PSD calculations

The advantage of the PSD calculation method outlined in this subsection is that only one additional differential equation need be integrated, a fact which considerably simplifies matters and saves in computation time. A drawback of the present approach (and also of the other approaches of Table II) is that the statistical variation of the distribution arising from the variation of the number of radicals per particle in a class of a certain size is missing. The PSD s thus calculated indicate the average particle diameter growth of each particle. 

In the current work, the aforementioned novel approaches for determination of PSDs will be discussed. A critical comparison of the different methods for the PSD calculation will be presented in order to make some recommendations for practical applications. Advances in the analysis of surface properties of as-synthesized MMSs by means of comparative methods will also be discussed. 

Figure 4. (a) Capillary condensation/evaporation pressures as functions of the pore diameter for argon adsorption at 87 K. The solid line corresponds to Eq 5. See Fig. la for additional explanations, (b) PSDs calculated from nitrogen adsorption data at 77 K and argon adsorption data at 87 K using the BJH algorithm with the KJS relations (Eqs. 2 and 5, respectively). 

The PSDs calculated by the CONTIN procedure and the Micromeritics DFT are similar i.e., represented estimations of f(x) are quite reliable, which are in agreement with the structural parameters calculated by the DS method. Notice that the availability of broad mesopores x < 50 nm and transport pores at x > 50 nm (Fig. 1, a and b) is important with respect to adsorption of such biopol5mier molecules cis endotoxins or inflammatory c)nokines of peptide/protein nature. 

Figures 6 and 7 show PSDs calculated from nitrogen adsorption method and-thermo-desorption method.

As is seen PSDs calculated from the data obtained by using different techniques are close together. It should be noted that in the case of the nitrogen desorption data corrections with respect to the surface film thickness were introduced. 

It is shown that the position of PSDs calculated by the HK method depend strongly on the A(w)-relationship used. For instance, the pore widths at the maxima of PSDs obtained by the HK method with the Saito-Foley expression for cylindrical pores are underestimated about 1.4 nm. However, the HK method with the relationship between A and w established on the basis of good-quality MCM-41 materials [26] provides an accurate estimation of the pore widths of mesoporous silicas. While the position of PSD may be improved by a proper selection of the A(w)-relation, its unphysical features remain. The height of main peak is significantly reduced in order to compensate the appearance of an artificial small peak and tail in the micropore-mesopore transition range. These artifacts arise from the condensation approximation used in the HK method, which does not provide a good representation for the volume filling of micropores. 

The PSD curves in the case of nitrogen adsorption, were calculated by the BJH method [43] with corrections of the pore radii in respect of the surface film thickness d, where d = 4.3" 5/(ln p/Po). Points in Figures 11 and 12 represent the PSDs calculated from nitrogen adsorption data. 

As shown, in the case of acetone, the location of the peak of PSD, rpeak is shifted toward smaller radii in comparison with the PSD calculated from nitrogen adsorption data. However, in the case of n-butanol and carbon tetrachloride the difference rp/ j - r e j is small. It should be noted that the total pore volumes calculated from TG curves for both adsorbents are very similar to those estimated from the nitrogen adsorption isotherms. 

This would allow performing accurate PSD calculations using these simple algorithms. Theoretical considerations [13], nonlocal density functional theory (NLDFT) calculations [62, 146], computer simulations [147], and studies of the model adsorbents [63, 88] strongly suggested that the Kelvin equation commonly used to provide a relation between the capillary condensation or evaporation pressure and the pore size underestimates the pore size. 

The analysis program takes into account the PSD correction when it calculates zeta potential. It uses either PSD calculated from the attenuation spectra or a priori known PSD. The analysis routine also makes a correction for aden-uation of the sound pulse. 

Nanosilica A-400 (S BKr=409 mVg and bulk density Pb=0.061 g/cm ) was selected as a material with small nanoparticles (d, =6J nm) relatively strongly aggregated (pore volume 1 =0.86 cmVg) and having a broad PSD calculated using the nitrogen adsorption isotherm with the model of a... 

FIGURE 1.206 Comparison of the pore (voids between nanoparticles) size distributions calculated using four methods (1) NMR cryoporometry and (2) NMR relaxometry, (3) and (4) TSDC cryoporometry (aqueous suspensions at Cj, qq=3-1 wt%), and (5) PSD calculated using the nitrogen adsorption isotherm (SCV/SCR model, see Section 1.1.1). 

FIGURE 1.207 Comparison of the PSD calculated for 5 wt% suspension of A-300 using nonintegral (1) and integral (2) Gibbs-Thomson equations. 

FIGURE 1.256 PSDs calculated with the NMR cryoporometry with (a, b) GT and (c, d) IGT equations for A-300, modified silica (MS) and their mixture. (Adapted from J. Colloid Interface ScL, 308, Gun ko, V.M., Turov, V.V., Zarko, V.I. et al. Comparative characterization of polymethylsiloxane hydrogel and silylated fumed silica and silica gel, 142-156, 2007h. Copyright 2007, with permission from Elsevier.)... 

Notice that the PSD calculated from the NMR data overlaps with the PSDs calculated from water adsorption and mercury intrusion porosimetry (Valckenborg 2001). 

FIGURE 4.31 PSD (calculated with GT equation and corresponding to the size distributions of clusters of unfrozen bound water) for initial silica gel Si-60, AC-1, and carbosils with Ti, Zn, and Zr oxides (N2 NLDFT PSD is shown for Si-60, some PSD are displaced along the F-axis for better view). 

FIGURE 4.32 PSD calculated using (a, b) cryoporometry ((a) differential and (b) incremental PSD) and (c) nitrogen adsorption data for initial and modified Si-60 (IPSD). 

FIG U R E 5.37 PSDs calculated from the DSC data for HEM A-AGE (A, B, C, and D samples) and G gels (hydration h=mjm where is the weight of water evaporated on DSC measurements to 160°C and is the residual weight of heated sample) at (a) high and (b, c) low hydration. (Adapted from Savina, IN., Gun ko, V.M., Turov, V.V. et al., Noninvasive structural characterisation of macroporous cross-linked polymer and protein hydrogels. Soft Matter, 7,4276-4283,2011, Copyright 2011, with permission from The Royal Society of Chemistry.)... 

FIGURE 6.19 The size distributions of pores filled by unfrozen water for (a) aqueous suspension of A-300 and the PSD calculated on the basis of the nitrogen adsorption/desorption isotherm (with the model of voids between spherical particles) (b) 0.15 mol NaCl or HPF solution at 1.25 and 2.5 wt% and (e) HPF/A-300 at different concentrations of protein and siliea. (Adapted with kind permission from Springer Seience+Business Media Cent. Eiir. J. Chem., Interaction of fibrinogen with nanosilica, 5, 2007, 32-54, Rugal, A.A., Gun ko, V.M., Barvinchenko, V.N., Turov, W., Semeshkina, T.V., and Zarko, V.I.)... 

The shape of the nitrogen adsorption isotherm corresponds to type IV according to lUPAC classification (Figure 20.16). The brusque increase of the adsorbed volume at low relative pressure (<0.05) is characteristic of microporosity. The PSD, calculated from the BJH algorithm shows a narrow peak (inset in Figure 20.16) with its maximum at 3 nm. 

The radius of the cylindrical pore can thus be defined as tr +1. The widely used BJH (30) and DH (31) methods apply this concept in PSD computation. The shape of the hysteresis loop may help to decide which branch of the isotherm is suitable for the PSD calculation. For H2 and H3 the adsorption branch, and in the case of the H 1-type hysteresis the desorption branch is recommended (32). 

Fig. 2 (a) Nitrogen adsorption/desorption isotherm at 77 K and (b) PSD calculated by DFT method of the isotherm for typical activated carbon (YP50F, steam activated coconut shell carbon) for EDLC... 

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