Experimental Thermodynamics Volume X: Non-equilibrium Thermodynamics with Applications: 10 - Hardcover

Hardcover

ISBN 10: 1782620249 ISBN 13: 9781782620242

Publisher: Royal Society of Chemistry, 2015

Synopsis

Covering recent developments in the theory of non-equilibrium thermodynamics and its applications, this title is aimed at a predominantly, but not exclusively, academic audience of practitioners of thermodynamics and energy conversion.

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The book covers recent developments in the theory of non-equilibrium thermodynamics and its applications. Four chapters are devoted to the foundations; an overview chapter is followed by recent results addressing the underlying principles of the theory. The applications are concerned with bulk systems, with heterogeneous systems where interfaces are central and with process units in industry where entropy production minimization is useful. There is also a collection of chapters under the heading mesoscopic non-equilibrium thermodynamics, giving in the end an overview of extensions of the theory into the non-linear regime. Bringing the literature up to date and detailing new approaches in this area of research, it is aimed at a predominantly, but not exclusively, academic audience of practitioners of thermodynamics and energy conversion.

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Experimental Thermodynamics Volume X

Non-equilibrium Thermodynamics with Applications

By Dick Bedeaux, Signe Kjelstrup, Jan V. Sengers

The Royal Society of Chemistry

Copyright © 2016 International Union of Pure and Applied Chemistry
All rights reserved.

ISBN: 978-1-78262-024-2

Contents

Chapter 1 Basis and Scope Dick Bedeaux, Signe Kjelstrup and Jan V. Sengers, 1, 
Chapter 2 Fluctuating Hydrodynamics and Fluctuation–Dissipation Theorem in Equilibrium Systems José M. Ortiz de Zárate and Jan V. Sengers, 21, 
Chapter 3 Thermal Fluctuations in Non-equilibrium Steady States Jan V. Sengers, José M. Ortiz de Zárate and Theodore R. Kirkpatrick, 39, 
Chapter 4 Local Equilibrium in Non-equilibrium Thermodynamics Signe Kjelstrup and Dick Bedeaux, 61, 
Chapter 5 Diffusion in Liquids: Experiments, Molecular Dynamics, and Engineering Models Christine Peters, Ludger Wolff, Thijs J. H. Vlugt and André Bardow, 78, 
Chapter 6 Non-equilibrium Molecular Dynamics Fernando Bresme, Anders Lervik and Jeff Armstrong, 105, 
Chapter 7 Non-equilibrium Molecular Self-assembly Ger Koper and Wouter Hendriksen, 134, 
Chapter 8 Non-equilibrium Thermodynamics for Evaporation and Condensation Dick Bedeaux and Signe Kjelstrup, 154, 
Chapter 9 Non-equilibrium Thermodynamics Applied to Adsorption Jean-Marc Simon, 178, 
Chapter 10 Non-equilibrium Thermodynamics of Aqueous Solution–Crystal Interfaces F. Elif Genceli Güner, 204, 
Chapter 11 Membrane Transport Juan P. G. Villaluenga and V. María Barragán, 221, 
Chapter 12 Electrochemical Energy Conversion Signe Kjelstrup and Dick Bedeaux, 244, 
Chapter 13 Entropy Production Minimization with Optimal Control Theory Øivind Wilhelmsen, Eivind Johannessen and Signe Kjelstrup, 271, 
Chapter 14 Mesoscopic Non-equilibrium Thermodynamics Ignacio Pagonabarraga and J. Miguel Rubí, 290, 
Chapter 15 Non-equilibrium Thermodynamics for Nucleation Kinetics David Reguera and J. Miguel Rubí, 314, 
Chapter 16 Mesoscopic Non-equilibrium Thermodynamics in Biology Anders Lervik and J. Miguel Rubí, 338, 
Chapter 17 Dynamics of Complex Fluid–Fluid Interfaces Leonard M. C. Sagis, 356, 
Subject Index, 382, 

CHAPTER 1

Basis and Scope

DICK BEDEAUX, SIGNE KJELSTRUP AND JAN V. SENGERS

1.1 Short Historic Overview

Non-equilibrium thermodynamics describes transport processes in systems that are not in global equilibrium. The now classical field, which we briefly review here, resulted from efforts of many scientists to find a more explicit formulation of the second law of thermodynamics. This had started already in 1856 with Thomson's studies of thermoelectricity. Onsager is, however, counted as the founder of the field with his papers published in 1931, because these put earlier research by Thomson, Boltzmann, Nernst, Duhem, Jauman and Einstein into a systematic framework. Onsager was given the Nobel prize in chemistry in 1968 for this work.

The second law is reformulated in terms of the entropy production, σ. In Onsager's formulation, the entropy production is given by the product sum of so-called conjugate fluxes, Ji, and forces, Xi, in the system. The second law then becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

The entropy production is per unit of volume. Each flux is taken to be a linear combination of all forces,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Onsager showed that the reciprocal relations

Lji = Lij, (1.3)

apply. They now bear his name. In order to use the theory, one first has to identify a complete set of extensive independent variables, αi, like, for instance, the internal energy and the mass densities per unit of volume. The resulting conjugate fluxes and forces are Ji = dαi/dt and Xi = [partial derivative]S/[partial derivative]αi respectively. Here t is the time and S is the entropy of the system. The three equations above contain then all information on the non-equilibrium behaviour of the system. For cases where dαi/dt is equal to minus the divergence of a flux density, this flux density replaces Ji and the gradient of [partial derivative]S/[partial derivative]αi. For surfaces and contact lines the densities are per unit of surface area or length, respectively.

Following Onsager, a consistent theory of non-equilibrium processes in continuous systems was set up in the forties by Meixner and Prigogine. They calculated the entropy production for a number of physical problems. Prigogine received the Nobel prize for his work on dissipative structures in systems that are out of equilibrium in 1977, and Mitchell the year after for his application of the (driving) force concept to transport processes in biology.

The most general description of classical non-equilibrium thermo-dynamics is still the 1962 monograph of de Groot and Mazur reprinted in 1985. Haase's book, also reprinted, contains many results for electro-chemical systems and systems with temperature gradients. Katchalsky and Curran developed the theory for biophysical systems. Their analysis was carried further by Caplan and Essig. Førland and co-workers gave various applications in electrochemistry and biology, and they treated frost heave. Their book presented the theory in a way suitable for chemists. Newer books on equilibrium thermodynamics or statistical thermodynamics often include chapters on non-equilibrium thermodynamics, see, e.g., Carey. Kondepudi and Prigogine presented a textbook which integrated texts on basic equilibrium and non-equilibrium thermodynamics. Jou et al. published a book on extended non-equilibrium thermodynamics. Öttinger gave a non-equilibrium description which also extends to the nonlinear regime.

Non-equilibrium thermodynamics is constantly being applied in new contexts. Fitts gave an early presentation of viscous phenomena. In 1994 Kuiken wrote the most general treatment of multicomponent diffusion and rheology of colloidal systems. Rubí and co-workers used internal (molecular) degrees of freedom to explore the development towards equilibrium within a system. This allows us to deal with chemical reactions within the framework of non-equilibrium thermodynamics. Bedeaux and Mazur extended the theory to quantum mechanical systems. Kjelstrup and Bedeaux wrote a book dealing with transport into and across surfaces, presenting non-equilibrium thermodynamics for heterogeneous systems. Doi used the variational principle of Onsager at constant temperature to derive equations of motion for colloids.

A fundamental assumption in non-equilibrium thermodynamics is that of local equilibrium. In recent years the nature and conditions of local equilibrium have been investigated with computer simulations. Kjelstrup et al. pointed out that there can be local equilibrium in volume elements exposed to large fields, with as few as 10 particles. We have reasons to expect that the same holds true in surfaces and in systems at the mesoscale (see Chapter 4). Non-equilibrium thermodynamics has also been extended to include thermal non-equilibrium fluctuations as reviewed by Ortiz de Zárate and Sengers. Local equilibrium is no longer valid for hydrodynamic fluctuations, however. Correlations of the densities and temperature are much larger and longer-ranged in a system exposed, for instance, to temperature gradients than at equilibrium.

The linear relations in eqn (1.2) apply to a volume element, which according to the assumption of local equilibrium, can be very small. Mesoscopic non-equilibrium thermodynamics has been developed to deal with shorter space and time scales (cf., Chapters 14 to 16). Integration from the local to the macroscopic level leads to highly non-linear flux–force relations. To refer to non-equilibrium thermodynamics as a linear theory is misleading. Quoting the preface of the Dover edition of de Groot and Mazur, the theory is non-linear for a variety of reasons, because it includes (i) the presence of convection terms, (ii) quadratic source terms in the energy equation, (iii) the non-linear character of the equation of state, and (iv) the dependence of the Onsager coefficients on the state variables, and so on.

Chemical and mechanical engineering needs theories of transport for systems with gradients in velocity, pressure, concentrations, and temperature, see Denbigh and Bird et al. In isotropic systems there is no coupling between tensorial (viscous), vectorial (diffusional) and scalar (reactions) phenomena, so they can usually be dealt with separately. Simple vectorial transport laws have long worked well in engineering, but there is now an increased effort to be more precise. The need for more accurate flux equations in the modelling of non-equilibrium processes increases the need for non-equilibrium thermodynamics. The books by Taylor and Krishna, Cussler, Demirel, and Kjelstrup et al., which present Maxwell–Stefan's formulation of the flux equations, are important books in this context. Krishna and Wesselingh and Kuiken, analysing an impressive amount of experimental data, have shown that the diffusion coefficients in the Maxwell–Stefan equations depend less on the concentrations.

Non-equilibrium thermodynamics is necessary for a precise description of all systems that have transport of heat, mass, charge and momentum.29 There is also a need in mechanical and chemical engineering to design systems that waste less work. Fossil energy sources, as long as they last, lead to global warming. Better and more efficient use of energy resources is therefore central. It is, then, not enough to only optimize the first-law efficiency. One should minimize the entropy production, which defines the second-law efficiency. The total entropy production should be used as an appropriate measure of sustainability. Through non-equilibrium thermodynamics, one can obtain control of the local entropy production, develop more precise descriptions of the transport processes, and improve the second-law efficiency.

We believe that we are now in a situation where potential new users are looking to the field and asking themselves how non-equilibrium thermo-dynamics possibly can add to the understanding, the description, or the experimental design of their transport or energy conversion problems. The present collection of chapters is meant to help such readers, by giving them inspiration for additional applications, similar to the ones presented here. We expect that the reader is familiar with the basic elements that are used. For a pedagogical presentation of non-equilibrium thermodynamics theory we refer to textbooks, e.g. ref. 29 and 35. For the basis of hydrodynamic fluctuations, we refer to Ortiz de Zárate and Sengers. The seventeen chapters of this book, will update published works, and give state-of-the art mini-reviews of the field of non-equilibrium thermodynamics. We have chosen topics where the theory can add to the present description and open new lines of research and application.

In the remainder of Chapter 1 we provide the essence of the foundations for eqn (1.1) to (1.3) and give advice on how to derive the equations and use them in particular situations.

1.2 The Entropy Production in a Homogeneous Phase

In order to illustrate the points raised in the end of the previous section, we repeat how the entropy production in a homogeneous system can be derived for transport of heat, mass, charge, momentum and chemical reactions, aiming to make eqn (1.1) as concrete as possible. A homogeneous system can be described by the same variable set over its entire extension. The examples covered by this book extend from coupled transport of heat and mass, coupled transport of heat, mass and charge and chemical reactions, to all phenomena present in hydrodynamic flow. We will therefore give the most general expressions for hydrodynamic flow, before we introduce simplifying conditions that apply for the other cases.

The start in any case is to write the entropy balance. In the presence of flow, the best frame of reference is the barycentric (center-of-mass) frame of reference. The change in the entropy density in a volume element by net flow of entropy into the volume element and entropy production inside it is then given by:

[partial derivative]ρ(r, t)s(r, t)/[partial derivative]t = - [nabla] · [ρ(r, t)s(r, t)v(r, t) + Js(r, t)] + σ(r, t). (1.4)

Here ρ(r,t) is the mass density, s[(r,t) is the specific entropy, v(r,t) is the barycentric velocity, Js(r,t) is the entropy flux and σ(r,t) is the entropy production. The position vector is denoted r = x,y,z) and t is the time. We use italic bold symbols for vectors. The entropy density and flux depend on position as well as time, so we use partial derivatives. The volume element has a sufficient number of particles to give a statistical basis for thermo- dynamic calculations, i.e., local equilibrium. The state is given by the temperature T(r,t), the specific chemical potential, μj(r,t) for n neutral components and the pressure p(r,t). We suppress the explicit dependence on (r,t) from now on. It is straightforward to use standard IUPAC symbols and units for homogeneous systems, see however the special situation for heterogeneous systems (cf., Chapters 4, 8 and 12).

The explicit expression for σ is always found by introducing relevant balance equations into the Gibbs equation and comparing the resulting expression with eqn (1.4). This will identify the entropy flux as well as the entropy production, provided that the Gibbs equation holds, or in other words, provided that the system is in local equilibrium, see Chapter 4 for a discussion. The Gibbs equation for a co-moving volume element is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

where u is the specific internal-energy and wj [equivalent to] ρj/ρ are the mass fractions. The total differentials are time derivatives in the barycentric frame of reference, which will be defined below.

The balance equation for the mass density of species j is

[partial derivative]ρj/[partial derivative]t = [nabla] · (ρjv + Jj) + Mjvjr for j = 1, ..., n, (1.6)

where Jj = ρj(vj - v) are the diffusion fluxes in the barycentric frame of reference, and Mj and vj are the molar mass and the stoichiometric coefficient of species j in the chemical reaction, respectively. Furthermore, r(r,t) is the rate of the reaction. The coefficient vj is positive if j is a product and negative if j is a reactant. For simplicity, we consider only one reaction. The reaction Gibbs energy is ΔrG(r,t) = [summation]Mjvjμj. Independent components are, according to the phase rule, the number of species minus the number of restrictions between them. A chemical reaction at equilibrium may pose such a restriction.

We sum eqn (1.6) over all components and use conservation of mass in the chemical reaction. This gives the continuity equation

[partial derivative]ρ/[partial derivative]t = - [nabla] · ρv, (1.7)

where ρ = [summation]ρj and ρv = [summation]ρjvj. It follows from the definitions that the sum of all diffusive fluxes Jj = ρj(vj - v) is zero. The continuity equation for an arbitrary density, a, is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

The last identity defines the time derivative of a quantity that moves with the flow, a co-moving quantity. This defines then the derivatives in Gibbs eqn (1.5).

With the help of eqn (1.8) we can write terms in eqn (1.4) and (1.6) alternatively as

ρ ds/dt = - [nabla] · Js + σ, (1.9)

ρ dwj/dt = - [nabla] or [gradient] · Jj + Mjvjr for j = 1, ..., n. (1.10)

The momentum balance (the equation of motion) in the absence of external forces is given by

ρ dv/dt = - [nabla] p - [nabla]Π, (1.11)

where Π is the viscous pressure tensor. The momentum balance enters the Gibbs equation via the change in internal-energy density. According to the first law of thermodynamics, the change in internal-energy density per unit of time is equal to:

ρ du/dt = - [nabla] · Jq - p[nabla] · v - Π: [nabla] v + E · j, (1.12)

where J]qITL in the absence of external forces is the total heat (energy) flux in the barycentric frame of reference. The total heat flux across the volume element is the sum of the measurable heat flux, J']qITL, and a latent heat flux (the specific partial enthalpies, hj, carried by the diffusive fluxes, Jj of the neutral components)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.13)

(Continues...)

Excerpted from Experimental Thermodynamics Volume X by Dick Bedeaux, Signe Kjelstrup, Jan V. Sengers. Copyright © 2016 International Union of Pure and Applied Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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Edition details

• Publisher: Royal Society of Chemistry
• Publication date: 2015
• Language: English
• ISBN 10: 1782620249
• ISBN 13: 9781782620242
• Binding: Hardcover
• Number of pages: 457
• Editor: Dick Bedeaux