Unimodal pore size distribution

In the case of a chromatographic column packed with porous adsorbent particles showing a unimodal pore size distribution, the differential mass balance of the solute in a porous adsorbent particle can be described by... 

In the case of columns packed with spherical porous absorbent particles showing unimodal pore-size distribution the theoretical plate is... 

The commercial catalyst used in this work contains 12 wt% Ni and 83 wt% a-Al203. It has a BET total surface area of 3.4m /g and a unimodal pore size distribution with volume 0.155 cc/g, mean pore radius 1600 A and void fraction 0.362. Its activation required a reduction which was carried out under atmospheric pressure in situ, for 72 hrs at 850°C by means of a pure dried hydrogen flow of lOO Nl/hr. These severe reduction conditions were required because 20 wt% of the Ni was present as NiAl204-spinel phase, which could only be reduced above 770°C. It led to a very active catalyst, with a specific Ni-surface area of 0.68 m Ni/g.cat. 

For unimodal pore size distribution the effective diffusivity inside a catalyst particle is given by Bosanquet s formula ... 

Here Cq = c(z = 0,t) and Dg is the effective diffusion coefficient (porosity and tortuosity effects are incorporated in Dg). If the upstream (high) pressure is constant and much larger than the downstream (low) pressure, the slope of the asymptote will correspond to the steady state and so it is possible to determine the diffusivity under both steady state and transient conditions from a single permeation experiment. With a narrow and unimodal pore size distribution both methods yield reasonable consistent values. Large discrepancies point to strong microstructural effects (bimodal broad distribution, many dead ends, many defects). 

In summary, a fairly narrow unimodal pore-size distribution can be adequately described by the simple mean pore-size model. A broad pore-size distribution, /(r), requires a more extensive treatment, such as the parallel path model. A bimodal pore-size distribution can also be described by the micro-macro random pore model. 

Finally, the pol mer (B) and the compatibUizer (C) is extracted at least partially, from the polymer blend by dissolving (B) and (C) in a solvent that is a non-solvent of polymer (A). In this way, the article becomes essentially continuously porous. A void volume of 10-90% can be obtained. The pore diameters show a unimodal pore size distribution. 

In this figure, it is possible to see that the sample prepared without AC, which here is considered as the standard preparation, led to a unimodal pore size distribution with pore sizes of the order of 39 A in diameter. On the other hand, when 2.88 g. of AC are used in the preparation (experiment 1.2 in table 1), pores of larger diameter, around 130 A, begin to appear, and then, with even more AC, the pore size distribution turns to be clearly bi-modal with two maximums in the pore size distribution curve around 40 and 130 A. 

All the samples exhibited unimodal pore size distributions except the one with surface area of 39.7m /g exhibited double humped pore size distributitMis. With increasing temperature the pore size distribution and consequently also the most fiequently pore radii are seen to towards higher values. 

The NiAl (1 1) catalyst showed a high surface area (186 m /g), similar to that reported in the literature, when using an expensive alkoxide precursor and a very long preparation time by the sol-gel method [96, 100, 111, 114]. The NiAl (1 1) presented also unimodal pore size distribution in the range of 3-6 nm and pore volume of 0.18 mL/g, indicating mesoporous materials. On the contrary, the samples with excess of urea, NiAl (2 1), presented lower surface area (10 mVg) and nonporous material crystallite sizes of 13.2 and 16.6 nm, respectively. Nanosized crystallites of NiAl204 were obtained by Han et al. [98, 113] applying the sonochemical method, but with very poor crystallinity. 

Pore size optimization is one area where developmental efforts have been focused. Unimodal pore (NiMo) catalysts were found highly active for asphaltene conversion from resids but a large formation of coke-like sediments. Meanwhile, a macroporous catalyst showed lower activity but almost no sediments. The decrease of pore size increases the molecular weight of the asphaltenes in the hydrocracked product. An effective catalyst for VR is that for which average pores size and pore size distribution, and active phase distribution have been optimized. Therefore, the pore size distribution must be wide and contain predominantly meso-pores, but along with some micro- and macro-pores. However, the asphaltene conversion phase has to be localized in the larger pores to avoid sediment formation [134],... 

Novel general expressions were developed for the description of the behaviour of the height equivalent of a theoretical plate in various chromatographic columns such as unpacked (open capillary), packed with spherical nonporous particles and packed with spherical porous adsorbent particles. Particles may have unimodal or bimodal pore size distribution. The expression describing the mass balance in open capillaries is... 

Figure 1 shows the pore size distribution of primary supports and the immobilized derivatives in silica matrix on an incremental (derivative) basis, dV/d Log (D), to highlight the differences among the immobilized derivatives in silica gels. The computation of pore size distribution, from gas adsorption, was based on the BJH method using the desorption branch. From these plots it was observed that all solid samples (PS, SPS, ADS, CB1, CB2, EN1, and EN2) exhibit a unimodal distribution of pores, and the smaller mean pore diameters were observed with the samples PS, ADS, CB1, and CB2 (Table 2). 

This relatively large surface area and the fact that the isotheim is of Type IV indicate that this film has porosity in the mesoporous range. A pore size distribution using the desorption branch of the isotherm is shown in Figure 4.16 (page 190). A fairly unimodal distribution is obtained with a median pore diameter of... 

One further difference exists between HDS and HDM. Bridge [37] has shown, very clearly, that HDS is not limited by diffusion while HDM is. Using a nickel-molybdate based catalyst with a unimodal microporous size distribution, the demetalation of Arabian heavy atmospheric residuum was found to be affected by catalyst particle size, while HDS was not. As the diameter of the pore was decreased, the maximum in the metals deposition profile moved closer to the external surface of the pellet, agmn indicating difiusional limitations for FIDM. 

Figure 6.15 Pore size distribution in plasma-sprayed CaO-TiOj-ZrOj Oj coatings, (a) Unimodal distribution, mean porosity P = 10.8%. (b) Multimodal distribution, mean porosity P > 15.5% (Heimann, 2006b).

Taki, K. Okumura, S., Production of Porous Polymer Foams with Unimodal or Bimodal Pore-Size Distributions via Depressurization and Photopolymerization-Induced Bubble Nucleation in Low-Viscosity UV-Curable Monomer/High Pressure CO Solutions. Macromolecules 2010,43, 9899-9907. 

Relative density versus reduced time for particulate gels. Data (symbols) disagree with cylinder model (solid curve), possibly because gel contains bimodal pore size distribution and data are compared to model for unimodal pores. From Johnson e/a/. [42], reprinted by permission of the American Ceramic Society. 

Roth et al. [80] proposed a method to determine the state of membrane wear by analyzing sodium chloride stimulus-response experiments. The shape of the distribution of sodium chloride in the permeate flow of the membrane revealed the solute permeation mechanisms for used membranes. For new membranes the distribution of sodium chloride collected in the permeate side as well in the rejection side was unimodal. For fouled membranes they noted the presence of several modes. The existence of a salt leakage peak, as well as an earlier detection of salt for all the fouled membranes, gave evidence of membrane stmcture modification. The intensive use of the membranes might have created an enlargement of the pore sizes. Salt and solvent permeabilities increased as well. While this is a difficult paper to follow, it may be of use to those who want to develop new methods for measuring membrane degradation. 

If a unimodal pore network of arbitrary size is considered then, if the spatial distribution of pore sizes is non-random, the desorption percolation transition would be apparently smeared out (in addition to any finite size effect). It is possible that particular pores occupied by liquid-like phase might gain access to the vapour phase before would be expected to be the case for a purely random system because the actual layout of the pores might provide a convenient access route that would not have existed at that bond occupation level in a random system. The simulations of the nitrogen sorption... 

Controlled pore glass (CPG) has application as media for size exclusion and afQnity chromatographic media, commonly used to separate proteins. Nominally, the pore diameter should be 10 times the protein diameter. Most proteins [51] have a diameter below 3 nm. CPGs are required to have sharp, unimodal pore distributions, that is a high volume of pores of similar size. Clearly, a means of investigating pore size, pore... 

Madkour, T. Mark, J. E., Some Evidence on Pore Sizes in Poly(dimethylsiloxane) Elastomers Having Unimodal, Bimodal, or Trimodal Distributions of Network Chain Lengths. Polym. Bull. 1993, 31, 615-621. 

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