Porous solids desorption from

Vinten et al. (1983) demonstrated that the vertical retention of contaminated suspended particles in soils is controlled by the soil porosity and the pore size distribution. Figure 5.8 illustrates the fate of a colloidal suspension in contaminated water during transport through soil. Three distinct steps in which contaminant mass transfer may occur can be defined (1) contaminant adsorption on the porous matrix as the contaminant suspension passes through subsurface zones, (2) contaminant desorption from suspended solid phases, and (3) deposition of contaminated particles as the suspension passes through the soil. 

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... 

In the following discussion we will consider the application of percolation theory to describing desorption of condensate from porous solids. In Sections III,A-III,C we briefly recall types of adsorption isotherms, types of hysteresis loops, and the Kelvin equation. The matter presented in these sections is treated in more detail in any textbook on adsorption [see, e.g., the excellent monographs written by Gregg and Sing (6) and by Lowell and Shields (49) Sections III,D-III,H are directly connected with percolation theory. In particular, general equations interpreting the hysteresis loop are... 

A characteristic feature of type IV isotherms measured on porous solids is the hysteresis loop. The exact shape of the loop varies from one adsorption system to another, but the amount adsorbed is always larger at any given relative pressure along the desorption branch than along the adsorption branch. 

Application of percolation theory to describing the desorption process from porous solids is based on the identification of network sites with voids, and bonds with necks. A bond is considered to be unblocked if the neck radius r > Kp. Unblocked sites belonging to the percolation cluster correspond to voids containing nitrogen vapor. 

At present, data on the correlation effects in the void and neck arrangements are rather scarce. For this reason, the existing approaches to describing desorption from porous solids usually take into account only the simplest effect, i.e., the fact that the neck between the two nearest-neighbor voids should be lower than or equal to the size of the smaller void. The latter fact may be incorporated into the theory in two slightly different ways. The first approach outlined in this section was originally proposed by Zhdanov et al. (23). The second approach, which is presented in Section III,E, was developed by Mason (18-20) and Palar and Yortsos (26,27). 

Mason (18-20) and Palar and Yortsos (26,27) have employed another way of describing desorption from porous solids. Their approach is based on the assumption that the neck arrangement is random, i.e., the probability for an arbitrary neck to have a given value of the radius does not depend on the sizes of adjacent voids and necks. In this case, one can apply the percolation theory data obtained for the bond problem to all the voids. In particular, the probability for an arbitrary void to be empty during the desorption process is precisely 9 b(zo ), where the parameter z is given by Eq. (23). The latter probability is calculated for all the voids. We, however, know for a fact that voids with r < rp are filled. Thus the probability for a void with r > rp to be empty is just 9, (zoq)/F(rp), where F(rp) is the fraction of voids with r > rp [Eq. (33)]. Then, by analogy with Eq. (20), we derive... 

It is of interest to note that in the case under consideration (i.e., when the pore volume is concentrated in necks) the desorption process is described [Eqs. (38) or (39)] by employing only the neck-size distribution and the mean coordination number for voids. The neck-size distribution can be calculated from the adsorption branch of the isotherm. Thus, the analysis of the desorption branch of the isotherm allows one to obtain the mean coordination number. In fact, Zo can be obtained from the position of the desorption knee. In addition, using Eq. (39) and the scaling expression for 9 b2 [Eq. (12)], it is possible to estimate the average linear dimension L of the microparticles in porous solids 34). 

The results of simulations demonstrating the effect of various factors on nitrogen desorption from porous solids are presented in Figs. 16-18. To describe the desorption process, we use Eqs. (24) and (30). The adsorption branch of the isotherm is described by Eq. (18). The size distributions of necks and voids are assumed to be lognormal ... 

To describe mercury intrusion, one can use the same approaches as were employed in Section III for simulating condensate desorption from porous solids. In particular, if the pore volume is concentrated in voids (this model, shown in Fig. 2, has been analyzed in Refs. 14,37-41), the fraction of pore volume filled by mercury, C/in(rp), can be represented as (cf. Section III,D or Ref. 38)... 

The use of adsorption isotherms is subject to both theoretical and experimental limitations. There is effectively a minimum relative pressure value specific to each adsorbate (e.g. P/Pq = 0.42 for nitrogen, 0,2 for CCI4) which corresponds to the minimum value of the surface tension for the phase to remain in liquid form. Below this critical value, the liquid adsorbate is unstable and vaporises spontaneously, an effect represented on the desorption curves by a sharp drop in the adsorbed volume. Depending on the significance of this variation, the porous distribution calculated from the desorption data may show an artefact in the pore size domain corresponding to this process (3-4 nm in diameter). For a porous solid where this phenomenon occurs, it is advisable to study the adsorption curve. 

The onset of the hysteresis loop indicates the start of the capillary condensation mechanism. The desorption curve (AB C) is always above the adsorption branch (ABC), that is for a given loading adsorbate desorbs from a porous solid at a lower pressure than that required for adsorption. Before proceeding with the analysis of the isotherm, we first start with the basic capillary condensation theory of Lord Kelvin, the former William Thompson. 

Recent progress in the theory of adsorption on porous solids, in general, and in the adsorption methods of pore structure characterization, in particular, has been related, to a large extent, to the application of the density functional theory (DFT) of Inhomogeneous fluids [1]. DFT has helped qualitatively describe and classify the specifics of adsorption and capillary condensation in pores of different geometries [2-4]. Moreover, it has been shown that the non-local density functional theory (NLDFT) with suitably chosen parameters of fluid-fluid and fluid-solid interactions quantitatively predicts the positions of capillary condensation and desorption transitions of argon and nitrogen in cylindrical pores of ordered mesoporous molecular sieves of MCM-41 and SBA-15 types [5,6]. NLDFT methods have been already commercialized by the producers of adsorption equipment for the interpretation of experimental data and the calculation of pore size distributions from adsorption isotherms [7-9]. 

In this paper, we present a new vapour desorption set-up based on a microcalorimeter that provides a full control of temperature gradients. We demonstrate that is possible to carry out capillary condensation studies in the nanometer-micrometer range. In a first part the setup and the procedure established to study the desorption of a vapour from a porous solid that is initially immersed in an excess of liquid are described. In the second part a few results are shovra and compared to existent techniques. 

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