Non-equilibrium Thermodynamics of Heterogeneous Systems. Signe Kjelstrup

      Chapter 1

      Scope

      The aim of this book is to present a systematic theory of transport for heterogeneous systems. The theory is an extension of non-equilibrium thermodynamics for transport in homogeneous phases, a field that was established in 1931 and developed during the 1940s and 1950s. The foundation to describe transports across surfaces in a systematic way was laid in the 1980s. In this chapter, we set the theory into context and give perspectives on its application.

1.1What is non-equilibrium thermodynamics?

      Non-equilibrium thermodynamics describes transport processes in systems that are out of global equilibrium. The field resulted from the work of many scientists with the aim to find a more useful formulation of the second law of thermodynamics in such systems. The effort started already in 1856 with Thomson’s studies of thermoelectricity [10]. Onsager is, however, counted as the founder of the field with his papers from 1931 [11, 12], see also his collected works [13], because he put earlier research by Thomson, Boltzmann, Nernst, Duhem, Jauman and Einstein into proper perspective. Onsager was given the Nobel Prize in Chemistry in 1968 for these works.

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

(1.1)

      Each flux is a linear combination of all forces,

(1.2)

      The reciprocal relations

(1.3)

      were proven for independent forces and fluxes, cf. Chapter 7, by Onsager [11, 12]. They now bear his name. All coefficients are essential, as explained in Chapter 2. In order to use non-equilibrium thermodynamics, one first has to identify the complete set of extensive, independent variables, Ai. We shall do that in Chapters 4–6, for homogeneous phases, surfaces and three-phase contact lines, respectively. The conjugate fluxes and forces are

(1.4)

      Here, t is the time and S is the entropy of the system. Some authors have 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)–(1.4) contain all information on the non-equilibrium behavior of the system.

      Following Onsager, a systematic theory of non-equilibrium processes was set up in the 1940s by Meixner [14–17] and Prigogine [18]. They found the entropy production for a number of physical problems. Prigogine received the Nobel Prize in 1977 for his work on the structure of systems that are not in equilibrium (dissipative structures), and Mitchell the year after for his application of the driving force concept to transport processes in biology [19].

      Short and essential books were written early by Denbigh [20] and Prigogine [21]. The most general description of classical non-equilibrium thermodynamics is still the 1962 monograph of de Groot and Mazur [22], reprinted in 1985 [23]. Haase’s book [24], also reprinted [25], contains many experimental results for systems in temperature gradients. Katchalsky and Curran developed the theory for biological systems [26]. Their analysis was carried further by Caplan and Essig [27], and Westerhoff and van Dam [28]. Førland and coworkers’ book gave various applications in electrochemistry, in biology and geology [29]. This book, which presents the theory in a way suitable for chemists, has also been reprinted [30]. A simple introduction to non-equilibrium thermodynamics for engineers was given by Kjelstrup, Bedeaux and Johannessen [31, 32], and by Kjelstrup, Bedeaux, Johannessen and Gross [2, 3].

      Non-equilibrium thermodynamics