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Non-Equilibrium Thermodynamicsof Heterogeneous Systems

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Series on Advances in Statistical Mechanics -- Volume 16

Non-Equilibrium Thermodynamicsof Heterogeneous Systems

SIGNE KJELSTRUP DICK BEDEAUX

Norwegian University of Science and Technology

World ScientificNEW JERSEY . LONDON . SINGAPORE . BEIJING . SHANGHAI . HOND KONG . TAIPEI . CHENNAI

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Series on Advances in Statistical Mechanics – Vol. 16NON-EQUILIBRIUM THERMODYNAMICS OF HETEROGENEOUS SYSTEMS

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We dedicate this book to our teacher and friend Peter Mazur

v

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Preface

What the book is about

This book describes transport through complex, heterogeneous media.There are large coupling effects between transports of heat, mass, chargeand chemical reactions in such media, and it is important to know howone should properly integrate across heterogeneous systems where differentphases are in contact. Transports perpendicular to planar interfaces aretreated.

How we were inspired to write it

Waldmann [1] was the first to point out that an analysis of boundary con-ditions should be incorporated in the framework of non-equilibrium ther-modynamics. He did not include excess densities to describe the propertiesof the surface, however. In the 1970s and 1980s Bedeaux, Albano andMazur [2–4] extended his work and laid the theoretical foundation for thedescription of non-equilibrium processes, including a full description of thethermodynamic properties of the surface. The work of Bedeaux, Albanoand Mazur addressed not only transports into and through a surface, butalso transports along surfaces as well as the motion of the surface itself.Their methods were inspired by concurrent work on the optical propertiesof surfaces by Bedeaux and Vlieger [5, 6], using generalized functions forthe description of surface properties. The present work takes advantage ofthese methodologies.

Who it is meant for

The purpose of this book is to encourage the use of non-equilibriumthermodynamics to describe transport in complex, heterogeneous media.The book is written for a graduate level course for physicists, physicalchemists, chemical or mechanical engineers. The book requires knowledgeof basic thermodynamics corresponding to that given by Atkins, Physical

vii

viii Preface

Chemistry (Oxford University Press), or Moran and Shapiro, Fundamentalsof Engineering Thermodynamics (John Wiley & Sons).

Acknowledgments

A course on non-equilibrium thermodynamics has been taught at the Nor-wegian University of Science and Technology for many years. We aregrateful to the graduate students who worked with us on related subjectsand commented on preliminary versions of the text. In alphabetical order,we would like to mention Odne Stokke Burheim, Belinda Flem, AndreasGrimstvedt, Kirill Glavatskiy, Ellen Marie Hansen, Torleif Holt, IsabellaInzoli, Eivind Johannessen, Einar Eng Johnsen, Gelein de Koeijer, LarsNummedal, Anne-Kristine Meland, Steffen Møller-Holst, Magnar Ottøy,Audun Røsjorde, Erik Sauar, Kristin Syverud, Preben Joakim Svela Vie,Jing Xu and Anita Zvolinschi. Discussions and collaborations with col-leagues Bjørn Hafskjold, Jean-Marc Simon, Fernando Bresme and EdgarBlokhuis were appreciated.

During our peaceful and inspiring stay in Kyoto University in 2000,a large step forward was made. We are grateful to our host Prof. Ya-suhiko Ito, and to coworkers, Profs. Yoichi Tomii, Yasuhiro Fukunaka, Drs.Toshiyuki Nohira and Koji Amezawa. The next large step was made inbeautiful Barcelona in 2004, with our considerate host, Prof. Miguel Rubiof University of Barcelona.

We thank the Research Council of Norway for financial support overmany years. Marian Palcut contributed the cover image, while TharaldTharaldsen, Jaques van der Ploeg and Michel Uiterwijk contributed manyfigures.

Signe Kjelstrup∗ and Dick Bedeaux March 2008

Address for correspondence:

Department of Chemistry,Norwegian University of Science and Technology,NO-7491 Trondheim, Norway

∗The author changed her name from Ratkje to Kjelstrup in 1996.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Scope 1

1.1 What is non-equilibrium thermodynamics? . . . . . . . . . 11.2 Non-equilibrium thermodynamics in the context of

other theories . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 The purpose of this book . . . . . . . . . . . . . . . . . . . 4

2 Why Non-Equilibrium Thermodynamics? 7

2.1 Simple flux equations . . . . . . . . . . . . . . . . . . . . . 82.2 Flux equations with coupling terms . . . . . . . . . . . . . 92.3 Experimental designs and controls . . . . . . . . . . . . . . 112.4 Entropy production, work and lost work . . . . . . . . . . 122.5 Consistent thermodynamic models . . . . . . . . . . . . . . 14

3 Thermodynamic Relations for Heterogeneous Systems 17

3.1 Two homogeneous phases separated by a surface in globalequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 The contact line in global equilibrium . . . . . . . . . . . . 223.3 Defining thermodynamic variables for the surface . . . . . 233.4 Local thermodynamic identities . . . . . . . . . . . . . . . 293.5 Defining local equilibrium . . . . . . . . . . . . . . . . . . 323.A Appendix: Partial molar properties . . . . . . . . . . . . . 35

3.A.1 Homogeneous phases . . . . . . . . . . . . . . . . . 363.A.2 The surface . . . . . . . . . . . . . . . . . . . . . . 383.A.3 The standard state . . . . . . . . . . . . . . . . . . 40

Part A: General Theory 45

4 The Entropy Production for a Homogeneous Phase 47

4.1 Balance equations . . . . . . . . . . . . . . . . . . . . . . . 494.2 The entropy production . . . . . . . . . . . . . . . . . . . . 51

ix

x Contents

4.2.1 Why one should not use the dissipation function . 564.2.2 States with minimum entropy production . . . . . 57

4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4 Frames of reference for fluxes in homogeneous systems . . 64

4.4.1 Definitions of frames of reference . . . . . . . . . . 644.4.2 Transformations between the frames of reference . 66

4.A Appendix: The first law and the heat flux . . . . . . . . . 67

5 The Excess Entropy Production for the Surface 73

5.1 The discrete nature of the surface . . . . . . . . . . . . . . 745.2 The behavior of the electric fields and potential through the

surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3 Balance equations . . . . . . . . . . . . . . . . . . . . . . . 775.4 The excess entropy production . . . . . . . . . . . . . . . . 79

5.4.1 Reversible processes at the interface and the Nernstequation . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.2 The surface potential jump at the hydrogenelectrode . . . . . . . . . . . . . . . . . . . . . . . 86

5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6 The Excess Entropy Production for a Three Phase

Contact Line 91

6.1 The discrete nature of the contact line . . . . . . . . . . . 926.2 Balance equations . . . . . . . . . . . . . . . . . . . . . . . 946.3 The excess entropy production . . . . . . . . . . . . . . . . 956.4 Stationary states . . . . . . . . . . . . . . . . . . . . . . . 966.5 Concluding comment . . . . . . . . . . . . . . . . . . . . . 97

7 Flux Equations and Onsager Relations 99

7.1 Flux-force relations . . . . . . . . . . . . . . . . . . . . . . 997.2 Onsager’s reciprocal relations . . . . . . . . . . . . . . . . 1007.3 Relaxation to equilibrium. Consequences of violating

Onsager relations . . . . . . . . . . . . . . . . . . . . . . . 1047.4 Force-flux relations . . . . . . . . . . . . . . . . . . . . . . 1057.5 Coefficient bounds . . . . . . . . . . . . . . . . . . . . . . 1067.6 The Curie principle applied to surfaces and contact lines . 108

8 Transport of Heat and Mass 111

8.1 The homogeneous phases . . . . . . . . . . . . . . . . . . . 1128.2 Coefficient values for homogeneous phases . . . . . . . . . 1148.3 The surface . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.3.1 Heats of transfer for the surface . . . . . . . . . . 119

Contents xi

8.4 Solution for the heterogeneous system . . . . . . . . . . . . 1228.5 Scaling relations between surface and bulk resistivities . . 125

9 Transport of Heat and Charge 127

9.1 The homogeneous phases . . . . . . . . . . . . . . . . . . . 1289.2 The surface . . . . . . . . . . . . . . . . . . . . . . . . . . 1309.3 Thermoelectric coolers . . . . . . . . . . . . . . . . . . . . 1329.4 Thermoelectric generators . . . . . . . . . . . . . . . . . . 1339.5 Solution for the heterogeneous system . . . . . . . . . . . . 135

10 Transport of Mass and Charge 139

10.1 The electrolyte . . . . . . . . . . . . . . . . . . . . . . . . . 14010.2 The electrode surfaces . . . . . . . . . . . . . . . . . . . . 14310.3 Solution for the heterogeneous system . . . . . . . . . . . . 14610.4 A salt power plant . . . . . . . . . . . . . . . . . . . . . . 14710.5 Electric power from volume flow . . . . . . . . . . . . . . . 14810.6 Ionic mobility model for the electrolyte . . . . . . . . . . . 15010.7 Ionic and electronic model for the surface . . . . . . . . . . 154

Part B: Applications 155

11 Evaporation and Condensation 157

11.1 Evaporation and condensation in a pure fluid . . . . . . . 15811.1.1 The entropy production and the flux equations . . 15811.1.2 Interface resistivities from kinetic theory . . . . . 165

11.2 The sign of the heats of transfer of the surface . . . . . . . 16711.3 Coefficients from molecular dynamics simulations . . . . . 16911.4 Evaporation and condensation in a two-component fluid . 176

11.4.1 The entropy production and the flux equations . . 17611.4.2 Interface resistivities from kinetic theory . . . . . 179

12 Multi-Component Heat and Mass Diffusion 183

12.1 The homogeneous phases . . . . . . . . . . . . . . . . . . . 18412.2 The Maxwell–Stefan equations for multi-component

diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18612.3 The Maxwell–Stefan equations for the surface . . . . . . . 18812.4 Multi-component diffusion . . . . . . . . . . . . . . . . . . 192

12.4.1 Prigogine’s theorem . . . . . . . . . . . . . . . . . 19212.4.2 Diffusion in the solvent frame of reference . . . . . 19312.4.3 Other frames of reference . . . . . . . . . . . . . . 19512.4.4 An example: Kinetic demixing of oxides . . . . . . 200

xii Contents

12.5 A relation between the heats of transfer andthe enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . 202

13 A Nonisothermal Concentration Cell 205

13.1 The homogeneous phases . . . . . . . . . . . . . . . . . . . 20713.1.1 Entropy production and flux equations for

the anode . . . . . . . . . . . . . . . . . . . . . . . 20713.1.2 Position dependent transport coefficients . . . . . 21013.1.3 The profiles of the homogeneous anode . . . . . . 21113.1.4 Contributions from the cathode . . . . . . . . . . 21213.1.5 The electrolyte contribution . . . . . . . . . . . . 213

13.2 Surface contributions . . . . . . . . . . . . . . . . . . . . . 21413.2.1 The anode surface . . . . . . . . . . . . . . . . . . 21413.2.2 The cathode surface . . . . . . . . . . . . . . . . 217

13.3 The thermoelectric potential . . . . . . . . . . . . . . . . 218

14 The Transported Entropy 221

14.1 The Seebeck coefficient of cell a . . . . . . . . . . . . . . . 22214.2 The transported entropy of Pb2+ in cell a . . . . . . . . . 22614.3 The transported entropy of the cation in cell b . . . . . . . 22714.4 The transported entropy of the ions cell c . . . . . . . . . 22814.5 Transformation properties . . . . . . . . . . . . . . . . . . 23014.6 Concluding comments . . . . . . . . . . . . . . . . . . . . . 232

15 Adiabatic Electrode Reactions 235

15.1 The homogeneous phases . . . . . . . . . . . . . . . . . . . 23615.1.1 The silver phases . . . . . . . . . . . . . . . . . . . 23615.1.2 The silver chloride phases . . . . . . . . . . . . . . 23615.1.3 The electrolyte . . . . . . . . . . . . . . . . . . . . 237

15.2 The interfaces . . . . . . . . . . . . . . . . . . . . . . . . . 23715.2.1 The silver-silver chloride interfaces . . . . . . . . . 23715.2.2 The silver chloride-electrolyte interfaces . . . . . . 239

15.3 Temperature and electric potential profiles . . . . . . . . . 240

16 The Liquid Junction Potential 249

16.1 The flux equations for the electrolyte . . . . . . . . . . . . 25016.2 The liquid junction potential . . . . . . . . . . . . . . . . . 25316.3 Liquid junction potential calculations compared . . . . . . 25516.4 Concluding comments . . . . . . . . . . . . . . . . . . . . . 258

Contents xiii

17 The Formation Cell 261

17.1 The isothermal cell . . . . . . . . . . . . . . . . . . . . . . 26317.1.1 The electromotive force . . . . . . . . . . . . . . . 26317.1.2 The transference coefficient of the salt in

the electrolyte . . . . . . . . . . . . . . . . . . . . 26317.1.3 An electrolyte with a salt concentration gradient . 26517.1.4 The Planck potential derived from ionic fluxes

and forces . . . . . . . . . . . . . . . . . . . . . . . 26717.2 A non-isothermal cell with a non-uniform electrolyte . . . 268

17.2.1 The homogeneous anode phase . . . . . . . . . . . 26917.2.2 The electrolyte . . . . . . . . . . . . . . . . . . . . 27017.2.3 The surface of the anode . . . . . . . . . . . . . . 27217.2.4 The homogeneous phases and the surface of

the cathode . . . . . . . . . . . . . . . . . . . . . . 27317.2.5 The cell potential . . . . . . . . . . . . . . . . . . 275

17.3 Concluding comments . . . . . . . . . . . . . . . . . . . . . 275

18 Power from Regular and Thermal Osmosis 277

18.1 The potential work of a salt power plant . . . . . . . . . . 27718.2 The membrane as a barrier to transport of heat

and mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27918.3 Membrane transport of heat and mass . . . . . . . . . . . 28118.4 Osmosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28318.5 Thermal osmosis . . . . . . . . . . . . . . . . . . . . . . . . 285

19 Modeling the Polymer Electrolyte Fuel Cell 289

19.1 The potential work of a fuel cell . . . . . . . . . . . . . . . 29019.2 The cell and its five subsystems . . . . . . . . . . . . . . . 29119.3 The electrode backing and the membrane . . . . . . . . . . 293

19.3.1 The entropy production in the homogeneousphases . . . . . . . . . . . . . . . . . . . . . . . . . 293

19.3.2 The anode backing . . . . . . . . . . . . . . . . . . 29519.3.3 The membrane . . . . . . . . . . . . . . . . . . . . 29819.3.4 The cathode backing . . . . . . . . . . . . . . . . 300

19.4 The electrode surfaces . . . . . . . . . . . . . . . . . . . . 30119.4.1 The anode catalyst surface . . . . . . . . . . . . . 30419.4.2 The cathode catalyst surface . . . . . . . . . . . . 306

19.5 A model in agreement with the second law . . . . . . . . . 30719.6 Concluding comments . . . . . . . . . . . . . . . . . . . . . 310

xiv Contents

20 Measuring Membrane Transport Properties 311

20.1 The membrane in equilibrium with electrolyte solutions . . 31220.2 The membrane resistivity . . . . . . . . . . . . . . . . . . . 31220.3 Ionic transport numbers . . . . . . . . . . . . . . . . . . . 31620.4 The transference number of water and the

water permeability . . . . . . . . . . . . . . . . . . . . . . 31920.5 The Seebeck coefficient . . . . . . . . . . . . . . . . . . . . 32220.6 Interdiffusion coefficients . . . . . . . . . . . . . . . . . . . 323

21 The Impedance of an Electrode Surface 327

21.1 The hydrogen electrode. Mass balances . . . . . . . . . . . 32821.2 The oscillating field . . . . . . . . . . . . . . . . . . . . . . 33121.3 Reaction Gibbs energies . . . . . . . . . . . . . . . . . . . 33221.4 The electrode surface impedance . . . . . . . . . . . . . . . 332

21.4.1 The adsorption-diffusion layer in front ofthe catalyst . . . . . . . . . . . . . . . . . . . . . . 332

21.4.2 The charge transfer reaction . . . . . . . . . . . . 33621.4.3 The impedance spectrum . . . . . . . . . . . . . . 337

21.5 A test of the model . . . . . . . . . . . . . . . . . . . . . . 33821.6 The reaction overpotential . . . . . . . . . . . . . . . . . . 339

22 Non-Equilibrium Molecular Dynamics Simulations 341

22.1 The system . . . . . . . . . . . . . . . . . . . . . . . . . . 34422.1.1 The interaction potential . . . . . . . . . . . . . . 346

22.2 Calculation techniques . . . . . . . . . . . . . . . . . . . . 34722.3 Verifying the assumption of local equilibrium . . . . . . . . 351

22.3.1 Local equilibrium in a homogeneous binarymixture . . . . . . . . . . . . . . . . . . . . . . . . 351

22.3.2 Local equilibrium in a gas-liquid interface . . . . . 35322.4 Verifications of the Onsager relations . . . . . . . . . . . . 356

22.4.1 A homogeneous binary mixture . . . . . . . . . . . 35622.4.2 A gas-liquid interface . . . . . . . . . . . . . . . . 358

22.5 Linearity of the flux-force relations . . . . . . . . . . . . . 35922.6 Molecular mechanisms . . . . . . . . . . . . . . . . . . . . 359

23 The Non-Equilibrium Two-Phase van der Waals Model 361

23.1 Van der Waals equation of states . . . . . . . . . . . . . . 36323.2 Van der Waals square gradient model for the interfacial

region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36623.3 Balance equations . . . . . . . . . . . . . . . . . . . . . . . 36923.4 The entropy production . . . . . . . . . . . . . . . . . . . . 37123.5 Flux equations . . . . . . . . . . . . . . . . . . . . . . . . . 372

Contents xv

23.6 A numerical solution method . . . . . . . . . . . . . . . . . 37323.7 Procedure for extrapolation of bulk densities and fluxes . . 37623.8 Defining excess densities . . . . . . . . . . . . . . . . . . . 37823.9 Thermodynamic properties of Gibbs’ surface . . . . . . . . 37923.10 An autonomous surface . . . . . . . . . . . . . . . . . . . . 38023.11 Excess densities depend on the choice of dividing surface . 384

23.11.1 Properties of dividing surfaces . . . . . . . . . . . 38423.11.2 Surface excess densities for two dividing

surfaces . . . . . . . . . . . . . . . . . . . . . . . . 38523.11.3 The surface temperature from excess

density differences . . . . . . . . . . . . . . . . . . 38623.12 The entropy balance and the excess entropy production . . 38823.13 Resistivities to heat and mass transfer . . . . . . . . . . . 39023.14 Concluding comments . . . . . . . . . . . . . . . . . . . . . 392

References 393

Symbol Lists 415

Index 423

About the Authors 433

Chapter 1

Scope

The aim of this book is to present a systematic theory of transport for het-erogeneous systems. The theory is an extension of non-equilibrium thermo-dynamics for transport in homogeneous phases, a field that was establishedin 1931 and developed during the nineteen forties and fifties. The founda-tion to describe transports across surfaces in a systematic way was laid inthe nineteen eighties. In this chapter, we put the theory in context and giveperspectives on its application.

1.1 What is non-equilibrium thermodynamics?

Non-equilibrium thermodynamics describes transport processes in systemsthat are out of global equilibrium. The field resulted from the work ofmany scientists with the overriding aim to find a more useful formulationof the second law of thermodynamics in such systems. The effort started in1856 with Thomson’s studies of thermoelectricity [7]. Onsager however, iscounted as the founder of the field with his papers from 1931 [8,9], see alsohis collected works [10], because he put earlier research by Thomson, Boltz-mann, Nernst, Duhem, Jauman and Einstein into the proper perspective.Onsager was given the Nobel prize in chemistry in 1968 for these works.

In non-equilibrium thermodynamics, the second law is reformulated interms of the local entropy production in the system, σ, using the assumptionof local equilibrium (see Sec. 3.5). The entropy production is given by theproduct sum of so-called conjugate fluxes, Ji, and forces, Xi, in the system.The second law becomes

σ =∑

i

JiXi ≥ 0 (1.1)

1

2 Chapter 1. Scope

Each flux is a linear combination of all forces,

Ji =∑

j

LijXj (1.2)

The reciprocal relations

Lji = Lij (1.3)

were proven for independent forces and fluxes, cf. Chapter 7 by Onsager[8, 9]. They now bear his name. All coefficients are essential, as explainedin Chapter 2. In order to use non-equilibrium thermodynamics, one firsthas to identify the complete set of extensive, independent variables, Ai. Weshall do that in Chapters 4–6 for homogeneous phases, surfaces and threephase contact lines respectively. The conjugate fluxes and forces are

Ji = dAi/dt and Xi = ∂S/∂Ai (1.4)

Here t is the time and S is the entropy of the system. Some authorshave erroneously stated that any set of fluxes and forces that fulfil (1.1)also obeys (1.3). This is not correct. We also need Eq. (1.4). Equations(1.1) to (1.4) contain all information on the non-equilibrium behavior ofthe system.

Following Onsager, a systematic theory of non-equilibrium processes wasset up in the nineteen forties by Meixner [11–14] and Prigogine [15]. Theyfound the rate of entropy production for a number of physical problems.Prigogine received the Nobel prize in 1977 for his work on the structure ofsystems that are not in equilibrium (dissipative structures), and Mitchellthe year after for his application of the driving force concept to transportprocesses in biology [16].

Short and essential books were written early by Denbigh [17] and Pri-gogine [18]. The most general description of classical non-equilibrium ther-modynamics is still the 1962 monograph of de Groot and Mazur [19],reprinted in 1985 [20]. Haase’s book [21], also reprinted [22], contains manyexperimental results for systems in temperature gradients. Katchalsky andCurran developed the theory for biological systems [23]. Their analysis wascarried further by Caplan and Essig [24], and Westerhoff and van Dam [25].Førland and coworkers’ book gave various applications in electrochemistry,in biology and geology [26]. This book, which presents the theory in a waysuitable for chemists, has also been reprinted [27]. A simple introductionto non-equilibrium thermodynamics for engineers is given by Kjelstrup,Bedeaux and Johannessen [28, 29].

1.1. What is non-equilibrium thermodynamics? 3

Non-equilibrium thermodynamics is all the time being applied innew directions. Fitts showed how to include viscous phenomena [30].Kuiken [31] gave a general treatment of multicomponent diffusion andrheology of colloidal and other systems. Kinetic theory has been cen-tral for studies of evaporation [32–35], and we shall make a link be-tween this theory and non-equilibrium thermodynamics [36–40] (seeChapter 11). A detailed discussion of the link between kinetic the-ory and non-equilibrium thermodynamics was made by Roldughin andZhdanov [41]. The relation to the Maxwell–Stefan equations are givenin Chapter 12, following Kuiken [31] and Krishna and Wesselingh [42].Bedeaux and Mazur [43] extended non-equilibrium thermodynamics toquantum mechanical systems. All these efforts have together broadenedthe scope of non-equilibrium thermodynamics, so that it now appearsas a versatile and strong tool for the general description of transportphenomena.

All the applications of non-equilibrium thermodynamics mentionedabove use the linear relation between the fluxes and the forces given inEq. (1.2). For chemical reactions the rate is given by the law of mass ac-tion, which is a nonlinear function of the driving force. This excludes alarge class of important phenomena from a systematic treatment in termsof classical non-equilibrium thermodynamics. Much effort has thereforebeen devoted to solve this problem, see [44] for an overview. A promisingeffort seems to be to expand the variable set by including variables that arerelevant for the mesoscopic level. The description on the mesoscopic levelcan afterwards be integrated to the macroscopic level. This was first doneby Prigogine and Mazur, see the monograph by de Groot and Mazur [20],but Rubi and coworkers have pioneered the effort since then [45–47]. Inthis manner it has been possible to describe activated processes like nucle-ation [48]. Results compatible with results from kinetic theory [49] werederived [50], and a common thermodynamic basis was given to the Nernstand Butler–Volmer equations [51]. The systematic theory was also ex-tended to active transport in biology [52, 53] and to describe single RNAunfolding experiments [54]. This theoretical branch, called mesoscopic non-equilibrium thermodynamics, is however outside the scope of the presentbook.

Newer books on equilibrium thermodynamics or statistical thermody-namics often include chapters on non-equilibrium thermodynamics, seee.g. [55]. In 1998 Kondepudi and Prigogine [56] presented an integratedapproach to equilibrium and non-equilibrium thermodynamics. Öttingeraddressed the non-linear regime in his book [57]. An excellent overviewof the various extensions of non-equilibrium thermodynamic was given byMuschik et al. [44].

4 Chapter 1. Scope

1.2 Non-equilibrium thermodynamics in the context of

other theories

Non-equilibrium thermodynamics is a theory describing transport on amacroscopic level. Let us compare theories of transport on the particlelevel, the mesoscopic level, and the macroscopic level. Consider first equi-librium systems on all three levels, see Fig. 1.1. Classical mechanics andquantum mechanics describe the system on the particle level. Statisticalmechanics provides the link from the particle level to both the mesoscopiclevel and the macroscopic level (indicated by arrows in the figure). Themesoscopic level describes the system on an intermediate time and lengthscale. Equilibrium correlation functions are calculated at this level. Themacroscopic level is described by equilibrium thermodynamics, see Chapter3 in this book. Many of these macroscopic properties can be calculated asintegrals over equilibrium correlation functions. The same quantities canalso be calculated using ensemble theory.

The description of non-equilibrium systems can be illustrated similarly,see Fig. 1.2. On the particle level, the time-dependence is again describedby classical and quantum mechanics, while non-equilibrium statistical me-chanics provides the link between this level and the mesoscopic and macro-scopic levels. Theories for the particle- and mesoscopic level make it pos-sible to calculate macroscopic transport coefficients. They use expressionsfor transport coefficients as integrals over equilibrium correlation functionsof the fluxes, the Kubo relations. Transport coefficients can also be cal-culated using non-equilibrium statistical mechanics. The macroscopic levelis now described by non-equilibrium thermodynamics. The present bookis devoted to this description on the general level (Chapters 7–10) and inmany applications (Chapters 11–23).

1.3 The purpose of this book

Most references mentioned above concern transport in homogeneous sys-tems. It is known, however, that much entropy is produced at surfacesbetween homogeneous phases [41, 58], and the nature of the equations oftransport for this region is also unexplored in many ways.

The aim of the present book is to further develop the theory of non-equilibrium thermodynamics to transport in heterogeneous systems. Thefoundation for the work was laid by Bedeaux, Albano and Mazur [2,3], seealso Bertrand and Prud’homme [59, 60] and a review by Bedeaux [4]. Wegave a short overview of the objectives and the methods used in [61].

Heterogeneous systems distinguish themselves from homogeneous sys-tems by their inclusion of interfaces, or phase boundaries. The interface

1.3. The purpose of this book 5

Classical mechanics Statistical mechanics

Pair correlation functionMaxwell distribution

Structure factor

Equation of state

Ensemble theoryGibbs

S = S(U,V,N)

Particle level Particle level

Mesoscopic level Macroscopic level

BoltzmannNewton

Quantum mechanicsSchrodinger

Equilibrium correlationfunctions

Phase equilibria

ThermodynamicsIntegral

identities

Schrodinger:::

BS = k lnW

Figure 1.1 Theories for equilibrium systems on the particle-, mesoscopic- andmacroscopic level.

Classical mechanics

Particle level Particle level

Quantum mechanics

Kuborelations

Time dependent

statistical mechanics

Liouville equationLiouville von Neumann

equation

thermodynamics

..

Mesoscopic level Macroscopic level

Fokker Planck equationLangevin equation

Navier Stokes equations

Maxwell Stefan equations

Onsager relationsLinear flux force relations

correlation functionsKinetic theory

Master equation

Schrodinger

Newton

Boltzmann equation

Non equilibrium

Non equilibrium

Figure 1.2 Theories for non-equilibrium systems on the particle-, mesoscopic-and macroscopic level.

6 Chapter 1. Scope

can, according to Gibbs [62], be regarded as a separate two-dimensionalthermodynamic system within the larger three-dimensional system. Thisimplies the introduction of excess densities along the surface. The purposeof this book is to extend the work of Gibbs to non-equilibrium systems. Thisimplies the introduction of not only time dependent excess densities for thesurface but also of time dependent excess fluxes along the surface. In thismanner we shall describe transport of heat, mass and charge and chemicalreactions in heterogeneous systems.

Isotropic homogeneous systems have no coupling between vectorial phe-nomena (transport of heat, mass and charge) and scalar phenomena (chem-ical reactions). In these systems, the two classes can therefore be dealt withseparately [20]. In a two-dimensional surface, all transport processes in thedirection normal to the surface become scalar. This means that couplingcan occur between transport of heat, mass and charge in that direction, andchemical reactions. The coupling is typical for electrode surfaces, phasetransitions and membrane surfaces, and the coupling coefficients are oftenlarge. This coupling is special for heterogeneous systems, and is studied inmany examples throughout the book. The flux equations at the surfacesgive jumps in intensive variables, and define in this manner boundary condi-tions for integration through the surface. Such integrations can be found inseveral chapters. Electric potential changes shall be described by a changein the electromotive force, or in the electrochemical potential difference ofthe ion, which is reversible to the electrode pair [23, 27, 63].

With a short exception for the contact line (Chapter 6), we shall notconsider transport along the surface. While this is an extremely impor-tant phenomenon, it would be too much new material to cover. We fur-ther neglect viscous phenomena. We shall not deal with systems whereit is necessary to take fluxes along as variables [64]. We shall not intro-duce internal variables [45–47]. The transports considered in this bookare almost all in the direction normal to the surface, and are thereforeone-dimensional.

The book aims to make clear that non-equilibrium thermodynamics isa systematic theory that rests on sound assumptions. The assumption oflocal equilibrium is valid for systems in large gradients [65, 66], includingsurfaces [40, 67, 68], and does not depend on stationary state conditions,constant external forces, absence of electric polarization or isotropy, as hasbeen claimed, see for instance [22].

Chapter 2

Why Non-Equilibrium

Thermodynamics?

This chapter explains what the field of non-equilibrium thermodynamicsadds to the analysis of common scientific and engineering problems. Accu-rate and reliable flux equations can be obtained. Experiments can be welldefined. Knowledge of transport properties gives information on the sys-tem’s ability to convert energy.

The most common industrial and living systems have transport of heat,mass and charge, alone or in combination with a chemical reaction. Theprocess industry, the electrochemical industry, biological systems, as well aslaboratory experiments, all concern heterogeneous systems, which are notin global equilibrium. There are four major reasons why non-equilibriumthermodynamics is important for such systems. In the first place, the the-ory gives an accurate description of coupled transport processes. In thesecond place, a framework is obtained for the definition of experiments.In the third place; the theory quantifies not only the entropy that is pro-duced during transport, but also the work that is done and the lost work.Last but not least, this theory allows us to check that the thermody-namic equations we use to model our system are in agreement with thesecond law.

The aim of this book is to give a systematic description of transport inheterogeneous systems. Non-equilibrium thermodynamics has so far mostlybeen used in science [20,21,27]. Accurate expressions for the fluxes are nowrequired also in engineering [29, 42, 69]. In order to see immediately whatnon-equilibrium thermodynamics can add to the description of real systems,we compare simple flux equations to flux equations given by non-equilibriumthermodynamics in the following four sections.

7

8 Chapter 2. Why Non-Equilibrium Thermodynamics?

2.1 Simple flux equations

The simplest descriptions of heat, mass and charge transports are the equa-tions of Fourier, Fick and Ohm. Fourier’s law expresses the measurable heatflux in terms of the temperature gradient by:

J ′q = −λdT

dx(2.1)

where λ is the thermal conductivity, T is the absolute temperature, andthe direction of transport is along the x-axis. Fick’s law gives the flux ofone of the components in terms of the gradient of its concentration c:

J = −D dcdx

(2.2)

where D is the diffusion coefficient. Similarly, Ohm’s law gives the electriccurrent in terms of the gradient of the electric potential:

j = −κdφdx

(2.3)

where κ is the electrical conductivity, and φ is the electric potential.In a stationary state, there is no accumulation of internal energy, mass or

charge. This means that the heat, molar and electric fluxes are independentof position. The derivatives of the above equations with respect to x arethen zero:

d

dxλdT

dx= 0 (2.4)

d

dxDdc

dx= 0 (2.5)

d

dxκdφ

dx= 0 (2.6)

These equations can be used to calculate the temperature, concentrationand electric potential as a function of the position, when their values on theboundaries of the system and λ, D and κ are known. Such a calculation isillustrated by the following exercise.

Exercise 2.1.1. Calculate the temperature as a function of positionbetween two wall’s at a distance of 10 cm which are kept at constanttemperatures of 5 and 25 oC, assuming that the thermal conductivity isconstant.

2.2. Flux equations with coupling terms 9

• Solution: According to Eq. (2.4), d2T/dx2 = 0. The general solutionof this equation is T (x) = a+ bx. The constants a and b follow fromthe boundary condition. We have T (0) = 5oC and T (10) = 25oC. Itfollows that T (x) = (5 + 2x)oC.

Equations (2.1)–(2.3) describe a pure degradation of thermal, chemical andelectrical energy. In reality there is also conversion between the energyforms. For an efficient exploitation of energy resources, such a conversion isessential. This is captured in non-equilibrium thermodynamics by the so-called coupling coefficients. Mass transport occurs, for instance, not onlybecause dc/dx 6= 0, but also because dT/dx 6= 0 or dφ/dx 6= 0.

Chemical and mechanical engineers need theories of transport for in-creasingly complex systems with gradients in pressure, concentration andtemperature. Simple vectorial transport laws have long worked well in engi-neering, but there is now an increasing effort to be more precise. The booksby Taylor and Krishna [69], Kuiken [31] and by Cussler [70], which useMaxwell–Stefan’s formulation of the flux equations, are important books inthis context. A need for more accurate flux equations in modeling [42, 69]makes non-equilibrium thermodynamics a necessary tool.

2.2 Flux equations with coupling terms

Many natural and man-made processes are not adequately described bythe simple flux equations given above. There are, for instance, always largefluxes of mass and heat that accompany charge transport in batteries andelectrolysis cells. The resulting local cooling in electrolysis cells may lead tounwanted freezing of the electrolyte. Electrical energy is frequently used totransport mass in biological systems. Large temperature gradients acrossspace ships have been used to supply electric power to the ships. Saltconcentration differences between river water and sea water can be used togenerate electric power. Pure water can be generated from salt water byapplication of pressure gradients. In all these more or less randomly chosenexamples, one needs transport equations that describe coupling betweenvarious fluxes. The flux equations given above become too simple.

Non-equilibrium thermodynamics introduces coupling among fluxes.Coupling means that transport of mass will take place in a system notonly when the gradient in the chemical potential is different from zero, butalso when there are gradients in temperature or electric potential. Couplingbetween fluxes can describe the phenomena mentioned above.

For example, in a bulk system with transport of heat, mass, and electriccharge in the x-direction, we shall find that the linear relations (1.2) take


the form:

J ′q = Lqq

(d

dx

1

T

)+ Lqµ

(− 1T

dµTdx

)+ Lqφ

(− 1T

dφ

dx

)

J = Lµq

(d

dx

1

T

)+ Lµµ

(− 1T

dµTdx

)+ Lµφ

(− 1T

dφ

dx

)

j = Lφq

(d

dx

1

T

)+ Lφµ

(− 1T

dµTdx

)+ Lφφ

(− 1T

dφ

dx

)

(2.7)

where the forces of transport conjugate to the fluxes J ′q, J and j arethe thermal force d (1/T )/dx, the chemical force [− (1/T ) (dµT /dx)], andthe electrical force [− (1/T ) (dφ/dx)], respectively. The subscript T of thechemical potential, µT , indicates that the differential should be taken atconstant temperature. The electric field is given in terms of the electricpotential gradient by E = −dφ/dx.

The L-coefficients are so-called phenomenological coefficients, or On-sager coefficients for conductivity, as we shall call them. The diagonalOnsager coefficients can be related to λ, D, and κ. They are called maincoefficients. The off-diagonal L-coefficients describe the coupling betweenthe fluxes. They are called coupling coefficients. Another common nameis cross coefficients. According to Eq. (1.3), we have here three reciprocalrelations or Onsager relations for these coupling coefficients

Lµq = Lqµ, Lqφ = Lφq, Lφµ = Lµφ (2.8)

The Onsager relations simplify the system. Here, they reduce the numberof independent coefficients from nine to six. Coupling coefficients are smallin some cases, but large in others. We shall see in Chapter 7 that largecoupling coefficients may lead to a low entropy production. This is whythe coupling coefficients are so important in the design of industrial sys-tems, see Sec. 2.4. The relation Lqφ = Lφq is Thomson’s second relation,which dates from 1854, see Ref. [7]. Miller [71–74] and Spallek et al. [75]gave experimental evidence for the validity of the Onsager relations in elec-trolytes. Hafskjold and Kjelstrup Ratkje [65] and Xu and coworkers [67]proved their validity for heat and mass transfer, in homogeneous phases andat surfaces, using non-equilibrium molecular dynamics simulations, respec-tively. Annunziata et al. [76] took advantage of the Onsager relations toobtain thermodynamic data for an aqueous mixture of enzyme and salt nearthe solubility limit of the salt. Onsager’s proof of the reciprocal relationsshall be discussed in Chapter 7.

2.3. Experimental designs and controls 11

2.3 Experimental designs and controls

The importance of equilibrium thermodynamics for the design of exper-iments is well known. The definition of, say, a partial molar property,explains what shall be varied, and what shall be kept constant in the ex-perimental determination of the quantity. Also, there are relations be-tween thermodynamic variables that offer alternative measurements. Forinstance, the enthalpy of evaporation can be measured in a calorimeter, butit can also be determined by finding the vapor pressure of the evaporatinggas as a function of temperature.

Non-equilibrium thermodynamics is in a similar way, instrumental fordesign of experiments that aim to find transport properties, cf. Chapter 20.To see this, consider the exercise below.

Exercise 2.3.1. Find the electric current in terms of the electric field,E = −dφ/dx, using Eq. (2.7), in a system where there is no transport ofheat and mass, J ′q = J = 0.

• Solution: It follows from Eqs. (2.7a) and (2.7b) that

LqqT

d

dxT + Lqµ

dµTdx

= LqφE andLµqT

d

dxT + Lµµ

dµTdx

= LµφE

Solving these equations, using the Onsager relations, one finds

1

T

d

dxT =

LqφLµµ − LµφLqµLqqLµµ − L2qµ

E anddµTdx

=LqqLµφ − LµqLqφLqqLµµ − L2qµ

E

Substitution into Eq. (2.7c) then gives

j =E

T

[Lφφ − Lφq

LqφLµµ − LµφLqµLqqLµµ − L2qµ

− LφµLqqLµφ − LµqLqφLqqLµµ − L2qµ

]

This exercise shows that the electric conductivity, that one measures asthe ratio of measured values of j and E, is not necessarily given by Lφφ/Tas one might have thought, considering Eq. (2.7c). In the stationary state,the coupling coefficients lead to temperature and chemical potential gradi-ents, which again affect the electric current. Mathematically speaking, theelectric conductivity then becomes a combination of the conductivity of ahomogeneous conductor, found if one could measure with zero chemical po-tential and temperature gradients, and additional terms. The combinationof coefficients divided by the temperature, is the stationary state conductiv-ity. The stationary state electric conductivity, measured with J ′q = J = 0,is experimentally distinguishable from the Ohmic electric conductivity of


the homogeneous conductor, measured with dT/dx = dµT /dx = 0. Non-equilibrium thermodynamics helps define conditions that give well definedexperiments.

One important practical consequence of the Onsager relations is to offeralternative measurements for the same property. For instance, if it is dif-ficult to measure the coefficient Lqφ, one may rather measure Lφq [77–80].A valuable consistency check for measurements is provided if one measuresboth coupling coefficients in the relation. So, similar to the situation inequilibrium, also the non-equilibrium systems have possibilities for controlof internal consistency.

2.4 Entropy production, work and lost work

Non-equilibrium thermodynamics is probably the only method that can beused to assess how energy resources are exploited within a system. Thisis because the theory deals with energy conversion on the local level in asystem, i.e. at the electrode surface [81] or in a biological membrane [52].By integration to the system level, we obtain a link to exergy analysis [82].To see the relation between non-equilibrium thermodynamics and exergyanalysis, consider again the thermodynamic fluxes and forces, derived fromthe local entropy production

σ =∑

i

JiXi ≥ 0 (2.9)

The total entropy production is the integral of σ over the volume V of thesystem

dSirrdt

=

∫σdV (2.10)

The quantity dSirr/dt makes it possible to calculate the energy dissipatedas heat or the lost work in a system (see below).

In an industrial plant, the materials undergo certain transformationsin a time interval dt. Materials are taken in and are leaving the plant atthe conditions of the environment. The environment is a natural choice asframe of reference for the analysis. It has constant pressure, p0 (1 bar), andconstant temperature, T0 (for instance 298 K) and some average composi-tion that needs to be defined. The first law of thermodynamics gives theenergy change of the process per unit of time:

dU

dt=dQ

dt− p0

dV

dt+dW

dt(2.11)

Here dQ/dt is the rate that heat is delivered to the materials, p0dV/dt is thework that the system does per unit of time by volume expansion against the

2.4. Entropy production, work and lost work 13

dW/dt

dU/dt

T

dQ /dt

0 0

0

p

Figure 2.1 A schematic illustration of thermodynamic variables that areessential for the lost work in an industrial plant.

pressure of the environment, and dW/dt is the work done on the materialsper unit of time.

The minimum work needed to perform a process, is the least amount ofenergy that must be supplied, when at the conclusion of the process, theonly systems which have undergone any change are the process materials[83]. To see this, we replace dQ/dt by minus the heat delivered to theenvironment per unit of time, dQ/dt = −dQ0/dt:

dU

dt= −dQ0

dt− p0

dV

dt+dW

dt(2.12)

The second law givesdS

dt+dS0dt

≥ 0 (2.13)

where dS/dt is the rate of entropy change of the process materials anddS0/dt is the rate of entropy change in the environment. For a completelyreversible process the sum of these entropy rate changes is zero. For anon-equilibrium process, the sum is the total entropy production, dSirr/dt,

dSirrdt

≡ dSdt

+dS0dt

≥ 0 (2.14)

The entropy change in the surroundings is dS0/dt = (dQ0/dt) /T0. By in-troducing dQ0/dt into the expression for the entropy production, and com-bining the result with the first law, we obtain after some rearrangement:

dW

dt= T0

dSirrdt

+dU

dt+ p0

dV

dt− T0

dS

dt(2.15)


The left hand side of this equation is the work that is needed to accomplishthe process with a particular value of the entropy production, dSirr/dt.Since dSirr/dt ≥ 0, the ideal (reversible) work requirement of this processis:

dWidealdt

=dU

dt+ p0

dV

dt− T0

dS

dt(2.16)

We see that dS/dt changes the ideal work. By comparing Eqs. (2.15) and(2.16), we see that T0(dSirr/dt) is the additional quantity of work per unitof time that must be used in the actual process, compared to the work inan ideal (reversible) process. Thus, T0(dSirr/dt) is the wasted or lost workper unit of time:

dWlostdt

= T0dSirrdt

(2.17)

The lost work is the energy dissipated as heat in the surroundings. Therelation is called the Gouy–Stodola theorem [84, 85]. The right hand sideof Eq. (2.16) is also called the time rate of change of the availability orthe exergy [62, 82]. We shall learn how to derive σ in subsequent chap-ters. One aim of the process operator or designer should be to minimizedWlost/dt [86], which is equivalent to entropy production minimization, seeRef. [85] page 227 and Ref. [29] Chapter 6. Knowledge about the sourcesof entropy production is central for work to improve the energy efficiencyof processes. In a recent book “Elements of irreversible thermodynamicsfor engineers” [28, 29], we discussed how irreversible thermodynamics canbe used to map the lost work in an industrial process. The aluminiumelectrolysis was used as an example to illustrate the importance of sucha mapping. Furthermore the first steps in a systematic method for theminimization of entropy production was described, with its basis in non-equilibrium thermodynamics. Second law optimization [85, 87–92] in me-chanical and chemical engineering may become increasingly important inthe design of systems that waste less work. Non-equilibrium thermody-namics will play a unique role in this context. For details on how to mapthe second law efficiency of an industrial system or how to systematicallyimprove this efficiency, we refer to Bejan [84,85] and to the book “Elementsof irreversible thermodynamics for engineers” [28, 29].

2.5 Consistent thermodynamic models

The entropy balance of the system and the expression for the entropy pro-duction can be used for control of the internal consistency of the thermo-dynamic description. The change of the entropy in a volume element in ahomogeneous phase is given by the flow of entropy in and out of the volume

2.5. Consistent thermodynamic models 15

element and by the entropy production inside the element:

∂s

∂t= − ∂

∂xJs + σ (2.18)

where we take the entropy flux in the x-direction. In the stationary state,there is no change in the system’s entropy and

σ =∂

∂xJs (2.19)

By integrating over the system’s extension, keeping its cross-sectional areaΩ constant, we obtain

dSirrdt

= Ω(Jos − J is) (2.20)

The left hand side is obtained from σ, which is a function of transportcoefficients, fluxes and forces, Eqs. (1.1)–(1.3). The right hand side of thisequation, the difference in the entropy flows, Jos −J is, can be calculated fromknowledge of the heat flux, and of the entropy carried by components intothe surroundings. These quantities do not depend on the system’s transportproperties. The two calculations must give the same result. Chapter 19gives an example of such a comparison. Other examples were given byKjelstrup et al. [29], see also Zvolinschi et al. [92].

Chapter 3

Thermodynamic Relations for

Heterogeneous Systems

This chapter gives thermodynamic relations for heterogeneous systems thatare in global equilibrium, and discusses the meaning of local equilibrium inhomogeneous phases, at surfaces and along three phase contact lines.

The heterogeneous systems that are treated in this book exchange heat,mass and charge with their surroundings. The systems have homogeneousphases separated by an interfacial region. They are electroneutral, butpolarizable. The words surface and interface will be used interchangeablyto indicate the interfacial region. Gibbs [62] calls the interfacial region“the surface of discontinuity”. Thermodynamic relations for homogeneousphases, as well as for surfaces, are mandatory for the chapters to follow.Such equations are therefore presented here [62,93]. Equations for the threephase contact line are also given, but are less central. The equations aregiven first for systems that are in global equilibrium, and next for non-equilibrium systems where only local equilibrium applies.

A thermodynamic description of equilibrium surfaces in terms of excessdensities was constructed by Gibbs [62]. This description treats the surfaceas an autonomous thermodynamic system. We use this description in termsof excess densities as our basis also in non-equilibrium systems. This impliesfor instance that the surface has its own temperature. All excess densities ofa surface depend on this temperature alone and not on the temperatures inthe adjacent phases∗. We present evidence for this assumption using non-equilibrium molecular dynamics simulations (Chapter 22) and the squaregradient model of van der Waals (Chapter 23).

∗This property of the surface has been defended by some authors [94,95], but rejectedby others [96].

17

18 Chapter 3. Thermodynamic Relations for Heterogeneous Systems

We extend Gibbs’ formulation for the surface to the contact line. Equi-librium relations for the three phase contact lines can be formulated forexcess line densities. The line, described in this manner, is also an au-tonomous thermodynamic system.

3.1 Two homogeneous phases separated by a surface in

global equilibrium

Consider two phases of a multi-component system in equilibrium with eachother. The system is polarizable in an electric field. The last propertyis relevant for electrochemical systems. In systems that transport heatand mass only, this property is not relevant. The homogeneous phases, iand o, have their internal energies, U i and Uo, entropies, Si and So, molenumbers, N ij and N

oj of the components j (see for instance [97]), and their

polarizations in the direction normal to the surface, Pi and Po. For thetotal system, we have:

U = U i + U s + Uo (3.1)

The internal energy for the total system, U, minus the sum of the two bulkvalues gives the internal energy of the interfacial region U s [62,93]. For theentropy, the mole numbers and the polarization, we similarly have:

S = Si + Ss + So

Nj = Nij +N

sj +N

oj

P = Pi + Ps + Po

(3.2)

The surface consists of layer(s) of molecules or atoms, with negligible vol-ume, so

V = V i + V o (3.3)

The surface area, Ω, is an extensive variable of the surface, comparable tothe volume as a variable of a homogeneous phase. The surface tension, γ,plays a similar role for the surface as the pressures, pi and po, do for thehomogeneous phases.

The polarizations of the homogeneous phases can be expressed in termsof the polarization densities (polarization per unit of volume), P i and P o,by

Pi = P iV i and Po = P oV o (3.4)

The polarization of the surface can similarly be expressed as the polarizationper unit of surface area, P s:

Ps = P sΩ (3.5)

3.1. Two homogeneous phases separated by a surface in global equilibrium 19

The polarization density is particular for both homogeneous phases andthe surface. The system may contain polarizable molecules (apolar), polarmolecules and free charges. An electric field leads to polarization of theapolar molecules and to orientation of the polar molecules. Furthermore, itleads to a redistribution of free charges. In all cases, polarization occurs ina manner that keeps homogeneous phases and surfaces electroneutral. Fora double layer, we define a surface polarization. For a conductor, we definea polarization density. The divergence of the polarization of a conductorequals the charge distribution, but the induced charge distribution in anelectric field integrates to zero net charge.

In this book, we restrict ourselves to surfaces that are planar and con-tact lines that are straight. From the second Maxwell equation, and thecondition of electroneutrality, we have

d

dxD = 0 (3.6)

where D is the displacement field normal to the surface. The normal com-ponent of the displacement field across an electroneutral surface is contin-uous, and has accordingly no contribution for the surface; Ds = 0. Thenormal component of the displacement field is therefore constant through-out a heterogeneous system. Its value can be controlled from the outsideby putting the system between a parallel plate capacitor. In this book, wedo not consider electric potential gradients along the surface. Assumingthe system furthermore to be invariant for translations along the surface,rotations around a normal on the surface and reflection in planes normalto the surface, it follows that electric fields, displacement fields and polar-izations are normal to the surface. The displacement field is related to theelectric fields in the two phases by

D = ε0Ei + P i = εiE

i and D = ε0Eo + P o = εoEo (3.7)

where εi and εo are the dielectric constants of the phases i and o respectively.Furthermore, ε0 is the dielectric constant of vacuum. A similar relation canbe written for the electric field of the surface

Ds = 0 = ε0Es + P s → P s = −ε0Es (3.8)

For more precise definitions of the interfacial densities we refer to the nextsection and to Sec. 5.2. For an in-depth discussion of the excess polarizationand electric field, we refer to the book by Bedeaux and Vlieger, OpticalProperties of Surfaces [5, 6].

Remark 3.1. The surface polarization is equal to minus ε0 times the ex-cess electric field of the surface. This follows from the following argument:


The displacement field is the sum of ε0 times the electric field plus the po-larization density. The displacement field is continuous through the surface.The sum of the electric field times ε0 plus the surface polarization densitymust therefore be zero.

For heterogeneous, polarizable systems, we can then write the internalenergy as a total differential of the extensive variables S, Nj, V , Ω, Pi, Po

and Ps:

dU = TdS − pdV + γdΩ +n∑

j=1

µjdNj +Deqd

(P

i

εi+

Ps

ε0+

Po

εo

)(3.9)

where Deq is the displacement field that derives from a reversible trans-formation. This is the Gibbs equation [62], extended with terms due topolarization. By integrating with constant intensive variables, the energyof the system becomes

U = TS − pV + γΩ +n∑

j=1

µjNj +Deq

(P

i

εi+

Ps

ε0+

Po

εo

)(3.10)

Since we are dealing with state functions, this result is generally valid. Thecontributions to the internal energy from the polarization are DeqPi/εi andDeqP

o/εo in the homogeneous phases and DeqPs/ε0 [3] for the surface.

Remark 3.2. The contributions to the total polarization, i.e. from polar-izable molecules, from polar molecules and from free charges, have verydifferent relaxation times. They should therefore be counted as separatecontributions Pα =

∑k P

αk , where α equals i, o or s. The corresponding

equilibrium displacement fields, Deq,k, found from a reversible transforma-tion, are also different for the different contributions to the polarization.We shall not make the formulae more cumbersome by distinguishing thesedifferent contributions.

The surface tension is, like the pressure, a function of T , µj , and Deq.The most directly observable effect of the surface tension is found whenthe interface between two phases is curved. Consider for instance a spher-ical bubble in a liquid. The pressure inside the bubble is higher than thepressure outside. The same is true for a liquid droplet in air. The pressuredifference, called the capillary pressure, is given in terms of the surface ten-sion by 2γ/R , where R is the radius of the droplet. The capillary pressureis the reason why water rises in a capillary. The following exercise demon-strates that the energy represented by a surface can be compared to theenergy of a homogeneous phase.

3.1. Two homogeneous phases separated by a surface in global equilibrium 21

Exercise 3.1.1. Use the expression for the capillary pressure to derive aformula for the rise of liquid in a capillary as function of the diameter dof the capillary. The liquid wets the surface. Calculate the capillary rise ofwater in a capillary with a diameter of a micron. The surface tension ofwater is γw = 75 × 10−3 N/m.

• Solution: If the diameter of the capillary is not too large, and whenthe liquid wets the surface, the interface between the water and theair is a half sphere with radius d/2. This gives a capillary pressureof 4γ/d. This would cause an under-pressure in the liquid below thesurface. In order to have a normal atmospheric pressure at the bottomof the capillary, the water rises until the weight of the column gives thecapillary pressure. This implies that π(d/2)2hgρ/π(d/2)2 = hgρ =4γ/d where h is the height of the column, ρ is the density in kg/m3

of water and g is the acceleration of gravity. The capillary rise istherefore h = 4γ/dgρ. In a capillary with a diameter of a micron, thecapillary rise of water is 30 m.

In this book we restrict ourselves to surfaces that are flat. The equilibriumpressure is therefore always constant throughout the system, pi = po = p.

The Gibbs equation (3.9) gives the energy of both homogeneous phasesplus the surface. We need also equations for the separate homogeneousphases and for the surface alone. With Eqs. (3.1)–(3.3), we have for phasei:

dU i = TdSi − pdV i +n∑

j=1

µjdNij +Deqd

Pi

εi(3.11)

This is the Gibbs equation for phase i. We integrate the equation forconstant composition, temperature, pressure and displacement field. Thisgives:

U i = TSi − pV i +n∑

j=1

µjNij +Deq

Pi

εi(3.12)

The energy is a state function, so the result is generally valid (compareEq. (3.10)). By differentiating this expression and subtracting Eq. (3.11),we obtain Gibbs–Duhem’s equation:

0 = SidT − V idp+n∑

j=1

N ijdµj +P

i

εidDeq (3.13)

Similar equations can be written for phase o.


The Gibbs equation for the surface is likewise:

dU s = TdSs + γdΩ +n∑

j=1

µjdNsj +

(Deqε0

)dPs (3.14)

By integration for constant surface tension, temperature, composition anddisplacement field, we obtain

U s = TSs + γΩ +

n∑

j=1

µjNsj +

(Deqε0

)P

s (3.15)

Gibbs–Duhem’s equation for the surface follows by differentiating this equa-tion and subtracting Eq. (3.14):

0 = SsdT + Ωdγ +

n∑

j=1

N sjdµj + Psd

(Deqε0

)(3.16)

The thermodynamic relations for the autonomous surface, Eqs. (3.14)-(3.16), are identical in form to the thermodynamic relations for the ho-mogeneous phases, compare Eqs. (3.11)–(3.13).

Global equilibrium in a heterogeneous, polarizable system in a displace-ment field can now be defined. The chemical potentials and the temperatureare constant throughout the system

µij = µsj = µ

oj = µj

T i = T s = T o = T(3.17)

Furthermore, the normal component of the pressure and of the displacementfield are the same in the adjacent homogeneous phases:

pi = po = p and Deq constant (3.18)

For curved surfaces the normal component of the pressure is not constantand we refer to Blokhuis, Bedeaux and Groenewold [98–103].

The surface energy, and therefore also the surface entropy and surfacemole numbers, have so far been defined from the total values minus thebulk values. We shall see in Sec. 3.3, how more direct definitions can beformulated.

3.2 The contact line in global equilibrium

Consider three phases of a multi-component system in equilibrium with eachother. The three surfaces separating these phases come together in a contact

3.3. Defining thermodynamic variables for the surface 23

line. For simplicity, we assume that the contact line is not polarizable.In the previous section we have seen that the thermodynamic relationsfor an autonomous surface has the same form as for homogeneous phases.The same is true for the autonomous three-phase contact line. The Gibbsequation for the contact line is:

dU c = TdSc + γcdL+n∑

j=1

µjdNcj (3.19)

Superscript c indicates a contribution of the contact line, γc is the linetension and L is the length of the line. The curvature of the line is assumedto be negligible, so that there are no contributions to dU c from changes inthe curvature. By integration for constant line tension, temperature andcomposition, we obtain

U c = TSc + γcL+

n∑

j=1

µjNcj (3.20)

The line energy, and therefore also the line entropy and the line mole num-bers, have here been defined as the total values minus the bulk and thesurface values.

Gibbs–Duhem’s equation for the line follows by differentiating this equa-tion and subtracting Eq. (3.19):

0 = ScdT + Ldγc +n∑

j=1

N cj dµj (3.21)

When the contact line is part of a heterogeneous system in global equilib-rium, the chemical potentials and the temperature are constant throughoutthe system. For the contact line, we therefore also have

µcj = µj and Tc = T (3.22)

More direct definitions of thermodynamic properties of the line, can beobtained, following the procedure for the surface in the next section.

3.3 Defining thermodynamic variables for the surface

The relations for global equilibrium cannot be used to describe systems withgradients in µ, p, T or the electric field. We must therefore also formulaterelations for local equilibrium. We need local forms of the Gibbs and Gibbs–Duhem equations. These will be given after we have first defined the surfacevariables.


Figure 3.1 Variation in the molar density of a fluid if we go from the gas tothe liquid state. The vertical lines indicate the extension of the surface. Thescale of the x-axis is measured in nanometer.

An interface is the thin layer between two homogeneous phases. Werestrict ourselves to flat surfaces, that are perpendicular to the x-axis. Thethermodynamic properties of the interface shall now be given by the valuesof the excess densities of the interface. The value of these densities andthe location of the interface will be defined through the example of a gas-liquid interface. We shall correspondingly indicate these phases with thesuperscripts g and l.

Figure 3.1 shows the variation in concentration of A, in a mixture ofseveral components, as we go from the gas to the liquid phase. The surfacethickness is usually a fraction of a nm. The x-axis of Fig. 3.1 has coordinatesin nm, it has a molecular scale. A continuous variation in the concentrationis seen. Gibbs [62] defined the surface of discontinuity as a transition regionwith a finite thickness bounded by planes of similarly chosen points. In thefigure two such planes are indicated by vertical lines. The position a isthe point in the gas left of the closed surface where cA(x) starts to differfrom the concentration of the gas, cgA, and the position b is the point inthe liquid right of the closed surface, where cA(x) starts to differ from theconcentration of the liquid, clA. The surface thickness is then δ = b − a.It refers to component A. Other components may yield somewhat differentplanes.

Gibbs defined the dividing surface as “a geometrical plane, goingthrough points in the interfacial region, similarly situated with respect to


conditions of adjacent matter”. Many different planes of this type can bechosen. While the position of the dividing surface depends on this choice;it is normally somewhere between the vertical lines in Fig. 3.1. The planesthat separate the closed surface from the homogeneous phases are parallelto the dividing surface. The continuous density, integrated over δ, givesthe excess surface concentration of the component A as a function of theposition, (y, z), along the surface:

ΓA(y, z) =

∫ b

a

[cA(x, y, z) − cgA(a, y, z)θ(d− x) − clA(b, y, z)θ(x− d)

]dx

(3.23)where d is the position of the dividing surface. The surface concentrationis often called the adsorption (in mol/m2). The Heaviside function, θ,is by definition unity when the argument is positive and zero when theargument is negative. It is common to choose d between a and b. Otherexcess variables than ΓA are defined similarly. All excess properties of asurface can be given by integrals like Eq. (3.23). The excess variables arethe extensive variables of the surface. They describe how the surface differsfrom adjacent homogeneous phases.

Remark 3.3. It is clear from Fig. 3.1 that one may shift the position ato the left and b to the right without changing the adsorption. This showswhy the precise location of a and b is not important for the value of theadsorption.

Gibbs defined the excess concentrations for global equilibrium. Equa-tion (3.23) will be used in this book also for systems which are not in globalequilibrium. In fact, ΓA, cA, c

gA and c

lA as well as a, b and d, may all depend

on the time. For ease of notation this was not explicitly indicated.The equimolar surface of component A is a special dividing surface. The

location is such that the surplus of moles of the component on one side ofthe surface is equal to the deficiency of moles of the component on the otherside of the surface, see Fig. 3.2. The position, d, of the equimolar surfaceobeys:

∫ d

a

[cA(x, y, z) − cgA(a, y, z)]dx = −∫ b

d

[cA(x, y, z) − clA(b, y, z)

]dx

(3.24)According to Eqs. (3.23) and (3.24), ΓA = 0, when the surface has thisposition, see Fig. 3.2. The shaded areas in the figure are equal. In amulti-component systems, each component has its “equimolar” surface, butwe have to make one choice for the position of the dividing surface. Weusually choose the position of the surface as the equimolar surface of the


Figure 3.2 Determination of the position of the equimolar surface of compo-nent A. The vertical line is drawn so that the areas between the curve and thebulk densities are the same.

reference component. This component has then no excess concentration,while another component B, like the one sketched in Fig. 3.3, has an excessconcentration.

Remark 3.4. In homogeneous phases it is common to use densities permole or per unit of mass. For the surface this is possible, but not practical.The reason for this is that this would imply dividing the extensive propertiesof the surface by the excess molar density of the reference component. Aswe have just explained it is common to use the equimolar surface definedby this reference component. In that case the excess molar density of thereference component is zero and we cannot divide by it. To avoid confusionwe will always use excess densities per unit of surface area.

Thermodynamic properties of homogeneous systems are usually plottedon scales with greater dimension than nm. In Fig. 3.4, cA(x) is plotted ona µm scale. The fine details of Fig. 3.1 disappear, and the surface appearsas a discontinuity. When plotted on a macroscopic scale, a non-zero excesssurface density, like that of component B, appears as a singularity at theposition of the dividing surface. On this scale the possible choices of thedividing surface can no longer be distinguished, and the surface can beregarded as a two-dimensional thermodynamic system. The system hasproperties that are integrated out in the x -direction and are given persurface area. Dependence on the coordinates y and z remains. When excess


Figure 3.3 Variation in the density of component B across the surface. Theexcess surface concentration of component B is the integral under the curve inthe figure.

Figure 3.4 The equimolar surface plotted on a micrometer scale appears as ajump between bulk densities.


surface densities are defined in this manner, the normal thermodynamicrelations, like the first and second laws are valid [62].

Remark 3.5. Excess densities for the surface can be much larger than thetypical value of the integrated quantity times the thickness of the surfaceof discontinuity. The surface tension of a liquid-vapor is for instance afactor between 10 3 to 10 4 larger than the typical pressure of 1 bar times 3Angstroms. This implies that contributions from the surface play a muchmore important role than what the volume of the surface of discontinuitywould suggest.

Van der Waals constructed a continuous model for the liquid-vapor in-terface at equilibrium, around the end of the 19th century [104]. In thismodel one adds to the Helmholtz energy density per unit of volume a con-tribution proportional to the square of the gradient of the density. It istherefore often called the square gradient model. The model was rein-vented by Cahn and Hilliard in 1958 [105]. It has become an importantmodel for studies of the equilibrium structure of the interface [93,102]. Themodel gives a density profile through the closed surface and enables one tocalculate, for instance, the surface tension. An extension of his model tonon-equilibrium systems [106–108] is discussed in Chapter 23. It is foundin the context of this model [107] that the surface, as described by excessdensities, is in local equilibrium.

Exercise 3.3.1. The variation in density of A across a vapor-liquid inter-face between a = 0 and b = 5 nm is given by cA(x) = Cx

3 + 50 mol/m3.The vapor density is cgA = 50 mol/m

3 and the liquid density is clA = 50000mol/m3. Determine the value of C and the position of the equimolarsurface.

• Solution: The density function at b gives

C(5 × 10−9)3 + 50 = 5 × 104 ⇒ C = 4 × 1029mol/m6

For the equimolar surface, we have:

∫ d

0

Cx3dx = −∫ b

d

(Cx3 − 49950)dx

⇒ 14Cd4 = −1

4C(b4 − d4) + 49950(b− d)

3.4. Local thermodynamic identities 29

This results in the position d = 3.75 nm. The equimolar surface isclose to the liquid side.

3.4 Local thermodynamic identities

In the first section of this chapter, we considered a heterogeneous system inglobal equilibrium. The temperature, the chemical potentials, the pressureand the displacement field were constant throughout the system, Eqs. (3.17)and (3.18). In the previous section, we defined excess densities using ex-pressions that can also be used in non-equilibrium systems. In order to usethe thermodynamic identities from the first section for the excess densities,we need to cast the Gibbs equation and its equivalent forms into a localform.

For the homogeneous phases we introduce the extensive variables perunit of volume. In the i-phase we then have ui = U i/V i, si = Si/V i,cij = N

ij/V

i and the polarization density P i. By dividing Eq. (3.12) by V i

the internal energy density becomes

ui = Tsi − p+n∑

j=1

µjcij +Deq

P i

εi(3.25)

By replacing U i by uiV i etc. in Eq. (3.11), differentiating and usingEq. (3.25), we obtain:

dui = Tdsi +

n∑

j=1

µjdcij +Deqd

P i

εi(3.26)

Gibbs–Duhem’s equation becomes

0 = sidT − dp+n∑

j=1

cijdµj +P i

εidDeq (3.27)

All quantities in these equations refer now to a local position in space.All expressions for phase i, are also true for phase o. (Replace all the

super- and subscripts i by o). Other thermodynamic relations can also bedefined. We give below the i-phase internal energy, Gibbs energy, Helmholtzenergy, internal energy density, Gibbs energy density and the Helmholtz


energy density:

U i = TSi − pV i +n∑

j=1

µjNij +Deq

(P

i/εi)

Gi ≡ U i − TSi + pV i =n∑

j=1

µjNij +Deq

(P

i/εi)

F i ≡ U i − TSi = −pV i +n∑

j=1

µjNij +Deq

(P

i/εi)

ui = Tsi − p+n∑

j=1

µjcij +Deq

(P i/εi

)

gi = ui − Tsi + p =n∑

j=1

µjcij +Deq

(P i/εi

)

f i = ui − Tsi = −p+n∑

j=1

µjcij +Deq

(P i/εi

)

(3.28)

For the surface, the local variables are given per unit of surface area.These are the excess internal energy density us = U s/Ω, the adsorptionsΓj = N

sj/Ω, the excess entropy density, s

s = Ss/Ω, and the surface polar-ization, P s. When we introduce these variables into Eq. (3.14), and useEq. (3.15), we obtain the Gibbs equation for the surface:

dus = Tdss +

n∑

j=1

µjdΓj +DeqdP s

ε0(3.29)

The surface excess internal energy density is

us = Tss + γ +

n∑

j=1

µjΓj +DeqP s

ε0(3.30)

and Gibbs–Duhem’s equation becomes

0 = ssdT + dγ +n∑

j=1

Γjdµj +P s

ε0dDeq (3.31)

We give below the surface internal energy, surface internal energy density,Gibbs equation, Gibbs–Duhem’s equation, surface Gibbs energy density

3.4. Local thermodynamic identities 31

and the surface Helmholtz energy density:

U s = TSs + γΩ +

n∑

j=1

µjNsj +Deq (P

s/ε0)

us = Tss + γ +

n∑

j=1

µjΓj +Deq (Ps/ε0)

dus = Tdss +

n∑

j=1

µjdΓj +Deqd (Ps/ε0)

0 = ssdT + dγ +

n∑

j=1

Γjdµj + (Ps/ε0) dDeq

gs = us − Tss − γ =n∑

j=1


f s = us − Tss = γ +n∑

j=1


(3.32)

For the contact line, local variables are given per unit of length. Theseare the excess internal energy density uc = U c/L, the adsorptions Γcj =N cj /L and the excess entropy density, s

c = Sc/L. When we introduce thesevariables into Eq. (3.19), and use Eq. (3.20), we obtain the Gibbs equationfor the line:

duc = Tdsc +

n∑

j=1

µjdΓcj (3.33)

The contact line excess internal energy density is

uc = Tsc + γc +

n∑

j=1

µjΓcj (3.34)

and Gibbs-Duhem’s equation becomes

0 = scdT + dγc +

n∑

j=1

Γcjdµj (3.35)

Thermodynamic relations for the contact line are given below. We give theline internal energy, line internal energy density, Gibbs equation, Gibbs–Duhem’s equation, contact line Gibbs energy density and the contact line


Helmholtz energy density:

U c = TSc + γcL+

n∑

j=1

µjNcj

uc = Tsc + γc +

n∑

j=1

µjΓcj

duc = Tdsc +

n∑

j=1

µjdΓcj

0 = scdT + dγc +

n∑

j=1

Γcjdµj

gc = uc − Tsc − γc =n∑

j=1

µjΓcj

f c = uc − Tsc = γc +n∑

j=1

µjΓcj

(3.36)

3.5 Defining local equilibrium

We have so far considered global and local thermodynamic relations forheterogeneous systems that are in global equilibrium. We shall now moveone step further and introduce the assumption of local equilibrium. For avolume element in the homogeneous phase i, we say, following Fitts [30],Bedeaux and Kjelstrup [61] and Ortiz and Sengers [109], that there is localequilibrium when the thermodynamic relations (3.25)–(3.27) are valid:

ui(x, t) = T i(x, t)si(x, t) − pi(x, t) +n∑

j=1

µij(x, t)cij(x, t) +Deq(t)

P i(x, t)

εi

dui(x, t) = T i(x, t)dsi(x, t) +

n∑

j=1

µij(x, t)dcij(x, t) +Deq(t)d

P i(x, t)

εi

0 = si(x, t)dT i(x, t) − dpi(x, t) +n∑

j=1

cij(x, t)dµij(x, t) +

P i(x, t)

εidDeq(t)

(3.37)The position and time dependence of all the quantities are now explicitlyindicated. We used that Deq is independent of the position in electroneu-tral systems. The subscript “eq” of the electric field indicates that this

3.5. Defining local equilibrium 33

is the value of the electric field when the system is in global equilibrium,with entropy density, concentrations and polarization density constant andequal to the values of the volume element. The actual displacement fieldin the volume element is not equal to Deq(t) when the system is not inglobal equilibrium. Local equilibrium is defined in the same manner forthe o-phase.

The use of the word local in this context means that the quantities areaverages over sensibly chosen volume elements and time intervals. By asensibly chosen volume element, we mean a volume element that is largecompared to microscopic distances, and small compared to the distanceover which the averaged quantities vary. A sensibly chosen time interval islong compared to microscopic times, but small compared to the time rate ofvariation of the averaged quantities. The choices of the size of the volumeelement and the time interval are not independent. With a slowly varyingprocess, the volume element may be chosen small, and the time intervallarge. These statements can be made precise by non-equilibrium moleculardynamics simulations, see Chapter 22.

For a surface element, we say that there is local equilibrium when thethermodynamic relations (3.29)–(3.31) are valid:

us(t) = T s(t)ss(t) + γ(t) +

n∑

j=1

µsj(t)Γj(t) +Deq(t)P s(t)

ε0

dus(t) = T s(t)dss(t) +n∑

j=1

µsj(t)dΓj(t) +Deq(t)dP s(t)

ε0

0 = ss(t)dT s(t) + dγ(t) +

n∑

j=1

Γj(t)dµsj(t) +

P s(t)

ε0dDeq(t) (3.38)

The intensive thermodynamic variables for the surface, indicated by super-script s, are derivatives:

T s =

(dus

dss

)

Γj ,P sand µsj =

(dus

dΓj

)

ss,Γk,P s(3.39)

The temperature and chemical potentials, defined in this manner, dependonly on the surface excess variables, not on bulk variables. By introducingthese definitions, we therefore allow for the possibility that the surfacehas a different temperature and/or chemical potentials than the adjacenthomogeneous systems have. The equilibrium displacement field for thesurface is defined by

Deq = ε0

(dus

dP s

)

ss,Γj

(3.40)


The actual displacement field, D(t), which is externally controlled, andwhich is also independent of x, is not equal to Deq(t). In the surface, asuitable surface element and time interval must be chosen for the averagingprocedure, when thermodynamic quantities are determined.

For a contact line element, we say that there is local equilibrium whenthe thermodynamic relations (3.33)–(3.35) are valid:

uc(t) = T c(t)sc(t) + γc(t) +

n∑

j=1

µcj(t)Γcj(t)

duc(t) = T c(t)dsc(t) +

n∑

j=1

µcj(t)dΓcj(t)

0 = sc(t)dT c(t) + dγc(t) +

n∑

j=1

Γcj(t)dµcj(t) (3.41)

The intensive thermodynamic variables for the contact line, indicated bysuperscript c, are derivatives:

T c =

(duc

dsc

)

Γcj

and µcj =

(duc

dΓcj

)

sc,Γck

(3.42)

The temperature and chemical potentials, defined in this manner, dependonly on the excess variables for the contact line, not on bulk or surfacevariables. By introducing these definitions, we therefore allow for the pos-sibility that the contact line has a different temperature and/or chemicalpotentials than the adjacent homogeneous phases and surfaces. Along thecontact line, suitable line elements and time intervals must be chosen for theaveraging procedure, when the thermodynamic quantities are determined.

The assumption of local equilibrium, as formulated above, does notimply that there is local chemical equilibrium [20, 68]. According to theoriginal formulation of Prigogine [110], the Gibbs equation remains valid fora large class of irreversible processes, provided that the Maxwell distributionof molecular velocities are perturbed only slightly. The class includes alsochemical reactions slow enough not to disturb the equilibrium form of thedistribution to an appreciable extent.

The thermodynamic variables are, with the above relations, positiondependent. In particular, the thermodynamic variables for the surface de-pend on the position along the surface. Most excess densities depend onthe choice of the dividing surface. The surface tension of a flat surface,however, does not depend on the position of the surface when the systemis in mechanical equilibrium.

3.5. Defining local equilibrium 35

An essential and surprising aspect of the local equilibrium assumptionfor the surface and the adjacent homogeneous phases is that the temper-ature and chemical potentials on both sides of the surface may differ, notonly from each other, but also from the values found for the surface.

The surface temperature represents the thermodynamic state of the sur-face. This temperature depends in principle on the choice of the dividingsurface, but we have found that this dependence, both in molecular dy-namics si

non-equilibrium thermodynamics of heterogeneous …pasko/javni/nastava...non-equilibrium...

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