Radius distribution

Figure 3 The radius distribution function between the centers of the P4VP domains of (a) the block copolymer BCl, and (b) the microsphere MCI [24].

Figure 14 Radius distribution functions (a) between the P4VP cores of the microspheres, (b) from P4VP core to P2VP spherical microdomains, (c) from P2VP spherical microdomain to P4VP core, and (d) between P2VP spherical microdomains [37].

Anderson (A2) has derived a formula relating the bubble-radius probability density function (B3) to the contact-time density function on the assumption that the bubble-rise velocity is independent of position. Bankoff (B3) has developed bubble-radius distribution functions that relate the contacttime density function to the radial and axial positions of bubbles as obtained from resistivity-probe measurements. Soo (S10) has recently considered a particle-size distribution function for solid particles in a free stream ... 

In contrast, the hydrodynamic radius distribution recorded for aqueous solutions of copolymers with a low grafting density is bimodal, with a contribution from small entities of = 5-30 nm, assigned to single polymer chains, and a contribution from larger particles of = 80 = 150 nm (Fig. 24b, c). The relative importance of the two populations depends on copolymer concentration the relative amount of the larger particles increases with increasing copolymer concentration. 

Rgure 2.1. Pore radius distribution curve for a porous Vycor glass heat-treated at 500 C for 5 h (McMillan 1980). 

When we wanted to numerically fit experimental PFGE data of water diffusion in a water-in-oil emulsion, we found that for a beginner in this field the literature is quite confusing. First, all three expressions for diffusion in a sphere with reflecting walls are somewhat different and lead to very different fitting results, especially when the formulas are combined with a radius distribution function. Since the derivation of the published expressions needs some tedious algebra (which has not been published), it is not trivial to check the derivation in order to establish which expression is the correct one. Here we use a numerical approach to decide which expression is correct. 

Figure 2. Typical hydrodynamic radius distributions (/(RjO) of individual triblock PMMA-Z>-PS-Z>-PMMA copolymer chain end-capped with oxalyl chloride in a solvent mixture of methyl acetate and acetonitrile (10/1, v/v) at 45 °C and the aggregates formed via the self-assembly of the triblock copolymer chains at 29 °C, where the triblock copolymer concentration is 1 x 10 4 g / mL.[35]...

Before the coupling reaction, the self-assembly of PI-Z>-PS-Z>-PI triblock copolymer chains in w-hcxane was investigated by LLS. Figure 7 shows typical hydrodynamic radius distributions (/(Rh)) of individual PI-Z>-PS-Z>-PI triblock chains in THF, a good solvent for both the PI and PS blocks, and the core-shell micelles formed via the self-assembly of the triblock copolymer chains in -hexane, a solvent selectively good for the PI block. The shifting of the peak from... 

Figure 7. Typical hydrodynamic radius distributions (/(Rh)) of individual PI-6-PS-6-PI triblock copolymer chains end-capped with butyl bromide group (SI44) in THF and their self-assembled coreshell micelle in w-hexane, where C = 1.0 x 10 2 g/mL and T = 25.0 °C.

Figures 22,23 and 24, respectively, depict the apparent hydrodynamic radius distributions f(Rh) for STVPh-3, STVPh-9, and STVPh-15, as well as their blends with PEMA in toluene. Here, the STVPh unit fraction Fr is defined as the total moles of styrene and vinylphenol monomer units of STVPh relative to those of PEMA plus STVPh. In LLS experiments done, PMMA cannot be seen , because its dn/dc in toluene is close to zero. As seen in Fig. 22, the f(Rh) of the STVPh-3/PEMA (50 50, w/w) blend is similar to that of pure STVPh-3, indicating that...

Fig- 22. Hydrodynamic radius distributions f(%) for STVPh-3 and STVPh-3/PEMA blends (50 50, w/w) in toluene, determined at a total polymer concentration 1.0x10" g/ml and a scattering angle of 15° [147]... 

Fig. 23. Hydrodynamic radius distributions f(R ) for STVPh-9 and STVPh-9/PEMA with various blend compositions in toluene. The measuring conditions are the same as in Fig. 22 [147]...

Figure 2 shows the cumulative pore volume vs. pore radius for AC-ref SC-100 and SC-155 obtained by mercury intrusion technique. The curve corresponding to AC-ref shows a wide pore radius distribution instead, the curves assigned to SC-100 and SC-155 showed sharpened zones with maximum slope in 459A and 524A respectively, denoting a small increase of these values with the increase of the synthesis temperature. This phenomenon is probably produced by the growing of the big pores of the silica network at the expense of the... 

Mass transport inside the catalyst has been usually described by applying the Fick equation, by means of an effective diffusivity Deff a Based on properties of the interface and neglecting the composition effect, composite diffusivity of the multi-component gas mixture is calculated through the simplified Wilke model [13], The effect of pore-radius distribution on Knudsen diffusivity is taken into account. The effective diffusivity DeffA is given by... 

Fig. 1 Typical angular dependence of KC/Ryy(q) of PNIPAM in water at two different temperatures, where the weight-average molar mass (Mw) and concentration (C) of PNIPAM are 1.3 x 107 g/mol and 6.7 x 10-7 g/mL, respectively. The insert shows the corresponding hydrodynamic radius distributions f(R ) of the PNIPAM chains respectively in the coiled and the globular states [38]...

Fig. 23 Angular dependence of Rayleigh ratio (Rw(q)) of segmented PNIPAM-seg-St copolymer chains in water measured from static LLS, where K is a constant, q is the scattering vector and polymer concentration (C) was 7.2 x 10-7 g/mL. The inset shows the temperature dependence of the hydrodynamic radius distribution /(Rh) determined from dynamic LLS [94]...

Fig. 37 Typical hydrodynamic radius distributions (/(i h)) of resultant P(DEA-co-DMA) mesoglobules formed under different heating rates [139]...

The relation between the catalytic and aggregation properties of the copolymers was shown using the dynamic light scattering method. Hydro-dynamic radius distributions obtained by processing light scattering data showed that at temperatures below 35 °C the copolymers existed in the form... 

The volume and radius distribution functions are not equal, but can be related by the equation ... 

Figure 3. A typical imaae analysis output, (a) Particle radius distribution, (b) Particle area distribution.

The nanotube network was successfully investigated in our laboratory by Dalmas et al. (63), in terms of distribution of radius of curvature. Indeed, the 2D apparent nanotube segment curvature radius distribution was then measured on TEM bright field images acquired on the composites (see Figure 3.8). The experimental curvature... 

Fig. 24 Hydrodynamic radius distributions f(Rh) of different microgels in each step of the incorporation of TOPO-stabilized CdSe NPs into microgel column (a) NIPAM/VIm(2.78%), column (b) VCL/AAEM/VIm(2.88%). Right TEM images of single VCL/AEM/VIm(2.88%) microgel particle before and after loading with QDs. Taken from [158]...

Here and below, the radius distribution of voids, /(r), is assumed to be defined so that f r) dr is the number of voids with the radius from r to r + dr per unit volume (or mass) of the sample. The radius distribution of necks, first derived by Wall and Brown 14) and were then employed by Kheifets and Neimark (15-17). 

Equations (24) and (30) solve the problem they relate the fractions of pore volume emptied from the condensate during adsorption and desorption with the percolation probability and the radius distributions of voids and necks. If these distributions do not overlap, i.e., < C (Fig. 13a), one can... 

Mayagoitia et al. (28-32) have analyzed the desorption process from the pore space described by a joint site-bond radius distribution with a correlation function that carries structure information about the network. The integral equations derived (29) are based on the Bethe tree model. 

Analyzing adsorption (Section III,C) and desorption (Sections ni,D and III,E), we assumed that the pore volume is concentrated in voids, whereas necks do not possess volumes of their own (Fig. 2). Seaton (34) has recently considered an alternative model of porous solids assuming the pore volume to be concentrated in necks (Fig. 3). In the framework of this model, the adsorption process is described by analogy with Eq. (18) (one should only replace the void radius distribution by the neck radius distribution), and consequently the analysis of the adsorption branch of the isotherm allows one to obtain the neck-size distribution. The desorption process can be described by using the same ideas as in Sections III,D and III,E because this process is mathematically equivalent to the bond problem in percolation theory, even if the pore volume is concentrated in the necks. In particular, the volume fraction of emptied necks under desorption [1 - C/des(fp)] can... 

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