Non-Conventional Transport Phenomena

In this chapter, we wish to exploit the availability of the bracket formalism in the description of complex, non-conventional transport phenomena. In the first section, §10.1, we analyze relaxational phenomena in heat and mass transfer. The next section, §10.2, includes the description of phase transitions in inhomogeneous media. The last section, §10.3, contains a first effort to describe inertial effects in viscoelasticity. These problems have rarely been considered in the past, and when they have it has always been from a phenomenological perspective. We explore the availability of the bracket formalism here to provide a more systematic basis for these systems than has heretofore been available, and hence we characterize the models in this chapter as semi-phenomenological. The basic approach that we use is to first establish an appropriate internal variable for the system in consideration, and then to divine an appropriate Hamil-tonian which does, in some limits, produce available phenomenological models. (The latter step indicates why we characterize the models deve-loped in this chapter as “semi-phenomenological.”) As we shall see, describing the models on this more fundamental basis clears up a number of inconsistencies, as well as extending their range of validity without unduly sacrificing their simplicity. In most engineering applications of heat and mass transfer, the simple linear constitutive relations of (6.4-12) are adequate in order to describe the respective transport processes. A couple of very simple examples are the heat flux, when the affinity is the temperature gradient (giving Fourier's law of heat conduction), and the mass diffusion flux, when the affinity is the chemical potential (giving Pick's law of mass diffusion). The importance of such relationships in engineering practice cannot be overestimated. The validity of the linearized equations is generally established by steady-state experiments, so the question that naturally arises is whether or not the same constitutive relationship will hold for transient phenomena. This question cannot be answered as long as only steady-state experiments are performed. From physical considerations alone, it is obvious that the linearized constitutive relationships cannot be complete, in and of themselves.

Symplectic Geometry in Optics

The scope of this book is to address the fundamental problem of modeling transport processes within complex systems, i.e., systems with internal microstructure. The classical engineering approach involves the modeling of the systems as structured continua and the subsequent use of the models in order to derive (if possible) analytical results, exact or approximate. The advent of powerful computers and the promise through parallel processing of even more substantial computational gains in the near future have introduced yet another paragon to the established engineering practice: that of the numerical simulation. Numerical simulation has emerged as a viable alternative to experiments (contrast Computational Fluid Dynamics (CFD) simulations versus wind tunnel experiments); however, the key limitation to a wider application of numerical simulations in engineering practice lies in the reliability of the models (as well as in their simplicity). CFD applications are successful since the Navier/Stokes equations which they employ are quite capable of describing accurately enough the hydrodynamics of air and water. However, as we move our emphasis to materials of such internal complexity as polymer melts, liquid crystals, suspensions, etc., the development of reliable continuum models becomes an increasingly arduous task. The main objective of this treatise is to investigate a more systematic approach through which continuum models may be developed and analyzed. The key issue that the modeler has to cope with is how to construct models which describe more of the underlying physics without, at the same time, becoming excessively complex so that they either require a prohibitively large, experimentally determined number of adjustable parameters (such as current phenomenological theories) or a prohibitively large computational time (such as required for a detailed “brute force” description of the molecular dynamics). It is the thesis of the present work that a lot of effort can be saved if the appropriate formulation is used in deriving model equations, a formulation which is capable of exploiting to a maximum degree the inherent symmetry and consistency of the collective phenomena exhibited by a large number of internal degrees of freedom.

Introduction

The investigation of dynamical phenomena in gases, liquids, and solids has attracted the interest of physicists, chemists, and engineers from the very beginning of the modern science. The early work on transport phenomena focussed on the description of ideal flow behavior as a natural extension to the dynamical behavior of a collection of discrete particles, which dominated so much of the classical mechanics of the last century. As far back as 1809, the mathematical techniques which later came to be known as Hamiltonian mechanics began to emerge, as well as an appreciation of the inherent symmetry and structure of the mathematical forms embodied by the Poisson bracket. It was in this year that S. D. Poisson introduced this celebrated bracket [Poisson, 1809, p. 281], and in succeeding years that such famous scholars as Hamilton, Jacobi, and Poincaré laid the foundation for classical mechanics upon the earlier bedrock of Euler, Lagrange, and d'Alembert. This surge of interest in Hamiltonian mechanics continues well into the waning years of the twentieth century, where scholars are just beginning to realize the wealth of information to be gained through the use of such powerful analytic tools as the Hamiltonian/Poisson formalism and the development of symplectic methods on differential manifolds. Specifically, the study of the dynamics of ideal continua, which is analogous to the discrete particle dynamics studied by Hamilton, Jacobi, and Poisson, has recently benefited significantly by the adaptation of the equations of motion into Hamiltonian form. The inherent structure and symmetry of this form of the equations is particularly well suited for many mathematical analyses which are extremely difficult when conducted in terms of the standard forms of the dynamical equations, for instance, stability and perturbation analyses of ideal fluid flows. Thus, classical mechanics and its outgrowth, continuum mechanics, seem to be on the verge of some major developments. Yet, further progress in this area was hindered by the fact that the traditional form of the Hamiltonian structure can only describe conservative systems, thus placing a severe constraint on the applicability of these mathematically elegant and computationally powerful techniques to real systems.

The Dynamical Theory of Liquid Crystals

Liquid crystals (LCs) present a state of matter with properties—as the name suggests—intermediate between those of liquids and crystalline solids. Liquid-crystalline materials, as all liquids, cannot support shear stresses at static equilibrium. Their molecules are characterized by an anisotropy in the shape and/or intermolecular forces. Thus, there is the potential for the formation of a separate phase(s), called a “mesophase(s),” where a partial order arises in the molecular orientation and/or location, which extends over macroscopic distances. This partial long-range molecular order, reminiscent of (but not equivalent to) the perfect order of solid crystals, in addition to the material fluidity, is primarily responsible for the many properties which are inherent characteristics of liquid-crystalline phases, such as a rapid response to electric and magnetic fields, anisotropic optical and rheological properties, etc.—see, for examples, the reviews by Stephen and Straley [1974] and Jackson and Shaw [1991], the monographs by de Gennes [1974], Chandrasekhar [1977], and Vertogen and de Jeu [1988], and the edited volumes by Ciferri et al. [1982] and Ciferri [1991]. The variety of the liquid-crystalline macroscopic properties is such that trying to derive a theory capable of describing the principal liquid-crystalline dynamic characteristics can be a very frustrating task if one does not approach the issue in a systematic fashion. Characteristically, the main two theories that have been advanced over the last thirty years for the description of the liquid-crystalline flow behavior—the Leslie/ Ericksen (LE) theory and the Doi theory—are essentially models developed from a set theoretical frame work—continuum mechanics and molecular theory, respectively. Nevertheless, each one of these theories has a limited domain of application. The description of the dynamic liquid-crystalline behavior through the bracket formalism, as seen in this chapter, leads naturally to a single conformation tensor theory with an extended domain of validity. This conformation theory consistently generalizes both previous theories, which can be recovered from it as particular cases. This offers additional evidence that the wealth of inherent information in LCs can only be appropriately handled when pursued in a systematic, fundamental manner.

Transport Phenomena in Viscoelastic Fluids

In chapter 8, we applied the bracket formalism to the relatively old problem of incompressible and isothermal viscoelastic fluids. In addition to these assumptions concerning the state of the fluid medium, we therein assumed that the polymer concentration (in the case of polymer solutions) was constant. Although even in this case we were able to find some new results, it is through new applications altogether that the major advantages of this technique will be applied to the fullest extent. Thus, in this chapter we wish to study three new applications of the generalized bracket to outstanding problems concerning viscoelastic fluids. The first section of this chapter is concerned with the complications induced in viscoelastic-fluid modeling by considering compressible and non-isothermal systems. In the second section, we present the analysis of simultaneous concentration and deformation changes associated with the bulk flow of dilute polymer solutions in a form also suitable for the description of flow-induced phase separation. In the third and final section we focus our attention on the solid-surface/polymer interactions which may lead to an apparent “slip velocity” or “adsorption layer” at the interface. We consider §9.3 as the culmination of chapters 8 and 9, and therefore we present it last despite the fact that the natural order for chapter 9 would have been the reverse of what is given below, going from the simplest to the most complex. This last section is the perfect example of our theme of consistently abstracting microscopic information to the macroscopic level of description. Because of the abundance and variety of thought on the issue of flow-induced polymer migration, §9.2 is very inconclusive at this point in time. Its presence here is solely to stimulate additional thought upon this issue. In industrial applications involving polymers, rarely does the engineer deal with an isothermal, and, consequently, incompressible fluid. Most processes are performed at extremely high temperatures, and much heating and cooling design goes into the successful process. Indeed, even if industrial processes were performed at constant temperature, one would still need to handle non-isothermalities since polymers produce large degrees of viscous heating during flow.

Non-Equilibrium Thermodynamics

After having devoted five chapters of this book to the discussion of equilibrium thermodynamics and conservative dynamic phenomena, it is now high time that we entered into the realm of irreversible transport processes. As mentioned in chapter 1, most of the physical systems which engineers wish to model exhibit dissipative phenomena. Therefore, although the techniques touched upon in the previous chapters are mathematically profound and well-suited for diverse analyses for conservative systems, it is in this chapter and the next that the major engineering applications will find their foundation. Granted, in describing irreversible phenomena on the continuum level a certain amount of phenomenology is necessarily introduced; yet we hope to illustrate here how the application of thermodynamic knowledge to the irreversible system can reduce this phenomenology to the bare minimum. The objective of this chapter is similar to that of chapter 4; we wish to present a brief, yet sufficiently thorough, discussion concerning the theory of non-equilibrium thermodynamics applied to irreversible processes. There already exist several outstanding references on the subject [De Groot and Mazur, 1962; Yourgrau et al., 1966; Prigogine, 1967; Gyarmati, 1970; Woods, 1975; Lavenda, 1978; Truesdell, 1984]. Thus, the objective of our discussion here is mainly to introduce the principles that are subsequently used to formulate the dissipative bracket, as outlined in the next chapter. Moreover, the presentation of the subject is biased towards the presentation of the concepts that we consider as most helpful to continuum modeling. For example, the notion of internal variables is introduced early on, in §6.2. As we shall see, the inclusion of internal variables in the non-equilibrium description of the system has profound implications concerning the roles of the various thermodynamic variables and the definitions of the various state functions, in particular, the entropy. Indeed, the definitions of these functions hinge upon the notion of time scales which become of chief importance in the discussion of irreversible thermodynamics. In the philosophy of equilibrium thermodynamics, it is assumed that the time scale for changes in the system is sufficiently large as compared to the intrinsic time scales of any internal variables within the system.

Poisson Brackets in Continuous Media

Now that we have defined the necessary thermodynamic quantities in chapter 4, we can turn back to the consideration of the dynamics of various physical systems. In order to apply a Poisson bracket to macroscopic transport phenomena, it is first necessary to rewrite the bracket (3.3-3) in a form which is suitable for continuum-mechanical considerations. As the number of particles increases to infinity, the transition is made from the specification of a very large number of discrete particle trajectories, xi(t), i=1,2,...,N, N→∞, to the determination of a single, continuous, vector function, Y(r,t), indicating the position of a fluid particle at time t which at a reference time t=0 was at position r, i.e., Y(r,0)=r. This is called a Lagrangian or material description. Alternatively, an Eulerian or spatial description can be used according to which the flow kinematics are completely specified through the determination of the velocity vector field, v(x,t), indicating the velocity of a fluid particle at a fixed spatial position, x, and time, t. (Truesdell [1966, p. 17] notes that the Lagrangian/Eulerian terminology is erroneous, however.) In this chapter, we shall use both descriptions to tackle the problem of ideal (inviscid) fluid flow and to arrive at a Poisson bracket for each case. The dissipative system will be considered in chapter 7. Once again, the concept of time and length scales is very important in determining when the experimenter views the system in consideration as a continuum entity. In chapter 3, we studied the dynamics of a system of discrete particles bouncing around with our time scale implicitly set on the order of the mean free time of the particles between collisions, ζ, and the length scale on the order of the mean free path, λ. As the number of particles approaches infinity, however, we note that certain averages, such as the velocity and the energy of the system, are practically constant on such a small time scale. Since the number of particles is so large, it is almost impossible to get any detailed information about the system as a whole by looking at individual particles because the number of the degrees of freedom is horrendous.

Multi-Fluid Transport/Reaction Models with Application in the Modeling of Weakly Ionized Plasma Dynamics

The industrial use of low-ambient-temperature, weakly ionized plasmas as a reaction environment is growing rapidly. This is primarily evident in the manufacturing technologies of advanced materials, such as the ones used in micro-electronic devices [Jensen, 1987]. The advantages of the plasma environment are due primarily to the presence of high energy electrons which allow high energy chemistry to take place at low ambient temperatures. An example is the successful plasma-enhanced chemical vapor deposition of silicon nitride at temperatures as low as 250-350°C versus temperatures in the range of 700-900°C required for thermal deposition [Reif, 1984]. Thus emerges a need for modeling of the reaction chemistry and the transport phenomena within complex, multicomponent, charged-particle systems, under the influence of externally-imposed electric and magnetic fields. The present chapter addresses this need within the framework of a multi-fluid reactive continuum [Woods, 1975, ch. 9]. Multi-fluid continuum descriptions have arisen as a natural generalization of multicomponent systems in order to account for the absence of momentum and/or energy equilibria between different species populations within the same system [Enz, 1974; Woods, 1975, ch. 9]. The key underlying assumption is that of interpenetrating continua: each one of the mutually interacting, constituent subsystems is characterized as a separate continuum with its own (macroscopic) state variables. Hidden within this assumption is the local equilibrium hypothesis, not between different subsystems—that would have resulted in the more traditional multicomponent description—but within each subsystem in order for the description of each subsystem using (equilibrium) state variables to be meaningful. This is both an asset and a liability of the multi-fluid approach: an asset, because the whole framework of equilibrium thermodynamics is still applicable at the subsystem level, resulting, among other things, in a description requiring only a few well-defined macroscopic state variables; a liability, because it places very stringent requirements on the type of systems to which this theory can be applied. The multi-fluid approach is valid only for phenomena with characteristic time scales much larger than the time scale for each subsystem to reach internal (local) thermodynamic equilibrium.

Incompressible Viscoelastic Fluids

In Part I, we discussed in detail the foundations of the bracket description of dynamical behavior, demonstrating how the generalized bracket is linked to the theories of both Hamiltonian mechanics and irreversible thermodynamics. Now it is time to discuss the various applications towards seemingly complex systems which are the main focus of this book. Specifically, we want to look at a variety of microstructured media of immediate concern in science and industry, and to illustrate the advantages of using the generalized bracket formalism over traditional techniques when developing system-particular models. As we shall also see, there are certain advantages to be gained even when we are simply expressing existing models in Hamiltonian form. The first subject that we wish to address is that of viscoelastic fluid dynamics. As the name implies, viscoelasticity characterizes the materials that possess properties intermediate to those of an elastic solid and a viscous fluid. The most characteristic property is that of limited (“fading”) memory: viscoelastic materials partially resume their previous deformation state upon removal of the externally applied forces; the smaller the duration of the application of the forces, the better the recovery. Materials of this type contain a certain degree of internal microstructure (e.g., polymeric solutions and melts, advanced composites, liquid crystals, etc.), and are very important in the processing industry where one wishes to combine the “processability” of the medium's fluidity with the “structural quality” of the internal architecture to obtain high strength/ low-weight final products. We can distinguish two types of viscoelasticity: viscoelastic solids and viscoelastic fluids characterized by the ability or lack of ability respectively, to support shear stresses at finite deformations. In the following we shall focus on the analysis of viscoelastic fluids although the approach followed applies and/or can be extended in a straightforward fashion to viscoelastic solids as well. For a description of solid viscoelasticity, the interested reader may consult one of the many excellent monographs in the area [Eringen, 1962, chs. 8, 10; Ferry, 1980; Sobotka, 1984; see also Tschoegl, 1989].

The Dissipation Bracket

In the opening chapters of the book, we saw how a variety of conservative phenomena could be described through the Poisson bracket of classical physics. As already mentioned, the majority of systems with which engineers and physicists must deal reveal dissipative phenomena inherent to their nature. In the last chapter we offered a concise overview of nonequilibrium thermodynamics as traditionally applied to describe, close to equilibrium, dissipative dynamic phenomena. In this chapter, we lay the groundwork for the incorporation of non-conservative effects into an equation of Hamiltonian form, and show several well-known examples in way of proof that such a thesis is, in fact, tenable. The unified formulation of conservative and dissipative processes based on the fundamental equations of motion is not a new idea. Among the most complete treatments is the one offered by the Brussels school of thermodynamics [Prigogine et al., 1973; Prigogine, 1973; Henin, 1974] based on a modified Liouville/von Neumann equation. This is a seminal work where, using the mathematical approach of projection operators and relying only on first principles, it is demonstrated how large isolated dynamic systems may present dissipative properties in some asymptotic limit. A key characteristic of their theory is that dissipative phenomena arise spontaneously without the need of any macroscopic assumptions, including that of local equilibrium [Prigogine et al., 1973, p. 6]. The main value, however, is mostly theoretical, demonstrating the compatibility of dissipative, irreversible, processes with the reversible dynamics of elemental processes through a “symmetry-breaking process.” The value of the theory in applications is limited since it relies on a quantum mechanical equation for the density matrix ρ which is, in general, very difficult to solve except for highly simplified problems [Henin, 1974]. In a nutshell, we hope to offer with the present work a macroscopic equivalent of the Brussels school theory which, at the expense of the introduction of the local equilibrium assumption, attempts to unify the description of dynamic and dissipative phenomena from a continuum, macroscopic viewpoint. The main tool to achieve that goal is the extension of the Poisson bracket formalism, analyzed in chapter 5, to dissipative continua.