
PBE
TIP4P/2005
SPC/E
TIP4P-pol2
3
5
5.5
0
2
2.5
3
3.5
r [Å]
4
4.5
5
5.5
Figure 2.3: Oxygen-oxygen PCFs from (left) experimental sources and (right) simulation
models. There is large spread in the experimental diffraction results, but consistency is reached
when XD data are included in the structure modeling (EPSR and RMC results). Most MD
models are seen to give an overstructured first O-O peak compared to XD+ND experimental
results. Experimental results: EXAFS from ref. [5], Fu et al. XD results from ref. [47], Soper
ND 2000 results from ref. [56], joint XD+ND Soper 2007 EPSR results from ref. [44], RMC
results from ref. [46] (Paper II). Simulation results: vdW-DF2 and PBE from ref. [71] (Paper
XIV), TIP4P-pol2 from J.I. Siepmann (private communication), TIP4P/2005 and SPC/E from
this work.
gOH(r)
solution or provide essential and realistic physics to the structure modeling
which is not contained in the diffraction data, or indeed a combination of
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.2
0
1.5
2
2.5
3
3.5
r [Å]
4
4.5
vdW-DF2
PBE
TIP4P/2005
SPC/E
TIP4P-pol2
0.6
RMC-FREE ND+XD(PCCP)
RMC-SYM ND+XD(PCCP)
Soper 2007 ND+XD(Narten)
Soper 2007 ND+XD(PCCP)
Soper 2000 ND
0.4
0.4
0.2
5
5.5
0
1.5
2
2.5
3
3.5
r [Å]
4
4.5
5
5.5
Figure 2.4: Oxygen-hydrogen PCFs from (left) experimental sources and (right) simulation
models. There is approximate agreement between different experimental results regarding the
height of the peaks, but differences are seen in the exact position of the first (H-bonding)
peak. Similarly to the O-O PCF, the first O-H peak comes out overstructured from the MD
models. Experimental results: Soper ND 2000 results from ref. [56], joint XD+ND 2007 EPSR
results from ref. [44], RMC results from ref. [46] (Paper II). Simulation results: vdW-DF2 and
PBE from ref. [71] (Paper XIV), TIP4P-pol2 from J.I. Siepmann (private communication),
TIP4P/2005 and SPC/E from this work.
23
2 Structure of water
the two. How large the biasing effect is can be difficult to estimate due
the similarity of the most commonly used MD models with respect to
their PCFs, but as can be seen in fig. 2.3 the reference potential does bias
the O-O correlation when XD data are not included in the modeling (the
“Soper 2000 ND” curve). It is however clear that also RMC suffers from
inherent problems. Resulting RMC structures tend to be overly disordered
if important geometrical constraints are left out, and another small but
sometimes important artifactual feature is the inherent discreteness in the
structure models caused by the use of distance histograms in modeling the
diffraction data. The latter effect can result in unphysical distributions of
interatomic distances within the bins which is simply outside the resolution
of the data. From these considerations it can be concluded that the optimal
approach may be to compare results RMC and EPSR investigations. This may
provide estimates of the uncertainties and yield more reliable conclusions.
With regards to the O-H PCF, it seems at present that further experiments
and structure modeling are required to reduce the uncertainties in the
position of the important first O-H peak.
It can be seen that similarly to the O-O correlation, common MD
models give a higher first O-H peak compared to experimental results.
Again, however, the recently developed vdW-DF2 density functional gives
a distinctly different behavior where the first peak is shifted outwards and
the peak height considerably reduced, leading to improved agreement with
RMC modeling results. Dispersion forces in ab initio simulations apparently
significantly impact the local coordination. The vdW-DF2 model may
present an important step towards more realistic simulations of liquid water,
but fig. 2.3 suggests that further work may be required to better balance
the isotropic dispersion forces and directional electrostatic forces before
quantitative agreement with experiments is obtained. Other effects in AIMD
simulations should furthermore be addressed, primarly the use of a very
small number of molecules and limited simulation time due to the excessive
computational cost, and the influence of nuclear quantum effects.
2.6
Tetrahedrality and local structure index of
simulated water
Both the tetrahedrality parameter Q(i ) defined in eq. (2.33) and the
local structure index (LSI) I (i ) defined in eq. (2.35) have previously been
applied to understand the anomalous properties of supercooled water (e.g.
refs. [28, 32, 33]). Because of the important role played by MD simulations
of the TIP4P/2005 model in the present work, a brief discussion on similar
structural analyses of this model will be given.
Figure 2.5 shows the distributions of Q(i ) and I (i ) at various temperatures
at ambient pressure (1 bar). A qualitatively different behavior is seen where
24
2.7 Long-range correlations
the Q distribution becomes bimodal at higher temperatures and sharply
peaked around a large tetrahedral value at supercooled temperatures,
whereas the opposite occurs for the I distribution which becomes bimodal
at lower temperatures but sharply peaks around a disordered low value
at higher temperatures. The respective distributions shown in the insets
have been calculated on inherent structures, IS, obtained through energy
minimation. Interestingly, the bimodality in the Q distribution vanishes in the
IS while a clear bimodality emerges with a temperature invariant minimum
(isosbestic point) for the I distribution. This indicates that only in terms
of the LSI is there a qualitative difference between energy minimized local
geometries for molecules with different values of I; for the tetrahedrality
value however the distinction between high-Q and low-Q molecules vanishes
when the thermal motion is removed in the IS. The LSI parameter is therefore
likely a better descriptor of the structural differences between the low-density
liquid (LDL) like and high-density liquid (HDL) like local structures which
have been hypothesized to coexist and interconvert through fluctuations
in liquid water. Chapter 4 will introduce the thermodynamical scenario in
which the physical meaning of LDL and HDL becomes evident.
Paper IX presents results that reinforce this assertion. The bimodal
distribution of I values in the IS persists for temperatures between
supercooled and warm, with a rapid conversion from low-I to high-I
species in the deeply supercooled region, and application of pressure shifts
the balance from high-I to low-I. Furthermore, structural differences between
species classified as LDL-like (high-I) and HDL-like (low-I) in the IS are seen
to remain in the real thermally excited structure, as will be discussed further
below.
2.7
Long-range correlations
Small-angle scattering techniques provide information on long-range density
fluctuations. The SAXS data presented in Papers III and XIII provided
experimental evidence for an increase in small-angle scattering intensity
as water is supercooled. In Paper VI, TIP4P/2005 simulations were used
to investigate the origin of the enhanced scattering. Since the model
qualitatively reproduces the experimental data a connection could be
established between the experimental observations and the existence of a
Widom line in supercooled water, which will be discussed in Chaper 4. The
wide-angle x-ray diffraction data presented in Paper VII provided evidence
for structural correlations in ambient water up to around r ∼ 11 Å which
have only been discussed before in the context of supercooled water. Here,
structural aspects of the density fluctuations indicated by SAXS data and
the long-range correlations evidenced by the new XD data will be discussed,
25
2 Structure of water
195K
205K
215K
225K
230K
240K
250K
260K
270K
280K
290K
310K
320K
0.05
P(Q)
0.04
0.03
0.02
0.06
0.04
0.02
0.35
200K
230K
253K
278K
320K
360K
0.2
0.4
0.6
0.8
230K
253K
278K
320K
360K
0.04
0.03
0.25
0.02
0.2
0.01
0.1
0.15
0.2
0.3
0.4
200K
210K
220K
230K
240K
253K
278K
320K
360K
0.1
0.01
0
0.05
0.3
P(I)
0.06
0.05
0
0.2
0.4
0.6
0.8
1
0
0
0.05
Q
0.1
0.15
0.2
I [Å2 ]
0.25
0.3
0.35
0.4
Figure 2.5: Distributions of the tetrahedrality parameter Q (left) and the local-structure
index I (right) from simulations of TIP4P/2005 at different temperatures. The insets show
the distributions obtained in the energy-minimized inherent structures. Arrows indicate the
change with increasing temperature.
based on analysis of TIP4P/2005 simulations which qualitatively reproduce
both experimental observations.
The main conclusions from the large system-size simulations to
investigate the small-angle scattering region will be presented in Chapter 4.
For the present discussion it is sufficient to note that maximal small-angle
scattering intensity at atmospheric pressure is observed for TIP4P/2005
around 230 K which coincides with the Widom line of the model. Can the
enhanced small-angle intensity be related to the large-scale structure of
TIP4P/2005? The small-angle enhancement is related to an increase of the
Ornstein-Zernike correlation length ξ, implying that density fluctuations
in larger regions become correlated. In an attempt to visualize density
fluctuations, a coarse-graining procedure was applied to replace the
atomistic description of the system by a smooth density field calculated on
a numerical grid in the simulation box. A method suggested in ref. [72]
was implemented which “smears” the atoms by summing up Gaussians,
centered on each atom, into a density field. The microscopic density defined
in eq. (2.1) is thus replaced by
N
ρ̄(r) =
∑ φ(|r − ri |)
(2.36)
i =1
where φ(r ) = (2πd2 )−3/2 exp −r2 /2d2 and d is a parameter which
determines the breadth of the atomic contributions to ρ(r). Figure 2.6 (left)
26
2.7 Long-range correlations
shows an isosurface of the density field from a simulation snapshot at 230 K.
A low field value was chosen in order for the enclosed volumes to represent
low-density regions. An inhomogeneous distribution of low-density regions
of various sizes and shapes can be seen, and the presence of voids (highdensity regions) is apparent. However, this does not reflect a bimodality
in the distribution of local molecular densities. Shown in fig. 2.6 (right)
is the temperature dependence of the distribution of density field values.
No signatures of density inhomogeneities can be seen from the unimodal
Gaussian distribution at 230 K, but the rapid change of equilibrium density
and the transition to a LDL-like liquid as the simulations are cooled below
230 K can be seen from the shift and narrowing of the Gaussian peaks,
respectively.
1.2
200K
210K
220K
230K
240K
250K
260K
270K
288K
320K
Probability
1
0.8
0.6
T
0.4
0.2
0
9
9.5
10
10.5
11
11.5
12
Figure 2.6: Left: Isosurface of the smooth density field generated using the definition in
eq. (2.36). A low field value is chosen to display the spatial distribution of low-density regions.
Right: Distributions of field values at various temperatures. The shift with temperature is
indicated by the arrow.
An underlying heterogeneity can nonetheless be detected by investigating
the bimodal distribution of LSI values in the inherent structures discussed
above. Figure 2.7 shows partial PCFs and structure factors for low- and highLSI species. The classification into low- and high-I species is performed in
the IS where a clear bimodality exists and the cutoff value I = 0.13 Å2
suggests itself (see fig. 2.5), but the calculation of the correlation functions
is performed in the corresponding real structure at 298 K and 1 bar. There
are clear structural differences betwen the two species, and the enhanced
small-angle scattering evidences a heterogeneity in their spatial distribution.
Remarkably, the spatial heterogeneity that can be expected in the supercooled
region, where LDL-HDL fluctuations become prominent, persists to ambient
temperatures and above. Paper IX discusses these results in more detail.
27
2 Structure of water
Inherently low-LSI
Inherently high-LSI
1.6
298K 1bar
1.4
0.25 Å
2
1.5
B
1.8
298K 1bar
3
2.5
1
0.5
0
2
Inherently low-LSI
Inherently high-LSI
Cross correlation
A
SOO(k)
gOO(r)
5
4.5
4
3.5
1.2
1
0.8
0.6
0.4
2
3
4
5
6
7
8
r [Å]
0.2
0
2
4
k [Å-1]
6
8
10
Figure 2.7: Partial correlation functions in r- and k-space for subensembles of water molecules
classified in the IS as HDL-like (low-LSI value) and LDL-like (high-LSI value); the correlation
functions are calculated in the real structure at 298 K and 1 bar. A: O-O PCFs for the two
subspecies, and the cross correlation between them. The shift in the second peak position is
shown by arrows. B: O-O structure factors for the two species, showing pronounced smallangle scattering. The partial number densities have been used in calculating S(k) according to
eq. 2.6
Intermediate between the nearest-neighbor regime around 2-7 Å and
the length scale of long-range density fluctuations are the large-distance
correlations around 10 Å. Two previous x-ray diffraction studies which
focused on the supercooled regime have detected the presence of 4th and 5th
peaks in gOO (r ) around 9 and 11 Å, respectively. In ref. [73] the existence
of these peaks was attributed to the presence of clathrate-like structures
in supercooled water. Although ref. [48] confirmed that the large-distance
peaks can be seen in supercooled water, they were not analyzed further.
In Paper VII new synchrotron-based XD data are presented which after a
Fourier transform to gOO (r ) show the presence of 4th and 5th peaks that
are discernible even at 66◦ C. Figure 2.8A shows the experimental PCFs at
different temperatures, scaled as 4πρ0 r2 [ gOO (r ) − 1] to enhance the large-r
region, compared in fig. 2.8B to MD simulations of the TIP4P/2005 model
at the same temperatures. The nearest-neighbor peaks of the experimental
gOO (r ) are much lower than for TIP4P/2005 but beyond the nearest neighbor
peak there is good agreement. Since the MD model is seen to reproduce the
4th and 5th peaks, the simulation trajectories can be used to provide insights
into their structural origin. The trajectories were analyzed in terms of the
LSI and the tetrahedrality parameter. Subensembles of water molecules
were defined based on cutoff values in Q(i ) and I (i ), and partial PCFs for
molecules with Q(i ) < Qcut (I (i ) < Icut ) and Q(i ) > Qcut (I (i ) > Icut ) were
calculated separately. Clear evidence can be found in fig. 2.8 for a connection
between the large-distance peaks and highly structured species present in the
simulations, characterized by high values of Q and I. Particularly the cutoff
28
2.7 Long-range correlations
in the LSI is seen to shed light on the origin of these peaks: high-I species
feature very sharp peaks and low minimas, characteristic for LDL-like
water, while low-I species have lower peaks and an almost collapsed second
shell, characteristic for HDL-like water. Paper VII provides a more detailed
account of these findings. Neutron diffraction has been used to obtain PCFs
for LDL and HDL water by extrapolating measured neutron structure factors
obtained at different pressures [70]. Interestingly, the O-O PCFs for the
subensembles of low- and high-I values, respectively, are quite similar to the
experimentally obtained HDL and LDL counterparts. This confirms again
that the LSI parameter is a highly useful measure in investigating structural
fluctuations in both supercooled and ambient water.
29
2 Structure of water
6
A
5
280 K
298 K
339 K
4
4π r 2 ρ0 [g(r)-1]
B
8
278 K
298 K
340 K
6
3
4
2
2
1
0
0
−1
−2
−2
−3
0
2
4
6
8
10
12
−4
0
2
4
r [Å]
C
15
12
278K
298K
340K
8
10
12
D
10
278K
298K
340K
8
10
4π r 2 ρ0 [g(r)-1]
6
r [Å]
I < 0.05 Ų
I > 0.05 Ų
Q < 0.7
6
Q > 0.7
4
5
2
0
0
-2
-5
0
2
4
6
r [Å]
8
10
12
-4
0
2
4
6
r [Å]
8
10
12
Figure 2.8: Oxygen-oxygen pair-correlation functions scaled as 4πρ0 r2 [ g(r ) − 1] where ρ0 is
the average number density. A: Experimental data showing that the 4th and 5th peaks around
9 and 11 Å are visible even at 339 K (66◦ C). B: Data from TIP4P/2005 simulations, seen to
reproduce the intensity and positions of the 4th and 5th peaks. C and D: Partial contributions
from subensembles to the O-O PCF defined by cutoff values in C: the local structure index
I (i ) (eq. (2.35)) and D: the tetrahedrality Q(i ) (eq. (2.33)). Only high-I (and mostly high-Q)
species contribute to the 4th and 5th peaks.
30
3
Dynamics of water
The previous chapter illustrated that the structure of liquid water still has
many unsolved key questions. As will become evident in the following, the
dynamics of water also shows a complex behavior which has attracted much
attention in the scientific literature. The focus will be on the supercooled
regime where dynamical properties deviate more strongly from normal
behavior.
Several thermodynamical scenarios have been suggested to explain the
unusual static and dynamic properties of water. These theories deal with
physical phenomena in the metastable supercooled state, but the phase
behavior of supercooled water is of high relevance for the understanding of
the ambient regime as well. A thorough discussion of these theories will be
postponed until Chapter 4.
3.1
Introduction
Below the freezing temperature, if freezing is avoided, the motion of
water molecules becomes increasingly sluggish. The structural relaxation
time τα increases quickly, the diffusion constant D drops and the shear
viscosity η goes up. Experiments show a power-law behavior of τα , D and
η with an apparent divergence around 228 K [74]. Due to the so called
homogeneous nucleation taking place around 235 K, where crystallization
occurs instantly on the experimental time-scale, no data exist for bulk water
to investigate the behavior close to the singular temperature. On the other
hand, simulations of deeply supercooled water indicate that instead of a
complete structural arrest at a singular temperature a dynamical transition
takes place where the behavior transforms from fragile, defined by a nonArrhenius temperature dependence of dynamical quantities and increasing
activation energy as temperature is lowered, to strong, characterized by
Arrhenius-like dependence and constant activation energy. Experiments on
31
3 Dynamics of water
confined water where crystallization can be avoided down to below the
homogeneous nucleation temperature and on protein-hydration water also
support the fragile-to-strong (FS) hypothesis [75, 76]. The FS crossover has
been linked to a structural transition from a high-density liquid (HDL) to a
low-density liquid (LDL) [77]. Results presented below from simulations of
the high-quality TIP4P/2005 water model reinforce this connection.
Relevant to the discussion of dynamical properties of supercooled
water is the temperature of the glass transition, Tg . As an alternative
to the FS transition hypothesis, Tg has been suggested to occur close to
228 K and be the source of all structural and dynamical anomalies of
supercooled water [78]. Most other estimates of Tg are close to 136 K [79],
indicating that the liquid phase in the temperature interval 150-235 K is
a “no-man’s land” due to the homogeneous nucleation around 235 K.
Simulations of the TIP4P/2005 water model, which accurately reproduces
many thermodynamic properties of real water (as will be shown in
Chapter 4), do not support this reassignment of Tg since they appear fully
ergodic even below temperatures where the dynamical anomalies first
appear. Instead, the structural and dynamical anomalies in the model are
connected to the Widom line which is associated with a liquid-liquid critical
point. A correspondance to glassy behavior may nevertheless exist in that
the LDL-like liquid below the Widom line may quickly evolve into a highly
viscous liquid with glass-like dynamics as the temperature is further lowered.
In the rest of this chapter a theoretical background to time-dependent
correlation functions will be presented, the molecular dynamics method
introduced, and results presented from dynamical studies of simulated
supercooled water, where the presence of low-frequency excitations, similar
to the so called boson peak, is indicated.
3.2
3.2.1
Time-dependent correlation functions
van Hove correlation functions
A powerful generalization of the static pair-correlation theory presented in
Chapter 2 to the time-dependent domain was introduced in 1954 by van
Hove [80]. In analogy with eq. (2.1) a time-dependent microscopic density
is defined as [12]
N
ρ(r, t) =
∑ δ (r − ri (t)) .
i =1
32
(3.1)
3.2 Time-dependent correlation functions
The corresponding generalization of the PCF in eq. (2.2) is the van Hove
correlation function G (r, t),
*
+
N N
1
G (r, t) =
(3.2)
∑ δ(r − ri (t) + r j (0)) .
N i∑
=1 j =1
At t=0 G (r, t) becomes, apart from a delta function at the origin, proportional
to g(r ),
G (r, 0) = δ(r) + ρg(r).
(3.3)
The van Hove function can be divided into a self- and a distinct-part,
G (r, t) = Gs (r, t) + Gd (r, t)
(3.4)
where Gs measures the correlation of a particle’s position with its position
at other times while Gd measures the correlation with other particles at
other times. This separation is highly useful since it provides connections to
coherent and incoherent inelastic neutron scattering spectra. To see this, we
first Fourier transform to obtain the intermediate scattering functions (ISFs):
F (k, t)
Fs (k, t)
=
=
Z
G (r, t)e−ik·r(t) dk
(3.5)
Z
Gs (r, t)e−ik·r(t) dk
(3.6)
The former is often termed the coherent ISF and the latter the incoherent ISF.
Fourier transforming once more gives the coherent and incoherent dynamic
structure factors,
S(k, ω )
Ss (k, ω )
=
=
Z
F (k, t)eiωt dt
(3.7)
Z
Fs (k, t)eiωt dt
(3.8)
As will be further described below, information on collective dynamics
contained in the coherent dynamic structure factor can be measured by
light scattering and inelastic neutron and x-ray scattering. Self-dynamics,
or single-particle dynamics, is on the other hand probed in the incoherent
dynamic structure factor which can be measured by quasi-elastic neutron
scattering.
33
3 Dynamics of water
3.2.2
Velocity and current autocorrelation functions
Direct information on the dynamics of a system can be obtained by analyzing
the velocity autocorrelation function, which is defined by
Cv (t) =
1
hv (t)vi (0)i
N i
(3.9)
where vi (t) is the velocity of particle i at time t and the angular brackets
denote averaging over the x, y and z components of the velocity, over particles
i and over initial times. The Fourier spectrum of the velocity autocorrelation
function is the vibrational density of states (VDOS),
D (ω ) =
Z
Cv (t)eiωt dt.
(3.10)
Particle currents associated with the microscopic density in eq. (3.1) can be
defined according to
N
j(r, t) =
∑ vi (t)δ [r − ri (t)]
(3.11)
i =1
where vi (t) is the single-particle velocity. It is useful to decompose the Fourier
components of j(r, t) into transverse and longitudinal components,
jkL (t)
jkT (t)
N
=
∑ k̂k · vi (t) exp [−ik · ri (t)]
(3.12)
∑ k̂⊥ · vi (t) exp [−ik · ri (t)]
(3.13)
i =1
N
=
i =1
where k̂k and k̂⊥ are unit vectors parallel and perpendicular to k. This allows
the definition of longitudinal and transverse spectra of current correlations,
Cα (k, ω ) =
Z
Cα (k, t)eiωt dt =
1
N
Z iωt
α
jkα (t) j−
dt
k (0) e
(3.14)
where α = L or T and the angular brackets denote averaging over initial
times and directions of k. A formal relation exists between the spectrum of
longitudinal current correlations and the coherent dynamic structure factor,
S(k, ω ) =
k2
CL (k, ω ).
ω2
(3.15)
Information on transverse currents cannot be directly obtained experimentally,
however.
34
3.3 Molecular dynamics
3.2.3
Experimental methods: inelastic scattering
Inelastic neutron and x-ray scattering (INS and IXS) and light scattering can
be used to measure the dynamic structure factor [12]. The principles behind
INS and IXS are similar to those presented in Chapter 2 for elastic scattering
techniques. The main difference is that in addition to the momentum
exchange between the beam of neutrons or x-rays, i.e. h̄k = h̄k1 − h̄k2 (where
k1 and k2 are the incoming and outgoing wave vectors), the transferred
energy h̄ω = h̄ω1 − h̄ω2 is also measured. In particular INS experiments
are useful to obtain dynamical information on liquids since it allows both
the coherent and incoherent dynamic structure factors to be obtained.
The measured double differential scattering cross section is related to the
dynamic structure factor as (shown here for a single-component system),
k
d2 σ
= Nb2 2 S(k, ω )
dΩdω
k1
(3.16)
where Ω is the solid angle of the scattering which provides the k-dependence
and b is the atomic scattering length composed of coherent and incoherent
contributions as described in Chapter 2. For a multicomponent system the
total dynamic structure factor is a linear combination of the partial structure
factors whose relative contributions are dictated by the atomic scattering
lengths, similarly to eq. (2.29) for elastic neutron scattering. By detecting
scattered neutrons with small energy transfers h̄ω to the sample, as in quasielastic neutron scattering (QENS), the incoherent dynamic structure factor
Ss (k, ω ) is obtained. For water, since hydrogen has a very large incoherent
scattering length, mostly the self-motion of hydrogen atoms is detected in
QENS. Inelastic light- and x-ray scattering techniques give information on
coherent scattering only. The former measures the fluctuations in the local
dielectric constant, which for most systems is proportional to local density
fluctuations, while the latter measures electron density fluctuations where
k-dependent atomic form factors replace the scattering lengths in eq. (3.16).
3.3
Molecular dynamics
Molecular simulations are invaluable as a source of information on the
dynamics of liquids, particularly since some dynamical quantities such
as transverse current fluctuations cannot be obtained experimentally.
“Computer experiments” furthermore allow the investigation of the
connection between microscopic and hydrodynamic theories for the
liquid state, as well as the connections between structure, dynamics and
thermodynamics.
Classical molecular dynamics (MD) simulations are based on solving
Newton’s equation of motion for a many-body system. The simulated
35
3 Dynamics of water
molecules are either rigid, with a fixed molecular geometry, or flexible,
defined by an equilibrium geometry and intramolecular force constants.
Potential parameters or force fields are defined for each molecule and
constructed to take all important interactions between molecules into
account. Intermolecular pair-potentials are commonly expressed as

!12
!6 
σij
qi q j
σij
.
U (rij ) =
+ 4eij 
−
(3.17)
rij
rij
rij
The first term accounts for electrostatic Coulomb interactions between chosen
interaction sites (usually the positions of the nuclei) and the second term is a
Lennard-Jones potential which accounts for attractive dispersive interactions
(where induced dipole interactions dominate) through the r −6 term and the
short-range repulsion (overlap of electron clouds) through the r −12 term.
Typically, the partial atomic charges and Lennard-Jones parameters are
calibrated to reproduce known thermodynamic properties, dipole moments,
pair correlation functions etc. The intramolecular geometry and positions of
so called virtual sites, i.e. positions that carry potential parameters but have
no mass, are also sometimes used as variables in the parametrization of a
force field. Through the assumed potential, e.g. that in equation (3.17) with
given parameters, pair-wise forces between all particles in the simulated
system are calculated as Fij = −∇Uij and the total force on each atom is
then used to propagate the atomic positions in time by solving Newton’s
second law, r̈i = ∑ N
j=1 Fij /mi . Stable numerical procedures are needed for
this purpose; a common algorithm employed in simulations reported here is
the so called leap-frog algorithm. To run simulations in the canonical (NVT)
ensemble one also needs thermostats to control the temperature. This can
be accomplished by coupling the atomic positions to artificial degrees of
freedom as in, for instance, the Nosé-Hoover method [81], thus simulating the
coupling of the system to a constant temperature reservoir [82]. Isothermalisobaric (NPT) simulations are obtained by introducing barostats, e.g. by
coupling the simulation box coordinates to the equations of motion of the
simulated particles as in the Parrinello-Rahman coupling scheme [83]. In
studying liquid dynamics, however, simulations in the microcanonical NVE
ensemble are preferred since the coupling to artificial degrees of freedom in
NVT and NPT simulations may alter the true (microcanonical) dynamics.
Mainly two force fields have been investigated in this thesis: the popular
SPC/E model [84] and the recent TIP4P/2005 model [85]. These two
models differ in one fundamental way; the former is a 3-site model while
the latter has 4 sites, where one is a so called virtual site which has no
mass but carries the negative charge in the molecule. While SPC/E has
provided useful information in a vast range of investigations of water or
water-containing systems, its thermodynamic properties are inferior to that
36
3.4 Low-frequency excitations in supercooled water
of more recent models like TIP4P/2005 or TIP4P/Ew [86]. This is partly
because of developments in computer hardware and simulation algorithms
which have allowed more extensive parameter fits to be performed, but it is
also inherently related to the fact that 4-site force fields often perform better
than 3-site models; this has been suggested to be related to a more realistic
quadrupole moment in 4-site models [87].
As discussed in Chapter 2, most classical force fields give poor
agreement with experimental results for the first peak of the oxygenoxygen pair correlation function gOO (r ) despite good agreement with many
thermodynamic properties. Previous investigations (for a review see ref. [42])
have furthermore shown that calculations of x-ray absorption spectra on
structures from hitherto investigated MD models lack spectral features
observed experimentally. Also dynamical properties of water are not always
very accurately described in MD simulations. In short, perfect agreement
with reality should of course not be expected for simple models of water,
and care must be taken so as not to over-interpret simulation results. With
these limitations in mind, however, much can be learnt from the atomistic
description of structure, thermodynamics and dynamics provided by MD
simulations, particularly if the simulated atomic trajectories are examined
for information on physical mechanisms behind experimentally observed
phenomena.
3.4
Low-frequency
water
excitations
in
supercooled
An anomaly encountered in many glasses is the onset of low-frequency
collective excitations beyond what is expected from the Debye model for
crystalline solids, which predicts D (ω ) ∝ ω 2 for the vibrational density of
states (VDOS) and C ( T ) ∝ T 3 for the specific heat. The frequency range in
which these excess excitations are manifested is around 10-100 cm−1 (note
that 1 THz=33.35 cm−1 =4.14 meV). Since the temperature dependence of
excess vibrations follows Bose-Einstein statistics this phenomenon has been
termed the boson peak. However, its physical mechanism is still insufficiently
understood [88].
The existence of boson-like low-frequency excitations in low-density
amorphous (LDA) ice has been inferred from inelastic neutron scattering
experiments [89, 90]. Since the Widom line in the supercooled regime, as
determined from experiments on confined water [37] and from numerous
MD simulations, is related to a rapid crossover to a low-density liquid (LDL)
regime which structurally resembles LDA ice, an interesting question is
whether collective low-frequency modes can exist in the LDL-like liquid as
well. Indeed, it has recently been suggested [91] based on IXS experiments
on liquid and polycrystalline sodium that in the frequency range above
37
3 Dynamics of water
10-15 cm−1 the liquid dynamics is indistinguishable from the dynamics of
the amorphous solid.
Two studies on amorphous ices have attempted to clarify the structural
origins of the excess low-frequency excitations in amorphous ices. By
comparing the low-frequency region and the librational band of differently
prepared LDA ices together with ices Ih and Ic , Yamamuro et al. [90] assigned
a low-frequency excess scattering around 40 cm−1 in LDA to broken and
distorted H-bonds. They warned however against using the term boson peak
for the excess low-frequency scattering since a peak around 50 cm−1 is found
also for the crystalline ices. Since amorphization enhances the scattering
below 50 cm−1 it was nonetheless suggested that the enhanced low-frequency
scattering was closely related to the same physical mechanism underlying
the boson peak [90]. Tse et al. [92] on the other hand investigated highdensity amorphous (HDA) ice using INS and MD simulations, and found
that excess low-frequency modes emerging below 20 K could also in this
case be traced back to H-bonding defects, assigned in the MD simulations
to dimer or chain-like structures. Of largest relevance in the present context
are the experiments by Chen et al. [93]. A distinct low-frequency, boson-like
peak was found in INS spectra of confined water below 230 K, which was
experimentally characterized as the temperature where a FS transition takes
place and tentatively related to the Widom line. The energy position of this
peak was close to 40 cm−1 , which, as will be seen below, is well reproduced
in bulk simulations of the TIP4P/2005 model.
In the following sections, selected results will be presented which indicate
a sudden onset of localized low-frequency modes as the Widom temperature
is crossed in simulations of supercooled water. All simulations have been run
for several nanoseconds in the NVE ensemble after an initial equilibration
first in the NPT ensemble and then in the NVT ensemble. Dynamic
correlation functions have been calculated using equations presented above,
and Fourier transforms are performed using the FFT algorithm. First, rather
strong finite-size effects observed in supercooled water simulations will be
discussed.
3.4.1
Finite-size effects in simulations
While studying the dynamics in simulations of supercooled water in the
present work, sharp peaks at very low excitation energies were found.
Figure 3.1A shows the VDOS for different temperatures for simulations of
512 molecules. These sharp peaks are only seen at very low temperatures; in
fact they only emerge below the Widom line in TIP4P/2005, defined as the
locus of correlation length maxima emanating from the LLCP (the Widom
line will be futher discussed in Chapter 4). However, as figure 3.1B shows, the
positions of the sharp low-frequency peaks shift with system size. Comparing
different system sizes reveals that in the limit of infinite simulation box size
38
3.4 Low-frequency excitations in supercooled water
the energy of the excitations extrapolate to ω=0. Evidently, these sharp
excitations are an artifact of the finite system size, but the underlying
physical mechanism is not trivially understood. Similar effects have been
reported for silica [94, 95] and ortho-terphenyl [96] based on analysis of the
incoherent ISF and dynamic structure factors, and suggested to stem either
from sound waves propagating through the system and crossing the periodic
boundary conditions [94, 96], or from the lack of wave-vectors below a cutoff
value (dictated by the finite box size) which, due to a coupling between
15
40
A
ωpeak [cm ]
D(ω)
30
−1
5
0
-5
-10
B
35
10
0
20
40
60
-1
ω [cm ]
80
20
15
10
100K ice
210K
230K
250K
CT (k,ω)
finite size peak 1
finite size peak 2
25
5
0
100
0
0.01
0.02
0.03
L −1 [Å −1]
0.04
0.05
0.06
Figure 3.1: A: Calculated VDOS according to eq. (3.10) for different temperatures in
simulations of 512 molecules. In the bottom part the lowest wave vector (k=0.25 Å−1 )
transverse current spectrum is displayed, showing that the sharp low-frequency peaks are due
to a coupling to low-k transverse modes. Vertical lines are a guide to the eye. B: Dependence
of the sharp low-frequency peaks on the inverse box-length, showing that in the limit of an
infinite box size the spurious peaks shift to zero frequency.
1
0.96
2
0.92
0.88
0.84
0.8
0.01
512 k=1.5 Å-1
45000 k=1.6 Å-1
2.5
45000 k=1.6 Å -1
Ss(k,ω)
Fs (k,t)
3
512 k=1. 5 Å -1
512 k=1.7 Å -1
1.5
1
0.5
0.1
1
t [ps]
10
100
0
0
20
40
ω [cm-1 ]
60
80
100
Figure 3.2: Comparison of Fs (k, t) and Ss (k, ω ) at different k between simulations of 512
and 45,000 molecules at 210 K. The position of the minimum around 0.3 ps in Fs (k, t) and
the small blip around 0.9 ps are unaffected by the system size, but the region just after 1 ps
differs. The periodic oscillations connected to the sharp finite-size peaks in Ss (k, ω ) shift to
larger times in the large system.
39
3 Dynamics of water
transverse acoustic modes at low k and longitudinal modes at higher k,
removes intensity from the dynamic structure factors below a certain cutoff
frequency [95]. In fig. 3.1A the transverse current spectra (defined according
to eq. (3.14)) for the lowest wave vector at each temperature available in
the 512 molecules’ simulations is shown superimposed on the VDOS. It is
clearly seen that the frequency positions of the sharp lowest-k peaks coincide
exactly with the first sharp peaks in the VDOS. This indicates that the
finite-size peaks originate from a coupling to low-k transverse modes, i.e. the
interpretation presented in ref. [95].
Figure 3.2 shows the comparison of the incoherent ISF and dynamic
structure factor between two system sizes. The finite-size effect is clearly
seen as periodic oscillations in Fs (k, t) which shift to larger times for the
larger simulation box. In Ss (k, ω ) it is manifested by sharp peaks at low
frequencies (shifting to lower frequencies in the large simulation box), and
a vanishing of intensity on the low-frequency side of a broad peak centered
around ω=40 cm−1 . Note that the ω=40 cm−1 peak is unaffected by system
size and can therefore tentatively be assigned as a boson peak. To emphasize
remaining uncertainties in the interpretation it will however be refered to as
a “boson-like” peak. Interestingly, a reduction of the Lamb-Mössbauer factor,
i.e. the height of the plateau in Fs (k, t) after the initial fast relaxation, is seen
for the large system size. This implies a shorter structural relaxation time in
the larger system, similarly to earlier studies on amorphous silica [95].
3.4.2
Dynamic correlation functions
Having clarified the influence of finite-size effects on the low-frequency
region allows for a discussion on the properties of the boson-like lowfrequency intensity around ω=40 cm−1 in supercooled water below the
Widom line. Figure 3.3 shows the incoherent ISFs at different temperatures
and the associated incoherent dynamic structure factors for a wave vector
corresponding to the first peak of the structure factor S(k ). The main
message of fig. 3.3 is the strong qualitative change observed as the Widom
temperature (230 K for TIP4P/2005 as discussed in Chapter 4) is crossed;
the ISFs start to exhibit a minimum around 0.3 ps and a broad bosonlike peak appears around 40 cm−1 . However, results from a simulation of
hexagonal ice Ih show that the low-frequency peak does not arise only in
the metastable liquid regime. Similarly to the above-mentioned experimental
investigation of LDA ice [90], the low-frequency peak for hexagonal ice is
found at somewhat higher frequencies and has a weaker low-frequency side
compared to the supercooled liquid simulations.
Lastly, in fig. 3.4 the k-dependence of the coherent and incoherent
dynamic structure factors is shown. Apparently, the self-motion manifested
in the incoherent part Ss (k, ω ) is affected by the boson-like low-frequency
modes down to very low k. It is found that for wave vectors below 2 Å2 all
40
3.4 Low-frequency excitations in supercooled water
curves collapse onto a single master curve when Ss (k, ω ) is divided by k2 ,
suggesting that a separable k-dependent part of Ss (k, ω ) is proportional to k2
as also found for the boson peak in amorphous silica [95]. At the largest wave
vectors the boson-like peak decreases in intensity. In the collective coherent
part S(k, ω ) the boson-like peak is strongest around k=2.5 Å−1 , which is
approximately in the middle of the split double peak in the structure factor,
S(k ), and therefore corresponds to distances intermediate between the first
and second coordination shells. A dispersing longitudinal acoustic mode is
superimposed on the low-frequency region which makes it impossible to
determine whether the boson-like peak exists down to the lowest k-vector in
S(k, ω ), but it is seen to appear already at k =0.5 Å−1 . A final observation
concerns the k-dependence of the sharp finite-size peaks in S(k, ω ). It can
be seen that at low k the intensity of the finite-size peak around 20 cm−1 is
strongly diminished. Keeping in mind that the finite-size peaks are due to
a coupling to transverse modes at low k, as shown in fig. 3.1, it therefore
appears that the coupling between low-k transverse modes and longitudinal
modes (as probed in S(k, ω )) is smaller for low-k longitudinal modes.
6
1
0.9
k=2.0 Å -1
5
-1
k=2.0 Å
0.6
0.5
210K
0.4
220K
0.3
230K
240K
0.2
250K
0.1
260K
0
0.01
0.1
Ss(k,ω)
F s(k,t)
0.8
0.7
4
3
T
2
T
1
100K Ice Ih
1
10
100
t [ps]
0
0
20
40
60
-1
ω [cm ]
80
100
Figure 3.3: Fs (k, t) and Ss (k, ω ) at k=2.0 Å−1 for temperatures between 210 and 260 K
(arrows show the trend with increasing temperature). Ss (k, ω ) for simulated hexagonal ice
at 100 K is also shown. Following an initial fast β-relaxation below 0.3 ps in Fs (k, t) a second
slower α-relaxation takes place. As the Widom temperature Tw =230 K is crossed there is a sharp
increase in the α-relaxation time. A broad low-frequency peak centered just below 40 cm−1
shows up below the Widom temperature, Tw =230 K, but it can be seen at larger frequencies in
the hexagonal ice simulations as well.
Final remarks
Much has yet to be understood around the emergence of boson-like lowfrequency excitations, observed experimentally in amorphous ice and in
supercooled confined water, and in simulations reported here for bulk
supercooled water below the Widom temperature, in all cases around
ω=40 cm−1 . Since a peak at slightly higher frequencies also appears in the
41
3 Dynamics of water
dynamic structure factors of crystalline ice one cannot simply equate this
feature to a boson peak, observed in many glasses and vigorously debated
in the literature, but the much stronger intensity in the deeply supercooled
liquid simulations suggests that similar physical mechanisms may be at
work. Further efforts are required to connect the low-frequency modes to
structural features in order to understand why the Widom line marks a
dynamical transition below which the modes appear
10
Incoherent
6.25 Å
10
-1
Coherent
1
1.25 Å
-1
512 molecules
0.1
1
Ss(k,ω)
3.25 Å
1
-1
1.397 Å
S(k,ω)
Ss(k,ω)
-1
2.5 Å
-1
0.1
0.1
0.75 Å
0.01
-1
-1
0.001
0.056 Å
-1
0.25 Å
45000 molecules
0
20
40
60
ω [cm-1 ]
80
100
0.01
-1
1.0 Å
0.5 Å
-1
512 molecules
0
50
100
150
ω [cm-1]
200
Figure 3.4: Wave vector dependence of the incoherent and coherent dynamic structure factors
at 210 K. To the left the incoherent part is shown for k varying from very small, obtained in
simulations with 45,000 molecules, to large, obtained in simulations with 512 molecules. To the
right, a longitudinal acoustic peak is seen to disperse with k in S(k, ω ), but the non-dispersing
boson-like peak around 40 cm−1 is present as well and becomes particularly pronounced at k =
2.5 Å−1 . Note that S(k, ω ) at low k shows less pronounced sharp finite-size peaks compared to
higher k.
42
250
4
Thermodynamics of water
The peculiar structural and dynamic properties of liquid water described in
the previous chapters may ultimately be inherently related to its unusual
thermodynamic behavior originating in the supercooled region. After a
brief introduction, this chapter will give a short summary of the competing
theoretical scenarios that have been proposed for the phase behavior of
supercooled liquid water. Extensive simulations of the TIP4P/2005 model,
which is one model out of many shown to possess a liquid-liquid critical
point (LLCP) [97], will then be discussed. These simulations connect the
thermodynamic behavior in the simulations to experimental SAXS data
which provides indirect evidence for the LLCP scenario in real water. A
plausible structural interpretation of the enhanced small-angle scattering
will furthermore be presented. In the next chapter, the focus will shift back
to structural properties and insights from x-ray spectroscopic techniques will
be discussed.
4.1
Introduction
Liquid water shows many thermodynamical anomalies, many of which
become strongly enhanced in the supercooled region such as the strong
increases in thermodynamic response functions. Water also displays an
unusually complex phase diagram, with 15 known crystalline ice phases and
at least two amorphous solids – low-density and high-density amorphous
ice (LDA and HDA respectively) – which appear to be separated by a
first-order phase transition [98]. Recently a new amorphous state, very-high
density amorphous ice (VHDA) was reported [99]. Whether VHDA is a
distinct amorphous solid state of water is still unclear, however. This so
called polyamorphism of amorphous solid water has been suggested to be
the glassy counterpart of first-order phase transitions taking place between
different liquid phases in the deeply supercooled region, first observed in
43
4 Thermodynamics of water
MD simulations [6]. Figure 4.1 shows an illustration of a part of the phase
diagram of water, where the metastable supercooled and glassy behavior
is superimposed on the stable (crystalline) regions. The metastable part of
the diagram is only hypothetical since conclusive evidence for the existence
of a second critical point has yet to be found. It illustrates nonetheless the
behavior found for the TIP4P/2005 model and a majority of other MD
models investigated in the literature.
106
105
LDA
103
Pressure (bar)
VII
HDA
104
HDL?
III
LDL?
C*?
102
V
VI
C
Widom line?
101
Widom line
Liquid
Ih
Saturation Line
0
10
10-1
10-2
Triple Point
10-3
-4
10
Sublimation Line
Vapor
-5
10
10-6
10-7
0
100
200
300
400
500
600
700
800
Temperature (K)
Figure 4.1: Phase diagram of water. In the upper left corner, the hypothetical liquidliquid critical point scenario for the metastable supercooled region is schematically depicted
superimposed on the stable phase diagram. The dashed horizontal line shows the atmospheric
pressure isobar, and unknown features are followed by question marks.
4.2
Theoretical scenarios
Several scenarios have been suggested to describe the anomalous properties
of supercooled water [79]. In an early attempt by Speedy in 1982 [100]
the increases in response functions as water is cooled were suggested to
stem from a retracing of the liquid-vapor spinodal line through negative
pressures into the supercooled region, known as the stability-limit (SL) or
retracing spinodal conjecture. There would thus be a connection between
the superheated and supercooled liquids, and the response functions
44
4.2 Theoretical scenarios
in supercooled water would be influenced by the divergence of density
and entropy fluctuations at the retraced spinodal. Most MD simulation
models however have been shown not to display re-entrant behavior of
the liquid-vapor spinodal, and thermodynamic arguments [79] show that
thermodynamic consistency requires the liquid-vapor coexistence line to
have both upper and lower critical points in the SL scenario, which has no
support from experiments.
The liquid-liquid critical point (LLCP) hypothesis postulates that two
distinct liquid phases are separated by a first-order phase transition in the
supercooled regime [101]. Initially discovered in simulations of the ST2
water model [6], other MD models have also been shown to contain one or
even several liquid-liquid transitions (e.g., [102, 103]). Characteristic for the
LLCP scenario, where the critical point is usually postulated to reside at
high positive pressures, is the Widom line emanating as an extension of the
liquid-liquid phase transition into the one-phase region at lower pressure
(see fig. 4.1). The Widom line TW ( P) is defined as the locus of correlation
length maxima, and response functions are expected to show maxima close
to TW [104] which become more pronounced as the critical pressure is
approached.
Experimental observations favor the existence of a LLCP in water. By
investigating kinks in the melting lines of high-pressure ice polymorphs
Mishima and Stanley [105] estimated the critical parameters to be
PC ≈1000 bar and TC ≈220 K. Most recently, Mishima [106] measured the
equilibrium volume of supercooled emulsified water and found agreement
with the LLCP scenario with PC ≈500 bar and TC ≈223 K. A likely location
of the LLCP, should it exist, can therefore be considered to be at PC =5001000 bar and TC ≈220 K.
Despite support for the LLCP scenario from both simulations and
experiments, other scenarios must be mentioned since the available evidence
is not yet conclusive. The singularity-free (SF) [107] model is similar to the
LLCP model in that two liquid forms, highly structured low-density liquid
(LDL) and a more disordered high-density liquid (HDL), are proposed, but
the transition between them is everywhere continuous, as would be the case
if the LLCP was found at zero temperature and the Widom line emanated
into the supercooled region. The critical-point free (CPF) scenario [108]
postulates an order-disorder (or λ-) transition between HDL and LDL,
with a possible critical point at negative pressures. A very recent proposal
by Swenson and Teixeira [78] asserts instead that most of the observed
anomalies of supercooled water are due to the glass transition, thus radically
revising the glass transition temperature Tg from 135 K (or 165 K according
to some authors) to around 228 K.
The SL, SF, CPF, LLCP (PC <0) and LLCP (PC >0) scenarios have been
unified into a single theoretical framework by Stokely et al. [109]. In this
elegant cell-model formulation two key parameters describing interactions
45
4 Thermodynamics of water
between water molecules, namely the strengths of the directional and the
cooperative components of the H-bond, could be tuned to yield phase
behavior corresponding to any of the scenarios. For the largest range of
parameter values, and in particular for the most realistic combinations,
the LLCP (PC >0) scenario was obtained. At present, the largest amount of
evidence thus supports the LLCP scenario with a critical point at positive
(high) pressure.
4.3
Thermodynamic response
molecular simulations
functions
from
Many thermodynamic properties are readily available from MD simulations.
Equilibrium densities can be obtained by running long simulations in the
NPT ensemble or by calculating the dependence of enthalpy on the density.
Force fields differ in how well they reproduce the density maximum in water.
SPC/E shows rather poor agreement, where the temperature of maximum
density (TMD) is around 240 K [110], while TIP4P/2005 displays a very
good, almost quantitative agreement [85]. From the equilibrium densities the
thermal expansion coefficient, a thermodynamic response function, can be
calculated as
∂V
α P = V −1
(4.1)
∂T P
where V is the volume. As with other response functions, α P also reflects
thermodynamic fluctuations:
Vk B Tα P = h(δSδV )i
(4.2)
where S is the entropy, δS = S − hSi (and similarly for δV) and angular
brackets denote the ensemble average. We see that α P is related to the
cross-correlations between entropy and volume fluctuations, and one of
the anomalous characteristics of water is that this correlation is negative
below the TMD. Another anomaly is that fluctuations in volume and
entropy increase upon decreasing the temperature. This leads to increasingly
negative values of α P , and also to increases in the values of two other
response functions, the isothermal compressibility κ T and the isobaric heat
capacity c P . The former describes the response of a system to changes in
pressure,
−1 δV
κ T = −V
(4.3)
δP T
and its corresponding fluctuation formula (or fluctuation-dissipation
theorem) is
D
E
Vk B Tκ T = (δV )2 .
(4.4)
46
4.4 Enhanced small-angle scattering and the Widom line
The isobaric heat capacity is the enthalpy response to temperature changes,
δH
cP =
(4.5)
δT P
and its fluctuation formula is
D
E
Nk B c P = (δS)2 .
(4.6)
As described above, results from experiments and theoretical simulations
suggest that the anomalous enhancements of the volume and entropy
fluctuations in the supercooled regime of liquid water may be related to
the presence of a liquid-liquid critical point at high pressures. In the onephase region, e.g. at atmospheric pressure, the Widom line would cause the
thermodynamic response functions to increase upon supercooling and reach
a finite maximum at the Widom temperature TW .
Critical phenomena are notoriously difficult to describe in molecular
simulations due to finite-size effects which become pronounced as the
correlation length of density fluctuations increases, and a related effect
called critical slowing down [111]. To observe phase transitions, prohibitively
long simulations can furthermore be required since the system undergoes
flips between metastable states in the coexistence region. Special simulation
methods have been devised to deal with phase transitions, for instance
Gibbs ensemble Monte Carlo simulations, where two boxes are simulated in
parallel and particles are allowed to jump between them, and Gibbs-Duhem
simulations where an integration of the Clapeyron relation is performed. The
latter approach has been used, e.g., to determine the liquid-vapor coexistence
lines and critical points of TIP4P-like water models [112]. Directly simulating
a liquid-liquid phase transition has proved difficult however, partly because
the relevant order parameters are unknown.
4.4
Enhanced small-angle
Widom line
scattering
and
the
Small-angle x-ray scattering (SAXS) experiments presented in Papers III
and XIII and shown in fig. 4.3 unambiguously prove that water displays
enhanced small-angle scattering and an increasing correlation length
described by a power law as water is cooled below room temperature. In
Paper III this behavior was seen to be absent in SPC/E simulations, in part
due to the inability of the model to describe the minimum in isothermal
compressibility κ T observed experimentally at 46◦ C. The recently suggested
TIP4P/2005 water model, on the other hand, has been shown to accurately
reproduce many thermodynamic properties of liquid water [85, 113, 114].
47
4 Thermodynamics of water
In particular the minimum in κ T is well reproduced, and this gives hope
that increasing density fluctuations at supercooled temperatures can be
described in simulations. Figure 4.2 shows a comparison between SPC/E,
TIP4P/2005 and experimental data for κ T . TIP4P/2005 shows a minimum
close to the experimental value whilst SPC/E does not. Experimental data
for ambient temperatures and an experimental power-law fit to κ T in the
supercooled regime are shown for comparison. An interesting observation
is that the two MD models shown in fig. 4.2, as well as several other
models [113], overestimate the compressibility at higher temperatures but
underestimate it at lower temperatures. Since the enhanced compressibility
at lower temperatures must be related to structural fluctuations while the
increase at higher temperatures is due to thermal fluctuations, this indicates
that these MD models underestimate the former but overestimate the latter.
Figure 4.2 also clearly shows the effect of the Widom line in the simulations,
in that a pronounced maximum appears close to 230 K.
Simulated small-angle structure factors of TIP4P/2005 at deeply
supercooled temperatures along two different isobars are shown in fig. 4.3
together with experimental data at ambient pressure. The model is indeed
seen to give small-angle enhancements in qualitative agreement with the
experimental data in fig. 4.3. As can be seen, the small-angle scattering
becomes significantly stronger at elevated pressure, presumably because of
the closer proximity to the LLCP of TIP4P/2005 which is located around
1350 bar [97].
What is the physical origin behind the enhanced small-angle intensity?
Can it be related to thermodynamical behavior and structural effects?
Figure 4.4 attempts to answer these questions. In part A, Ornstein-Zernike
correlation lengths are displayed obtained from decomposing the small-angle
structure factors into a “normal” part, represented by a Percus-Yevick hardsphere model explained in relation to eq. (2.19), and an “anomalous” part
represented by the Ornstein-Zernike expression for S(k ), eq. (2.15). A powerlaw behavior of the form shown in eq. (2.17) is seen which indicates the
presence of critical fluctuations, and the critical exponents furthermore agree
qualitatively with experimental results in Paper XIII (see Paper VI for further
details). Figure 4.4B shows isothermal compressibilies of TIP4P/2005 at
different pressures compared to experimental data. An excellent agreement
is found with the experiments at higher pressure, particularly at 1500 bar,
but at ambient pressure the simulations are seen to underestimate the
compressibility, indicating that density fluctuations are larger in real water
along the 1 bar isobar. This is consistent with the possibility that a LLCP
exists for real water at pressures lower than found for the TIP4P/2005 model,
i.e. 1350 bar [97]. Two experiments [105, 106] have indeed indicated that a
LLCP might be found in the interval 500-1000 bar, but due to the challenging
nature of these experiments their results have large error bars and do not
give conclusive evidence for the presence of the LLCP.
48
4.4 Enhanced small-angle scattering and the Widom line
0.75
TIP4P/2005 S(0)
Speedy-Angell power-law
Pi et al. TIP4P/2005
Experiment
SPC/E S(0)
0.7
-1
κ T [GPa ]
0.65
0.6
0.55
0.5
0.45
0.4
0.35
220
240
260
280
T [K]
300
320
340
Figure 4.2: Experimental and simulated isothermal compressibilities at ambient pressure. The
curve labeled “Experiment” is from ref. [115] while “Speedy-Angell power-law” is a powerlaw fit [74] to experimental data on supercooled water where experimental data points go
down to 247 K. TIP4P/2005 simulation data from ambient to mildly supercooled temperatures
from Pi et al. [113] are shown, along with data from Paper VI which go into the deeply
supercooled regime and where the extrapolated k →0 limit of S(k) was used to determine
κ T . Satisfactory agreement between the two are seen between 260 and 300 K, but perfect
agreement is not expected due to uncertainties in the two different methods used. SPC/E
results obtained from extrapolating S(k ) to k=0 are also shown (in good agreement with other
work [113]); in contrast to TIP4P/2005 no minimum is seen although it could be present at
a lower temperature. The Widom line of TIP4P/2005 is accompanied by a maximum of κ T at
230 K.
Clear maxima of κ T and ξ are found in the simulations at 230 K and
1 bar and the behavior at 1000 bar indicates that a maximum is present close
to 200 K. As fig. 4.4C shows the absolute value of the thermal expansion
coefficient also rises sharply and exhibits maxima at 230 and 200 K at 1 and
1000 bar, respectively. This clearly shows that the Widom line is present,
consistent with other work on the TIP4P/2005 model [97]. The enhanced
small-angle scattering shown in fig. 4.3, which is seen to fall off below 230 K
at 1 bar, can therefore be directly related to the Widom line. In turn, due
to the good agreement between simulated and experimental small-angle
enhancements and compressibilities, this strongly supports the view that the
same physical mechanism is at work in both cases; namely, the influence of
a metastable liquid-liquid critical point.
Part D of fig. 4.4 shows how the Widom line is related to a structural
transition. The temperature derivative of the ensemble averaged tetrahedrality
parameter along the different isobars reveals that the crossover to a
tetrahedral LDL-like liquid is maximally rapid at the Widom temperature
49
4 Thermodynamics of water
252 K
254 K
258 K
263 K
278 K
298 K
0.09
0.08
210 K
220 K
230 K
240 K
253 K
278 K
298 K
320 K
200 K
210 K
220 K
230 K
240 K
250 K
260 K
278 K
S(k)
0.07
0.06
0.05
0.04
Experiment
0.03
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0
k [1/Å]
1 bar
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0
k [1/Å]
1000 bar
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
k [1/Å]
Figure 4.3: Simulated small-angle structure factors of TIP4P/2005 compared with
experimental small-angle x-ray scattering data from Paper XIII. Experimental data are shown
to the left, followed by simulation results at 1 and 1000 bar.
TW at 1 and 1000 bar. Below TW an almost perfectly tetrahedral H-bonding
network has formed. An interesting analogy exists between the crossover
related to the Widom line in supercooled water and the behavior observed
in experiments on supercritical argon [116]. Measurements of the velocity
of acoustic waves by inelastic x-ray scattering showed that the supercritical
region beyond the liquid-vapor critical point is partitioned by the Widom line
into two distinctly different regions with liquid- and vapor-like properties,
respectively. As the present simulations and numerous other work suggests,
a similar division into HDL-like and LDL-like regions with qualitatively
different structural and dynamical properties exists in supercooled liquid
water.
4.5
Bimodality in inherent structures
Figure 2.6 in Chapter 2 illustrated how regions of low density are
inhomogeneously distributed in the simulation box at the Widom
temperature at 1 bar (230 K). In Paper III it was proposed that a similar
situation holds for water at ambient temperatures as well, and furthermore
that a large difference in H-bonding geometries exists between locally
LDL-like and HDL-like regions. The local structure index (LSI) in the
inherent structures (IS), shown in fig. 2.5 and discussed in Paper IX, indicates
50
4.5 Bimodality in inherent structures
Correlation length [Å]
A
1 bar
1000 bar
1500 bar
2000 bar
4
-1
3.5
3
2.5
2
1.5
1
160
0.001
180
200
260
220 240
T [K]
B
1.2
Isoth. Compressibility [GPa ]
5
4.5
280
1
0.8
0.6
0.4
0.2
160
300
0.035
C
1 bar S(0)
1,000 bar S(0)
1,500 bar S(0)
2,000 bar S(0)
Fluctuation formula
Experiment
0
180
200
T [K]
240
260
280
300
D
1 bar
0.03
1 bar
220
-0.001
0.025
1,000 bar
-0.004
1,5
r
240
260
ba
220
00
2,0
0.5
200
r
ba
0.6
0.55
2,000 bar
180
200
220
240
T [K]
260
280
0
160 180 200 220 240 260 280 300 320 340
-0.008
160 180 200 220 240 260 280 300 320 340
180
00
0.005
1 bar
960
r
ba
0.65
980
940
160
0.7
00
1,000 bar
1020
1,0
ρ [kg/m3]
0.75
ar
1040
1000
2,000 bar
0.85
0.8
0.01
1,500 bar
1060
-0.007
0.9
1,500 bar
2,000 bar
1080
-0.006
0.015
1100
1,000 bar
1b
-0.005
1,500 bar
0.02
<Q>
-0.003
d<Q>/dT
α P [K -1]
-0.002
280
T [K]
T [K]
Figure 4.4: Thermodynamic and structural quantities from TIP4P/2005 simulations of
supercooled water at four isobars: 1, 1000 1500 and 2000 bar. A metastable liquid-liquid
critical point has been found for TIP4P/2005 at 1350 bar and 193 K [97]. A: Ornstein-Zernike
correlation lengths ξ and fit power-laws. Error bars have been estimated from variations in
ξ from different time intervals from the simulation trajectories. A maximum is clearly seen
at the Widom line for the ambient pressure isobar. B: Isothermal compressibilities compared
to experimental values (black lines). The dashed sections of the experimentally determined
power-laws represent extrapolated values. C: Thermal expansion coefficients determined from
the equilibrium densities, obtained from long NPT simulations of 360 molecules, shown in the
inset. The negative maxima at 1 and 1000 bar coincide with the Widom line. D: Temperature
dependence of the average tetrahedrality value h Qi. The inset shows h Qi as function of
temperature while the main figure shows its temperature derivative, where maxima again
coincide with the Widom line.
that an underlying structural bimodality between species with low- and
high-LSI, respectively, does in fact exist in simulations of TIP4P/2005 water,
51
4 Thermodynamics of water
but is masked by thermal excitations that smear out the clearly bimodal
distribution in the IS.
As discussed in Chapter 2 and Paper IX, classifying all molecules in
the IS as HDL-like or LDL-like based on their LSI value, and using that
classification to investigate the structures in the real, thermally excited
structure, reveals that despite the thermal smearing the HDL-like and
LDL-like species are structurally different and spatially separated even in
the real structure at higher temperatures. Furthermore, counting the number
of low- and high-LSI (HDL-like and LDL-like) species shows that the Widom
line is associated with maximal structural fluctuations between the two since
it coincides almost exactly with a 50:50 ratio. Figure 4.5 shows the relative
fractions of HDL-like and LDL-like species as function of temperature at
various pressures. A transition from a HDL-dominated to LDL-dominated
liquid appears in a narrow temperature interval around the Widom line and
the transition becomes more rapid at higher pressures, closer to the LLCP
of TIP4P/2005. The very rapid transition at 1500 bar is consistent with the
possibility that a discontinuous jump would occur at the liquid-liquid phase
transition of TIP4P/2005, while the slow change at 2000 bar is consistent
with an HDL-dominated liquid at this pressure.
Interesting new information is gained by calculating the local volumes
of the HDL- and LDL-like species. Voronoi polyhedra are useful for this
purpose since they can be used to calculate the volume around each
molecule, from which the local density is directly obtained. Figure 4.6
shows the distributions of local densities, where the total distribution has
been separated into contributions from HDL- and LDL-like species defined
according to their LSI value in the IS. A very clear difference is observed
where the HDL-like low-LSI species have a higher mean local density by
about 50 kg/m3 , or around 5%. The difference in mean density decreases
slightly with temperature. Another interesting result is reported in the inset
of fig. 4.6, where the temperature dependence of the mean local densities
reveals that while LDL-like species have a density maximum close to the
TMD of TIP4P/2005 (i.e. 277 K), the HDL-like species are denser at 253 K
and decrease more in local density at higher temperatures than the LDL-like
species. This indicates that the “normal” thermal expansion begins at lower
temperatures for the HDL-like species, and furthermore that they become
more thermally excited with increasing temperature.
It is an intriguing possibility that in real water a similar bimodality as
observed here in the inherent structures of TIP4P/2005 would somehow
influence the structure of ambient water despite thermal excitations, as in the
picture proposed in Paper III. A possible explanation could be that classical
simulations underestimated structural fluctuations but overestimated
stochastic thermal fluctuations, which indeed appears to be the case as
seen by the systematic underestimation of κ T at low temperatures and
overestimation at high temperatures. This could potentially explain the
52
4.5 Bimodality in inherent structures
0.9
0.8
1500 bar
2000 bar
1 bar
0.7
Population
1000 bar
Low-LSI
0.6
TWidom(1 bar)�230 K
0.5
TWidom(1000 bar)�200 K
TLLPT(1500 bar)�190 K
0.4
0.3
High-LSI
0.2
0.1
160 180 200 220 240 260 280 300 320 340 360
T [K]
Figure 4.5: Temperature dependence of the relative populations of low- and high-LSI species in
energy-minimized inherent structures. A cutoff value corresponding to the isosbestic minimum
point in the distributions of LSI values (see fig. 2.5) is used to define the two species. TWidom is
the Widom temperature at the given pressures and TLLPT is the temperature of the liquid-liquid
phase transition.
experimental observations from x-ray spectroscopies reported in Paper III,
but further work is required to prove or refute this hypothesis.
53
4 Thermodynamics of water
5
230 K
P(ρ (1) )
4
253 K
3
Total
Inh. low-LSI
2
Inh. high-LSI
1
0
5
1.03
P(ρ (1) )
ρ(1) [g/cm3]
278 K
4
3
1.01
0.97
0.95
230
2
320 K
inh. low-LSI
inh. high-LSI
0.99
250
270
T [K]
290
310
1
0
0.7
0.8
0.9
1
1.1
ρ(1) [g/cm3]
1.2
1.3
0.7
0.8
0.9
1
1.1
ρ(1) [g/cm3]
1.2
1.3
Figure 4.6: Distributions of local molecular densities for four different temperatures at 1 bar.
Voronoi tesselation has been used to obtain the local molecular volumes. The dashed vertical
lines indicate the mean densities of the two partial contributions to the total local density
distribution, and the inset shows the mean densities as function of temperature.
54
5
Electronic structure of water
In this last chapter before turning to a summary of Papers I-IX, selected
results from calculations of x-ray spectroscopic properties of liquid water
will be presented. Such calculations aim to shed light on structural properties
of water and concepts introduced in Chapter 2 will frequently be applied
here. Because of the very different nature of methods dealing with electronic
structure, as opposed to the atomic structure discussed in Chapter 2, ample
room will be given to introducing theoretical concepts relevant for the
calculation of x-ray spectroscopic absorption spectra.
5.1
Theoretical background
Since the basic foundations of quantum mechanics were established in
the 1920’s by the introduction of the Schrödinger and Dirac equations,
wave-particle duality, Heisenberg’s uncertainty principle etc. a large number
of theoretical and numerical techniques have been developed to effectively
solve the electronic structure of atoms, molecules and condensed matter, to a
varying degree of approximation.
The following discussion is based on invoking the Born-Oppenheimer
approximation to separate the total wave function into nuclear and electronic
parts,
Ψtotal (r, R) = Ψel (r; R)Ψnuc (R)
(5.1)
where r and R are the electronic and nuclear coordinates, respectively, and
the parametric dependence of the electronic wave function on the nuclear
coordinates is explicitly shown. The energy of an interacting system of
electrons in a fixed frame of nuclear coordinates can then be written as
E = ET + EU + EV
55
(5.2)
5 Electronic structure of water
where ET is the electron kinetic energy and EU and EV are the electronelectron and electron-nucleus interaction energies, respectively. The
corresponding operators for these energy contributions can be written
as (in atomic units: h̄ = me = e = 1/4πe0 = 1)
T̂
= −
Û
=
1 N 2
∇i
2∑
i
N M
i a
N N
V̂
=
Za
∑ ∑ |ri − Ra |
∑ ∑ r
i j >i
1
−
rj i
(5.3)
(5.4)
(5.5)
where N is the number of electrons and M is the number of nuclei with
positions R a and charges Za . A basic concept in electronic structure theory is
the self-consistent field (SCF) approach which encompasses methods based
on both the Hartree-Fock (HF) ansatz and density-functional theory (DFT)
methods. The idea behind the SCF approach is to separate a many-body
problem into effective single-particle equations describing each particle
individually, in which the interaction with other particles is approximated
by a term describing the mean-field from all other particles, and solving the
set of one-particle equations self-consistently. Hartree-Fock theory is based
on a single-determinant ansatz for the wave function:
φ1 (r1 ) φ2 (r1 ) · · · φN (r1 ) 1 φ1 (r2 ) φ2 (r2 ) · · · φN (r2 ) Ψ(r1 , r2 , ..., rN ) = √ .
(5.6)
..
..
..
.
N! ..
.
.
φ1 (rN ) φ2 (rN ) · · · φN (rN )
The full time-independent Schrödinger equation, ĤΨ = EΨ, is then replaced
by the single-particle canonical eigenvalue equations
F̂i φi = ei φi
(5.7)
where F̂ is the Fock operator, composed of single-electron terms and electronelectron interaction terms:
F̂i = ĥi + ∑ Ĵi − K̂i
(5.8)
i
where Ĵ and K̂ are the Coulomb and exchange terms, respectively. HF theory
is explained in full detail in standard textbooks [117].
56
5.1 Theoretical background
5.1.1
Density functional theory
In density functional theory (DFT), the 3N-dimensional problem (3 degrees
of freedom for each electron) is replaced by the much simpler 3-dimensional
electron density n(r). DFT rests on two theorems put forward by Hohenberg
and Kohn [118]. The first states that if two external potentials V1 (r) and V2 (r)
(potentials felt by the electrons) share the same electron density n(r) then
they can at most differ by a constant, V1 − V2 = const. A corollary to this
theorem is that all other properties of the system are unique functionals of
the electron density. The second theorem proves the existence of a universal
energy functional F [n(r)] which, for a given external potential V (r), gives the
minimal value of the energy expression
E[n(r)] = F [n(r)] +
Z
n ( r )V ( r ) d3 r
(5.9)
only for the correct ground-state density n0 (r); this energy is furthermore
the correct ground state energy E0 . Even though the true functional
F [n(r)] is and will likely remain unknown, the two theorems provide the
theoretical justification for considering the density as the relevant variable
in electronic structure calculations instead of the complicated many-electron
wave function. Kohn and Sham [119] invented a practical way to obtain
approximate ground-state densities by introducing an artificial auxiliary
system of non-interacting electrons and decomposing the true kinetic energy
into a non-interacting part and a remaining correction due to electronelectron correlations. The functional can then be written as (all quantities are
functionals of the electron density n(r))
F = T + U = T0 + J + Exc .
(5.10)
T0 and J are the kinetic and Coulomb energies of the non-interacting system
while Exc = ( T − T0 ) + (U − J ) is the exchange-correlation energy functional
which accounts for all energy contributions neglected by the non-interacting
system. In practical calculations, one-electron Kohn-Sham orbitals φiKS are
introduced and the total wave function is written as a single determinant as
in eq. (5.6). Constraining the orbitals to be orthogonal and minimizing the
energy expression in eq. (5.10) gives the one-electron equations, similar to
HF theory:
1 2
− ∇ + ve f f (r) φiKS = ei φiKS
(5.11)
2
where
ve f f (r) =
Z
Za
n(r0 ) 0
dr − ∑
+ v xc (r)
r
−
Ra |
|r − r0 |
|
a
(5.12)
is the effective one-electron potential. The first and second terms on the
right hand side of eq. (5.12) are the electron-electron and electron-nucleus
57
5 Electronic structure of water
interactions, while the third is the (unknown) exchange-correlation potential:
v xc (r) =
δExc (r)
.
δn(r)
(5.13)
A vast number of different approximations for v xc (r) have been proposed.
The local density approximation (LDA) which is based on exchange and
correlation energies of a uniform electron gas constitutes the simplest
approximation. An improvement is obtained by also including the gradient
of the electron density in the functional expression, i.e. the so called
generalized gradient approximation (GGA). Functionals that include parts
of the exact exchange energy as obtained in the HF formalism (where
however dynamical correlation is completely neglected) are termed hybrid
functionals.
Many GGA and hybrid functionals can provide accurate energies for
various systems, but systems in which dispersion (or van der Waals, vdW)
interactions play an important role are often poorly described since non-local
effects are neglected. Efforts have been made to include empirical corrections
in DFT functionals [65, 66], but a complete ab initio generally applicable
treatment of non-local effects has been suggested only recently in the vdWDF [60], vdW-DF2 [61] and optPBE [120] models. The non-local vdW energy
in these functionals takes the form of a 6-dimensional integral,
Ecnl =
1
2
Z Z
n(r)φ(r, r0 )n(r0 )d3 rd3 r0
(5.14)
where φ(r, r0 ) is an interaction kernel which depends on the densities at r
and r0 and their gradients. vdW forces are isotropic and attractive, and their
inclusion in DFT simulations of liquid water can be expected to significantly
impact the resulting ensemble properties. This has indeed been found and
reported recently for the vdW-DF model [63], and structural analysis of vdWDF2 simulations are reported in ref. [71] (Paper XIV). In spite of excellent
agreement with accurate quantum chemical results for the energies of small
water clusters [121], indeed outperforming other common DFT functionals
that lack non-local interactions, the O-O PCF from vdW-DF2 simulations
shows poor agreement with experimental data as the first peak is shifted
to too large distances and a well defined second coordination shell is absent;
in fact it becomes similar to experimental results on liquid water under high
pressure. Much theoretical work is at present dedicated to the study of vdW
interactions in DFT, and it remains to be seen to what extent vdW-DF2 can
be improved.
58
5.1 Theoretical background
5.1.2
Theoretical spectrum calculations
Figure 5.1 shows the different energy regions for XAS and EXAFS
experiments and the physical processes involved in each technique. Note
that in the literature XAS is also referred to as NEXAFS (near-edge x-ray
absorption fine-structure) or XANES (x-ray absorption near-edge structure).
The 1s core states are strongly localized and since the electronic transition
ћω
Absorption
XAS
k2 χ(k)
EXAFS
0
400
500
600
1
700
800
E [eV]
2
3
4
-1
k [Å ]
900
5
6
7
1000
8
1100
Figure 5.1: Schematic picture of the x-ray absorption process. Close to the absorption edge the
x-ray photons excite core-electrons into unoccupied states, while far above the edge the excited
photoelectrons travel as free particles and scatter off neighboring atoms.
in XAS follows the dipole selection rule only the unoccupied p-character
of the local environment of the excited water molecule is probed. XAS on
water is furthermore strongly sensitive to molecular orbitals localized along
the H-bonds and thus has the potential to provide structural information
on order and disorder in the H-bond network. The x-ray absorption cross
section can be written according to the so called Fermi’s Golden Rule in the
following form:
σ (ω ) = 4π 2 αω ∑ |hc|ê · r| f i|2 δ( E f − h̄ω )
(5.15)
f
where α is the fine-structure constant, ê is the x-ray polarization vector, c is the
core state, f runs over the unoccupied final states and E f are the energies of
the final states relative to the initial state. The term e · r is the dipole operator
written in “length form” [122] and it provides an accurate approximation
to the absorption process when the condition k · r 1 holds, where k is
the incoming photon wave vector and r is roughly the size of the absorbing
electron’s wave function. At high photon energies, such as in x-ray Raman
59
5 Electronic structure of water
scattering at higher momentum transfer, higher order terms may have to be
considered to take into account s → s and s → d transitions. Both DFT and
multiple-scattering theory can be applied to calculate XAS and an outline of
both approaches will be given below.
EXAFS on the other hand probes free-electron states where the excited
core-electron progagates as a spherical wave out from the absorbing atom
and scatters off neighboring atoms. The relevant theoretical framework to
analyze EXAFS oscillations is multiple-scattering theory.
The transition-potential density-functional theory method
A well established approach for calculating XAS spectra is the transitionpotential (TP) method [42, 123] in DFT. It is based on Slater’s transition
state method [124], in which excitation energies are calculated as the orbital
energy difference between two levels of a variationally determined state,
where only one half electron has been excited into the higher-lying level.
It can be shown that the excitation energy is correct up to second order
in the change of occupation [42, 124, 125] for the transition state method.
Slater’s approach becomes impractical to apply for calculations of spectra,
however, due to the requirement that each excitation energy be variationally
computed independently. The TP-DFT approach simply neglects the half
excited electron and considers the core-hole potential from the half-occupied
core-state to provide a good balance between initial- and final-state effects.
Experience has shown that the TP-DFT method provides a good description
of XAS spectra of many different systems. Two academic DFT codes which
implement the TP-DFT method in a Gaussian basis-set cluster approach are
the StoBe-deMon [126] and deMon2k [127] programs. TP-DFT has also been
implemented in GPAW [128], which is based on the real-space grid-based
projector augmented wave method. In StoBe-deMon and deMon2k, which
have been applied in this work, a double-basis set approach is applied where,
after convergence of the SCF loop, the converged state with a half-occupied
core-hole is expanded in a large basis set augmented with diffuse basis
functions to better describe Rydberg and continuum states.
When computing spectra of disordered compounds such as liquids, it
is important to average over a relevant range of different local structures
to converge the spectrum towards the ensemble average. Each computed
spectrum then needs to be calibrated to an absolute energy scale. This is
accomplished with the delta-Kohn-Sham (∆K–S) approach [42,129], in which
the first core-excited state with a full electron excited into the first unoccupied
orbital is computed. The energy difference between the first core-excited
state and the ground state then defines a corrected onset of the spectrum
as calculated with the half core-hole potential. This corrects for the neglect
of relaxation effects from the interaction of the excited electron with the full
core-hole state.
60
5.1 Theoretical background
Multiple-scattering theory
A completely different approach to the electronic structure and x-ray
absorption calculation problem is provided by multiple-scattering (MS)
theory. Introducing this approach requires a basic outline of the underlying
mathematical concepts; for a full exposition of the theory see, e.g., ref. [130].
The first step is to recast Fermi’s golden rule for the core-electron absorption
process, eq. (5.15), into [131, 132]
σ(ω ) ∝ −Im c ê∗ · rG (r, r0 ; E)ê · r0 c θ ( E − EF )
(5.16)
where G is the Green’s function, G = 1/( E − Ĥ + iη ), expressed explicitly in
terms of matrix elements as
D −1 0 E
G (r, r 0 ; E) = r | G ( E)| r 0 = r E − Ĥ + iη
(5.17)
r .
H = H0 + V is the full Hamiltonian and η is a small positive quantity that
shifts the poles of G into the complex energy plane. θ in eq. (5.16) is a step
function (or a smeared Fermi distribution function) to exclude poles of G
below the Fermi level.
The total Green’s function is separated into contributions from the central
absorbing atom and neighboring scattering atoms, G = G c + G SC , and
each part is constructed using the regular and irregular solutions to Dirac’s
equation (note that the Hamiltonian of the absorbing atom contains the
effects of a core-hole and usually a fully screened core hole corresponding
to the final-state rule is used). A multiple-scattering expansion is then
performed on the scattering contribution by iterating the Dyson equation for
the Green’s function, G = G0 + G0 TG, as
G SC = G0 TG0 + G0 TG0 TG0 + . . .
(5.18)
where G0 = 1/( E − Ĥ0 + iη ) is the unperturbed Green’s function and
T = ∑i ti (where ti are scattering matrices at each site, calculated as
tl = exp(iδl ) sin(δl ) where δl is the scattering phase shift for angular
momentum l). The multiple-scattering expansion does not always converge,
in particular for energies close to the absorption edge in the XAS region.
Therefore, full multiple-scattering (FMS) is used for XAS calculations, where
eq. (5.18) is expressed as
h
i −1
G0 .
G SC = 1 − G0 T
(5.19)
The matrix inversion is equivalent to an infinite order scattering expansion.
61
5 Electronic structure of water
Having thus obtained the MS expansion or FMS inversion for G SC , the
x-ray absorption cross section can be written as
σ( E) = σ0 ( E) [1 + χ( E)]
(5.20)
where χ( E) is called the fine-structure, and is based on the scattering part
of the Green’s function G SC , and σ0 ( E) is the smooth atomic background,
calculated from the embedded absorbing atom’s Green’s function G c . Since
the Green’s function has all information about the system built into it,
other properties can be readily obtained.
The electronic density of states for
R
instance is calculated as ρ( E) ∝ −Im G (r, r; E)d3 r.
In EXAFS the fine-structure χ( E) is the main quantity of interest since
it contains the structural information about the system. Experimentally, it is
separated from the total x-ray absorption edge through (cf. eq. (5.20))
χ( E) =
σ( E) − σ0 ( E)
.
∆σ0 ( E)
(5.21)
The absorption coefficient σ( E) is experimentally defined from I = I0 e−σd , I0
and I are the incoming and transmitted photon intensities respectively, d is
the sample thickness, and ∆σ0 is the jump in the absorption coefficient at a
spectrum threshold energy E0 . Through the threshold E0 (spectrum onset) a
photoelectron wave number is defined as
s
2m( E − E0 )
k=
(5.22)
h̄2
from which the fine-structure can be expressed in k-space, χ(k ). Finally, the
so called EXAFS equation [122] is used to connect the experimental χ(k ) to
geometrical properties:
χ(k) =
∑
j
Nj f j (k )e
−2k2 σj2 −2R j /λ(k)
e
kR2j
sin 2kR j + δj (k ) .
(5.23)
The sum runs over all shells j of neighboring atoms around the absorbing
atom, Nj , R j and exp(−2k2 σj2 ) are the coordination number, distance and
Debye-Waller (DW) factor of the shell, and f j (k ) and δj (k ) are the effective
scattering amplitudes and the scattering phase shifts of the neighboring
atoms, respectively; λ(k) is the mean-free path of the photoelectron which is
determined by inelastic losses through interactions with other electrons and
phonons.
As can be seen above, a close correspondence exists between theoretical
and experimental definitions. Theoretical phase shifts, scattering amplitudes
and mean-free paths can be calculated using the real-space Green’s function
62
5.2 Extended x-ray absorption fine-structure
multiple-scattering program FEFF [132]. Structural refinement can then be
applied to obtain distances and DW factors through eq. (5.23) using, e.g., the
Ifeffit program [133].
Comparison between TP-DFT and FMS for XAS calculations
Both the transition-potential approach in DFT and the Green’s function
based approach in MS theory can be used to calculate x-ray absorption
spectra. The two methods complement each other in that they are suitable
for different systems. It has proved difficult to generalize the MS approach
to non-spherical scattering potentials; calculations performed with FEFF
use the muffin-tin approximation which represents the atoms by spherically
symmetric scattering potentials. This approximation breaks down near
absorption edges in systems where chemical effects strongly distort the
spherical symmetry of atomic charge densities, but many systems, such
as metals, are accurately described by muffin-tin potentials. TP-DFT
calculations have the advantage that the absorption spectrum can be
intuitively interpreted in a molecular orbital framework where spectral
features can be related to orbitals pertaining to particular structural motifs.
On the other hand, MS calculations are more appropriate for the energy
region farther away from the absorption threshold, and relativistic effects
such as spin-orbit coupling are readily accounted for by using relativistic
wave functions in the construction of the Green’s function. The latter allows
for a description of absorption edges split by relativistic effects, such as the
L I I and L I I I edges reflecting the separation between 2p1/2 and 2p3/2 levels
by the spin-orbit effect.
5.2
5.2.1
Extended x-ray absorption fine-structure
Experimental results
Since there are still contradicting claims on the structure of liquid water
even regarding the height, shape and position of the first peak in the O-O
pair correlation function gOO (r ), the recent introduction of synchrotronbased extended x-ray absorption fine-structure (EXAFS) measurements
as a probe of water structure [5] was highly welcomed. EXAFS has been
measured on water before [134, 135], but low signal-to-noise levels due to
the weak scattering power of water molecules precluded any quantitative
conclusions. The new measurements [5] overcame many difficulties by
utilizing a third-generation lightsource and the x-ray Raman scattering
mode. Furthermore, both liquid water and hexagonal ice could be measured
in the same experimental setup. The reported results, however, presented a
puzzle to the water research community; using multiple-scattering analysis
with the GNXAS package [136] resulted in a first peak in gOO (r ) which was
63
5 Electronic structure of water
significantly higher and at a shorter position than that found in analyses of
diffraction data [44, 46–48]. Figure 2.3 in Chapter 2 shows the magnitude of
this discrepancy in gOO (r ).
5.2.2
Theoretical calculations and structural information
The SpecSwap-RMC method described in Chapter 2 was employed to
investigate the discrepancy between EXAFS and diffraction-based results
on gOO (r ). Multiple-scattering calculations of the EXAFS fine-structure χ(k )
were performed on a large set of molecular clusters extracted from MD
and RMC simulations of water. The basis set required for SpecSwap-RMC
structure modeling was created from the cluster geometries together with
their associated computed χ(k ). An energy threshold E0 , shown in eq. (5.22),
must be determined to put the calculated χ(k ) on a calibrated k-scale. A
common procedure in EXAFS analysis is to fit the E0 of the investigated
system to obtain best agreement with experimental data. For the SpecSwap
fits reported in Paper V, a model structure of ice was instead used to obtain
E0 which was subsequently used for all calculations on water clusters.
Figure 5.2A shows the resulting agreement with experimental EXAFS
data on ice; the good agreement indicates that the calculations are reliable
and that structural information on liquid water can be obtained from fits
to the liquid water data. Inspection of the scattering paths that contribute
to the total EXAFS oscillations shows that many-body scattering, primarly
from so called focusing paths where a hydrogen atom aligns colinearly
between two oxygen atoms in a H-bond, contribute significantly. This can
also be seen in fig. 5.2A where the isolated single-scattering contribution in
ice gives poor agreement with experimental data. Further evidence for the
importance of many-body scattering paths in water is seen in fig. 5.2B, where
the SD and DD RMC models from Paper I are seen to give very different
EXAFS oscillations despite their equivalent agreement to diffraction data
and almost identical gOO (r ). Parts C and D of fig. 5.2 show the results from
SpecSwap-RMC fits to only the EXAFS data and simultaneously to both
EXAFS and diffraction-based gOO (r ), respectively. Fitting only experimental
EXAFS gives a poor agreement with gOO (r ) from diffraction, while the joint
fit gives a satisfactory agreement with both data sets. There are numerous
sources of uncertainties in these fits, including the use of a single E0 value for
all EXAFS cluster calculations and the removal of the weak intramolecular
EXAFS scattering from the calculations (due to an inherent problem in
multiple-scattering calculations in handling very short intramolecular
bonds), so a perfect agreement with experiment is not expected.
The main strength of the SpecSwap method is however the ability to
inspect structural trends, i.e. to use the acquired weights of each cluster
after an equilibrated SpecSwap run to compute reweighted geometrical
properties, as discussed in Chapter 2 in relation to eq. (2.31). In this spirit,
64
5.2 Extended x-ray absorption fine-structure
1
0.6
SPC/E ice, single scattering
SPC/E ice, multiple scattering
Experimental ice
0.8
0.6
0.4
0.2
0.2
k 2X(k)
k 2X(k)
0.4
0
-0.2
-0.4
0
-0.2
-0.6
-0.8
-1
2
A
-0.4
3
4
5
-1
6
7
8
2
k[Å ]
0.6
0.4
3
5
4
0.3
6
k [Å -1 ]
7
8
EXAFS+PCF fit
pure EXAFS fit
Unweighted basis set
Target PCF
3
2.5
0.2
0.1
g(r)
k 2X(k)
B
3.5
EXAFS+PCF fit
pure EXAFS fit
Unweighted
Expt data
0.5
0
-0.1
2
1.5
1
-0.2
-0.3
-0.4
-0.5
SD
DD
SPC/E
Expt
2
C
0.5
3
4
5
-1
k [Å ]
6
7
8
0
2
D
2.5
3
r [Å]
3.5
4
4.5
Figure 5.2: Ensemble averaged theoretical EXAFS oscillations compared to experimental data,
plotted as k2 χ(k) to enhance the oscillations. A: Calculated EXAFS on a model system
of hexagonal ice, taken from a SPC/E path-integral simulation, after calibration of the E0
parameter (E0 = 5 eV). Both the total multiple-scattering contribution and only the singlescattering contribution are shown. For agreement with experimental data, the large many-body
contribution mainly from focusing OH. . . O scattering paths must be included. B: Comparison
between two RMC structure models and the SPC/E MD model for liquid water. The SD
(single-donor) and DD (double-donor) RMC models from Paper I have almost identical OO PCFs and are in optimum agreement with XD and ND data, but contain very different
H-bonding networks which can be seen to greatly impact the resulting EXAFS oscillations.
Despite bad agreement with XD data due to the too sharp first O-O peak, the SPC/E model
is seen to agree well with experimental EXAFS data. C and D show the SpecSwap-RMC fits
to EXAFS and diffraction-based gOO (r ). A fit to only EXAFS (blue dashed line) gives bad
agreement with diffraction based gOO (r ), while a joint fit to EXAFS and gOO (r ) is seen to
give good agreement with both.
two unweighted and reweighted geometrical properties are shown in fig. 5.3,
namely the distribution of H-bonding angles and asphericity parameters
defined in eq. (2.34). The former yields direct information on the H-bonding
network and the importance of linear H-bonds giving rise to focusing
scattering paths (which enhance the EXAFS oscillations as seen in fig. 5.2A),
while the latter quantifies the degree of order or disorder in the oxygen
65
5 Electronic structure of water
positions of the first coordination shell (perfect hexagonal ice gives η=2.25
while disorder and interstitial molecules lead to lower values). As can be
seen in fig. 5.3, the joint fit to EXAFS and diffraction-based gOO (r ) leads to
opposite behavior in H-bonding angles and asphericity values. Specifically,
the weighted distribution of H-bond angles is shifted to more linear bonds
with angles between 0◦ and 10◦ , evidencing an enhanced contribution from
the focusing effect for nearly linear bonds, while the weighted distribution of
asphericity values is shifted to lower values, indicating larger disorder in the
first coordination shell. The former effect comes from the fit to experimental
EXAFS while the latter is due to the fit to the diffraction-based gOO (r ).
The main outcome of this study is the finding that the initially reported
contradicting result from EXAFS can be resolved if a large focusing effect
is accounted for, indicating that a certain fraction H-bonds in liquid water
are very close to linear. Further experiments and theoretical modeling will
potentially provide quantitative information on H-bonding angles in water
where the novel SpecSwap-RMC approach may prove to be a powerful tool.
0.14
0.12
0.1
A
0.14
0.12
Unweighted basis set
Weighted EXAFS+PCF fit
0.1
sin(θ)P(θ)
Unweighted basis set
Weighted EXAFS+PCF fit
0.08
0.08
P(η)
0.06
0.04
0.06
0.04
0.02
0.02
0
0
2 x diff
-0.02
-0.04
B
0
5
10
15
20
2 x diff
-0.02
25
30
35
HB-angle θ [˚]
40
-0.04
1.4
1.6
1.8
η
2
2.2
Figure 5.3: Geometrical results from the joint SpecSwap fit to experimental EXAFS and
diffraction-based gOO (r ). A: Unweighted and reweighted distributions of H-bonding angles
(illustrated in fig. 2.2), normalized by the solid angle. B: Unweighted and reweighted
distributions of aspericity values, defined in eq. (2.34).
5.3
5.3.1
X-ray absorption spectroscopy
Experimental results
In 2004 Wernet et al. [7] reported experimental x-ray absorption spectra of
liquid water and ice which, supported by model experiments and theoretical
density functional theory (DFT) calculations, indicated that a large fraction
of water molecules in the liquid are in asymmetric H-bonding environments
in the sense that one of the two possible donating bonds is distorted or
66
2.4
5.3 X-ray absorption spectroscopy
broken. Experimentally, this was shown by comparing the XAS spectrum of
the surface of ice, which features “dangling” unbonded O-H groups, and the
liquid spectrum; the similarity between the two in the pre-edge region (i.e.
the strong pre-peak around 535 eV) which is sensitive to structural disorder
was interpreted to indicate a large fraction of single-donor (SD) molecules
also in the liquid phase.
5.3.2
Importance of zero-point vibrational effects
A subtle but important aspect in the calculation of XAS spectra is the
influence of zero-point (ZP) vibrational effects. Neglecting this effect often
gives poor results since vibronic couplings and a vibrational broadening of
excited state resonances are ignored. A comparison between XAS spectra
computed on centroid and bead geometries from path-integral simulations
of hexagonal ice has shown [42] that the spectral shape and intensity ratio
between the main- and post-edge features are greatly improved by sampling
the zero-point vibrational motion, which is done implicitly in path-integral
simulations with flexible molecules.
Many MD models are however defined and parametrized using rigid
molecules, where the intramolecular degrees of freedom are constrained in
the simulations using constraint algorithms. An approach was developed
to approximately account for the ZP vibrational effects in a post-treatment,
where the intramolecular geometries from simulations of rigid molecules
are modified according to the quantum mechanical vibrational distributions
assuming a decoupled isolated stretch mode. Intramolecular potential
energy surfaces (PES) for the O-H stretch coordinates are first calculated
using the flexible, polarizable and ab initio-parametrized potential model
TTM2.1F [137]. A harmonic potential, 2k (rOH − req )2 where req is the
equilibrium
p position, is fit to the PES of each O-H bond and the frequency
ω = 2π k/µ (where µ is the reduced mass of the oxygen and hydrogen) is
used to construct the ground-state harmonic oscillator wave function and the
probability distribution from its absolute square. Finally, new O-H distances
are sampled from the calculated probability distributions for each O-H bond
separately. The resulting distribution of intramolecular O-H distances in the
modified simulation box gives a physical representation of the zero-point
vibrational effect.
Figure 5.4A shows calculated XAS spectra from a TIP4PQ/2005
simulation of hexagonal ice 1 compared to experimental data [41], where
the ZP treatment is seen to shift intensity from the post-edge to the preand main-edge regions, providing better agreement with experiment. Also,
including augmentation basis functions (to improve the description of
diffuse states) on the four nearest-neighbor oxygen atoms is seen to improve
the agreement in the post-edge region, compared to using augmentation
1 Carlos
Vega, private communication
67
5 Electronic structure of water
functions only on the absorbing atom. However, although the main- and
post-edge positions and the overall spectral shape is well reproduced, the
agreement is not quantitative. Spectral intensity in the pre-edge region is
lacking and the main- to post-edge intensity ratio is still underestimated.
The discrepancies are undoubtedly in part due to structural details in the
simulated ice which may differ in important aspects from the experimentally
prepared ice, but further developments of theoretical spectrum calculations
are nonetheless required to understand in detail the XAS spectrum of
model systems such as hexagonal ice. This is particularly important for a
quantitative interpretation of the XAS spectrum of liquid water since the
interpretation in terms of large structural disorder and broken H-bonds [7, 8]
has been intensely debated and often questioned in the literature [138–140].
In spite of the disagreement on the precise interpretation of the XAS
spectrum of water there is a general accord on many qualitative aspects.
The pre-edge region has been connected to broken H-bonds by several
independent studies and the post-edge region has an unambiguous
connection to ice-like, four-coordinated molecules [139, 141, 142]. In the
following, a pragmatic viewpoint is adopted and theoretically computed
spectra are discussed based on observed qualitative trends only.
5.3.3
Structural sensitivity of XAS
As reported in Paper I and refs. [143, 144] the cone criterion for H-bonding,
originally used to define asymmetrically H-bonded species to explain the
experimentally observed spectrum of liquid water, has turned out to be
an insufficient requirement to obtain theoretical spectra in quantitative
agreement with XAS experiments. The asymmetrical SD model and the
symmetrical DD model, generated by RMC modeling of diffraction data
reported in Paper I, gave rather similar XAS spectra. An interpretation of
this finding is that the structural sensitivity of XAS is such that not only
H-bonding geometries but also other aspects of the structure are important
for computed spectra. Various many-body correlations that emerge from
the interactions between molecules in the liquid are neglected in RMC
simulations, simply due to the fact that no intermolecular potential is
used. MD simulations, or the hybrid EPSR method [23] which combines
structure modeling of diffraction data with effective pair-potentials, provide
structures from well-defined thermodynamical ensembles where many-body
correlations are included through the interaction potential, but the possible
downside of these methods is the reliance on an approximate potential
which biases the resulting structure.
Structures derived from MD simulations nevertheless provide a
convenient testing ground for investigating the sensitivity of computed
XAS spectra to various many-body correlation functions. Parts B, C and
D of fig. 5.4 show computed spectra using the TP-DFT approach on a
68
5.3 X-ray absorption spectroscopy
snapshot from a TIP4P/2005 simulation at 310 K decomposed based on
the number of donating H-bonds, the tetrahedrality parameter Q and the
local structure index I, respectively (Chapter 2 defines these parameters).
Since the simulations were performed using rigid intramolecular geometries
the ZP treatment described above was applied. Comparable behavior is
seen for the Q and I parameters; namely, species with lower values of
Q and I, reflecting less tetrahedral local structure and more interstitial
neighbors, respectively, give stronger pre- and main-edge features but a
weaker post-edge. The difference spectra are closely similar to the difference
between experimentally measured high-temperature and low-temperature
spectra [7]. This is an anticipated trend since both Q and I are expected to
decrease with temperature (see fig. 2.5 for the temperature dependence for
the TIP4P/2005 model), and it confirms that TP-DFT calculations realistically
capture this aspect. By comparing with fig. 5.4B, however, it becomes
evident that computed spectra show far stronger sensitivity to the number
of donated H-bonds of the absorbing water molecule. The pre-edge intensity
is absent for double-donor (DD) molecules, shows up for single-donor
(SD) molecules and becomes comparable to the experimental intensity for
non-donor (ND) molecules. On the other hand, the contribution from ND
species overestimates the main-edge intensity and underestimates the postedge intensity, while the SD contribution appears to give the best overall
agreement with experiment.
What conclusions can be drawn from the observed trends? Firstly, there
are clearly structural environments present in the simulation which after
averaging give a sharp pre-edge peak in good agreement with experimental
spectra; namely, the weakly H-bonded ND species. There is thus no inherent
deficiency in the TP-DFT approach in reproducing this peak for appropriately
defined structural species. However, judging from the comparison between
theoretical and experimental hexagonal ice spectra, it appears that either the
pre-edge peak intensity is underestimated by the TP-DFT method, or the
experimentally prepared ice structure differs from the simulated structure
due to experimental contributions from structural defects, heterogeneities
and grain-boundaries. The latter explanation has been suggested based on
experimental grounds [41], and future theoretical and experimental work
on hexagonal ice and other ice polymorphs will indubitably shed further
light on this issue. Clearly, the experimental observation of a strong pre-edge
peak in liquid water is of considerable interest for a deeper understanding of
the complex H-bond network. Secondly, the post-edge intensity after the ZP
vibrational post-treatment is seen to be farther away from the experimental
intensity for the liquid compared to the ice calculations. This indicates that
even if the intensity ratio between the main- and post-edge features were to
be improved in the calculations, rendering the ice spectrum in quantitative
agreement with experiment through, e.g., a better description of quantum
effects, the MD simulations of liquid water would still feature a too strong
69
5 Electronic structure of water
ND
SD
DD
Expt
Absorption
Absorption
Expt iceIh
iceIh rigid
OH ZP-distrib.
OH ZP-distrib. extra-aug.
A
534
536
538
540
Energy [eV]
542
544
534
546
536
538
542
544
540
542
544
Absorption
2x diff
Expt
C
D
x2
x2
534
540
Energy [eV]
I < 0.05
I > 0.05
Q < 0.7
Q > 0.7
2x diff
Expt
Absorption
B
536
538
540
Energy [eV]
542
534
544
536
538
Energy [eV]
Figure 5.4: Calculated XAS spectra compared with experiment. The ice calculations have been
performed on a snapshot taken from a simulation of hexagonal ice and the liquid calculations
on a snapshot from a liquid simulation at 310 K, in both cases the TIP4P/2005 potential was
used. A: Hexagonal ice. The “iceIh rigid” curve has been calculated with fixed intramolecular
geometries, “OH ZP-distrb” is the same structure after the zero-point vibrational treatment
and “OH ZP-distrb extra-aug” is where extra augmentation basis functions have additionally
been added. B: Calculated liquid water spectra separated according to H-bonding statistics
max =3.2 Å. ND means non-donor
where the cone criterion in eq. (2.32) was used with rOO
molecules, SD is single-donors and DD is double-donors. C and D: same liquid water
snapshot where the separation is based on the tetrahedrality and local-structure index, using,
respectively the cutoffs Qc =0.7 and I=0.05. The difference spectra have been scaled by a factor
2.
post-edge peak. But this should not be surprising since, as discussed in
Chapter 2, the pair-correlation functions of TIP4P/2005 and many other MD
models are overstructured in the nearest-neighbor environment.
70
6
Summary of main results
In this chapter the main conclusions of Papers I-IX will be briefly
summarized. When needed, reference will be made to figures presented in
the papers using the convention that fig.I-1 refers to the first figure of Paper
I, etc.
6.1
Water, tetrahedral or not - Paper I
Neutron and x-ray diffraction experiments are powerful tools to obtain
structural information on pair-correlations in liquids and solids. In a
complex molecular liquid such as water where many-body correlations play
a key role, an important question is whether the structural information
contained in the PCFs is sufficient to pin down the topology of the complex
H-bonding network. Previous diffraction studies have often been interpreted
to support the continuum tetrahedral model. This conclusion has mainly
been based on the observed position of the second maximum in the oxygenoxygen PCF around 4.5 Å, which coincides approximately with the distance
√
between second-nearest neighbors in a tetrahedral network, i.e. a factor 8/3
times the first neighbor distance. However, not only is the second peak in
gOO (r ) very broad, which shows that local structures in water must vary
greatly between tetrahedral and disordered, but the form of gOO (r ) also
lacks information on local H-bonding patterns.
A critical investigation using RMC simulations into the sensitivity of
diffraction data to H-bonding topologies in water was therefore conducted
to see what limits on H-bonding topologies could be derived from the
most recent x-ray and neutron diffraction data. The RMC method was
ideally suited for this task since geometrical constraints can be applied
to investigate the structural content of the data. In addition to 5 neutron
diffraction data sets [44] on isotopically substituted samples and one x-ray
diffraction data set [58], an approximative scheme for fitting the O-H stretch
71
6 Summary of main results
Raman spectrum of dilute HOD in D2 O was also included. This scheme
relies on a correlation [145, 146] between the electric field projected onto
the intramolecular O-H bond vector, calculated using the SPC/E force field,
and the O-H stretch frequency where the stretch is assumed to be a local
uncoupled mode, as in dilute HOD in D2 O.
To investigate the structural sensitivity of the diffraction and IR/Raman
data to H-bonding geometries, the RMC++ [147] code was extended to
allow for constraints on the H-bonding geometries. Specifically, the Hbonding cone criterion suggested in ref. [7] was used to enforce, in two
different structure models, the largest possible number of double donor
(DD) species and the largest number of single donor (SD) species with
one broken donating H-bond. The two models represent on one hand
the continuum tetrahedral model and on the other the asymmetrical Hbonding model suggested by Wernet et al. [7]. Geometrical constraints on
the intramolecular geometry were furthermore implemented to ensure
that the resulting ensembles of RMC structure models had intramolecular
O-H distances and HOH angles that satisfied distributions derived from
path-integral simulation studies [148]. Intramolecular constraints of this
sort appear not to have been used in RMC simulations before, but since
neutron scattering structure factors of many compounds are dominated by
intramolecular contributions it is important to have physical distributions
of intramolecular distances; the RMC algorithm finds the necessary means
to reproduce any given experimental constraints, including deforming the
molecular geometries to compensate a lack of agreement for the sought-after
intermolecular distances.
Two widely different structure models could be found that reproduced the
x-ray and neutron diffraction experiments and the electric field distribution
equally well. The tetrahedral DD model contained, according to the Hbonding cone criterion, 74% DD species and 21% SD species, while the
asymmetrical SD model contained 81% SD species and 18% DD species. It
is important to note that tetrahedral refers here to the H-bonding topology
and not to the oxygen-triplet angles as in the tetrahedrality parameter
Q. The DD model has most of its molecules in fully bonded tetrahedral
environments as shown to the left in fig.I-1 while the asymmetrical model
has mostly single-donor molecules, as shown in fig.I-1 right. That the two
structure models really differ in their H-bonding topologies can be seen in
fig.I-4 where geometrical distributions relating to the H-bonding network are
shown for the two RMC models, together with the TIP4P-Pol2 MD model.
The DD model is seen to be qualitatively similar to the MD model while the
SD model is very different.
The main conclusions of Paper I can be summarized as follows.
1. Two structure models with widely different H-bonding topologies
give equivalent fits to the most recent diffraction data. The fits are
shown in fig.I-2. This proof of existence shows that diffraction data are
72
6.2 Bounds on liquid water structure from diffraction - Paper II
inconclusive regarding the relative populations of differently H-bonded
species, a conclusion that indeed has been found also using the EPSR
method [149].
2. The IR/Raman spectrum, represented by an electric field distribution
from an MD model, is also well reproduced by both structure models as
shown in fig.I-8. Decomposing the electric field into contributions from
different H-bonding species, fig.I-9, shows that ultrafast pump-probe
experiments, where an ultrafast spectral shift to redshifted frequencies
when exciting on the blue side of the spectrum has been interpreted to
show that H-bonds are broken only fleetingly [38, 150], has a different
interpretation in the SD model; the interpretation is therefore modeldependent.
3. MD models that have been claimed to reproduce XD data sets well do
in fact, upon closer inspection as in figs.I-6 and I-7, show discrepancies
with experimental XD data in the high-k region, indicating that the first
gOO (r ) peak is too high and narrow and centered at a somewhat too
small O-O distance.
4. XAS spectra of liquid water, connected by Myneni et al. [151] and
Wernet et al. [7] to asymmetrical H-bond coordination in water which
the SD model intentionally tried to mimick, are sensitive to other
aspects of the structure than only the H-bond populations according to
the cone criterion. This is seen in fig.I-10 where the SD and DD models
are shown to give very similar theoretically computed XAS spectra
after summation.
6.2
Bounds on liquid water structure from diffraction
- Paper II
A follow-up study to Paper I investigated the structural content of diffraction
data in more detail, with particular focus on the neutron diffraction data
sets on 5 different isotopic mixtures of H2 O and D2 O and the three partial
correlation functions in water. Using the RMC method, four unique structure
models were generated that equally well reproduced neutron and x-ray
diffraction data. In this study the electric field distribution, used in Paper I
as a representation of the IR/Raman O-H stretch spectrum of dilute HOD in
D2 O, was left out as it was deemed to bias RMC structural solutions towards
the TIP4P-Pol2 structure from which it had been obtained and furthermore
to contain uncertainties [152], thus complicating the extraction of structural
information from the diffraction data. The four RMC models had different
H-bonding geometrical constraints, as follows: the FREE model had no Hbonding constraints and thus represented an unbiased fit of a water structure
73
6 Summary of main results
model to XD and ND data, the ASYM model had asymmetrical constraints to
enforce asymmetrical H-bonds, the SYM model had symmetrical constraints,
and the MIX model had a combination of symmetrical and asymmetrical
constraints to create a mixture model with both highly tetrahedral and
highly asymmetrical species. Due to the removal of the E-field fit, the ASYM
and SYM models differ structurally from the SD and DD models reported in
Paper I. Furthermore, all models reproduced the intramolecular geometrical
distributions from path-integral simulations [148]; this was important due to
the large intramolecular “background” in the five neutron structure factors
which were carefully investigated.
The main conclusions of Paper II can be summarized as follows.
1. The importance of including XD data in structure modeling in order
to extract reliable O-O PCFs was demonstrated by fitting only the 5
neutron diffraction data sets and comparing it with joint fits to XD
and ND data. Without the XD data the resulting O-O PCF featured a
very low first peak, in contrast to earlier determinations using the EPSR
method [56] to fit only ND data. The O-H and H-H PCFs, however, were
unaffected by the inclusion of XD data (fig.II-2).
2. In accordance with Paper I, it was found that structure models with
very different H-bonding topologies can be made equally consistent
with diffraction experiments by using RMC simulations (fig.II-1).
By analyzing the tetrahedrality in the oxygen positions (as opposed
to the H-bonding geometries), it was furthermore found that only
with the inclusion of geometric constraints enforcing double-donor
(DD) H-bonding geometries did the distribution of oxygen-oxygenoxygen angles (θOOO ) resemble that of the SPC/E and TIP4P-Pol2
MD models (fig.II-8). Without constraints for two intact donating
H-bonds per molecule the tetrahedral peak around θOOO =109◦ was
greatly diminished and a strong peak around θOOO =60◦ showed up
instead, reflecting the presence of a large number of non-tetrahedrally
coordinated interstitial molecules.
3. An important observation was made in the comparison of calculated
ND structure factors from MD simulations with ND experiments. The
conclusion from Paper I concerning the first peak position in gOO (r )
from the SPC/E and TIP4P-Pol2 models, namely that it is too short to
be consistent with XD data, could be confirmed in the comparison with
data on the 75% D2 O, 25% H2 O isotopic mixture. This data set contains
a large fraction of O-O correlations as can be seen in Table 2.1, and the
phase shift seen in the XD structure factor caused by the too short O-O
distance can be detected on close inspection in the neutron data set as
well (fig.II-6)
74
6.3 Inhomogeneous structure of water - Paper III
6.3
Inhomogeneous structure of water - Paper III
In recent years, observations from x-ray spectroscopic techniques have
indicated that liquid water displays complex spectra that differ qualitatively
from the hexagonal ice counterparts. Myneni et al. in 2002 [151] and Wernet
et al. in 2004 [7] showed that water gives a sharp pre-edge peak around
535 eV in the XAS spectrum and a weak post-edge feature, similar to the
surface of ice where a large fraction of molecules have uncoordinated
dangling H-bonds; from this it was concluded that a large asymmetry
must exist on average around each water molecule where one of the two
possible donating H-bonds is strongly distorted or broken, in contrast to
the picture of water structure obtained in, e.g., MD simulations. Soon after
the publication by Wernet et al., competing interpretations of XAS were
suggested [138] that supported the conventional view of water structure; this
was however disputed by Wernet et al. [153] where experimental difficulties
were pointed out. In 2008, Tokushima et al. [154] presented evidence from
x-ray emission spectroscopy (XES) on liquid water showing that the emission
peak corresponding to the occupied lone-pair 1b1 orbital, which gives a
single peak both in gasphase water and in ice, exhibits a split-peak nature in
the liquid. Interpreting this split to be due to a core-level shift induced by
distinct chemical environments results in a picture where water is seen to
participate either in strong, tetrahedral and ice-like H-bonds, or to reside in
disordered regions with distorted H-bonds. Again, this interpretation was
challenged in the literature [155, 156], but new data [157] and computational
techniques [158] provide further support for the original interpretation.
Alongside the debate around the XAS and XES results, experiments at
the Stanford synchrotron radiation lightsource were performed to measure
new, greatly improved small-angle structure factors of water using the smallangle x-ray scattering (SAXS) technique. The results of these experiments,
together with MD simulations of SPC/E showing that this model lacks the
anomalous small-angle enhancement observed experimentally, provided a
new framework in which to interpret the XAS and XES results. In short, the
two coexisting structurally distinct species indicated by XES experiments
are interpreted to be related to the high-density and low-density liquid
(HDL and LDL) phases in the supercooled regime, manifested at higher
temperatures far away from the possible liquid-liquid critical point (LLCP) as
fluctuations around predominantly LDL-like and HDL-like local structure.
A characteristic size of around 1 nm was suggested based on the SAXS data
analysis as a measure of the length-scale of structural inhomogeneities in
water. In this picture, new light could also be shed on the XAS spectra of
water; as the temperature is increased, both the pre-edge peak in XAS and
the high-energy side of the split 1b1 peak in XES (termed 1b100 , with emission
energy closer to the gasphase value and assigned to disordered species) shift
to energies closer to the gasphase values, indicating that species contributing
75
6 Summary of main results
to these spectral features become thermally excited with temperature. On the
other hand, the energy position of the low-energy side of the split 1b1 peak
(termed 1b10 ), connected to ice-like strongly bonded species, is unchanged
with temperature while the peak intensity goes down. This was interpreted
to indicate that the strongly bonded component is less thermally excited with
temperature, primarly due to its larger energy spacing between librational
levels. Another rather compelling experimental observation could be made
using resonant excitation in the XES measurements. By tuning the photon
energy to the pre-, main- or post-edge regions of the XAS energy interval, the
XES response of selected species with absorption energies in the respective
energy regions could be detected. When exciting with photon energies
in the pre-edge region, the low-energy 1b10 peak associated with strongly
bonded ice-like species in the XES spectrum was perfectly quenched, while
resonant excitation in the post-edge energy region on the contrary gave
significantly enhanced 1b10 peak intensity; this is shown in fig.III-4. Since the
pre-edge peak arises from species with broken or weakened H-bonds and
the post-edge reflects strong intact bonds, this provides evidence for the
assignment of the 1b10 peak to strongly bonded species and the 1b100 peak to
disordered species.
Viewed separately, important information can be derived from the x-ray
spectroscopies on one hand and the SAXS results on the other. The enhanced
small-angle scattering observed in SAXS, which increases in magnitude
in the supercooled regime as discussed in Papers VI and XIII, indicates
anomalous density fluctuations possibly originating from a LLCP, but the
central question is whether the split 1b1 peak observed in XES is related to
LDL-HDL fluctuations that survive far into the ambient temperature region.
Paper III proposes that this indeed is the case, and Paper IX provides some
support for this hypothesis. Contesting opinions have been expressed against
this interpretation [159–161], and the debate can be expected to continue for
some time to come.
6.4
SpecSwap-RMC: a new multiple-data set
structure modeling technique - Paper IV
The reverse Monte Carlo (RMC) method, while useful for generating
structure models consistent with neutron and x-ray diffraction data, is not
applicable to experimental data sets such as x-ray absorption spectroscopy
where the required theoretical calculations from atomic coordinates are very
time-consuming. In TP-DFT calculations, for instance, the XAS spectrum of
a single molecule in a condensed phase requires a full electronic structure
calculation of a cluster with ∼40 molecules, a task that requires hours of
computer time. Considering that a conventional RMC simulation requires
at least on the order of 106 atomic or molecular moves, with an updating
76
6.5 Inconsistency between diffraction and EXAFS on the first coordination
shell in water - Paper V
of the calculated signal after each move, it is easy to see that XAS spectra
could never be modeled in conventional RMC. A novel approach to
structure modeling of computationally intensive data sets is provided by
the SpecSwap-RMC method. Its name derives from the way in which the
modeling is performed: instead of performing atomic/molecular moves
in a periodic simulation box in which the complete configurational phase
space (to within machine precision) is accessible, a large set of clusters is
used to represent a discretized version of configurational phase space. Paper
IV introduces the SpecSwap-RMC method and gives examples of its use.
By comparing SpecSwap fits to experimental EXAFS data on copper with
the conventional approach (based on non-linear least-squares minimization
of structural parameters in the EXAFS equation, eq. (5.23)) implemented
in the FEFFIT program [133] it was found that SpecSwap gives reliable
results in good agreement with FEFFIT fits (figs.IV-5 and IV-6). The chief
advantage of the SpecSwap method is that no limit is imposed on what type
of experimental data sets can be modeled, apart from the requirement that
they can be calculated on atomic structures. It thus holds promise to resolve
many key questions regarding, for instance, the real structural content in
both XAS and XES experimental data on liquid water and work along those
lines is being planned.
6.5
Inconsistency between diffraction and EXAFS
on the first coordination shell in water - Paper
V
Surprising results were presented based on experimental EXAFS data on
water in ref. [5]. As shown in fig. 2.3 the reported pair-correlation function
gOO (r ) was in strong disagreement with results from diffraction experiments.
The height of the first peak was almost 3.5, as compared to 2.3 from the most
recent diffraction results (refs. [44, 47, 48] and Papers I, II and VII), and the
center position of the peak was 2.7 Å, compared to around 2.8 Å as derived
from diffraction.
This puzzle was tackled using the SpecSwap-RMC method introduced
in Paper IV. 7633 clusters of water molecules from various MD and RMC
structure models were used as a basis set in configurational phase space,
and their corresponding EXAFS oscillations, χ(k ), were computed using
the FEFF8.4 multiple-scattering program. If x-ray diffraction and EXAFS
experiments really were in stark contradiction, a joint structural solution
reproducing both the EXAFS data and gOO (r ) determined from diffraction
would not be possible. EXAFS is however sensitive to other parts of the
structure than only pair correlations. This opens up the possibility that
structural features which diffraction experiments are insensitive to but which
77
6 Summary of main results
influence EXAFS, namely many-body correlations, could be enhanced in
the SpecSwap fits which would lead to an agreement between diffraction
and EXAFS. And this was indeed found; simultaneous fits to χ(k ) from
EXAFS and gOO (r ) from diffraction gave good agreement to both data sets,
and it was found that three-body OH-O focusing paths from linear H-bonds
provided the physical mechanism for this to be possible. Conclusions drawn
in Paper V can be summarized as follows.
1. Good agreement with experimental χ(k ) for hexagonal ice is obtained
by multiple-scattering calculations on clusters from path-integral MD
simulations of ice, as seen in Fig.V-2. Two different MD models of ice
were compared, SPC/E and TIP4P/2005, but only small differences
were seen. A single parameter, the energy onset E0 of χ(k ) which
optimized the agreement with experiment, was extracted from the
calculations on hexagonal ice and used for all subsequent calculations
on liquid water clusters. The good agreement for the ice calculations
gave confidence that the calculations on liquid water clusters were
reliable.
2. Multiple-scattering contributions, where three-body OH-O focusing
paths play the dominant role, were found to contribute significantly to
χ(k ) from both liquid water and ice. Figure V-2 shows the calculated
χ(k ) from an MD simulation of hexagonal ice, with and without
multiple-scattering paths beyond single-scattering. The multiplescattering paths both enhance the magnitude of the oscillations and
produce a phase-shift. Both of these effects are important to bring the
resulting theoretical χ(k ) into agreement with the experimental data
on ice.
3. A fit of only the experimental χ(k ) on liquid water leads to a structural
solution with a very sharp first peak in gOO (r ) which is centered at a
very short distance compared with results from diffraction, as shown in
Fig.V-6. Furthermore, as shown in figs.V-7, V-8 and V-9, distributions of
structural parameters such as H-bonding angles, tetrahedrality Q and
asphericity η shift towards more structured values when modeling only
χ ( k ).
4. When a diffraction-determined gOO (r ) is included in the modeling,
the agreement with the experimental χ(k ) is slightly impaired while
the agreement with gOO (r ) is excellent. The structural solution is very
different from the one where only χ(k ) is fit since distributions of Q
and η shift to more disordered values, evidenced in figs.V-8 and V-9.
Contrary to these trends, the distribution of H-bonding angles shifted
towards more linear H-bonds (fig.V-7). Thus, despite the disordering
effect of gOO (r ) on order parameters relating to the oxygen atom
positions, the importance of the focusing effect in reproducing the
78
6.6 Enhanced small-angle scattering and the Widom line in supercooled
water - Paper VI
amplitude and phase of χ(k ) increases the influence of linear H-bonds
in the SpecSwap structural solution.
6.6
Enhanced small-angle scattering and the
Widom line in supercooled water - Paper
VI
After having studied the SPC/E model in Paper III and finding poor
agreement with experimental compressibility and small-angle scattering
data, the good agreement of TIP4P/2005 with experimental compressibilities
gave hope that this model could perform better in the small-angle region
of S(k) and help shed light on the origin of the experimentally observed
enhancement. These hopes were indeed fulfilled since an anomalous smallangle enhancement could be detected below room temperature in large
(45,000 molecules) simulations using this force field. Upon supercooling the
simulations furthermore gave increasingly stronger small-angle scattering,
but only down to a certain temperature below which the intensity decreased
again. At higher pressures up to around 1500 bar, the small-angle
enhancement was also seen to become more and more pronounced. A
parallel simulation effort [97] by the developers of TIP4P/2005 located a
liquid-liquid critical point of TIP4P/2005 at Pc ≈1350 bar and Tc ≈193 K; this
was of immediate value for the small-angle scattering study in Paper VI since
it confirmed both the thermodynamic behavior of the response functions
and the location of the Widom line reported in Paper VI, and furthermore
shed light on the increase in small-angle scattering at higher pressures.
A summary of the main conclusions of Paper VI follows.
1. TIP4P/2005 gives good qualitative agreement with experimental
SAXS measurements, shown in fig.VI-1, but a temperature offset
is observed in the comparison with experiments at 1 bar and the
maximum enhancement in the model (at 230 K) is smaller than
that from experiment (at 252 K, the lowest achieved temperature).
Since simulations can reach lower temperatures than in experiment,
the experimental observations can be extrapolated into the deeply
supercooled regime to shed light on the origin of the enhanced
scattering.
2. Extracted correlation lengths ξ feature closely similar power-law
behavior as experimental data, and the Widom line can be directly
observed as a maximum of ξ at 1 bar (fig.VI-2) Interestingly, the
correlation lengths do not grow very significantly upon approaching
the LLCP. Simulations in the vicinity of the liquid-vapor critical point
of TIP4P/2005 shed light on this observation since even though clear
79
6 Summary of main results
critical fluctuations are seen in S(k ) at low k, ξ only reaches about 6 Å
(fig.VI-3).
3. Both the isothermal compressibility κ T and coefficient of thermal
expansion α P show extrema that coincide with the Widom line, as
shown in figs.VI-4 and VI-5. The former shows excellent agreement
with experiment at high pressures, while at 1 bar a sharper rise of κ T is
found experimentally indicating that stronger critical fluctuations are
seen in real water compared to TIP4P/2005 at 1 bar.
4. A structural transition, quantified by the change in the average
tetrahedrality value h Qi, takes place as the Widom line is crossed
(fig.VI-6), indicating that Q may be a relevant structural order
parameter for the HDL-LDL transition. At higher temperatures the
distribution of Q per molecule becomes bimodal (fig.VI-7). Computing
XAS spectra and separating the contributions from low-Q and high-Q
species indeed shows a sensitivity of XAS to Q; lower-Q species give
stronger pre- and main-edge spectral intensities (fig.VI-8).
6.7
Improved x-ray diffraction experiments on
liquid water - Paper VII
The importance of reliable structural information on water from diffraction
experiments can hardly be overstated. Due to remaining controversies
regarding the height, shape and position of the first peak in gOO (r ) [19, 43,
45, 47, 48, 56], an experimental study was initiated in an effort to measure as
accurately as possible the x-ray diffraction pattern of liquid water at different
temperatures. The third-generation synchrotron x-ray facility at the Stanford
Synchrotron Radiation Lightsource was used and the experimental setup is
schematically shown in fig. 2.1. The main conclusions of Paper VII are the
following.
1. The new XD data confirm earlier studies showing that the height of
the first peak in gOO (r ) is close to 2.3 at room temperature, as seen in
fig.VII-5.
2. The presence of long-range correlations, reflected in visible 4th and 5th
peaks of the experimental gOO (r ), is detected in liquid water even at
ambient and warm temperatures, fig.VII-6. In itself this indicates that
very ordered regions with structural correlations up to 11 Å exist in
water.
3. MD simulations using the TIP4P/2005 potential provide a way to
interpret the long-range correlations. The simulations reproduce the 4th
and 5th peaks and their temperature dependence rather well (fig.VII-6).
80
6.8 The boson peak in supercooled water - Paper VIII
An analysis of partial contributions to the simulated gOO (r ), obtained
using the structural order parameters Q and I (defined in eqs. 2.33
and 2.35) to define subensembles of water molecules, reveal that a
class of very structured molecules lie behind the large-distance peaks,
as seen in fig.VII-7. In particular the subensemble of water molecules
possessing a large value of the I parameter contribute strongly to
the large-distance peaks. As discussed in Paper IX, the distribution
of molecular I values shows a unimodal peak at low values in the
instantaneous snapshots from the simulations, but it becomes bimodal
with a temperature invariant minimum, or isosbestic point, between
low-I and high-I peaks in the so called inherent structure (IS). High-I
values are firmly connected to LDL-like local structure, and the new
XD results with resolved 4th and 5th peaks in gOO (r ) thus imply the
presence of LDL-like molecules up to high temperatures.
6.8
The boson peak in supercooled water - Paper
VIII
Peculiar dynamical properties were observed in simulations performed on
supercooled water. It was found that strong finite-size effects appeared as
the Widom line of the TIP4P/2005 model was approached, manifested in
sharp low-frequency peaks in the vibrational density of states (VDOS) and
the incoherent and coherent dynamic structure factors. Simulations of much
larger systems with 45,000 molecules revealed however that a system-size
independent low-frequency excitation exists close to ω=40 cm−1 , which
however is shifted to ω=50 cm−1 in the TIP5P model indicating that the peak
has structural origins. This is interesting because experiments on confined
and protein-hydration water have detected a very similar low-frequency
excitation [93] at the same energy as for the TIP4P/2005 model, and this
feature has been proposed to be a boson peak, a well known but poorly
understood phenomenon in glasses and glass-forming liquids. The main
conclusions of Paper VIII can be summarized as follows.
• Distinct boson-like peaks appear in the VDOS and dynamic structure
factors of the TIP4P/2005 and TIP5P models as they are cooled
below their respective Widom temperatures. In the former model the
temperature onset and frequency position agree well with experimental
results on confined and protein-hydration water. Simulations of
hexagonal ice are also seen to show a similar peak, although at a higher
frequency.
• Finite-size effects, manifested by sharp peaks in the incoherent and
coherent dynamic structure factors as well as in the VDOS, are seen
to be due to a strong coupling between density fluctuations and low
81
6 Summary of main results
wave vector transverse modes. This indicates that transverse modes
contribute significantly to the low-frequency side of the boson-like
peak. In the limit of an infinitely large simulation box the sharp peaks
extrapolate to zero frequency.
• Molecules with low tetrahedrality (Q parameter) values are seen to
contribute more to the boson-like peak compared to highly tetrahedral
molecules.
• The frequency position of the boson-like peak in the reduced VDOS
(D (ω )/ω 2 ) is seen to shift towards the frequency position found in
hexagonal ice as the temperature is lowered to 150 and 100 K. This fact,
together with the observation that non-tetrahedral molecules contribute
strongly to the boson-like peak, suggest that defective HDL-like regions
confined in a strongly bonded matrix of LDL-like molecules support the
soft vibrational modes that constitute the boson-like peak.
6.9
Bimodal inherent structure
ambient water - Paper IX
in
simulated
This last paper builds upon simulations reported in Paper VI. Inspired
by other interesting work on the I parameter, i.e. the local-structure index
(LSI) [32, 33], defined in eq. 2.35, a thorough investigation of the significance
of the LSI parameter for the TIP4P/2005 simulations in Paper VI was
performed. High LSI values reflect a nearest-neighbor environment which is
very ordered, with well separated first and second coordination shells, while
low values reflect structural disorder with interstitial molecules between the
first and second shells. The inherent structure (IS) approach was adopted, i.e.
snapshots taken from the large 45,000 molecules’ simulations were energy
minimized and thereafter analyzed. An intriguing picture emerged from
these investigations, which can be described as follows.
1. When calculated on inherent structures at temperatures ranging from
deeply supercooled to warm, the distribution of LSI values is clearly
bimodal with a temperature and pressure invariant minimum (an
isosbestic point), for pressures ranging between 1 bar and 2000 bar, as
shown in fig.IX-1
2. Higher pressure shifts the balance from high- to low-LSI species,
consistent with an expected correlation between high LSI values
and LDL-like local structure, and low LSI values and HDL-like local
structure.
3. The relative population of species with low- and high-LSI values,
respectively, changes rapidly in the supercooled region, but stabilizes
82
6.9 Bimodal inherent structure in simulated ambient water - Paper IX
around a 75:25 value for ambient pressure at higher temperatures,
shown in fig.IX-2.
4. The Widom line of TIP4P/2005 coincides almost perfectly with the
temperature where a 50:50 population of low- and high-LSI species
occurs. Also the expected liquid-liquid phase transition at 1500 bar
close to 190 K is associated with a very sharp transition from low-LSI
dominated to high-LSI dominated structure; indeed the transition
would likely become discontinuous if a liquid-liquid phase transition
could be directly simulated (fig.IX-2)
5. A classification into inherently low-LSI and high-LSI species, i.e.
where the LSI value in the IS is used to separate into subspecies
in the real structure, reveals an inherent heterogeneity in the real
structure. Inherently low-LSI species are disordered and HDL-like
while inherently high-LSI species are well ordered and LDL-like,
as seen in their O-O PCFs shown in fig.IX-3. The two species are
furthermore spatially separated and their local densities, quantified
by calculating molecular Voronoi polyhedra, form two distributions
displaced by about 50 kg/m3 where the high-LSI species have lower
density (fig.IX-5).
6. A connection to the structural bimodality reported from XAS and
XES experiments in Paper III may be provided by these observations
from classical MD simulations. In the MD simulations, however, the
clear bimodality found in the inherent structure is almost completely
washed out by thermal excitations in the real structures, at odds with
the interpretation of XAS and XES experimental data. Whether this is
due to deficiencies in the simulations remains an open question.
83
6 Summary of main results
6.10
A note on my participation in Papers I-IX
Papers I-IX are all the outcome of close collaborations between the listed
coauthors. My participation will here be briefly summarized. In Papers I
and II some of the RMC simulations were performed by me while Mikael
Leetmaa performed others. I analyzed the experimental x-ray and neutron
structure factors and calculated structural properties of the MD models.
The writing of the papers was done collaboratively and I contributed
significantly. For Paper III I performed all MD simulations and analyzed
their thermodynamic and structural properties, discussed the interpretation
of the results with coauthors and participated in writing the paper. Mikael
Leetmaa implemented most of the SpecSwap-RMC method presented in
Paper IV, but I was heavily involved in discussing the idea, developing
it and implementing several geometrical tools, and I performed the test
calculations on copper EXAFS using both SpecSwap and the FEFF and
FEFFIT programs. Most calculations in Paper V were performed by me, but
some calculations in the early stage of the work were done by Amber Mace
in close collaboration with me. I also wrote the largest part of the paper,
together with my supervisors Lars G. M. Pettersson and Anders Nilsson.
All calculations in Paper VI were performed by me and I wrote most of the
paper, again together with my supervisors. In Paper VII I participated in
early experiments leading up to the final published x-ray diffraction data,
performed all MD simulations, discussed the results with coauthors and
wrote parts of the paper. For Paper VIII, I implemented the tools to calculate
dynamic correlation functions of TIP4P/2005, performed the calculations on
TIP4P/2005 together with Daniel Schlesinger, discussed all results with the
coauthors and wrote parts of the manuscript. For Paper IX, I performed all
simulations, calculations and analysis for and wrote most of the paper in
collaboration with my supervisors.
84
7
Conclusions and Outlook
This thesis has dealt with several facets of the problem to understand the
physical nature of liquid water. Although each facet is an independent
research topic with specialized theory, experiments and scientific literature,
it can be fruitful to study them together in order to better grasp the complex
behavior displayed by this most important liquid. Indeed, an increasing
amount of evidence points to deep connections between structure, dynamics
and thermodynamics of liquid water – connections that, in particular,
influence water’s anomalous behavior in the metastable supercooled region
of the phase diagram. In turn, an understanding of deeply supercooled
water may be key to understanding the properties of liquid water at ambient
conditions.
Structural properties of water were discussed in Chapter 2. The main
contributions to this field, contained in this work, can be briefly summarized
as follows. First, the structural information content of diffraction data
was thoroughly investigated. The complementarity of neutron and x-ray
diffraction became evident in structure modeling using the reverse Monte
Carlo (RMC) method. It was found that oxygen-oxygen (O-O) correlations
were mainly determined from x-ray data, and that many commonly used
molecular dynamics (MD) models overestimate nearest-neighbor peak
heights in both the O-O and O-H partial pair-correlation functions (PCFs).
Long-range O-O correlations in TIP4P/2005 simulations, on the other hand,
were seen to agree well with experimental x-ray data, and the important
experimental observation of large-distance correlation peaks at around 9
and 11 Å could be connected to highly structured “low-density liquid”like (LDL-like) species. Another important insight came from the use of
geometrical constraints in RMC. By enforcing different constraints on the
hydrogen (H-) bonding geometries in RMC structure models, it was found
that diffraction data are not sufficiently sensitive to many-body correlations
to discriminate between structure models ranging from very tetrahedral
85
7 Conclusions and Outlook
and strongly H-bonded to very disordered and weakly H-bonded. Second,
insights on the puzzling disagreement between diffraction results and recent
EXAFS measurements regarding the first peak of the O-O PCF in ambient
water were obtained by using the novel SpecSwap-RMC approach. This
method can be seen as a generalization of the conventional RMC method,
which for the first time makes it possible to fit structure models directly to
experimental data that are very time-consuming to calculate theoretically.
The main conclusions from this work were, firstly, that the single-scattering
approximation fails in water and ice due to a substantial contribution from
the many-body focusing effect, which arises when an H-bond between two
molecules is close to linear, and secondly that diffraction and EXAFS can in
fact be reconciled if a large contribution from this focusing effect is accounted
for.
With respect to the thermodynamic properties of water, the main
contribution presented in this work was the relation established between
experimentally observed small-angle x-ray scattering (SAXS) intensities
and the presence of a Widom line in supercooled water. Relying on MD
simulations, it was found that the TIP4P/2005 water model, in good
agreement with experimental data, produces an enhanced small-angle
scattering signal which increases upon decreasing the temperature. The
Widom line is the locus of correlation length maxima in the phase diagram
emanating from a critical point, and the simulated small-angle intensity was
seen to increase continuously down to the Widom temperature of the model,
below which it suddenly dropped. Since both extracted Ornstein-Zernike
correlation lengths and isothermal compressibilities for the TIP4P/2005
model agree well with experimental measurements, it appears likely that the
same physical mechanism governs the increased small-angle scattering in the
model and in real water. This provides strong support for the liquid-liquid
critical point thermodynamic scenario for liquid water.
An intriguing link between thermodynamics and dynamics in supercooled
liquid water has received much attention in recent years. The energylandscape perspective [162] offers physical insights into this relationship,
but many experimental observations require direct simulation studies to
explore the underlying mechanisms. One such observation is the onset of
low-frequency excitations (LFEs) resembling a boson peak as the temperature
is crossed where a dynamical transition in confined and protein-hydration
water takes place [93]. A deeper understanding of this phenomenon was
obtained in work presented in Chapter 3. It was found that the onset of
well-resolved LFEs coincided with the Widom line of the TIP4P/2005 model,
thus establishing a relation between the two and proving that the boson-like
LFEs are not exclusive to confined and protein-hydration water. Consistent
with experimental results [90], however, a similar LFE at slightly higher
frequencies was also found in simulations of crystalline hexagonal ice. In
the time-domain, i.e. in the incoherent incoherent scattering function Fs (k, t),
86
the feature generally related to boson peaks, namely the emergence of a
minimum after the initial fast relaxation regime followed by a small peak
around 1 ps, was found also in the simulations of ice. These observations
challenge common definitions of boson peaks. However, since the LFEs
shift to lower frequencies and are considerably stronger in amplitude in the
supercooled liquid simulations below the Widom line, it can be argued that
the LFEs may share features with boson peaks found in strong glass-forming
substances such as silica. A possible interpretation of the observations is that
structurally disordered HDL-like regions confined in a highly structured
LDL-like matrix support soft vibrational modes that constitute the distinct
boson-like peak.
Finally, the main findings from x-ray absorption spectrum (XAS)
calculations presented in this work can be briefly summarized. These efforts
were rather limited in scope and did not address several key questions,
such as the precise interpretation of experimental spectra which is a topic
of active and intense debate. A few conclusions could nevertheless be
drawn. First, it was found that a method to include zero-point vibrational
effects to simulations performed using rigid molecules greatly improved
the agreement with experimental data in terms of the relative intensities
of spectral features. Theoretical XAS calculations are very sensitive to
intramolecular geometries, and since many molecular simulations of water
and other substances are performed using rigid intramolecular geometries
this method will be useful in enabling more meaningful comparisons with
experiments by approximately including the intramolecular zero-point
vibrational effect. Second, insights into the structural sensitivity of XAS
were obtained by analyzing spectral contributions from molecular species
with different structural properties. The sensitivity to local structure as
quantified by the tetrahedrality value or the local structure index was
found be very similar to temperature effects found experimentally. A much
larger sensitivity was however found for H-bonding geometries, where only
non-donating and single-donating molecules were found to contribute to
the much debated pre-edge feature in the spectrum. This highlights the
connection between the pre-edge feature and structurally disordered species
with broken H-bonds.
Many challenges related to the topics presented in this thesis remain for
future investigations. With respect to the partial pair-correlation functions
of liquid water, it would be desirable for a consensus to be reached
regarding the peak heights and positions. Work presented in this thesis,
and also elsewhere [44, 48], points to a rather severe overstructuring in
nearest-neighbor correlations in many MD models, but the abundance in
the literature of different experimentally determined PCFs makes it difficult
for MD simulators to make the proper comparisons. It would also be
highly valuable to explore further the potentially powerful approach of
combining EXAFS measurements on water with structure modeling using
87
7 Conclusions and Outlook
the SpecSwap-RMC method presented in this thesis. Further experiments at
different temperatures, in particular at supercooled temperatures, along with
further developments of the SpecSwap-RMC methodology could provide
important new information on H-bonding geometries in water.
In connection to the phase behavior of supercooled water, it is likely that
the simulation approach presented in Chapter 4 could shed new light on the
possible location of a liquid-liquid critical point if experimental small-angle
scattering data studies, of a similar quality as presented in Paper III and
ref. [15] (Paper XIII), were performed both at higher pressures and at lower
temperatures. This will likely be accomplished in the near future.
The onset of a boson-like peak in the coherent and incoherent dynamic
structure factors at the Widom temperature, in both simulations reported
in Chaper 3 and in experiments on confined and protein-hydration water,
is an important topic which requires further work. Since the TIP4P/2005
simulations reproduce the experimental observations rather well, it is likely
that detailed analyses of the simulation data will bring further clarity to the
connection between these low-frequency modes and the structural transition
which takes place at the Widom line.
To end, the future prospects for x-ray spectroscopies in investigating
liquid water can be summarized. The structural sensitivity of both x-ray
absorption and x-ray emission spectra holds great promise to provide
detailed insights into the H-bonding network in water beyond what can be
obtained by other experimental probes. Currently, however, the interpretation
of the x-ray spectroscopic results presented in Paper III is rather controversial.
The main challenge for future investigations will be to establish agreement
between experiments and theoretical calculations on model systems for both
x-ray absorption and emission spectra.
When consistency and general agreement is reached regarding the
interpretion of the various experimental and theoretical techniques available
to investigate water, our understanding of this most important liquid will
have come a long way.
88
Acknowledgements
This thesis would never have come into being if it wasn’t for the support,
guidance and encouragement of my supervisor, Professor Lars G.M.
Pettersson. For this I am highly grateful. I am also grateful to my cosupervisor, Professor Anders Nilsson, for inspiring conversations and for
sharing with me his expert knowledge on experimental x-ray science.
Working in this group where theoretical methods and experiments are
so closely intertwined, and where the ambition to “get things right” is
impressive, has been a great privilege. A crucial ingredient in my work has
been the rewarding interaction with fellow PhD students and other group
members. The first years were strongly colored by a very close collaboration
with my two “elders”, Mikael Leetmaa and Mathias Ljungberg, from whom
I learned so many things and with whom I share many lifelong memories.
I also want to thank Michael Odelius for his kind assistance and for many
useful discussions. Becoming an elder myself when Henrik Öberg, Jonas
Sellberg and Daniel Schlesinger joined the group presented a new but no less
rewarding experience, and I greatly appreciate the good times we’ve had,
both the scientific and the absolutely non-scientific ones. I also want to thank
Amber Mace for a good collaboration and friendship when she joined the
group as a master’s student. As already mentioned, the close connection to
the experimental activity at Stanford University proved highly valuable and
I especially want to thank Congcong Huang for an outstanding collaboration
and for the innumerable long conversations. I have also benefited a great
deal from interacting with a wider circle of competent and interesting
people, among whom I can mention (in no particular order) Hannes Jónsson,
Andreas Møgelhøj, Pradeep Kumar, Gene Stanley, Jan Swenson, László
Pusztai, Alexander Lyubartsev, Ling Fu, Sean Brennan, Arthur Bienenstock,
Uwe Bergmann, Henrik Öström, Dennis Nordlund, Lars-Åke Näslund,
Klas Andersson, Theanne Schiros, Ning Dong Huang, Ira Waluyo, Daniel
Friebel, Daniel Miller, Toyli Anniyev, Hirohito Ogasawara, Chris O’Grady,
Osamu Takahashi, John Rehr, Josh Kas, Ingmar Persson and many others.
For making this long journey a more social and fun experience I want to
thank Holger, Katta, Pernilla, Jörgen, Micke S., Rasmus, Ali, Prakash, Elias
and many more whom I am not forgetting. Special thanks go to my loving
parents who I know are always there for me. I also want to acknowledge the
rest of my family, in Iceland, Sweden and Pakistan. This thesis is dedicated
to the memory of my dear uncle, Lars Wikfeldt, who I wish was still here.
Finally, I want end by stating something that cannot adequately be put in
words, namely my gratitude and love for my wife Saroosh.
89
Sammanfattning
Vatten är en komplex vätska med flera ovanliga egenskaper. Vår förståelse
av dess fysiska, kemiska och biologiska egenskaper har utvecklats mycket
sedan systematiska vetenskapliga studier började genomföras för mer än
ett sekel sedan, men många viktiga frågor är fortfarande obesvarade. En
ökad förståelse skulle på sikt kunna leda till framsteg inom viktiga områden
så som medicinutveckling och vattenrening. Denna avhandling presenterar
resultat kring vattnets struktur, dynamik och termodynamik. Fokusen ligger
på teoretiska simuleringar som använts för att tolka experimentella data
från huvudsakligen röntgen- och neutronspridning samt spektroskopier.
Den strukturella känsligheten i röntgen- och neutrondiffraktionsdata
undersöks via reverse Monte Carlo metoden och information om de partiella
parkorrelationsfunktionerna erhålls. En ny metod för strukturmodellering
av beräkningsintensiva data presenteras och används för att lösa en
motsägelse mellan experimentell diffraktion och EXAFS angående syresyre parkorrelationsfunktionen. Data från röntgensmåvinkelspridning
modelleras med storskaliga klassiska molekyldynamiksimuleringar, och den
observerade förhöjda småvinkelspridningen vid underkylda temperaturer
kopplas till existensen av en Widomlinje härrörande från en vätskevätske kritisk punkt i det djupt underkylda området vid höga tryck.
En undersökning av inherenta strukturer i simuleringarna påvisar en
underliggande strukturell bimodalitet mellan molekyler i oordnade
högdensitetsregioner respektive ordnade lågdensitetsregioner, vilket
ger en tydligare tolkning av den experimentella småvinkelspridningen.
Dynamiska anomalier i underkylt vatten som har observerats i inelastisk
neutronspridning, speciellt förekomsten av lågfrekventa excitationer som
liknar en bosontopp, undersöks och kopplas till den termodynamiskt
definierade Widomlinjen. Slutligen presenteras densitetsfunktionalberäkningar
av röntgenabsorptionsspektra för simulerade vattenstrukturer. En approximation
av intramolekylära nollpunktsvibrationseffekter förbättrar relativa intensiteter
i spektrumen avsevärt, men en strukturanalys visar att klassiska simuleringar
av vatten underskattar andelen brutna vätebindningar.
91
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