Pore size distribution, definition

Porous materials have attracted considerable attention in their application in electrochemistry due to their large surface area. As indicated in Section I, there are two conventional definitions concerning with the fractality of the porous material, i.e., surface fractal and pore fractal.9"11 The pore fractal dimension represents the pore size distribution irregularity the larger the value of the pore fractal dimension is, the narrower is the pore size distribution which exhibits a power law behavior. The pore fractal dimensions of 2 and 3 indicate the porous electrode with homogeneous pore size distribution and that electrode composed of the almost samesized pores, respectively. 

Fig. 1.18A shows the pore size distribution for nonporous methacrylate based polymer beads with a mean particle size of about 250 pm [100]. The black hne indicates the vast range of mercury intrusion, starting at 40 pm because interparticle spaces are filled, and down to 0.003 pm at highest pressure. Apparent porosity is revealed below a pore size of 0.1 pm, although the dashed hne derived from nitrogen adsorption shows no porosity at aU. The presence or absence of meso- and micropores is definitely being indicated in the nitrogen sorption experiment. 

N2 adsorption-desorption isotherms and pore size distribution of sample II-IV are shown in Fig. 4. Its isotherm in Fig. 4a corresponds to a reversible type IV isotherm which is typical for mesoporous solids. Two definite steps occur at p/po = 0.18, and 0.3, which indicates the filling of the bimodal mesopores. Using the BJH procedure with the desorption isotherm, the pore diameter in Fig. 4a is approximately 1.74, and 2.5 nm. Furthermore, with the increasing of synthesis time, the isotherm in Fig. 4c presents the silicalite-1 material related to a reversible type I isotherm and mesoporous solids related to type IV isotherm, simultaneously. These isotherms reveals the gradual transition from type IV to type I. In addition, with the increase of microwave irradiation time, Fig. 4c shows a hysteresis loop indicating a partial disintegration of the mesopore structure. These results seem to show a gradual transformation... 

Variants of preparation have been proposed [135, 248] including sintering [391] or co-electrodeposition of the precursors [138, 407], and aluminization of the surface of Ni at high temperature whose nature has a definite effect on the resulting electrocatalytic activity [408]. The main features of Raney Ni have been evaluated, including the pore size distribution and the real surface area [93, 135]. It has been found that the composition of the precursor alloys and their particle size have important influence on the adsorption properties of the resulting Raney metal, hence on its electrocatalytic properties [409]. 

Unsteady state diffusion in monodisperse porous solids using a Wicke-Kallenbach cell have shown that non-equimolal diffusion fluxes can induce total pressure gradients which require a non-isobaric model to interpret the data. The values obtained from this analysis are then suitable for use in predicting effectiveness factors. There is evidence that adsorption of the non-tracer component can have a considerable influence on the diffusional flux of the tracer and hence on the estimation of the effective diffusion coefficient. For the simple porous structures used in these tests, it is shown that a consistent definition of the effective diffusion coefficient can be obtained which applies to both the steady and unsteady state and so can be used as a basis of examining the more complex bimodal pore size distributions found in many catalysts. 

The available transport models are not reliable enough for porous material with a complex pore structure and broad pore size distribution. As a result the values of the model par ameters may depend on the operating conditions. Many authors believe that the value of the effective diffusivity D, as determined in a Wicke-Kallenbach steady-state experiment, need not be equal to the value which characterizes the diffusive flux under reaction conditions. It is generally assumed that transient experiments provide more relevant data. One of the arguments is that dead-end pores, which do not influence steady state transport but which contribute under reaction conditions, are accounted for in dynamic experiments. Experimental data confirming or rejecting this opinion are scarce and contradictory [2]. Nevertheless, transient experiments provide important supplementary information and they are definitely required for bidisperse porous material where diffusion in micro- and macropores is described separately with different effective diffusivities. 

G64) have described these methods. In both cases, the material must first be dried to water contents lower than those likely to be encountered in the normal environment of the material, and the effects can be serious and difficult to allow for. The amount of water considered to form part of the solid obviously affects the definition of porosity, but the extent and manner of drying also affect the pore size distribution. 

For the five mixtures, the cumulative mesoporous volume, Feds, and mesoporous surface area, S edB, and are both linear decreasing functions of the micropore content y (Figure 2b). The cumulative specific surface area SedB is definitely a better estimator of the mesoporous surface than the specific surface S xt computed Ifom the t-plot. The lUPAC classification states that mesopores are pores whose width is larger that 2 nm. In the case of the cylindrical pore model retained for the pore size distribution, this is equivalent to radii larger than 1 nm. It should however be stressed that the calculation of the cumulative surface and volume of the mesopores must not be continued at lower pressures than the closing of the hysteresis loop (gray zones of Figures 3a and 3b). If a black box analysis tool is used and if the calculation is systematically continued down to 1 nm, severe overestimation of the mesopores surface and volume may occur. 

Usually the pores in a material do not have the same size but exist as a distribution of size which can be wide or sharp. We can characterise a film by a nominal or an absolute pore size. In fact this definition rather characterises the size of the particles or molecules retained by the layer. Pore size distribution is classically represented by the derivatives dSp/dfp or dUp/drp as a function of Fp (pore radius) where Sp and Vp are respectively the wall area and volume of the pores. The size in question is here the radius, which implies that the pores are known to be, or assumed to be, cylindrical. In other cases, Fp should be replaced by the width. 

The customary definition of this distribution is that 80% of pore volume consists of pores whose size fall within 10% of the average pore size. Such narrow pore size distribution results from the narrow size distribution of alkali-borate heterogeneities formed in alkali-boro-silicate glasses during their thermal treatment. The uniformity of the kinetics... 

Fig. 2. Two-dimensional illustration of the geometric definition of the pore size distribution [25]. Point Z may be overlapped by all three circles of differing radii, whereas point Y is accessible only to the two smaller circles and point X is excluded from all but the smallest circle. The geometric pore size distribution is obtained by determining the size of the largest circle that can overlap each point in the pore volume. (Reproduced with permission from S. Ramalingam, D. Maroudas. and E. S. Aydil. Interactions of SiH radicals with silicon surfaces An atomic-scale simulation study. Journal of Applied Physics, 1998 84 3895-3911. Copyright 1998, American Institute of Physics.)...

Size can refer to volume, area, or length, and therefore pore-size distribution may be defined in terms of any one of these properties. In practice, the definition of size adopted is highly dependent upon the method of measurement. For example, the area size distribution of pores is often measured by image analysis of soil thin sections, while water retention data are usually interpreted in terms of the distribution of pore diameters (Bullock Thomasson, 1979). For consistency with the definition of the Peclet number, we have chosen to define size in terms of length, L. Dullien (1991) has proposed the following interrelationships between the different definitions of size L = VIS in three-dimensions or L=AJP in two-dimensions, where V is volume, S is surface area, A is cross-sectional area and P is perimeter. These relations can be used to compare pore-size distributions measured using different methods. 

Theoretically the use of the desorption branch of the isotherm is applicable for materials with pores predominandy comprised of independent capillaries, because this was the system used for the deduction of the Kelvin equation. If bottle-hke pores are anticipated, where the Hquid in the narrow opening prevents evaporation of the condensate from larger compartments, it is better to calculate pore size distribution using the adsorption branch of the hysteresis loop. Since the shape of pores is usually a priori unknown, the selection of the adsorption or desorption branch of the hysteresis loop largely remains arbitrary. Concerning the choice between dWo/dD or dfFo/dlog(D) plots, which by definition are different functions, one may note that the difference may be small for materials with narrow pore size distribution, but become more significant for a broad and unsymmetrical pore size distribution. 

For applications in modem technology fields, not only high surface area and large pore volume, but also a sharp pore-size distribution at a definite size and control of strrface natirre of pore walls are strongly required. In order to control the pore stmcture in carbon materials, studies on the selection of prectrrsors and preparation conditions have been extensively carried out and certain successes have been achieved [ 1-3]. Pore sizes and their distributions in adsorbents have to comply with requirements from different applications. Thus, relatively small pores are needed for gas adsorption and relatively large pores for liquid adsorption, and a very narrow PSD is reqtrired for molecular sieving applications. Macropores in carbon materials were fotmd to be effective for sorption of viscous heavy oils. Recent novel techniques to control pore stmcture in carbon materials can be expected to contribute to overcome this limitation [41-46]. 

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