Porous solids theory

Conventional bulk measurements of adsorption are performed by determining the amount of gas adsorbed at equilibrium as a function of pressure, at a constant temperature [23-25], These bulk adsorption isotherms are commonly analyzed using a kinetic theory for multilayer adsorption developed in 1938 by Brunauer, Emmett and Teller (the BET Theory) [23]. BET adsorption isotherms are a common material science technique for surface area analysis of porous solids, and also permit calculation of adsorption energy and fractional surface coverage. While more advanced analysis methods, such as Density Functional Theory, have been developed in recent years, BET remains a mainstay of material science, and is the recommended method for the experimental measurement of pore surface area. This is largely due to the clear physical meaning of its principal assumptions, and its ability to handle the primary effects of adsorbate-adsorbate and adsorbate-substrate interactions. 

Porous-Electrode Models. The porous-electrode models are based on the single-pore models above, except that, instead of a single pore, the exact geometric details are not considered. Euler and Nonnenmacher and Newman and Tobias were some of the first to describe porous-electrode theory. Newman and Tiedemann review porous-electrode theory for battery applications, wherein they had only solid and solution phases. The equations for when a gas phase also exists have been reviewed by Bockris and Srinivasan and DeVidts and White,and porous-electrode theory is also discussed by New-man in more detail. 

Figure 10. Resistor-network representation of porous-electrode theory. The total current density, i, flows through the electrolyte phase (2) and the solid phase (1) at each respective end. Between, the current is apportioned on the basis of the resistances in each phase and the charge-transfer resistances. The charge-transfer resistances can be nonlinear because they are based on kinetic expressions.

In this edition, we have incorporated new material in all the chapters and updated references to the literature. New sections dealing with porous solids, fullerenes and related materials, metal nitrides, metal tellurides, molecular magnets and other organic materials have been added. Under preparative strategies, we have included new types of synthesis reported in the literature, specially those based on soft chemistry routes. We have a new section covering typical results from empirical theory and electron spectroscopy. There is a major section dealing with high-temperature oxide superconductors. We hope that this edition of the book will prove to be a useful text and reference work for all those interested in solid state chemistry and materials science. 

It is extremely difficult to model macroscopic transport of mass, energy, and momentum in porous media commonly encountered in various fields of science and engineering based on microscopic transport models that account for variation of velocity and temperature as well as other quantities of interest past individual solid particles. The basic idea of porous media theory, therefore, is to volume average the quantities of interest and develop field equations based on these average quantities. 

Further development of theory of diffusion in porous solids. 

Measurements of the velocity and attenuation, usually as a function of frequency, can be used to provide valuable information about a system, e.g. microstructure. Theories are available which describe ultrasonic propagation in emulsions, suspensions, bubbly liquids, laminated solids, porous solids, fibrous materials and a number of other materials [20-29],... 

The capillary condensation theory provides a satisfactory explanation of the phenomenon of adsorption hysteresis, which is frequently observed for porous solids. Adsorption hysteresis is a term which is used when the desorption isotherm curve does not coincide with the adsorption isotherm curve (Figure 5.8). 

The theory of Brunauer, Emmett and Teller167 is an extension of the Langmuir treatment to allow for multilayer adsorption on non-porous solid surfaces. The BET equation is derived by balancing the rates of evaporation and condensation for the various adsorbed molecular layers, and is based on the simplifying assumption that a characteristic heat of adsorption A Hi applies to the first monolayer, while the heat of liquefaction, AHL, of the vapour in question applies to adsorption in the second and subsequent molecular layers. The equation is usually written in the form... 

Biot, M.A. (1972) Theory of finite deformations of porous solids. Indiana University... 

Abstract A simplified quintuple model for the description of freezing and thawing processes in gas and liquid saturated porous materials is investigated by using a continuum mechanical approach based on the Theory of Porous Media (TPM). The porous solid consists of two phases, namely a granular or structured porous matrix and an ice phase. The liquid phase is divided in bulk water in the macro pores and gel water in the micro pores. In contrast to the bulk water the gel water is substantially affected by the surface of the solid. This phenomenon is already apparent by the fact that this water is frozen by homogeneous nucleation. 

Taking into account the aforementioned effects of ice formation in porous materials, a macroscopic quintuple model within the framework of the Theory of Porous Media (TPM) for the numerical simulation of initial and boundary value problems of freezing and thawing processes in saturated porous materials will be investigated. The porous solid is made up of a granular or structured porous matrix (a = S) and ice (a = I), where it will be assumed that both phases have the same motion. Due to the different freezing points of water in the macro and micro pores, the liquid will be distinguished into bulk water ( a = L) in the macro pores and gel water (a = P, pore solution) in the micro pores. With exception of the gas phase (a = G), all constituents will be considered as incompressible. 

In the following investigation, we use this theory allowing a continuum thermomechanical approach to the two-phase flow problem in a porous solid. The transient and stationary motion of liquids in porous solids with small pores is complex and not all related problems have been solved yet. The main internal... 

Bluhm, J. (2002) Modelling of saturated thermo-elastic porous solids with different phase temperatures, in Porous Media Theory, Experiments and Numerical Applications, W. Ehlers, J. Bluhm (eds.), Springer-Verlag, Berlin/Heidelberg/New York... 

In this chapter, we present in some detail gas adsorption techniques, by reviewing the adsorption theory and the analysis methods, and present examples of assessment of PSDs with different methods. Some examples will show the limitations of this technique. Moreover, we also focus on the use of SAXS technique for the characterization of porous solids, including examples of SAXS and microbeam small-angle x-ray scattering (pSAXS) applications to the characterization of activated carbon fibers (ACFs). We remark the importance of combining different techniques to get a complete characterization, especially when not accessible porosity exists. 

In this section, after a brief review of the SAS theory, some examples of SAXS (mainly centered on ACF) will be discussed to show its application for the characterization of porous solids. 

In 1911 Zsigmondy pointed out that the condensation of a vapour can occur in very narrow pores at pressures well below the normal vapour pressure of the bulk liquid. This explanation was given for the large uptake of water vapour by silica gel and was based on an extension of a concept originally put forward by Thomson (Lord Kelvin) in 1871. It is now generally accepted that capillary condensation does play an important role in the physisorption by porous solids, but that the original theory of Zsigmondy cannot be applied to pores of molecular dimensions. 

In an early discussion of the adsorption of gases by solids, Rideal (1932) had stressed the fundamental importance of the nature and extent of the solid surface and had pointed out that it was necessary to evaluate the accessible area rather than the true specific area. Many attempts have been made to check the accuracy of the values of surface area derived from adsorption data (in particular from die BET theory), but the concept of surface area still remains problematical. In attempting to assess the magnitude of the surface area of a fine powder or porous solid, one is faced with a similar problem to that of a cartographer required to evaluate a coastal perimeter. Obviously, the answer must depend on the scale of the map and the manner in which the measurement is made. 

For pores smaller than 10 m, a molecular sieving effect can be present and the movement of one or more species inside the porous solid occurs due to the molecular interactions between the species and the network of the porous body here, for the description of species displacement, the theory of molecular dynamics is frequently used. The affinity between the network and the species is the force that controls the molecular motion at the same time, the affinity particularities, which appear when two or more species are in motion inside the porous structure, explain the separation capacity of those solids. We can use a diffusive characterisation of species motion inside a porous solid by using the notion of conformational diffusion. 

Porous electrode theory assumes that medium is a superposition of continuous solid and electrolyte phases with a known vo-Inme fraction. The solid phase potential of the positive electrode is because of electronic conduction ... 

Over the years, vapour adsorption and condensation in porous materials continue to attract a great deal of attention because of (i) the fundamental physics of low-dimension systems due to confinement and (ii) the practical applications in the field of porous solids characterisation. Particularly, the specific surface area, as in the well-known BET model [I], is obtained from an adsorbed amount of fluid that is assumed to cover uniformly the pore wall of the porous material. From a more fundamental viewpoint, the interest in studying the thickness of the adsorbed film as a function of the pressure (i.e. t = f (P/Po) the so-called t-plot) is linked to the effort in describing the capillary condensation phenomenon i.e. the gas-Fadsorbed film to liquid transition of the confined fluid. Indeed, microscopic and mesoscopic approaches underline the importance of the stability of such a film on the thermodynamical equilibrium of the confined fluid [2-3], In simple pore geometry (slit or cylinder), numerous simulation works and theoretical studies (mainly Density Functional Theory) have shown that the (equilibrium) pressure for the gas/liquid phase transition in pores greater than 8 nm is correctly predicted by the Kelvin equation provided the pore radius Ro is replaced by the core radius of the gas phase i.e. (Ro -1) [4]. Thirty year ago, Saam and Cole [5] proposed that the capillary condensation transition is driven by the instability of the adsorbed film at the surface of an infinite... 

Section III is concerned with moisture movement through porous solids. The general theory of moisture distribution and the rate of moisture movement inside porous media is reviewed. The three theories of condensation— diffusion, capillarity, and vaporization—are discussed. The roles of various mechanisms causing liquid movement in solids are assessed. 

Section V deals with the drying of porous solids in continuous operations. The study of drying in rotary and tunnel dryers is presented based on the relationships derived from basic theory. The effect of the operating variables on drier performance is discussed. A suitable procedure is developed for sizing rotary and tunnel driers. 

To be valid, each theory has its specific requirements. The major factors that decide the mode of liquid movement through porous solids are the nature of the liquid, the structure of the solid, the concentration of liquid, and the temperature and pressure of the system. 

Fractal geometry has been used to describe the structure of porous solid and adsorption on heterogeneous solid surface [6-8]. The surface fractal dimension D was calculated from their nitrogen isotherms using both the fractal isotherm equations derived from the FHH theory. The Frenkel-Halsey-Hill (FHH) adsorption isotherm applies the Polanyi adsorption potential theory and is expressed as ... 

Application of Percolation Theory to Describing Kinetic Processes in Porous Solids... 

Most of the pore structures (e.g., spongy structures) consist of extensive three-dimensional networks in which there is a profusion of interconnections between voids within the structure. The latter interconnections affect considerably the kinetics of various processes in porous solids. This effect can adequately be described by employing the ideas developed in percolation theory 7-13). In the framework of this theory, the medium is defined as an infinite set of sites interconnected by bonds. Percolation theory can be applied to porous solids via identification of network sites with voids, and bonds with necks. Thus, the theory is applicable primarily to spongy porous structures but in some cases also to corpuscular structures. 

During the past decade, percolation theory has been successfully used to analyze condensate desorption from porous solids (14-34), mercury penetration into porous solids (35 -43), and the kinetics of catalytic deactivation... 

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