Non-Equilibrium Thermodynamics Of Heterogeneous Systems (Series on Advances in Statistical Mechanics) (Series on Advances in Statistical Mechanics)

the Purpose Of This Book Is To Encourage The Use Of Non-equilibrium Thermodynamics To Describe Transport In Complex, Heterogeneous Media. With Large Coupling Effects Between The Transport Of Heat, Mass, Charge And Chemical Reactions At Surfaces, It Is Important To Know How One Should Properly Integrate Across Systems Where Different Phases Are In Contact. No Other Book Gives A Prescription Of How To Set Up Flux Equations For Transports Across Heterogeneous Systems. the Authors Apply The Thermodynamic Description In Terms Of Excess Densities, Developed By Gibbs For Equilibrium, To Non-equilibrium Systems. The Treatment Is Restricted To Transport Into And Through The Surface. Using Local Equilibrium Together With The Balance Equations For The Surface, Expressions For The Excess Entropy Production Of The Surface And Of The Contact Line Are Derived. Many Examples Are Given To Illustrate How The Theory Can Be Applied To Coupled Transport Of Mass, Heat, Charge And Chemical Reactions; In Phase Transitions, At Electrode Surfaces And In Fuel Cells. Molecular Simulations And Analytical Studies Are Used To Add Insight. Contents......Page 10 Preface......Page 8 1.1 What is non-equilibrium thermodynamics?......Page 18 1.3 The purpose of this book......Page 21 2 Why Non-Equilibrium Thermodynamics?......Page 24 2.1 Simple flux equations......Page 25 2.2 Flux equations with coupling terms......Page 26 2.3 Experimental designs and controls......Page 28 2.4 Entropy production, work and lost work......Page 29 2.5 Consistent thermodynamic models......Page 31 3 Thermodynamic Relations for Heterogeneous Systems......Page 34 3.1 Two homogeneous phases separated by a surface in global equilibrium......Page 35 3.2 The contact line in global equilibrium......Page 39 3.3 Defining thermodynamic variables for the surface......Page 40 3.4 Local thermodynamic identities......Page 46 3.5 Defining local equilibrium......Page 49 3.A Appendix: Partial molar properties......Page 52 3.A.1 Homogeneous phases......Page 53 3.A.2 The surface......Page 55 3.A.3 The standard state......Page 57 Part A: General Theory......Page 62 4 The Entropy Production for a Homogeneous Phase......Page 64 4.1 Balance equations......Page 66 4.2 The entropy production......Page 68 4.2.1 Why one should not use the dissipation function......Page 73 4.2.2 States with minimum entropy production......Page 74 4.3 Examples......Page 75 4.4.1 Definitions of frames of reference......Page 81 4.4.2 Transformations between the frames of reference......Page 83 4.A Appendix: The first law and the heat flux......Page 84 5 The Excess Entropy Production for the Surface......Page 90 5.1 The discrete nature of the surface......Page 91 5.2 The behavior of the electric fields and potential through the surface......Page 92 5.3 Balance equations......Page 94 5.4 The excess entropy production......Page 96 5.4.1 Reversible processes at the interface and the Nernst equation......Page 101 5.4.2 The surface potential jump at the hydrogen electrode......Page 103 5.5 Examples......Page 104 6 The Excess Entropy Production for a Three Phase Contact Line......Page 108 6.1 The discrete nature of the contact line......Page 109 6.2 Balance equations......Page 111 6.3 The excess entropy production......Page 112 6.4 Stationary states......Page 113 6.5 Concluding comment......Page 114 7.1 Flux-force relations......Page 116 7.2 Onsager’s reciprocal relations......Page 117 7.3 Relaxation to equilibrium. Consequences of violating Onsager relations......Page 121 7.4 Force-flux relations......Page 122 7.5 Coe cient bounds......Page 123 7.6 The Curie principle applied to surfaces and contact lines......Page 125 8 Transport of Heat and Mass......Page 128 8.1 The homogeneous phases......Page 129 8.2 Coe cient values for homogeneous phases......Page 131 8.3 The surface......Page 134 8.3.1 Heats of transfer for the surface......Page 136 8.4 Solution for the heterogeneous system......Page 139 8.5 Scaling relations between surface and bulk resistivities......Page 142 9 Transport of Heat and Charg......Page 144 9.1 The homogeneous phases......Page 145 9.2 The surface......Page 147 9.3 Thermoelectric coolers......Page 149 9.4 Thermoelectric generators......Page 150 9.5 Solution for the heterogeneous system......Page 152 10 Transport of Mass and Charge......Page 156 10.1 The electrolyte......Page 157 10.2 The electrode surfaces......Page 160 10.3 Solution for the heterogeneous system......Page 163 10.4 A salt power plant......Page 164 10.5 Electric power from volume flow......Page 165 10.6 Ionic mobility model for the electrolyte......Page 167 10.7 Ionic and electronic model for the surface......Page 171 Part B: Applications......Page 172 11 Evaporation and Condensation......Page 174 11.1.1 The entropy production and the flux equations......Page 175 11.1.2 Interface resistivities from kinetic theory......Page 182 11.2 The sign of the heats of transfer of the surface......Page 184 11.3 Coe cients from molecular dynamics simulations......Page 186 11.4.1 The entropy production and the flux equations......Page 193 11.4.2 Interface resistivities from kinetic theory......Page 196 12 Multi-Component Heat and Mass Di usion......Page 200 12.1 The homogeneous phases......Page 201 12.2 The Maxwell–Stefan equations for multi-component di usion......Page 203 12.3 The Maxwell–Stefan equations for the surface......Page 205 12.4.1 Prigogine’s theorem......Page 209 12.4.2 Di usion in the solvent frame of reference......Page 210 12.4.3 Other frames of reference......Page 212 12.4.4 An example: Kinetic demixing of oxides......Page 217 12.5 A relation between the heats of transfer and the enthalpy......Page 219 13 A Nonisothermal Concentration Cell......Page 222 13.1.1 Entropy production and flux equations for the anode......Page 224 13.1.2 Position dependent transport coe cients......Page 227 13.1.3 The profiles of the homogeneous anode......Page 228 13.1.4 Contributions from the cathode......Page 229 13.1.5 The electrolyte contribution......Page 230 13.2.1 The anode surface......Page 231 13.2.2 The cathode surface......Page 234 13.3 The thermoelectric potential......Page 235 14 The Transported Entropy......Page 238 14.1 The Seebeck coe cient of cell a......Page 239 14.2 The transported entropy of Pb2+ in cell a......Page 243 14.3 The transported entropy of the cation in cell b......Page 244 14.4 The transported entropy of the ions cell c......Page 245 14.5 Transformation properties......Page 247 14.6 Concluding comments......Page 249 15 Adiabatic Electrode Reactions......Page 252 15.1.2 The silver chloride phases......Page 253 15.2 The interfaces......Page 254 15.2.2 The silver chloride-electrolyte interfaces......Page 256 15.3 Temperature and electric potential profiles......Page 257 16 The Liquid Junction Potential......Page 266 16.1 The flux equations for the electrolyte......Page 267 16.2 The liquid junction potential......Page 270 16.3 Liquid junction potential calculations compared......Page 272 16.4 Concluding comments......Page 275 17 The Formation Cell......Page 278 17.1.2 The transference coe cient of the salt in the electrolyte......Page 280 17.1.3 An electrolyte with a salt concentration gradient......Page 282 17.1.4 The Planck potential derived from ionic fluxes and forces......Page 284 17.2 A non-isothermal cell with a non-uniform electrolyte......Page 285 17.2.1 The homogeneous anode phase......Page 286 17.2.2 The electrolyte......Page 287 17.2.3 The surface of the anode......Page 289 17.2.4 The homogeneous phases and the surface of the cathode......Page 290 17.3 Concluding comments......Page 292 18.1 The potential work of a salt power plant......Page 294 18.2 The membrane as a barrier to transport of heat and mass......Page 296 18.3 Membrane transport of heat and mass......Page 298 18.4 Osmosis......Page 300 18.5 Thermal osmosis......Page 302 19 Modeling the Polymer Electrolyte Fuel Cell......Page 306 19.1 The potential work of a fuel cell .......Page 307 19.2 The cell and its five subsystems......Page 308 19.3.1 The entropy production in the homogeneous phases......Page 310 19.3.2 The anode backing......Page 312 19.3.3 The membrane......Page 315 19.3.4 The cathode backing......Page 317 19.4 The electrode surfaces......Page 318 19.4.1 The anode catalyst surface......Page 321 19.4.2 The cathode catalyst surface......Page 323 19.5 A model in agreement with the second law......Page 324 19.6 Concluding comments......Page 327 20 Measuring Membrane Transport Properties......Page 328 20.2 The membrane resistivity......Page 329 20.3 Ionic transport numbers......Page 333 20.4 The transference number of water and the water permeability......Page 336 20.5 The Seebeck coe cient......Page 339 20.6 Interdi usion coe cients......Page 340 21 The Impedance of an Electrode Surface......Page 344 21.1 The hydrogen electrode. Mass balances......Page 345 21.2 The oscillating field......Page 348 21.4.1 The adsorption-di usion layer in front of the catalyst......Page 349 21.4.2 The charge transfer reaction......Page 353 21.4.3 The impedance spectrum......Page 354 21.5 A test of the model......Page 355 21.6 The reaction overpotential......Page 356 22 Non-Equilibrium Molecular Dynamics Simulations......Page 358 22.1 The system......Page 361 22.1.1 The interaction potential......Page 363 22.2 Calculation techniques......Page 364 22.3.1 Local equilibrium in a homogeneous binary mixture......Page 368 22.3.2 Local equilibrium in a gas-liquid interface......Page 370 22.4.1 A homogeneous binary mixture......Page 373 22.4.2 A gas-liquid interface......Page 375 22.6 Molecular mechanisms......Page 376 23 The Non-Equilibrium Two-Phase van der Waals Model......Page 378 23.1 Van der Waals equation of states......Page 380 23.2 Van der Waals square gradient model for the interfacial region......Page 383 23.3 Balance equations......Page 386 23.4 The entropy production......Page 388 23.5 Flux equations......Page 389 23.6 A numerical solution method......Page 390 23.8 Defining excess densities......Page 395 23.9 Thermodynamic properties of Gibbs’ surface......Page 396 23.10 An autonomous surface .......Page 397 23.11.1 Properties of dividing surfaces......Page 401 23.11.2 Surface excess densities for two dividing surfaces......Page 402 23.11.3 The surface temperature from excess density di erences......Page 403 23.12 The entropy balance and the excess entropy production......Page 405 23.13 Resistivities to heat and mass transfer......Page 407 23.14 Concluding comments......Page 409 References......Page 410 Symbol Lists......Page 432 Index......Page 440 About the Authors......Page 450 1. Scope. 1.1. What is non-equilibrium thermodynamics? 1.2. Non-equilibrium thermodynamics in the context of other theories. 1.3. The purpose of this book -- 2. Why non-equilibrium thermodynamics? 2.1. Simple flux equations. 2.2. Flux equations with coupling terms. 2.3. Experimental designs and controls. 2.4. Entropy production, work and lost work. 2.5. Consistent thermodynamic models -- 3. Thermodynamic relations for heterogeneous systems. 3.1. Two homogeneous phases separated by a surface in global equilibrium. 3.2. The contact line in global equilibrium. 3.3. Defining thermodynamic variables for the surface. 3.4. Local thermodynamic identities. 3.5. Defining local equilibrium -- 4. The entropy production for a homogeneous phase. 4.1. Balance equations. 4.2. The entropy production. 4.3. Examples. 4.4. Frames of reference for fluxes in homogeneous systems -- 5. The excess entropy production for the surface. 5.1. The discrete nature of the surface. 5.2. The behavior of the electric fields and potential through the surface. 5.3. Balance equations. 5.4. The excess entropy production -- 6. The excess entropy production for a three phase contact line. 6.1. The discrete nature of the contact line. 6.2. Balance equations. 6.3. The excess entropy production. 6.4. Stationary states -- 7. Flux equations and Onsager relations. 7.1. Flux-force relations. 7.2. Onsager's reciprocal relations. 7.3. Relaxation to equilibrium. Consequences of violating Onsager relations. 7.4. Force-flux relations. 7.5. Coefficient bounds. 7.6. The Curie principle applied to surfaces and contact lines -- 8. Transport of heat and mass. 8.1. The homogeneous phases. 8.2. Coefficient values for homogeneous phases. 8.3. The surface. 8.4. Solution for the heterogeneous system. 8.5. Scaling relations between surface and bulk resistivities -- 9. Transport of heat and charge. 9.1. The homogeneous phases. 9.2. The surface. 9.3. Thermoelectric coolers. 9.4. Thermoelectric generators. 9.5. Solution for the heterogeneous system -- 10. Transport of mass and charge. 10.1. The electrolyte. 10.2. The electrode surfaces. 10.3. Solution for the heterogeneous system. 10.4. A salt power plant. 10.5. Electric power from volume flow. 10.6. Ionic mobility model for the electrolyte. 10.7. Ionic and electronic model for the surface -- 11. Evaporation and condensation. 11.1. Evaporation and condensation in a pure fluid. 11.2. The sign of the heats of transfer of the surface. 11.3. Coefficients from molecular dynamics simulations. 11.4. Evaporation and condensation in a two-component fluid -- 12. Multi-component heat and mass diffusion. 12.1. The homogeneous phases. 12.2. The Maxwell-Stefan equations for multi-component diffusion. 12.3. The Maxwell-Stefan equations for the surface. 12.4. Multi-component diffusion. 12.5. A relation between the heats of transfer and the enthalpy -- 13. A Nonisothermal concentration cell. 13.1. The homogeneous phases. 13.2. Surface contributions. 13.3. The thermoelectric potential -- 14. The transported entropy. 14.1. The Seebeck coefficient of cell a. 14.2. The transported entropy of Pb[symbol] in cell a. 14.3. The transported entropy of the cation in cell b. 14.4. The transported entropy of the ions cell c. 14.5. Transformation properties -- 15. Adiabatic electrode reactions. 15.1. The homogeneous phases. 15.2. The interfaces. 15.3. Temperature and electric potential profiles -- 16. The liquid junction potential. 16.1. The flux equations for the electrolyte. 16.2. The liquid junction potential. 16.3. Liquid junction potential calculations compared -- 17. The formation cell. 17.1. The isothermal cell. 17.2. A non-isothermal cell with a non-uniform electrolyte -- 18. Power from regular and thermal osmosis. 18.1. The potential work of a salt power plant. 18.2. The membrane as a barrier to transport of heat and mass. 18.3. Membrane transport of heat and mass. 18.4. Osmosis -- 19. Modeling the polymer electrolyte fuel cell. 19.1. The potential work of a fuel cell. 19.2. The cell and its five subsystems. 19.3. The electrode backing and the membrane. 19.4. The electrode surfaces. 19.5. A model in agreement with the second law -- 20. Measuring membrane transport properties. 20.1. The membrane in equilibrium with electrolyte solutions. 20.2. The membrane resistivity. 20.3. Ionic transport numbers. 20.4. The transference number of water and the water permeability. 20.5. The Seebeck coefficient. 20.6. Interdiffusion coefficients -- 21. The impedance of an electrode surface. 21.1. The hydrogen electrode. Mass balances. 21.2. The oscillating field. 21.3. Reaction Gibbs energies. 21.4. The electrode surface impedance. 21.5. A test of the model. 21.6. The reaction overpotential -- 22. Non-equilibrium molecular dynamics simulations. 22.1. The system. 22.2. Calculation techniques. 22.3. Verifying the assumption of local equilibrium. 22.4. Verifications of the Onsager relations. 22.5. Linearity of the flux-force relations. 22.6. Molecular mechanisms -- 23. The non-equilibrium two-phase van der Waals model. 23.1. Van der Waals equation of states. 23.2. Van der Waals square gradient model for the interfacial region. 23.3. Balance equations. 23.4. The entropy production. 23.5. Flux equations. 23.6. A numerical solution method. 23.7. Procedure for extrapolation of bulk densities and fluxes. 23.8. Defining excess densities. 23.9. Thermodynamic properties of Gibbs' surface. 23.10. An autonomous surface. 23.11. Excess densities depend on the choice of dividing surface. 23.12. The entropy balance and the excess entropy production. 23.13. Resistivities to heat and mass transfer "The purpose of this book is to encourage the use of non-equilibrium thermodynamics to describe transport in complex, heterogeneous media. With large coupling effects between the transport of heat, mass, charge and chemical reactions at surfaces, it is important to know how one should properly integrate across systems where different phases are in contact. No other book gives a prescription of how to set up flux equations for transports across heterogeneous systems." "The authors apply the thermodynamic description in terms of excess densities, developed by Gibbs for equilibrium, to non-equilibrium systems. The treatment is restricted to transport into and through the surface. Using local equilibrium together with the balance equations for the surface, expressions for the excess entropy production of the surface and of the contact line are derived. Many examples are given to illustrate how the theory can be applied to coupled transport of mass, heat, charge and chemical reactions; in phase transitions, at electrode surfaces and in fuel cells. Molecular simulations and analytical studies are used to add insight."--Jacket