water in biological and chemical process

WATER IN BIOLOGICAL AND CHEMICAL PROCESSES

Building up from microscopic basics to observed complex functions, this insightfulmonograph explains and describes how the unique molecular properties of watergive rise to its structural and dynamical behavior, which in turn translates into itsrole in biological and chemical processes.The discussion of the biological functions of water details not only the stabilizing

effect of water in proteins and DNA, but also the direct role that water moleculesthemselves play in biochemical processes, such as enzyme kinetics, protein synth-esis, and drugDNA interaction. The overview of the behavior of water in chemicalsystems discusses hydrophilic, hydrophobic, and amphiphilic effects, as well as theinteractions of water with micelles, reverse micelles, microemulsions, and carbonnanotubes.Supported by extensive experimental and computer simulation data, highlighting

many of the recent advances in the study of water in complex systems, this is anideal resource for anyone studying water at the molecular level.

biman bagchi is a Professor at the Indian Institute of Science, Bangalore. He is aFellow of the Indian National Science Academy, the Indian Academy of Sciences,The National Academy of Sciences, India, and TWAS, The Academy of Sciencesfor the Developing World, Italy.

Cambridge Molecular Science

As we move further into the twenty-rst century, chemistry is positioning itself asthe central science. Its subject matter, atoms and the bonds between them, is nowcentral to so many of the life sciences on the one hand, as biological chemistrybrings the subject to the atomic level, and to condensed matter and molecularphysics on the other. Developments in quantum chemistry and in statisticalmechanics have also created a fruitful overlap with mathematics and theoreticalphysics. Consequently, boundaries between chemistry and other traditional sciencesare fading and the term Molecular Science now describes this vibrant area ofresearch.Molecular science has made giant strides in recent years. Bolstered by both

instrumental and theoretical developments, it covers the temporal scale down tofemtoseconds, a timescale sufcient to dene atomic dynamics with precision, andthe spatial scale down to a small fraction of an angstrom. This has led to a verysophisticated level of understanding of the properties of small molecule systems, butthere has also been a remarkable series of developments in more complex systems.These include protein engineering, surfaces and interfaces, polymers, colloids, andbiophysical chemistry. This series provides a vehicle for the publication of advancedtextbooks and monographs introducing and reviewing these exciting developments.

Series editors

Professor Richard SaykallyUniversity of California, Berkeley

Professor Ahmed ZewailCalifornia Institute of Technology

Professor David KingUniversity of Cambridge

WATER IN BIOLOGICALAND CHEMICAL PROCESSES

From Structure and Dynamics to Function

BIMAN BAGCHIIndian Institute of Science, Bangalore

University Printing House, Cambridge CB2 8BS, United Kingdom

Published in the United States of America by Cambridge University Press, New York

Cambridge University Press is part of the University of Cambridge.

It furthers the Universitys mission by disseminating knowledge in the pursuit ofeducation, learning, and research at the highest international levels of excellence.

www.cambridge.orgInformation on this title: www.cambridge.org/9781107037298

Biman Bagchi 2013

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2013

Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall

A catalog record for this publication is available from the British Library

Library of Congress Cataloging in Publication dataBagchi, B. (Biman)

Water in biological and chemical processes : from structure and dynamics tofunction / Biman Bagchi, Indian Institute of Science, Bangalore.

pages cm. (Cambridge molecular science)Includes bibliographical references.

ISBN 978-1-107-03729-81. Water in the body. 2. Water chemistry. I. Title

QP535.H1B34 20136120.01522dc232013013114

ISBN 978-1-107-03729-8 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy ofURLs for external or third-party internet websites referred to in this publication,and does not guarantee that any content on such websites is, or will remain,

accurate or appropriate.

To my mother and my father, Abha and Binay K. Bagchi. They taught me, from anearly age, to love poetry and science, opening doors to the wonders of Nature.

Understanding the role of water as the ubiquitous solvent for the chemical biologyand throughout molecular science remains one of the most active areas of currentscientic research. The puzzling issues that arise throughout this eld require aunied understanding of structure, dynamics and thermodynamics. This bookprovides a valuable resource in relating microscopic properties to complex phenom-enology, connecting diverse topics of contemporary interest.

David J. Wales, University of Cambridge

This book by Biman Bagchi covers an extremely broad range of topics on water,written with an eye to relating theory and experiment and by someone who hasinsight into both. Its use of recent references in the eld is a helpful attribute. Theauthor emphasizes that our understanding is not a closed subject and so there will befurther room for developments, debate on interpretation, and discussions. Forteachers of topics in equilibrium and nonequlibrium statistical mechanics there isalso, I believe, much useful material on interesting applications.

Rudy A. Marcus, California Institute of Technology

Water continues to both fascinate and confound those who study its properties andits vital roles in lifes structures and dynamical processes. In this unique book BimanBagchi has brought together an extraordinary range of experimental data and theresults of both theory and simulation studies at a level generally accessible to readerswith a background in chemistry at the rst year university level. He illuminates howthe remarkable properties of water are key to a multitude of chemical and biologicalprocesses and in doing so provides both insight and the springboard for newinvestigations of this endlessly fascinating liquid.

Graham R. Fleming, University of California, Berkeley

Contents

Preface page xvAcknowledgements xviii

Part I Bulk water 1

1. Uniqueness of water 31.1 Introduction 31.2 Molecular structure 41.3 Six unique features 71.4 Modeling of water 91.5 Conclusion 10

2. Anomalies of water 132.1 Anomalous properties 13

2.1.1 Density maximum 132.1.2 Isobaric specic heat (CP) 152.1.3 Isothermal compressibility (T) 152.1.4 Coefcient of thermal expansion (P) 162.1.5 Dynamic anomalies present at low temperature 17

2.2 Translational and orientational order 192.3 Temperaturedensity range of water anomalies 212.4 Conclusion 22Appendix 2.A Microscopic expressions of specic heat, isothermal

compressibility, and coefcient of thermal expansion 23Appendix 2.B Quantication of spatial order in water 24

3. Dynamics of water: molecular motions and hydrogen-bond-breakingkinetics 273.1 Introduction 273.2 Timescales of translational and rotational motion 28

vii

3.3 Jump reorientation motion in water 303.4 Effects of temperature on water motion 333.5 Translational diffusion 353.6 Hydrogen-bond lifetime dynamics 363.7 Vibrational dynamics of the OH bond 393.8 Dielectric relaxation 403.9 Solvation dynamics 423.10 Ionic conductivity of rigid ions in water 453.11 Electron transfer reactions in water 473.12 Motion becomes collective at low temperature 493.13 Conclusion 50Appendix 3.A Rotational time correlation functions 51Appendix 3.B Quantication of hydrogen-bond

lifetime dynamics 58

4. Inherent structures of liquid water 614.1 Introduction 614.2 Transition between inherent structures of water 664.3 Connected water cluster moves during transition 674.4 HB network restructuring 674.5 Coordination number uctuation in inherent structure and

corresponding dynamics in parent liquid 684.6 Low-energy excitations in liquid water 694.7 Conclusion 69

5. The pH of water 715.1 Introduction 715.2 Temperature and pressure dependence of pH 735.3 Mechanism of autoionization 745.4 pH of blood 755.5 Food and blood pH 765.6 pH of seawater 775.7 Conclusion 77

Part II Water in biology 79

6. Biological water 816.1 Introduction 816.2 Relaxation measurements 836.3 Unique characteristics of biological water 836.4 Phenomenological models and simple theories 846.5 Proteinglass transition and hydration-layer dynamics 88

viii Contents

6.6 Protein aggregation and biological water 906.7 Conclusion 90Appendix 6.A The dynamic exchange model 91

7. An essential chemical for life processes: water in biological functions 977.1 Introduction 977.2 Role of water in enzyme kinetics 997.3 Role of water in drugDNA intercalation 1017.4 Role of water in the biological function of RNA 1057.5 Water-mediated molecular recognition 1077.6 Protein folding and protein association: role of biological water 1097.7 Role of water in beta-amyloid aggregation in Alzheimer disease 109

7.7.1 Role of water in the early stages of oligomer formation 1107.7.2 Role of water in the late stages of bril growth 111

7.8 Role of water in photosynthesis 1127.9 Conclusion 114

8. Hydration of proteins 1178.1 Introduction 1178.2 What is the thickness of the hydration shell? 1188.3 How structured is the water in the hydration shell of a protein? 1218.4 Orientational arrangement of water molecules at the surface 1238.5 Dynamics of the protein hydration shell: experimental studies 124

8.5.1 Dielectric spectrum 1248.5.2 Nuclear magnetic resonance studies 1268.5.3 Quasi-elastic neutron-scattering experiments 1278.5.4 Vibrational spectroscopy 1288.5.5 Solvation dynamics 129

8.6 Conclusion 131Appendix 8.A Orientation of water molecules in the hydration layer 132

9. Understanding the protein hydration layer: lessons fromcomputer simulations 1359.1 Introduction 1359.2 Molecular motion in the hydration layer 1369.3 Hydrogen-bond lifetime dynamics 1409.4 Computer simulation of solvation dynamics 1429.5 Dielectric relaxation 1439.6 Explanation of anomalous dynamics in the hydration layer 1449.7 Proteinglass transition at 200 K: role of water dynamics 144

Contents ix

9.8 Free-energy barrier for escape of water molecules fromprotein hydration layer 146

9.9 Conclusion 146

10. Water in and around DNA and RNA 15110.1 Introduction: the unique role of water in stabilizing

DNA and RNA 15110.2 Hydration of different constituents 15210.3 Groove structure and water dynamics 15310.4 Translational and rotational dynamics of water molecules in the

grooves 15310.5 Solvation dynamics 15510.6 Entropy of groove water and dynamics 15610.7 Correlation between diffusion and entropy: AdamGibbs

relation 15710.8 Sequence dependence of DNA hydration: spine of hydration

in AT minor groove 15910.9 Effects of nanoconnement and surface-specic interactions 16110.10 Water around RNA 161

10.10.1 Structure of water around RNA 16210.10.2 Dynamics of water around RNA 162

10.11 Conclusion 162Appendix 10.A Hydrogen-bonding pattern around DNA 163

11. ProteinDNA interaction: the role of water as a facilitator 16711.1 Introduction 16711.2 Structural analysis of proteinDNA complex: classication of

hydration water 16811.3 Dynamics of water around a proteinDNA complex 16911.4 Role of water in thermodynamics of proteinDNA interactions 17011.5 Protein diffusion along DNA 17411.6 Conclusion 174

12. Water surrounding lipid bilayers: its role as a lubricant 17712.1 Introduction 17712.2 Hydration of different constituents: phospholipids and buried

proteins 17912.3 Rugged energy landscape for water motion 17912.4 Translational and rotational dynamics of water 18012.5 Solvation dynamics 18112.6 Transport of small molecules across the bilayer 182

x Contents

12.7 Transport of large molecules across the bilayer 18412.8 Electrostatic potential across the membrane 18412.9 Conclusion 185

13. The role of water in biochemical selection and protein synthesis 18713.1 Introduction 18713.2 Role of water in kinetic proofreading 188

13.2.1 Brief analysis of the HopeeldNinio approachto kinetic proofreading 190

13.2.2 Analysis of experimental results in the light of theHopeldNinio formulation 190

13.2.3 Aminoacylation of tRNA during protein synthesis 19213.2.4 tRNA selection in ribosome 19413.2.5 DNA replication 196

13.3 Water as a lubricant of life 19613.4 Conclusion 197

Part III Water in complex chemical systems 199

14. The hydrophilic effect 20114.1 Introduction 20114.2 Water near ions 20214.3 Water near an extended hydrophilic surface 20414.4 Aqueous hydrophilic binary mixtures 207

14.4.1 Waterurea binary mixture 20814.4.2 Waterguanidinium hydrochloride

binary mixture 20914.5 Aqueous salt solutions 209

14.5.1 Ionic conductivity 20914.5.2 Viscosity 211

14.6 Conclusion 212

15. The hydrophobic effect 21515.1 Introduction 21515.2 Hydrophobic hydration 21715.3 Temperature dependence of hydrophobicity: enthalpy

versus entropy stabilizations 21915.4 Hydropathy scale 22015.5 Pair hydrophobicity and potential of mean force between two

hydrophobic solutes 22115.6 Biological applications of potential of mean force 223

15.6.1 Protein folding 224

Contents xi

15.6.2 Hydrophobic association 22715.6.3 Pattern formation in chiral molecules 227

15.7 Hydrophobic collapse of polymers 22715.7.1 The FloryHuggins theory 228

15.8 Molecular-level understanding of hydrophobic interaction 23015.9 Hydrophobic force law 23415.10 Hydrophobicity at different length scales 23415.11 Conclusion 235Appendix 15.A PrattChandler theory 23615.A.1 Cavity distribution functions 23715.A.2 Theory for AWand AA pair correlations 239

16. The amphiphilic effect: the diverse but intimate world of aqueousbinary mixtures 24316.1 Introduction: the role of aqueous mixtures in chemistry

and biology 24316.2 Non-ideality of amphiphilic binary mixtures 24516.3 WaterDMSO binary mixture 24516.4 Wateralcohol binary mixture 249

16.4.1 Aqueous methanol solution 25016.4.2 Aqueous ethanol solution 25016.4.3 Watertertiary butyl alcohol 250

16.5 Wateracetone binary mixture 25216.6 Waterdioxane binary mixture 25216.7 Liquidliquid structural transformation in aqueous

binary mixtures: a generic phenomenon for amphiphilic solutes 25316.8 Theoretical development 25416.9 Biological applications 25616.10 Conclusion 258

17. Water in and around micelles, reverse micelles, and microemulsions 26117.1 Introduction: different self-assemblies in water 26117.2 Structure of micelles and reverse micelles 262

17.2.1 Micelles 26217.2.2 Reverse micelles 263

17.3 Dynamics of water surrounding micelles 26517.4 Free-energy landscape of hydrogen-bond arrangements at the

surface 26617.5 Reverse micelles and microemulsions: dynamics of water 26817.6 Orientational dynamics 26917.7 Coreshell model 270

xii Contents

17.8 Distance-dependent relaxation near the core of the reversemicelle: propagation of surface-induced frustration 273

17.9 Ising model description of the dynamics 27317.10 Conclusion 274

18. Water in a carbon nanotube: nature abhors a vacuum 27718.1 Introduction 27718.2 Type and structures of carbon nanotubes 27718.3 Structure of water inside a carbon nanotube 27818.4 Dynamics and transport of water 279

18.4.1 Translational motion of water inside a CNT 27918.4.2 Rotation of water molecules within a CNT 280

18.5 Nanotubes as a ltration device 28218.6 Conclusion 283

Part IV Advanced topics on water 285

19. The entropy of water 28719.1 Introduction 28719.2 Relation between entropy and diffusion 291

19.2.1 Diffusionentropy scaling relation:the Rosenfeld relation 291

19.2.2 The AdamGibbs relation 29319.3 Calculation of the entropy of water 295

19.3.1 From structure 29619.3.2 From dynamics 297

19.4 Entropy from cell theory 29819.5 Entropy of water in conned systems (reverse micelles, carbon

nanotubes, grooves of DNA) 29919.6 Conclusion 300Appendix 19.A Entropy for translational degree of freedom of

an ideal gas (SackurTerode equation) 301Appendix 19.B Entropy for vibrational degree of freedom 302Appendix 19.C Entropy for rotational degree of freedom 303

20. The freezing of water into ice 30520.1 Introduction 30520.2 Phase diagram of water and ice 30620.3 Ice formation in micro-droplets 30720.4 A lesson from the freezing of interacting spheres and

the difference from water 30820.5 The freezing of water 308

Contents xiii

20.6 Nucleation of an embryo 30920.7 The freezing of water in computer simulations 31020.8 Mechanism of ice formation 31120.9 Freezing inside nanotubes 31420.10 Conclusion 315

21. Supercritical water 31721.1 Introduction 31721.2 Inhomogeneous density uctuation in supercritical uids 31821.3 Crossing the Widom line 32021.4 Spectroscopic studies of supercritical uids 32021.5 Conclusion 322

22. Approaches to understand water anomalies 32322.1 Introduction 32322.2 Reason for density maximum 32722.3 Reason for large isobaric specic heat of water 32722.4 Percolation model of water 32722.5 Hydrogen-bond network rearrangement dynamics 330

22.5.1 Energy landscape view of hydrogen-bondrearrangement dynamics 331

22.5.2 Depolarized Raman scattering prole 33322.6 Low-temperature anomalies 33422.7 Conclusion 341

Epilog 345Index 349

The color plates will be found between pages 78 and 79.

xiv Contents

Preface

This book attempts to summarize the large body of experimental and simulationdata gathered recently on the structural and dynamic aspects of water in complexchemical and biological systems. In the process we try to present a unied view ofthis emerging eld. While most discussions on water focus on its role in complexsystems (like the role of water as a polar solvent stabilizing the native state of aprotein), I thought it would be equally, if not more, appropriate to study and ifpossible explain why water has so many unique properties and how it is able toplay important parts in so many diverse settings. For example, water moleculesthemselves need to change and adjust to the surface. In enzyme catalysis, theyparticipate actively and get consumed as a chemical not act just as a good solventfacilitating the catalysis a fact not often appreciated.Many important aspects of water have been discovered only in the last two

decades or so. For example, we came to know about the astonishingly fast rate ofsolvation of a polar solute by water only around the mid-nineteen nineties! Thedetailed role of water in chemical reactions, such as in electron transfer, has alsobecome clearer around the same time. It is therefore not surprising that it is only nowthat we have turned our attention towards understanding molecular aspects ofwaters role in biology. The specic role of water in most of the biological processesis far from well understood even today.Studies of unique properties of water have often followed two disjointed paths.

On the one side, detailed microscopic properties of water molecules, both in the bulkand in and around biomolecules, have been studied in vitro, such as water structureand arrangement around proteins and DNA. These studies have often remainedconned to their own domains of choice/focus, with hardly any attempt to connect itwith other properties and functions of water. The second line of studies has focusedon the utilitarian aspects of water. Here the approach is largely qualitative andfocused on the role of water in various aspects of life and nature. The latter have

xv

been popular since antiquity. Neither of these two approaches addresses the explicit(especially dynamic) role of water molecules in biological functions.Water that is present in biological cells, in the grooves of DNA, on the surfaces of

proteins is found to be quite different fromwater in the bulk, the water that we drink.The term biological waterwas coined to highlight this difference. In nature, wateris also found within rocks and conned systems, such as in tree leaves. Suchconned water also exhibits properties quite distinct from those in the bulk. Themain modication that occurs from the bulk state of water is the partial or even fullloss of the hydrogen-bond network that so uniquely denes water. In biological andmany natural systems, water faces a multitude of interactions from the surface.However, water seems to retain sufcient resources of its own to adjust to newenvironments and continue to perform its wide-ranging roles.We have placed special emphasis on properties that have been observed in bio-

molecules, such as proteins, DNA, and RNA, and in other complex systems such asmicelles, reverse micelles, and carbon nanotubes. As observed above, we tried tosee what happens to water due to the proximity to a foreign surface. Second, weattempted to provide a coherent explanation of properties observed from a modern,molecular, often dynamic, perspective. The latter relies heavily on recent advancesin the eld, often driven by computer simulations. Third, we spend considerableeffort to discuss biological functions of water. By biological function we do notimply only the stabilizing effect of water in proteins and DNA, but the direct rolethat water molecules themselves play in biochemical processes, such as in enzymekinetics and protein synthesis, that are essential for life. Thus, the third purpose ofthis book is to articulate such biological and chemical functions in the light of ourcurrent understanding of molecular aspects of water although, as stated above, thedevelopment in this area is largely incomplete.Throughout the monograph, we have attempted to avoid using mathematical

expressions and minute details of sophisticated theories in order to make the contentaccessible to a larger number of students and interested readers who are notprofessional researchers in the area. We believe that the properties of water are sointeresting, especially given the uniqueness of the liquid, that many scienticallyinclined people will nd the subject fascinating. Although in some places detaileddiscussions have been included to give a avor of the subject, we have attempted tokeep them at a minimum. We also address, towards the end of the book, certainadvanced topics of current research in water. They are not disjoint from the earlierchapters and substantiate our efforts to explain the uniqueness of water. But readers,if not interested in advanced topics, can avoid these chapters without much loss tothe completeness.Our focus on molecular explanations of the observed properties distinguishes the

present monograph from the others existing in the literature. At the same time, this

xvi Preface

approach also limits the range of topics that we could address here. But there aremany excellent books/monographs on water which can supplement this lacuna.Da Vinci called water Natural vehicle of change. We attempt to show here that

the detailed role of water in biological and chemical change can be fascinating andelusive at the same time. We hope this book (despite many lacunae) will bewelcomed by students and scientists at large, especially because it documentssome of the signicant progress that has been made in the last few decades.It is tting to end the preface of this book on water with the following well-known

quote of Mark Twain. My books are like water; those of great geniuses are likewine. (Fortunately) everybody drinks water. I hope this book on water qualies asMark Twains water.

Preface xvii

Acknowledgements

Many people, particularly my students, present and past, have helped during thewriting of this book. Without their support, this project would never have beencomplete. I am particularly thankful to Dr. Biman Jana, who started on this projectwith me and contributed signicantly to the initial stages of development. Ms.Susmita Roy helped enormously in preparing the gures, reading the manuscriptand correcting many errors, even adding paragraphs when needed. Mr. SaikatBanerjee and Dr. Mantu Santra helped in the writing of the hydrophobicity chapter.Mr. Rajib Biswas, Mr. Rakesh S. Singh, Ms. Sarmistha Sarkar, Mr. Milan Hazra,Mr. Rajesh Dutta, Ms. Rikhia Ghosh, Mr. Jonathan Furtado, Mr. Arpan Kundu,and Dr. Mantu Santra also read several of the chapters and offered correctionsand modications. Ms. Naina Vinayak helped in reproducing many gures. I amgrateful to Professor Kankan Bhattacharyya for many discussions, suggestions,and encouragement. Professor Iwao Ohmine and Professor Graham Fleming havealways been sources of encouragement in this long endeavor. Professor Shinji Saitohas been an incredible source of information and strength he helped with manygures that he generated from his own simulation data. Kaushik Bagchi offeredvaluable suggestions at critical stages and Kushal Bagchi read many pages. I amgrateful to my wife, Ms. Sukla Das, for support and encouragement. I also thank mymany students and collaborators who helped fashion my ideas and concepts in thisrapidly developing subject.

xviii

Part I

Bulk water

1Uniqueness of water

What makes water so unique? People have asked this question forcenturies. Yet the answer seems elusive. Despite all of its complex proper-ties, water has an amazing, deceptively simple chemical composition just two hydrogen and one oxygen atoms! Yet it exhibits highly unusualand puzzling physical properties. In this chapter we make a list of sixunusual molecular properties that together could be responsible for theunusual characteristics of water.

1.1 Introduction

It is fair to state that in the biologically relevant temperature and pressure range,none of the properties of water are like those observed in other common liquids(such as ethanol, benzene or acetonitrile). To begin with, water has remarkablyhigh boiling (100 degree Celsius) and melting (0 degree Celsius) temperatures thatare unusual for a liquid consisting of molecules that are so small in size and solight in molecular weight. Additionally, it exhibits a high critical temperature(374C), compared to the liquids with similar or comparable molecular structurethat are mostly in the gaseous state at room temperature and pressure (such ashydrogen sulde (H2S) and carbon dioxide (CO2)). It has large specic heat andexhibits many other thermodynamic anomalies to be discussed later in Chapter 2.

Understanding the origin of the anomalous properties of water has turned out tobe an extraordinarily difcult task a task that is only partly completed.Nevertheless, we need to make a beginning, with whatever understanding wehave acquired of bulk water [15], in our attempt to understand the diverse (andmyriad) roles that water plays in many complex environments, including biology.We discuss below six unique features that can be held responsible for many of theproperties of water. But rst we present a few of the essential details about a watermolecule so that those features can be understood and appreciated.

3

1.2 Molecular structure

A water molecule is made of two hydrogen atoms that are attached to an oxygenatom via covalent bonds, making it look V-shaped (as shown in Figure 1.1). TheOH bond length is about 0.1 nm (1 ) and the HOH bond angle is 104.51 (in thegas phase). The oxygen atom has two unused lone pairs of electrons on it and theycontribute in no small measure to the unusual properties of water.Apart from the covalent bonds between oxygen and hydrogen atoms, a water

molecule also has the ability to form hydrogen bonds with four other neighboringwater molecules. Two hydrogen atoms pair up with two oxygen atoms of twodifferent water molecules while the oxygen atom pairs up with two differenthydrogen atoms of two different water molecules. That is, one water molecule canform hydrogen bonds with four different water molecules, and the three-dimensional arrangement is tetrahedral, as shown in Figure 1.2.A typical hydrogen-bond (OHO) distance, that is, the distance between the

oxygen atom of one molecule and the hydrogen atom of the participating secondwater molecule, is 0.25 nm (2.5 ).It is worth noting here that Linus Pauling was the rst to mention the hydrogen

bond, in 1912. In 1935, he rst advanced the theory of hydrogen bonds betweenwater molecules. Using quantum mechanics and chemical bonds he evaluated thatboth covalent bonds and other electrostatic forces hydrogen bonds werecommencing in water. According to Pauling the covalency in a typical OHOhydrogen bond is about 5% [6].Although the arrangement of oxygen and hydrogen atoms is V-shaped with

positive charges localized on the hydrogen atoms and the negative charge on theoxygen atom, the situation is known to be more complex, with the distribution of theelectrons of the two lone pairs of electrons which have been recently described assmeared between two tetrahedral lobes. As a result, in an approximate sense, thelocal arrangement of water molecules in liquid water at low temperature can be

Figure 1.1. Molecular structure of water. The dark and light balls represent oxygenand hydrogen atoms, respectively. The sticks between the oxygen and thehydrogen atoms represent the OH chemical bond.

4 Uniqueness of water

regarded as a distorted tetrahedral, as shown in Figure 1.2, with each watermolecule, on average, forming four hydrogen bonds, as mentioned above.In fact, the delocalized nature of the electron density between the two lobes facilitates

a fth neighbor to approach a tagged water molecule with a hydrogen atom of theincoming molecule pointed towards the oxygen atom of the tagged molecule. Thisallows formation of a bifurcated hydrogen bond and the existence of a water moleculewith ve neighbors. The ve-coordinate water molecule so formed plays an importantrole in the thermodynamics and dynamics of liquid water. In Figure 1.3 we present aschematic representation to show how two 4-coordinated water molecules can beconverted to one 3-coordinated and one 5-coordinated water molecule.In Figure 1.3 we show the structure of a 5-coordinated species. Note that the

formation of a H bond with a fth neighbor is already possible even with localizedcharges on the two tetrahedrally placed lobes of the oxygen atom, and has beenobserved in many simulations with classical models of localized charges, but thedelocalized nature may further facilitate its formation.The quantum nature of electrons makes a water molecule more responsive and

discriminative to external perturbation than possible in the classical world. The

Figure 1.2. Typical molecular arrangement in the HB network around a centralwater molecule in liquid water. Four other water molecules form HBs (two donor(upper) and two acceptors (lower)) with the central water molecule. Theenvironment around the central water molecule is tetrahedral. (Adapted withpermission from http://mi-bitacora-diaria.blogspot.in/2009_02_01_archive.html.)

1.2 Molecular structure 5

lengths of the hydrogen bonds are also exible and they vary considerably at a giventime even among the three or four hydrogen bonds formed by the same molecule.

To summarize, the perfect picture given in Figure 1.2 represents an idealsituation. The tetrahedron around the central water molecule is often distorted,except in ice, where collective effects reinforce a tetrahedral geometry. We shalldiscuss later quantitative descriptions which do more justice to local arrangement ofwater molecules in the liquid state. But right now we ignore the detailed correctionsof the picture depicted above in Figure 1.2 and proceed with it.

In order to form an extended (that is, percolating) network that connects a largefraction of molecules of the entire system, there should be three or more hydrogenbonds per water molecule (unless molecules form large disconnected linearchains, which are unlikely and not seen in liquid water). Since each watermolecule can easily form four hydrogen bonds, it can support such a network.Indeed this very ability to form a hydrogen-bond network has always beenhypothesized to be the main reason for many anomalies exhibited by water (asshall be discussed later) [16].

Although bulk and conned water has been relentlessly studied both theoreticallyand experimentally for many decades [7], quantitative progress towards under-standing the mysteries of water began only after we could study the structure anddynamics of about a thousand molecules via computer simulations. The landmarkpapers by Stillinger and Rahman [3] in the 1970s marked a turning point in our

4

3

4

5

Figure 1.3. Pictorial representation of the conversion scheme from 4- to 5- and3-coordinated water. Here oxygen atoms are light gray and hydrogen atoms areblack.


understanding of liquid water. We note that these simulations employ classicalmechanics and use a rigid model of water where charges are xed at differentsites to mimic the charge distribution of water.Nevertheless, these simulations led to many important results. First, they conrm

the view that water contains an extensive hydrogen-bond network where the twohydrogen atoms participate in one hydrogen bond each while the lone oxygen atomforms two or three hydrogen bonds with the hydrogen atoms of nearest-neighborwater molecules (as shown in Figures 1.2 and 1.3). The average number of hydro-gen bonds per water molecule in liquid water under ambient conditions was found tobe about 3.5. As we have already mentioned, we need at least three hydrogen bondsby a water molecule to ensure a connected percolating network; these computersimulation studies established that liquid water is a giant gel consisting of watermolecules connected by hydrogen bonds. Doubly hydrogen-bonded water mole-cules are also present in the network, as they connect two extended networks. Somewater molecules can be even singly hydrogen bonded and are called danglingbonds. But the fraction of doubly and singly hydrogen-bonded water moleculesis small.However, compared to chemical bonds (like the ones between hydrogen and

oxygen atoms in water) these hydrogen bonds are weak, with dissociation energycomparable to thermal energy. As a result, these bonds continuously form and breakin liquid water. Hence the lifetime of a hydrogen bond is quite short, of the order oftwo to three picoseconds (ps) where 1 ps = 1012 s.Thus, the extended network of H2O molecules in liquid water is a uctuating

network [4]. This uctuation lets water be responsive to foreign solutes because itallows water to easily rearrange and solvate a large variety of solutes. This featurepartly allows water to act as a unique solvent.

1.3 Six unique features

As mentioned above, we can make a list of six unique features of water that areresponsible for many of its abilities and properties. We now list these properties.

(i) Awater molecule is small in size and low in molecular weight. The rst allowsit to occupy even relatively conned spaces, such as the grooves of DNA, orthe active sites of enzymes. In the latter case, often the presence of a singlewater molecule plays an important role.

(ii) Due to the large electro-negativity difference between oxygen and hydrogenatoms there is partial charge separation along the bond giving rise to (approxi-mately) 0.84e charge on the oxygen atom and (again, approximately) +0.42echarge on each hydrogen atom. This distribution of positive and negative

1.3 Six unique features 7

charges promotes its hydrogen-bonding ability. Since water is made up of twohydrogen atoms that can form two H bonds and one oxygen atom that canfurther hold two to three H bonds, each water molecule can form on the wholefour, and even ve, hydrogen bonds.In addition, water can act as both a donor and an acceptor of a hydrogen

bond. Thus, it can stabilize both a positively and a negatively charged atom/group. This property comes in handy at a protein surface. As mentioned earlier,the electron charge in the lone pairs might be smeared between the oxygenatoms, allowing water to react quite differently to an anionic or cationicligand.

(iii) An additional aspect of the waterwater hydrogen bond not captured in theclassical models is the transfer of electron density from the oxygen atom of theacceptor molecule to the hydrogen atom of the donor molecule [8]. SeeFigure 1.4 for an illustration of this phenomenon.This electron transfer gives the hydrogen bond a small measure of covalent

character, estimated to be about 5% [9].Note further that the OH stretching frequency of the donor molecule

decreases proportionally to the strength of the hydrogen bond. This purelyquantum effect allows additional cooperativity in hydrogen bonds.

(iv) Awater molecule is characterized by a large dipole moment which is reectedin the large dielectric constant of liquid water, about 80 at room temperature.This large dielectric constant is extremely useful in many chemical processes.Related to this feature, the two lone pairs of electrons on the oxygen atom

render a water molecule polarizable to electric elds from other molecules orfrom a charged surface. Thus, water molecules can respond to the changing ofcharge distribution in an external solute (or surface) to lower the energy of an

HB Donor

e

+

+

HB AcceptorFigure 1.4. Electron cloud migration from the hydrogen-bond (HB) acceptor to theHB donor water molecule. In a water dimer the hydrogen atoms (shown in white)of a better HB acceptor becomemore positive, while the oxygen (shown in gray) ofthe HB donor becomes more negative.


assembly or molecular arrangement. However, a water molecule is not highlypolarizable, and is quite low in polarizability.In neat liquid water, the polarizability effectively increases the dipole

moment of each individual water molecule, and is partly responsible for thelarge dielectric constant of water.

(v) A collection of water molecules can form many structures of nearly equalenergy. This is most evident from a study of water clusters [10]. Also, ice isknown to have many polymorphs. As many of these structures are of similarenergy this makes a collection of a small number of water molecules highlyadaptive to various complex environments [5]. For example, when a layer ofwater molecules faces a non-polar surface, the spatial arrangement and orien-tation of water molecules are quite different from those when the layer faces apolar or charged surface. This is because water molecules can adopt manydifferent structures.

(vi) The remarkable ability of water to sustain a uctuating extended hydrogen-bond network allows facilitation of many dynamic processes that wouldotherwise be impossible. The marginally stable nature of the hydrogen-bondnetwork arrangement makes it rather easy to initiate the molecular rearrange-ment. The hydrogen-bond network can easily be distorted. In addition,hydrogen-bond energy in water spans a wide range of energy (~39 kBT forliquid water at 25C temperature) [4].

We have made the above list such that each property operates at least somewhatindependently. There are obviously correlations among these. Actually the ability ofwater to sustain multiple timescales is a unique feature of this liquid. Water seems tobe able to respond according to the speed of perturbation. It responds slowly to slowperturbation and rapidly to fast perturbation.In the subsequent chapters we shall try to rationalize the properties of water in

complex systems by using these six features.

1.4 Modeling of water

Unfortunately, it has turned out to be exceedingly difcult to accommodate all theunique features of water within any given, classical model. This is reected in theabsence of satisfactory agreement between experiments and any given model [2].This is a bit unusual (and of course frustrating) because when one usually models agiven molecule, such as methane, it is adequate to use a simple functional form suchas a Lennard-Jones potential that incorporates a measure of size and a measure ofinteraction energy at an optimal separation between two molecules. In the case ofwater molecules, such a simple procedure does not work. Here we have to account

1.4 Modeling of water 9

for at least two length scales one for the molecular size and the other for hydrogen-bond length. Second, one needs to take into account the charge distribution. And aswe have already discussed, quantum effects (electron transfer) give rise to acooperativity in hydrogen-bonding that is hard to mimic within a classical model.Thus, the unique features discussed above prove to be particularly hard to model.As a result of these complexities, although more than 100 different potential

functions have been employed, no fully satisfactory model has yet been developed.But many of these models have been able to explain many of the experimentalobservables, such as density maximum, values of viscosity and self-diffusion coef-cient, specic heat and compressibility, and dynamics of electron transfer reactions.Thus, it is also not fair to state (as the statement is often made) that we do not

understand water. Although clearly perspectives differ, one should not lose sight ofthe successes that have been achieved.

1.5 Conclusion

Liquid water is different from other liquids. Unique (and often termed anomalous)properties of water originate ultimately from the unique molecular features of water.We have made a list of six such features which combine to give rise to the unusualproperties of water. The list itself may not be unique or exhaustive but we think thatit provides a starting point to rationalize the properties of water.In fact, an attempt to rationalize the diverse properties in terms of a few basic

features is a reductionist view which has a lot of advantages. Most importantly, it ispossible to get back to basics when one faces difculty in explaining the experi-mentally observed properties. For example, a lot of our difculties in understandingor describing the behavior of water molecules at the surface of proteins or chargedsurfaces arise from our difculty in handling the polarizability anisotropy of watermolecules.In the next chapter, we discuss a few of the well-known anomalies of liquid water.

We shall return to the discussion of those anomalies again in the penultimate chapterof the book. In the intervening chapters we shall discuss various properties of waterin diverse systems, with a close connection between theory and simulations.

References

1. J. H. Gibbs, C. Cohen, P. D. Fleming, and H. Porosoff, Toward a model for liquid water.J. Solution Chem., 2 (1973), 277; P. D. Fleming and J. H. Gibbs, An adaptation of thelattice gas to the water problem. J. Stat. Phys., 10 (1974) 157.

2. F. H. Stillinger, Effective pair interactions in liquid water. J. Phys. Chem., 74 (1970),3677; F. H. Stillinger, Theory and molecular models for water. Adv. Chem. Phys., 31(1975), 1.


3. F. H. Stillinger and A. Rahman, Improved simulation of liquid water by moleculardynamics. J. Chem. Phys., 60 (1974), 1545; A. Rahman and F.H. Stillinger, Hydrogen-bond patterns in liquid water. J. Am. Chem. Soc., 95:24 (1973), 79437948.

4. I. Ohmine and H. Tanaka, Fluctuation, relaxations and hydration in liquid water.Hydrogen-bond rearrangement dynamics. Chem. Rev., 93:7 (1993), 25452566;M. Matsumoto, S. Saito, and I. Ohmine, Molecular dynamics simulation of the icenucleation and growth process leading to water freezing. Nature, 416 (2002), 409.

5. Y. K. Cheng and P. J. Rossky, Surface topography dependence of bimolecular hydro-phobic hydration. Nature, 392 (1998), 696.

6. L. Pauling, B. Kamb, et al., Linus Pauling: Selected Scientic Papers, World Scienticseries in 20th century chemistry, vol. 10 (River Edge, NJ: World Scientic, 2001).

7. B. Bagchi, Molecular Relaxation in Liquids (New York: Oxford University Press,2012).

8. R. J. Gillespie and P. L. A. Popelier,Chemical Bonding andMolecular Geometry: FromLewis to Electron Densities (New York: Oxford University Press, 2001).

9. E. Arunan, G. R. Desiraju, R. A Klein, et al., Denition of the hydrogen bond. PureAppl. Chem., 83:8 (2011), 16371641.

10. F. N. Keutsch and R. J. Saykally, Water clusters: untangling the mysteries of the liquid,one molecule at a time. Proc. Natl. Acad. Sci. USA, 98 (2001), 1053310540.

References 11

2Anomalies of water

The unique features of individual water molecules (discussed in thepreceding chapter) give rise to many anomalous properties of liquidwater. Commonly attributed to the presence of an extensive hydrogen-bond network, these anomalies teach us a lot more about water itself.Anomalies are observed in many properties, ranging from a densitymaximum at 4C, the temperature dependence of isobaric specic heatand isothermal compressibility to a host of dynamic properties. Here wediscuss some of them, with the emphasis on collective properties that arerelevant to our study of complex systems discussed later. Understandingthese anomalies is still the subject of considerable research activity.

2.1 Anomalous properties

Water is most anomalous at low temperatures, with the remarkable density max-imum at 4C. When water is supercooled below its freezing/melting temperature of0C at ambient pressure, most thermodynamic properties exhibit strong anomalies.However, water exhibits many weak to strong anomalies even at room temperature,particularly in its interactions with solute molecules. The existence of a largenumber of anomalous properties makes water one of the most puzzling substancesknown to mankind. We next discuss some of these anomalies. The understanding ofthese anomalies is not only challenging and has bafed scientists for generations butit also holds the key to evolving a unied understanding of this liquid.

2.1.1 Density maximum

The density anomaly is one of the oldest known and one of the most quoted puzzlesin the behavior of water [1]. Unlike other simple liquids, which expand upon heating(density decreases), water contracts on heating above 277 K (4C), at atmosphericpressure. The density prole of liquid water is shown in Figure 2.1 as a function of

13

temperature at various pressures. The temperature of maximum density (popularlyknown as TMD) moves to lower temperature as the pressure increases, as it is moredifcult to expand at higher pressures. This TMD serves as a good measure of theorder in the liquid.As is well known, solid ice is of lower density than liquid water. It is because of

this anomaly that ice can oat on water and sh can survive in the warm liquid waterbelow a layer of ice, at temperatures well below 0C. Freezing of water and meltingof ice are still not well understood.There is a relatively simple explanation of the density maximum at 4C, in terms of

the average coordination number of water molecules. As we discussed in the rstchapter, a given water can form different numbers of hydrogen bonds, ranging fromzero to six, with the most probable number being close to three at ambient conditions.We can relate the volume of a given water molecule to the number of hydrogen bondsit forms. Thus, a given water molecule with two hydrogen bonds has a larger volumethan one having six hydrogen bonds (both examples are relatively rare).As temperature is decreased from above (say, from 20C) towards 0C, the number

of hydrogen bonds per water molecule increases as the 2- and 3-coordinated watermolecules get replaced predominantly by more stable 4-coordinated water molecules.In the process some 5-coordinated water molecules also form. Computer simulationstudies show that at 10C, about 70% of themolecules are 4-coordinated while 3- and5-coordinated are nearly equally populated at about 14% each. As mentioned above,conversion of 2- and 3-coordinated water molecules to 4-coordinated ones is themain reason for the increase in density on lowering the temperature ofwater. However, as we approach 4C, energetic reasons now favor 4-coordinatedwater molecules over 5- or 6-coordinated water molecules. These higher coordinated

0 5 10 15 20 25 30Temperature (C)

75

50

251 bar

1004

1002

1000

998

996

p.de

nsity

(kg/

m3 )

Figure 2.1. Temperature-dependent density of liquid water at various pressures.Note the shifting of TMD to the lower temperature as pressure is increased.(Adapted with permission from http://www.engineeringtoolbox.com/uid-density-temperature-pressure-d_309.html.)

14 Anomalies of water

water molecules begin to get replaced by energetically more stable 4-coordinatedwater molecules, leading to a fall in density. Thus, the density maximum is aninterplay between the natural strength of hydrogen bonds (which arise from chargedistribution in the water molecule) and the thermal energy.Detailed numerical calculations indeed justify the above simple logic. Thus, the

maximum density of water at 4C is not too much of a mystery. However, explana-tions of other anomalies are not that simple.

2.1.2 Isobaric specic heat (CP)

The specic heat of a substance provides a quantitative measure of the amount of heatnecessary to increase the temperature of the system by 1C. We now discuss how thisamount is related to the number of congurations (distinct molecular arrangements) thatare available to the system, within a small range of energy around a given energy. Asweprovide heat energy to the system, the energy gets divided into all themicroscopic statesof the system. All the microscopic states must get the energy. This energy is actually theenthalpy (H), which includes both the internal and themechanical energy (in the form ofPV, where P is the pressure and V is the volume of the system).Since the enthalpy of the system is a sum of contributions of many molecules, the

probability distribution of enthalpy H is a Gaussian function, with the width ofthe distribution naturally given by the mean-square deviation of enthalpy, or theuctuation of enthalpy. The exact relationship between specic heat and enthalpyuctuation is given in Appendix 2.A.Now the number of congurations available to a system at a given energy is

measured by the entropy of the system. As discussed in Appendix 2.A, CP is thusdirectly proportional to the entropy uctuation in the system.Since the thermal uctuation should generally decrease with decreasing tempera-

ture, one would expect that CP should decrease with decreasing temperature. This isthe scenario for most simple liquids. However, in the case of water, it increases asthe temperature is decreased below T = 320K and at temperature below the freezing/melting temperature, the specic heat appears to diverge at a singular temperaturewith a power law (as shown in Figure 2.2) [2].

2.1.3 Isothermal compressibility (T)

The isothermal compressibility of liquids gives a measure of the change in volumeof the system due the change in pressure applied at a constant temperature. Amicroscopic expression of isothermal compressibility is given in Appendix 2.A.The expression shows that compressibility is related to the natural uctuations in thetotal volume of the system.

2.1 Anomalous properties 15

For most simple liquids, T decreases with decrease in temperature as the volumeuctuation decreases. However, in the case of water, it increases like CP below acertain temperature and appears to diverge with lowering temperature, as shown inFigure 2.3 [3].

2.1.4 Coefcient of thermal expansion (P)

The coefcient of thermal expansion P provides us with a measure of volumechange of the system due to a change in temperature at constant pressure.

Temperature (C) T

Cp

typical liquid

Tm 35 C

H2O

Spec

ific

Heat

Cap

acity

Figure 2.2. Temperature dependence of the isobaric heat capacity (CP) in liquidwater. The dashed line represents the behavior of typical liquids. Note the turn-around and divergence-like behavior for water at the melting temperature (Tm). Thegure is reproduced from the thesis of Dr. Pradeep Kumar. http://polymer.bu.edu/~hes/water/thesis-kumar.pdf.

Tm 46 C

typical liquid

H2O

T

Isot

herm

al C

ompr

essi

bilit

y

Temperature (C) T

Figure 2.3. Temperature dependence of isothermal compressibility (T) in liquidwater. The dashed line represents the behavior of typical liquids. Note the turn-around and divergence-like behavior for water. The gure is reproduced from thethesis of Dr. Pradeep Kumar. http://polymer.bu.edu/~hes/water/thesis-kumar.pdf.


For simple liquids, the volume of the system increases with temperatureand thus P is always positive. Also P decreases with decrease in temperatureas the volume and entropy uctuations in the system decrease. However, in thecase of water it becomes zero at the temperature where density is maximum(TMD) and then becomes negative with further decrease in temperature. Thissuggests that below TMD the entropy increases with decrease in volume. Like CPand T, P also seems to diverge with a power law at low temperature as shown inFigure 2.4 [2].Since the experiments on bulk liquids including water cannot be performed below

the homogeneous nucleation temperature (TH; for bulk water TH = 38C), wherecrystal formation is found to become inevitable, it is not possible to test whether theapparent divergences of the above three quantities at low temperature are indeeddivergences or something else. However, experiments on nano-conned water andextensive computer simulation studies (which have been possible since the forma-tion of crystals is difcult in such systems and we can study the liquid water wellbelow its homogeneous nucleation temperature) nd that these quantities do notdiverge but rather have a maximum at low temperature.

2.1.5 Dynamic anomalies present at low temperature

For simple liquids, the temperature dependence of dynamics is usually given by aform that is known as the Arrhenius equation. For relaxation time, the Arrhenius

equation is given by 0 exp AkBT

, where is the measured relaxation time and A

is the activation energy, which is usually weakly temperature-dependent, and 0 is atting parameter which can be regarded as a reference relaxation time. The

P

Tm

4C

(c)

T

typical liquid

H2O

Ther

mal

ex

pans

ion

coef

ficie

nt

Temperature (C)

Figure 2.4. Temperature dependence of coefcient of thermal expansion (P) inliquid water. The dashed line represents the behavior of typical liquids. Note theunusual behavior of liquid water below the melting temperature (Tm). The gure isreproduced from the thesis of Dr. Pradeep Kumar. http://polymer.bu.edu/~hes/water/thesis-kumar.pdf.

2.1 Anomalous properties 17

Arrhenius form is valid at high temperatures (above its freezing/melting tempera-ture) and its origin is attributed to the presence of energy barriers that restrict themotion of molecules. However, the temperature dependence of relaxation timebecomes non-Arrhenius at low temperature, below the freezing/melting tempera-ture. The reason for such crossover in dynamic behavior has been a subject ofintense discussions, and it is usually attributed to the emergence of a situation wheremotions of distinct molecules are correlated. Such correlated motions appear in coldliquids below their freezing/melting temperature, where they are termed super-cooled liquids. Here the liquids become increasingly more viscous and ultimatelysome liquids transform into glass if cooled sufciently fast.However, water behaves differently. In the case of water, correlated motions

appear even above the freezing temperature of 0C. Below the freezing temperature,the motion of molecules becomes increasingly slower. In addition to the rapidgrowth of specic heat and other response functions, the relaxation rates of watershow anomalous non-Arrhenius temperature-dependence.There have been several explanations of this behavior, in terms of an impend-

ing rst-order phase transition, or the existence of a second critical temperatureowing to a liquidliquid transition at lower pressure. Unfortunately, thesesuggestions cannot be veried in pure water as the liquid cannot be cooledbelow 40C.Inability to look at bulk water at low temperatures, say 3040C below the

freezing temperature, has motivated a different approach where water has beenstudied in conned small systems that do not seem to freeze to ice easily. Thedynamics of water conned in nanopores and water surrounding biomolecules(these water molecules can be cooled below 40C) are found to change rathersharply from non-Arrhenius at high temperature to Arrhenius at low temperatures.The logarithm of density relaxation time () of water conned in MCM-41-S pores(the MCM-41-S nanoporous silica matrix has 1D cylindrical pores arranged in 2Dhexagonal arrays, with pore diameters characterized by a narrow distribution) is afunction of 1/T. It has been shown that log () has a distinct crossover from non-Arrhenius (increasing activation energy with decrease in temperature) at high T toArrhenius (constant activation energy) at low T [4]. Analysis of molecular arrange-ment shows that the low-temperature phase is a low-density liquid/amorphousphase.We must alert the reader that it is not clear to what extent the above experiments

on narrow pores with a rather small number of water molecules can be used toexplain, or can be related to, the anomalous properties of bulk water at lowtemperatures. However, it does establish the existence of a low-density liquidphase with a free energy perhaps not too different from the high-density liquidphase (the normal liquid at room temperature).


2.2 Translational and orientational order

We now shift our attention towards more microscopic structural aspects of water.These aspects are discussed by quantication of local order inside a liquid.Translational or spatial order in a liquid provides information about the localarrangement of water molecules as a function of distance. This can be ascertainedby looking at the pair correlation function (g(r)) of the liquid. The pair correlationfunction (also known as radial distribution function) gives the probability of ndinga pair of molecules separated by distance r. They can be any pair. Since all the watermolecules are identical and the liquid is homogeneous, we can x our attention onany water molecule and look for the arrangement around it. The function g(r) thengives the probability of nding other water molecules around our central molecule,at a separation r. If the value of g(r) is more than 1, then it simply means that theprobability of nding the particles is more than what one would expect according tothe density of the liquid and vice versa.A typical g(r) of liquid water is shown below in Figure 2.5. Note that the radial

distribution function between oxygen atoms of two different water molecules give apeak at about the hydrogen-bonding distance (approximately 0.3 nm) as the neigh-boring two water molecules are hydrogen bonded through the hydrogen atom of oneof the water molecules, as shown in Figure 1.2 of Chapter 1.The rst of the two approximately equal height peaks of gOH(r) corresponds to the

hydrogen involved directly in the hydrogen (O- -H) bond (the smaller distance)while the second one at longer distance corresponds to the non-hydrogen-bondedsecond hydrogen of the molecule involved in the hydrogen bond with the centralwater molecule.

3

2

1

00 5 10

gOO gOH

r ()

g(r)

Figure 2.5. Typical radial distribution functions (gOO(r) and gOH(r)) in liquid water.The modulations at small separation distances indicate the short-range local orderin the liquid. Adapted with permission from Frontiers in Bioscience, 14 (2009),35363549. Copyright (2009) Frontiers in Bioscience.

2.2 Translational and orientational order 19

The initial modulation and the rst peak in g(r) are due to the formation of localstructure in the liquid. For completely uncorrelated systems, g(r) = 1, and thus theorder parameter is zero. For a system with long-range order, the modulation in g(r)persists over large distances, causing the translational order to grow.In order to further quantify the local translational (or, spatial) order around

molecules, one introduces a second quantity, tO, which is obtained by averaging g(r) over separation r. Thus, tO is a number which is a function of temperature anddensity (or pressure). The precise denition is given in Appendix 2.B of this chapter.A parameter such as tO can describe the variation of order when the temperature orpressure of the liquid is changed.Figure 2.6 depicts an interesting variation of this translational order parameter as

both temperature and density are varied but at constant pressure. As shown inFigure 2.6, tO shows both a maximum and a minimum at low temperatures as thedensity is lowered from a high value of 1.3 gcm3. Such a combined presence ofmaximum and minimum is not observed at higher temperatures and seems to appearfor the rst time close to the freezing temperature. The initial increase in spatialorder is found to be due to the formation of an increasing number of 4-coordinated(that is, hydrogen-bonded) water molecules. During this range the density of theliquid also decreases, facilitating the formation of an open 4-coordinated network.While the pair correlation function (or radial distribution function, g(r)) provides

information about two particle arrangements, it does not provide information aboutthe relative arrangement of three or four or more water molecules. Knowledge about

1.4

1.2

1

0.8

0.60.8 0.9 1 1.1 1.2 1.3

T=240KT=280KT=320K

(g/cm3)

tmax

qmax

Transl

atio

nal o

rder

t Te

tra

hedr

al o

rder

q

tmin

Figure 2.6. Density-dependent translational (tO) and tetrahedral order (q) of liquidwater at various temperatures. The maximum of translational order coincides withthe maximum of tetrahedral order. Note that the plot of translational order also hasa minimum. Adapted with permission from Phys. Rev. E, 76 (2007), 051201.Copyright (2007) American Physical Society.


such higher-order structural arrangements can provide information that is essentialto understand the properties of water. As discussed earlier, water lacks the perfecttetrahedral local structure (of ice) in the liquid state. There is a lot of disorder in thelocal arrangement of water. Description of such order and disorder requires con-sideration of angles between bonds formed by nearest-neighbor molecules. That is,one considers the relative arrangements of three water molecules. Such an arrange-ment involving three water molecules can be described by the angle made by thetwo bonds that connect one molecule with two others, as shown below.The molecular arrangement involving three water molecules can be described by

using the trigonometric function of the angle between three molecules (seeFigure 2.7). As discussed in Appendix 2.B, one usually denes a function q byaveraging over all three-particle neighboring molecules. The temperature depen-dence of q is shown in Figure 2.6 (where we have also shown the same for spatialorder parameter, tO). The parameter q tracks the behavior exhibited by tO.There is, however, an interesting aspect to this temperature dependence. For

simple liquids, q and tO increase with increasing density of the system. However,for liquid water both q and tO increase with decrease in density and go through amaximum at a certain temperature. Maxima of q and tO seem to coincide with eachother for a given temperature (as shown in Figure 2.6).

2.3 Temperaturedensity range of water anomalies

It helps if we categorize the anomalies discussed above into three different types:(1) thermodynamic anomalies (for example, in density, CP, T and P), (2) dynamicanomalies (relaxation time or diffusion, dynamic crossover), and (3) structuralanomalies (in translational and orientational order).However, interestingly, these anomalies do not persist over the entire temperature

and density (or pressure) range. Thus it is also important to know in which range

Figure 2.7. Molecular arrangement of three water molecules, with linear HBs. Theangle between three oxygen atoms (OOO) is indicated as .

2.3 Temperaturedensity range of water anomalies 21

these anomalies exist. The range of anomalies for three different types is shown inFigure 2.8 [5].This interesting gure shows that the region of thermodynamic anomalies is

bounded inside the region of dynamic anomalies which in turn is bounded inside theregion of structural anomalies. Thus, as a preliminary guess, it can be inferred thatthermodynamic and dynamic anomalies can be understood in terms of structuralanomalies [5].

2.4 Conclusion

The rapid variations (rise or fall) in the value of the thermodynamic response functions,namely the specic heat, the isothermal compressibility (both increase), and thecoefcient of thermal expansion (which decreases with temperature when the latter islowered below the freezing point), are some of the known spectacular anomalies ofliquid water. These variations have till now eluded a fully satisfactory understanding[6]. Many computer simulation studies have been done and several theoreticalapproaches have been developed but they are still not universally accepted.

Figure 2.8. Here three shaded regions in the densitytemperature plane showdifferent types of water anomalies. The structurally anomalous region is boundedby the loci of q (orientational order parameter) maxima (upward-pointing triangles)and tO (translational order parameter) minima (downward-pointing triangles).Inside this region, water becomes more disordered when compressed, as tO and qdecrease with increasing density. The loci of diffusivity (inverse of relaxation time)minima (circles) and maxima (diamonds) dene the region of diffusion (D)anomalies, where D increases with density. The thermodynamically anomalousregion is dened by the temperature of maximum densities, TMD (squares), insidewhich the density increases when water is heated at constant pressure. Adaptedwith permission from Nature, 409 (2001), 318321. Copyright (2001) NaturePublishing Group.


Denitions of these response functions in terms of the mean-square uctuationsor correlations among appropriate thermodynamic quantities are given in Appendix2.A. Thus, the increase of specic heat and compressibility is related to a rathersudden increase in these uctuations as temperature is lowered below the freezing/melting temperature of water/ice. Also, the increase in mean-square uctuations inentropy and volume is accompanied by a decrease in correlations between these twoquantities. The latter could happen if there is some degree of anti-correlationbetween the two uctuations. That is, increase in volume leads to decrease inentropy and vice versa.An age-old and qualitative explanation of these anomalies is provided by assum-

ing a two-state model of water. In this model, a large region of the liquid (muchlarger than the size of an individual water molecule) can exist either in a high-density liquid (which is the normal liquid, say at 10C) and a low-density liquidwhich consists mostly of randomly connected mostly 4-coordinated (by hydrogenbonds, of course) water molecules with a density only slightly higher than that of icebut less than that of the high-density liquid dened above. However, over a range oftemperature, say between 275 K and 240 K, these two states have similar freeenergy, with the low-density liquid (LDL) gaining stability over the high-densityliquid (HDL) as the temperature is lowered. However, these two regions can inter-convert. This can give rise to large uctuations in enthalpy, entropy, and volume.The HDL has higher entropy but lower enthalpy as it is a mixture of 3-, 4-, and5-coordinated water molecules, with a signicant fraction in each. Thus, when anHDL region converts to an LDL region, the volume increases but the entropydecreases. Repeated conversions such as this lead to large uctuations.While there is general agreement up to this point among different views of low-

temperature water anomalies, there is considerable disagreement about the progres-sion of the system when temperature is further lowered, say below 240 K. Weshall refrain from discussing these different approaches, as we shall hardly need todwell on temperatures below even 250 K in the book. However, we shall use thistwo-state picture, which is a generally accepted explanation of the anomalies below240 K. This is consistent with our view that water molecules can form many nearlyisoenergetic structures among themselves.

APPENDIX 2.A MICROSCOPIC EXPRESSIONS OFSPECIFIC HEAT, ISOTHERMAL COMPRESSIBILITY,

AND COEFFICIENT OF THERMAL EXPANSION

All the above three quantities (specic heat, isothermal compressibility, and coef-cient of thermal expansion) provide the response of the system to external perturba-tion of different kinds (clear from the names) and are called response functions of

2.4 Conclusion 23

the system. Statistical mechanics provides useful expressions for them in terms ofuctuations, which are given below. These expressions also provide insight into thestate of the system, as also discussed below.Specic heat at constant pressure, CP, is related to the microscopic properties of

the system by the following relations,

CP @H@T

P

T @S@T

P

DS 2kB

DH 2kBT

2 2:A:1

where H is the enthalpy, S is the entropy and kB is the Boltzmann constant. Hereh DS 2i denotes mean-square entropy uctuations and h DH 2i corresponds tomean-square enthalpy uctuations. A way to understand the above expression isas follows. At a constant temperature and pressure, the entropy of the system canuctuate (within a bound) because of inow and outow of heat from the reservoir(bath). The mean-square entropy uctuation is a measure of the heat that a systemcan naturally absorb due to its inherent capacity when temperature is raised. Formost liquids these uctuations decrease with lowering temperature, as expected.However, for water it is found to increase. Thus, entropy uctuations increase, forwater, on lowering the temperature.In a similar vein, the isothermal compressibility, T, can be written as,

T 1V@V

@P

T

DV 2kBTV

2:A:2

The last formula shows that isothermal compressibility is related to the uctuation inthe total volume (V) of the system. Again, the amplitude of this uctuation involume, measured by the second moment, is a measure of the volumes accessible tothe system. Nevertheless, Eq. (2.A.2) is really an elegant expression.The case of thermal expansion coefcient, P, is different. This is related to the

uctuation of volume and entropy of the system by the following relation,

P 1V@V

@T

P

1kBTV

hDVDSi 2:A:3

APPENDIX 2.B QUANTIFICATION OF SPATIALORDER IN WATER

Awater molecule is spatially correlated with its nearest neighbors due to hydrogen-bonding. That is, we expect to nd a number of water molecules at a nearly xeddistance and also at a relatively xed orientation, as discussed in Chapter 1. In orderto quantify this spatial order one often denes a translational order parameter as,


tO 4rc0

g r 1r2dr 2:B:1Here rc is the separation where g(r) reaches the value unity after the peak, that isbefore any other modulation due to higher separation. In most liquids, the tetra-hedral parameter increases with lowering of temperature but in water it reaches amaximum and then drops, as discussed in the text.The local tetrahedral arrangement of neighbors around a water molecule in liquid

water is best captured by the orientational order parameter q, which is dened as,

q 1 38

X3j1

X4kj1

cosjk 1

3

22:B:2

Here jk is the OOO angle formed by the two nearest-neighbor oxygen atoms(j and k) with the central oxygen atom for which the local order parameter q isbeing calculated. Here (j,k) indices are arranged so that the OOO bond angles arepicked up properly, without over-count. We then average the value of q over all theoxygen atoms of the liquid. For a perfect tetrahedral local structure (as in ice), thevalue of q is 1, while q is equal to zero for random rotational arrangements (such asin the gas phase). In liquid water at room temperature, this value is close to 0.5,which is pretty high.

References

1. R. Waller, Essayes of Natural Experiments (original in Italian by the Secretary of theAcademie del Cimento): (1964) Facsimile of English translation. Johnson Reprint, NewYork (1964).

2. C. A. Angell, J. Shuppert, and J. C. Tucker, Anomalous properties of supercooled water.Heat capacity, expansivity and proton magnetic resonance chemical shift from 038%.J. Phys. Chem., 77 (1973), 3092.

3. R. J. Speedy and C.A. Angell, Isothermal compressibility of supercooled water andevidence for a thermodynamic singularity at 45C. J. Chem. Phys., 65 (1976), 851858.

4. L. Liu, S.-H. Chen, A. Faraone, C.-W. Yen, and C. Y. Mou, Pressure dependence onfragile to strong transition and a possible second critical point in supercooled connedwater. Phys. Rev. Lett., 95 (2005), 117802.

5. J. R. Errington and P.G. Debenedetti, Relationship between structural order and theanomalies of liquid water. Nature, 409 (2001), 318321.

6. P. G. Debenedetti and H. E. Stanley, Supercooled and glassy water. Phys. Today, 56:6(2003), 4046.

References 25

3Dynamics of water: molecular motionsand hydrogen-bond-breaking kinetics

Molecular motions (rotation, translation, and vibration) of a watermolecule also turn out to be quite different from those of other commonliquids. Here all the six unique features of an individual water moleculeoutlined in Chapter 1 manifest themselves in diverse ways. As we discussbelow, not only is the mechanism of displacements of individual watermolecules different, but the collective dynamics and dynamical responseof bulk water are also different. For example, the rotational motion of anindividual water molecule contains a surprising jump component andvibrational energy relaxation of the OH mode involves a cascadingeffect mediated by anharmonicity of the bond. These motions are reectedin many important processes such as electrical conductivity, solvationdynamics, and chemical reactions in aqueous medium.

3.1 Introduction

It is natural to expect that the extensive hydrogen-bond network present in watermay substantially alter the nature of the molecular motion of individual watermolecules from those found in normal liquids where such a network is absent. Inthose non-hydrogen-bonded liquids individual molecules usually move by smallsteps. One such small step is mostly uncorrelated with the next one. Such a motionby random steps is called Brownian motion. Brownian motion is the erratic and tinymovement of small particles (often observable under optical microscope) whenlarge particles are suspended in a uid or gas of small particles. For example, if yousprinkle tiny grains of dust into water, and then look at the dust particles under amicroscope, the dust particles appear to dance around, continuously and quiterandomly. This zigzag random motion happens regardless of how still the surfaceof the water is kept.This interesting phenomenon of random motion of particles in a liquid was

discovered in 1827 by the British botanist Robert Brown [1]. He was investigating

27

pollen grains in water, and noticed that they would not remain still under hismicroscope. At rst he thought that the pollen was moving because it was alive.But even hundred-year-old pollen grains danced around, so he knew there had to besome other explanation. Later it was found that Brownian motion is exhibited notonly by particles suspended in liquids but also by the atoms and molecules con-stituting a liquid itself, at least for most liquids. This mode of motion holds for mostliquids, starting from simple liquids such as argon (which can be approximated byspheres), as shown below in Figure 3.1. In the same gure, we also show theposition displacements with time of several tagged molecules.The reason for the small-step randommotion of atoms and molecules in liquids is

the incessant collisions that they suffer with each other. In a beautiful series ofpapers, Einstein showed that this Brownian motion is a consequence of the naturalmotion of the molecules controlled by the temperature of the system [2]. Therefore,Brownian motion is also called thermal motion of the molecules. Obviously, themolecules of the liquid move faster when the liquid is heated, causing more agitatedBrownian movement of the big particles. Similarly, if you make the liquid lessviscous, the molecules canmovemore easily, also resulting in faster particle motion.Turning now to water, it was indeed believed for a long time that a similar small-

step Brownian motion is the primary mode of displacement of water molecules inthe liquid too. However, recent studies have shown that the situation can be quitedifferent. In addition to small-amplitude motion, a water molecule often rotates bylarge-amplitude jumps! A part of this chapter is devoted to telling the story ofanomalous water motion. However, even for large-amplitude motion surprisinglythe Einstein relation between diffusion and viscosity remains valid.

3.2 Timescales of translational and rotational motion

Translational and rotational diffusion coefcient of a molecule in a liquid provides aquantitative measure of the dynamic timescales in the liquid. These coefcients arerelated to viscosity by the StokesEinstein [2] and the DebyeStokesEinstein rela-tion [3], respectively. Using the denition of diffusion coefcient in terms of mean-square displacement [2] and the StokesEinstein relation, we can estimate the timeneeded by a water molecule to translate a distance equal to its molecular diameter

trans 2

6D3:1

Putting the value of molecular diameter ( = 2.75) and diffusion coefcient (D =2.5 105 cm2/s) of water, one gets an estimate of timescale trans as ~5 ps (psdenotes picosecond and 1 ps = 1012 s). That is, a water molecule in the liquid statemoves one molecular diameter in 5 ps. This is quite fast!

28 Dynamics of water: molecular motions and hydrogen-bond-breaking kinetics

(a)

2

10.5 0 0.5 1 1.5 2 2.5

3 10

1

32

45

6

01

3456789

10

(b)

X-axis

Y-axis

Z-axis

Figure 3.1. (a) Simulation box of argon atoms (interacting with each other via theLennardJones potential) extracted from their trajectories. The position of an argonatom is depicted as it executes its natural motion in the liquid state at temp = 183 K.(b) This shows the trajectory of one tagged argon atom.

3.2 Timescales of translational and rotational motion 29

We now discuss the speed of rotational motion of a single water molecule. At25C, the time constant of rotation of water is 2.5 ps. This is the time that a watermolecule takes to forget its initial state of rotation (determined by the angle it makeswith a laboratory frame). That is, water rotates also very quickly!As we strive here to understand motion of water molecules at a microscopic level,

we need to use a certain formalism developed in the area of statistical mechanics.This formalism is broadly known as time correlation formalism (TCF). While manyspecialized texts exist in the literature on this important topic, we have included inAppendix 3.A a brief discussion on the time correlation functions necessary tounderstand the dynamics of rotation of a molecule in liquid.As discussed in Appendix 3.A, we employ two kinds of time correlation func-

tions to describe rotational motion. They employ single particle and collectivequantities. While they can be quite different in some cases, usually they bothmeasure similar dynamics. Most of the experiments measure the collective responseof the liquid. It is, however, important to know the difference.In the case of rotational motion of water molecules, the dynamic quantity is

naturally the angle that the water molecule makes with a coordinate axis, usuallywith the z-axis in a space or laboratory xed frame. However, one cannot directlymeasure the angle. Instead, one measures the cosine of the angle because in experi-ments one sets a direction or axis by an external means, such as the electric eld of alight. Light, being an electromagnetic wave, interacts with the dipole moment or thepolarizability of the water molecule. Thus, the light incident on the medium at timet = 0 serves two purposes. First, it creates a direction or reference of measurement ofrotation. Second, it creates a disturbance or perturbation in the system. That is, theelectric eld of the light forces a rotation of some of the water molecules of the liquid.After the light passes through the medium, the disturbed water molecules start theirnatural Brownianmotion and rotate back to equilibrium. This process of restoring theisotropy can be measured optically by different techniques. One then constructs thetime correlation function and obtains the rate of rotation.At the present time one can measure fast rotation by using ultrafast laser pulses.

Laser pulses are now available with width less than even 1 femtosecond (fs) where1 fs = 1015 s. However, there are many technical difculties in using short pulses.But reliable experiments can be done with pulses of the order of 100 fs or so. As aresult one can measure the fast rotation of water.

3.3 Jump reorientation motion in water

Now we turn to a detailed discussion of rotational motion in water. As alreadymentioned, the Debye rotational diffusive model was initially widely employed todescribe water reorientation. As explained above, it describes the reorientation as an


angular Brownian motion, that is, a sequence of uncorrelated small-amplitude,angular steps [3]. Such a rotational Brownian motion picture is not a priori implau-sible if one considers that a water molecule interacts strongly with its neighbors viahydrogen bonds and that water rst has to break at least one H bond to reorient; theresulting dangling OH then performs a random search for a new H-bond acceptor,during which it reorients.Unfortunately, even sophisticated experiments cannot unravel the detailed micro-

scopic nature or mechanism of the rotational motion of a small molecule in theliquid state. Fortunately, however, one can examine this microscopic aspect of thereorientation of water by employing computer simulations. This method allows oneto tag individual water molecules and follow their rotational and translationalmotion over a period of time by solving Newtons equations of motion. Oneneeds to follow both the orientational and the translational (that is, the linear)motion of the water molecules. When plotted against time, the motion of a moleculeover time is called a time trajectory.Inspection of many such trajectories for the time-dependent orientation of the

individual water O-H bond shows that water molecules under normal conditions oftemperature and pressure move mainly by large-amplitude jumps [4,5]. This isshown in Figure 3.2. There is of course a constant motion of water molecules bysmall steps, but superimposed on such continuous motion are these large-amplitudejumps that are absent in most liquids. This is a relatively new insight, developedprimarily by Damien Laage and James Hynes, in the year 2006 [5]!

Figure 3.2. Fluctuation of the direction of the dipole moment vector of a taggedwater molecule in the course of its motion through the liquid. The places wherelarge-scale uctuations occur are indicated by arrows. Adapted with permissionfrom J. Phys. Chem. B., 112 (2008), 14230. Copyright (2008) American ChemicalSociety.

3.3 Jump reorientation motion in water 31

Laage and Hynes carried out detailed analysis and found the mechanism of suchjumps. As stated earlier, in order to reorient the tagged water molecule needs tobreak at least one hydrogen bond. After reorientation, this water molecule againforms an H bond to compensate for the loss of energy. Thus the mechanism of thereorientation of a water OH bond is a natural dynamic process where the averagenumber of hydrogen bonds is locally conserved and is a simple consequence of thetrading of H-bond acceptors receiving an H bond from the tagged water OH.Laage and Hynes analyzed molecular dynamics trajectories and recorded the

rotational dynamics of a water O*H* that was initially H-bonded to a water oxygenOa but became H-bonded to a different oxygen Ob (see Figure 3.3). For each of theseswitching events, they examined the sequence preceding it and the sequencefollowing it, as long as no other H-bond exchanges occurred. They monitored theoxygenoxygen distances, RO*O

a and RO*Ob, together with the angle between

the projection of the O*H* vector on the O*OaOb plane and the OaO*Ob anglebisector (Figure 3.3). When the bond breaks and re-forms, the angle rotates. It israther easy to understand the basics of the process. When = 0, then H* isequidistant from Oa and Ob, and that describes the transition state of the switchingevents.Figure 3.4 shows the exchange of distances between the three oxygen atoms

during the hydrogen-bond exchange process. The distance between Q* and Oa

undergoes a rather sharp increase while that between O* and Ob undergoes a sharpdecrease [5].

Roaob

Ro*ob

ob

Ro*ob

H*

O*

Figure 3.3. We show the key microscopic quantities used to determine the jumpreorientation motion in water. Notations O*, Oa, and Ob are dened in the text. TheHB involving hydrogen atom H* and oxygen atoms O* and Oa breaks (in theexample) and gets replaced by the one between O* and Ob. Adapted withpermission from J. Phys. Chem. B., 112 (2008), 1423014242. Copyright (2008)American Chemical Society.


3.4 Effects of temperature on water motion

To be historically fair, other people did observe the existence of jump motions in therotation of water molecules in the liquid state but detailed analysis of the dynamicsof an individual event was not carried out before. Given that perspective, the LaageHynes mechanism of water rotation by large-amplitude jumps is indeed a departurefrom conventional and prevailing wisdom that water rotation is Brownian; that is, itoccurs differently in water from in other liquids where motion by small stepsdominates. Experimental verication of the jump diffusion model came from abeautiful study of the temperature-dependent rate of water rotation. However, boththe experiments and the interpretation of results are quite involved. We shall discussthe results as simply as possible.Investigation of the effect of temperatures was carried out by Fayer and

co-workers of Stanford University by studying the time variation of the frequencyof the bond between oxygen and deuterium atoms (that is, the OD bond) of a HODmolecule [6]. HOD is water with one of the hydrogen atoms of water replaced by adeuterium atom. The advantage of this replacement is that now the frequency of theOD bo

water in biological and chemical process

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