Critical Behaviour in Confined Systems

The prediction of properties in complex materials is a problem of importance in many applications in chemical and materials engineering; by the term “complex material” we mean a heterogeneous substance, like a porous material containing a confined fluid. Such materials appear in many technological applications, including: (1) processes using supercritical fluids to dry porous aeorogels and thin films [1], (2) physical adsorption of trace components from gaseous effluents, (3) gas storage using microporous materials [2], and (4) chemical separation using inorganic membranes [3]. Inorganic membranes are often highly porous and randomly structured materials with large surface areas available for adsorption, a property that makes them useful in chemical separation and as catalyst supports. In addition to their heterogeneity, complex materials have another distinguishing characteristic that relates to the structure of the heterogeneity itself. Is it periodic, or is it dispersed throughout in some random fashion? These two situations are quite distinct and may, in each instance, show critical behavior for a confined fluid belonging to entirely different universality classes, an issue that to the present time is still unsettled in the literature. In this chapter, we investigate the critical properties of fluids confined in randomly structured host materials like that found in porous silicon. The main question we address is: how does confinement in a porous structure affect the critical point or phase behavior of a fluid mixture? Before investigating some of the more advanced ideas in this area, we look at the basic thermodynamics of interfaces, and the phenomenon of capillarity in a single idealized pore structure. This simple example provides the impetus for a more detailed study of confinement effects. Consider two phases in equilibrium separated by an interface. The total energy of the composite system is the sum of the energy of each phase plus the energy associated with the interface. In formulating the fundamental thermodynamic equation for energy in this system, we presume that the formation of an interface requires energy; therefore, the energy equation must reflect this fact.

Supercritical Adsorption

In this chapter, we discuss adsorption phenomena in supercritical systems, a situation that occurs in many application areas in chemical-process and materials engineering. An example of a commercial application in this area, which has achieved wide acceptance as a tool in analytical chemistry, is supercritical fluid chromatography (SFC). Not only is SFC a powerful technique for chemical analysis, but it also is a useful method for measuring transportive and thermodynamic properties in the near-critical systems. In the next section, we analyze adsorption-column dynamics using simple dynamic models, and describe how data from a chromatographic column can be used to estimate various thermodynamic and transport properties.We then proceed to discuss the effects of proximity to the critical point on adsorption behavior in these systems. The closer the system is to its critical point, the more interesting is its behavior. For very dilute solute systems, like those considered here, the energy balance is often ignored to a first approximation; this leads to a simple set of mass-balance equations defining transport for each species. These equations can be developed to various levels of complexity, depending upon the treatment of the adsorbent (stationary phase). The conceptual view of these phases can span a wide range of possibilities ranging from completely nonporous solids (fused structures) to porous materials with complicated ill-defined pore structures. Given these considerations, it is customary to make the following assumptions in the development of a simple model of adsorber-bed dynamics: . . .1. The stationary and mobile phases are continuous in the direction of the flow, with the fluid phase possessing a flat velocity profile (“plug” flow).. . . . . . 2. The porosity of the stationary phase is considered constant irrespective of pressure and temperature conditions (i.e., it is incompressible). . . . . . .3. The column is considered to be radially homogeneous, leading to a set of equations with one spatially independent variable, representing distance along the column axis. . . . . . . 4. The dispersion term in the model equation represents the combined effects of molecular diffusion and dispersion due to convective stirring in the bed. These effects are combined into an effective phenomenological dispersion coefficient, considered to be constant throughout the column. . . .

Scaling Near the Critical Point in Mixture

The critical point of mixtures requires a more intricate set of conditions to hold than those at a pure-fluid critical point. In contrast to the pure-fluid case, in which the critical point occurs at a unique point, mixtures have additional thermodynamic degrees of freedom. They, therefore, possess a critical line which defines a locus of critical points for the mixture. At each point along this locus, the mixture exhibits a critical point with its own composition, temperature, and pressure. In this chapter we investigate the critical behavior of binary mixtures, since higher-order systems do not bring significant new considerations beyond those found in binaries. We deal first with mixtures at finite compositions along the critical locus, followed by consideration of the technologically important case involving dilute mixtures near the solvent’s critical point. Before taking up this discussion, however, we briefly describe some of the main topographic features of the critical line of systems of significant interest: those for which nonvolatile solutes are dissolved in a solvent near its critical point. The critical line divides the P–T plane into two distinctive regions. The area above the line is a one-phase region, while below this line, phase transitions can occur. For example, a mixture of overall composition xc will have a loop associated with it, like the one shown in figure 4.1, which just touches the critical line of the mixture at a unique point. The leg of the curve to the “left” of the critical point is referred to as the bubble line; while that to the right is termed the dew line. Phase equilibrium occurs between two phases at the point where the bubble line at one composition intersects the dew line; this requires two loops to be drawn of the sort shown in figure 4.1. A question naturally arises as to whether or not all binary systems exhibit continuous critical lines like that shown. In particular we are interested in the situation involving a nonvolatile solute dissolved in a supercritical fluid of high volatility.

Mean-Field Theories

The most widely used analytic models for representing thermodynamic behavior in supercritical ßuids are of the mean-Þeld variety. In addition to the practical interest in studying this topic, this class of models is also the conceptual starting point for any microscopic discourse on critical phenomena. In this chapter we take up the basic ideas behind this approach, studying different physical models, showing how their mean-Þeld approximations can be constructed as well as investigating their critical behavior. A useful conceptual model for understanding mean-Þeld ideas is the Ising model whose properties we consider in some detail, especially its mean-Þeld approximation. The Ising model has the advantage of belonging to the same critical universality class as so-called simple fluids, deÞned as ßuids with short-range intermolecular potentials. Most supercritical ßuid solvent systems of practical interest fall within this class; hence results developed using the Ising model have important implications for understanding the critical behavior of this entire universality class. While we discuss universality and related ideas in more detail in subsequent chapters, sufÞce it to say here that the Ising system belongs to arguably the most important critical universality class from a process engineering standpoint. In its simplest form, the Ising model considers N spins arranged on a lattice structure (of 1, 2, or 3 dimensions) with each spin able to adopt one of two (up or down) orientations in its lattice position. A speciÞc state of the system is determined by a given conÞguration of all the spins. The model can be made more complex by considering additional degrees of freedom to the spin orientations. For example, the Heisenberg model considers a 3-dimensional lattice with the spin orientation at each lattice site described by a 3-dimensional vector quantity. All that is required to facilitate the use of statistical mechanics with this model is the deÞnition of the Hamiltonian (the systemÕs energy function) associated with a particular lattice state υ. This Hamiltonian usually consists of spinÐspin interaction terms, as well as a term representing the presence of a magnetic Þeld, which serves to orient the spins in its direction.

Thermodynamic Scaling Near the Critical Point

Thermodynamic scaling near the critical point is a signature of critical phenomena, and many useful applications of supercritical solvent fluids depend upon exploiting this behavior in some technologically interesting way. Near the critical point, many transport and thermodynamic properties show anomalous behavior which is usually linked to the divergence of certain thermodynamic properties, such as the fluid’s isothermal compressibility. In figures 3.1 and 3.2 we depict the near-critical behavior of both the density of xenon and the thermal conductivity of carbon dioxide, respectively, adapted from published data [1, 2]. The onset of what appear to be critical singularities in these properties is clearly evident in both instances. In this chapter, we focus upon the thermodynamic basis for this type of behavior. In the theory of critical phenomena, the limiting behavior of certain thermodynamic properties near the critical point assumes special significance. In particular, properties that diverge at the critical point are of interest, and this divergence is usually described in terms of scaling laws.

The Renormalization-Group Method

The renormalization-group (RG) method discussed in this chapter has assumed a pivotal role in the modern theory of critical phenomena. It attempts to relate the partition function of a given system to that of a “similar system” with decreased degrees of freedom through a process referred to as renormalization. Exactly how these degrees of freedom are removed from the system, what we mean by a “similar” system, and how successive systems are coupled to one another are essentially the questions we take up in the introductory treatment given in this chapter. The RG method is a topic with large scope and found widely disseminated in an extensive physics literature on the topic; however, it is seldom found in engineering journals. Our purpose here is to try and make sense of some basic ideas with the RG approach so that it is more accessible to this wider community. For this we often rely upon some prior exposes of the subject in more specialized settings [1, 2, 3, 5]. In its complete sense, the RG method has only been made to work, at least analytically, for a few simple statistical-mechanical models. But aside from these numerical results, many important and quite general insights about critical phenomena can be developed from studying this approach to the problem, especially the central role played by length scale as a factor in describing the phenomenology. These ideas have significantly enhanced our understanding of ideas like scale invariance, universality classes, relevant scaling fields (as opposed to irrelevant ones), Hamiltonian renormalization, and so on; these and related concepts lie at the center of modern discourse on the subject. The essential concepts of the approach can be well illustrated using the Ising system since, with this model, lattice spins are fixed in space, which makes the analytical work quite transparent. This approach, called real space renormalization, is the RG method studied in this chapter.

Fundamentals of Thermodynamic Stability

The second law of thermodynamics states that the entropy change in any spontaneous adiabatic process is greater than or equal to zero. It is a disarmingly simple statement but one that is a cornerstone of scientific theories. It is instrumental in describing the extent and direction of all physical and chemical transformations and contains within it the essential ideas for developing thermodynamic stability theory. Stability theory concerns itself with answering questions such as (1) What is a stable thermodynamic state? (2) Which conditions define the limit to this state beyond which the system becomes unstable? (3) How does the instability manifest itself ? In a real sense, stability theory provides the underlying framework for a macroscopic understanding of phase transitions and critical phenomena, the subject of this text. Many of the results of stability theory related to phase equilibria are well known; an example is the condition that, for a pure fluid in a stable state, the quantity −(∂P/∂V )T,N must be greater than or equal to zero, with the equality condition holding at the limit of stability. Many other facets of thermodynamic stability theory, however, are relatively unfamiliar. For example, any of the well-known thermodynamic potentials E, H, A, and G can be used to develop stability criteria for a given system. Are these criteria always equivalent, or do some take precedence over others? If so, what are the implications of this for understanding phase transformations in physicochemical systems? It is questions of this sort that we take up in this chapter, where we lay the macroscopic foundations for the material developed throughout the rest of the text. In this analysis, we rely heavily upon results taken from linear algebra, a branch of mathematics that provides an ideal tool for developing a comprehensive description of thermodynamic stability concepts. The combination of the first and second law of thermodynamics for a closed system leads to the well-known equation: . . . dE = T dS − PdV . . . . . . (1.1) . . . where E(S, V ) represents the system energy as a function of the independent variables S and V.

Solvation in Supercritical Fluids

The use of supercritical fluids as solvent media is driven mainly by the need to reduce the use of organic and halogenated solvents in chemical processes. In the future, one of the main aims of research in this area will be to supplant organic solvent use in many of these processes with solvents such as supercritical carbon dioxide, environmentally a much more acceptable alternative. One of the most common engineering requirements in this area is the need to predict solubility, and other thermodynamic behavior, in high-pressure mixtures where the solvent is close to its critical point and contains nonvolatile solute species of large molecular weight present in small amounts. In this chapter, we address this problem focusing upon solvation in organic solid–supercritical fluid systems which are among the most technologically interesting. The extension of the analyses presented here to situations where the condensed phase may be a mixture of miscible liquids, for example, is straightforward and left to a problem in the additional exercises.

Transport in the Critical Region

The behavior of dynamic properties in the critical region is important in many engineering applications, and in this chapter we investigate this topic, focusing upon diffusion. In the literature, the term critical slowing down is used to describe the long relaxation times that occur when criticality is approached. Does this mean that diffusion processes per se come to a halt and, if not, how does slowing down manifest itself in fluids? We see that, in spite of the nonequilibrium nature of this topic, equilibrium concepts still play a key role in describing dynamics in the critical region. To begin this discussion, we investigate the dynamic behavior of a tagged fluid molecule as it experiences random fluctuations in its position in the fluid. These fluctuations would be induced by random thermally induced collisions between the tagged species and other fluid molecules. This type of dynamics is referred to as Brownian motion or a random walk. A schematic showing two 10-step trajectories depicting a random walk in two dimensions is shown in figure 12.1, and its analysis leads naturally to the definition of the self-diffusion coefficient.