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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 vi 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 6 6 8 10 15 15 16 17 17 18 19 20 20 24 25 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 32 32 34 35 35 vii . . . . . . . . . . . . . . . . . . CONTENTS 3.4 Low-frequency excitations in supercooled water . . . . . . . . . . 3.4.1 Finite-size effects in simulations . . . . . . . . . . . . . . 3.4.2 Dynamic correlation functions . . . . . . . . . . . . . . . 37 38 40 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 43 43 44 46 47 50 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 55 57 59 63 63 64 66 66 67 68 . . . . . . . . . . . . . . . . . . . . 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 71 73 75 7 Conclusions and Outlook 85 Acknowledgements 89 Sammanfattning 91 References 93 viii 76 77 79 80 81 82 84 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. 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