Introduction and scope

R.C. Srivastava; R.P. Rastogi

1 INTRODUCTION

Classical thermodynamics or for that matter entire classical science relies heavily on equilibrium. Classical thermodynamics which is based on limited number of axioms; the three laws of thermodynamics, has been most successful in deriving the relationships between external measurements such as the exchanges of heat and other forms of energy and of matter between the system and its surroundings and the internal parameters of the systems e.g. equilibrium concentrations in a reacting mixture. In view of the reductionist approach and the success with which correlations between different parameters at equilibrium are obtained, classical thermodynamics has been described as one of the best established pillars of modern science and has been given the same status in physical science which logic is given in humanities. Although classical thermodynamics has been most successful in deriving the relationships that characterize systems at equilibrium where all processes are reversible, the basic premises, equilibrium and reversibility, confront us with a paradox. Concepts like equilibrium and reversibility though very important, reside only in our imagination and do not belong to the real world. The real world is in fact non-equilibrium and irreversible. In natural processes equilibrium is the exception rather than the rule. Living systems and biological processes are typical examples of non-equilibrium phenomena.

In the middle of twentieth century classical thermodynamics especially in the formulations of deDonder and Duhem has been extended to irreversible processes. Non-equilibrium thermodynamics has evolved as a discipline [1-4] for the treatment of non-equilibrium phenomena; thanks to the efforts made by people such as Onsager [5], Meixner [6], Casimir [7], de Groot [8] and members of the Brussels group, Prigogine [9], Glansdorff [10] and Nicolis [11]. The starting point of the extension of equilibrium thermodynamics to non-equilibrium situation was the demonstration of the fact that Gibb’s entropy equation, which was hitherto known to the valid for equilibrium is valid even outside equilibrium. This was accomplished by Prigogine [12] who showed that the Gibb’s entropy equation is valid up to first order perturbations beyond the equilibrium i.e. not very far away from equilibrium.

Equilibrium situation is different from non-equilibrium in that in the former there are no difference/gradients of potentials e.g. temperature, concentration etc within the system and hence there are no flows whereas in the system at non-equilibrium state differences of potentials and the flows induced by them exist in the system; differences or gradients of potentials are the driving forces of the flows. Interactions of the flows giving rise to certain cross phenomena also exist in the system. The conjugate pairs of flows (Ji) and thermodynamics driving forces (Xi) which are differences/gradients of potentials are picked up from the equation for entropy production obtained by transforming the Gibb’s entropy equation in the form Σ J,X, using the laws of conservation of mass and energy. Having picked up the conjugate pairs of fluxes (Ji) and thermodynamic forces (Xi), phenomenological rate equations are written as linear relationships of the form

i=∑j=lnLijXj

where Lij are the phenomenological coefficients which should remain constants as long as Eq. (1) is linear equation. In equation (1) the cross coefficients Lij = Lji which is well known Onsager’s relation. Finally steady states are evaluated either by using Prigogine’s theorem of minimum entropy production or by using physical conditions of steady state of no net flow.

Success in the treatment of non-equilibrium phenomena using the above-summarized procedure implies that the phenomena are in the linear non-equilibrium regime. This linear non-equilibrium regime is close to equilibrium and is within the domain of validity of Gibb’s entropy equation.

If one closely looks into the relative domains of validity of the Gibb’s entropy equation, linear laws (Eq.1) and Onsager’s reciprocal relations one would discover that the domain of validity of Gibb’s equation is larger than those of linear laws and Onsager’s relations. Since domain of validity of Gibb’s equation is larger than those of both linear phenomenological relations and Onsager’s relations [4,12] it is possible to enter the non-linear region by considering non-linear relationships between fluxes and forces, which of course are chosen using Gibb’s entropy equation. Such attempts have already been made [13,14]. Thus this nonlinear regime is within the domain of validity of Gibb’s equation; it may be called non-linear near equilibrium regime.

If we go beyond the domain of validity of Gibb’s equation we get exotic phenomena of dynamic instability [10,11,15]. This also is a non-linear regime where linear laws (Eq.1) and Onsager’s relations are no longer applicable and one may observe multiple steady states and oscillatory phenomena. This non-linear regime is naturally far from equilibrium. If one goes still farther i.e. very far away from equilibrium one comes across the phenomena like turbulence, polarization, chaotic oscillation etc.

The oscillatory phenomena are not obtained in the linear region. These are obtained in the non-linear region which lies in the far from equilibrium regime. Dynamics and stability theory, together with the subservient role of the thermodynamics of irreversible processes has helped in exploring the far from equilibrium region. Tools for the exploration of the non-linear regime very far away from equilibrium where one comes across the phenomena of turbulence and chaos, are still in the process of development and this is why this non-linear regime very far away from equilibrium is still largely unexplored.

Thus, we can classify non-equilibrium regimes as follows:

(i) Linear regime close to equilibrium

(ii) Non-linear regime close to equilibrium

(iii) Non-linear regime far from equilibrium

(iv) Non-linear regime very far equilibrium

The four non-equilibrium regimes listed above are tabulated in Fig.1 along with their charactistics.

1.1 Scope of the monograph

If one wants to probe into the different non-equilibrium regimes listed above he has to station his system at different distances from equilibrium. In chemical reaction systems it can easily be done using the device called continuously stirred tank reactor (CSTR). This is why the far from equilibrium regimes have been subjected to intense investigations in chemical reaction systems [16,17].

In the chemically non-reacting systems it is not so easy in general. In case of electro-kinetic phenomena, however, it is possible to hold the system at the desired distance from equilibrium by controlling the magnitude of driving forces (Xi) and of consequent fluxes (Ji). Therefore, electrokinetic phenomena are good candidates for investigating experimentally the four non-equilibrium regimes particularly the non-linear regimes in the far from equilibrium region where one comes across exotic phenomena like bistability and oscillations. In fact, experimental investigation of the far from equilibrium region in chemically non-reacting systems has been hindered due to non-availability of suitable experimental systems. Above all the study of oscillatory transport processes has assumed great significance from the view point of science of complexity which is considered to the science of 21st century [18].

In addition to conventional electro-kinetic phenomena which indeed are mediated by electrified interfaces we intend to include a few more phenomena mediated by charged liquid-liquid interfaces and solid-liquid interfaces. This is why we have titled this volume as “Transport mediated by electrified interfaces” in the broad ambit of which we may discuss conventional electro-kinetic and also the other transport phenomena mediated by charged interfaces.

In the next chapter we will discuss in general different non-equilibrium regimes.