Ideal 140 monodisperse

In applying this concept, the factor of particle size must be continuously borne in mind. A heterodisperse system can reach a steady state wherein the smaller particles are agglomerated and the larger particles are dispersed, giving the apparent effect of an equiUbrium. In ideal monodisperse systems under steady conditions, however, no such effects are noted. 

The specihc surface area of an ideal, monodisperse catalyst per unit of its mass is related to its diameter das S = 6/pd, where p is the density. In the case of platinum. 

There have been a number of computer simulations of block copolymers by Binder and co-workers (Fried and Binder 1991a,ft), and this work was reviewed in Binder (1994). Although computer simulations are limited due to the restriction on short chain lengths that can be studied, finite size effects and equilibration problems at low temperatures, the advantages are that the models are perfectly well characterized and ideal (monodisperse, etc.) and microscopic details of the system can be computed (Binder 1994). In the simulations by Binder and co-workers, diblocks were modelled as self- and mutually-avoiding chains on a simple cubic lattice, with chain lengths N = 14 to 60 for/ = 1.A purely repulsive pairwise interaction between A and B segments on adjacent sites was assumed. A finite volume fraction of vacancies was included to speed the thermal equilibration process (Binder 1994). 

Eq. (17) predicts that, when Pn is 100, the polydispersity is equal to 1.01, so that the polymer is virtually monodisperse. However, such ideal monodisperse polymers have scarcely been synthesized. The lowest values of polydispersity (Mw/Mn = 1.05-1.10) have been attained in homogeneous anionic polymerization 43). Gold 41) calculated the polydispersity of a polymer in the living polymerization with a slow initiation reaction and showed that the value of Pw/Pn increases slightly to a maximum (1.33) with an increase in polymerization time, followed by a decrease toward 1.00. Other factors affecting the molecular weight distribution of living polymer have been discussed in several papers 5S 60). 

A molecular theory for the effect of flow on the orientation of ideal monodisperse rigid rod-like polymers has been developed by Doi (1980) and by Hess (1976). In this theory, a Smoluchowski equation is derived for the probability (u) that a rod-like molecule is oriented parallel to a unit vector u ... 

Let us now examine in detail the equilibrium in such thermodynamically stable system. We will base our discussion on the analysis of the change in free energy, A 5 ", of idealized monodisperse system of constant composition, formed by dispersing a known volume of continuous phase 1 in another continuous phase 2 (the dispersion medium). Depending on the particle size, expressed either as radius or diameter, the number of particles, JTX, in the newly formed dispersed phase changes. If the total volume of substance forming the dispersed phase is constant, one may write... 

Both Eqs. (9.24) and (9.25) are for ideal monodispersity. As r]o is proportional to the first moment of the relaxation-time distribution (Eq. (4.65)) while Jg is proportional to the second moment (Eq. (4.66)), the former is far less sensitive to the molecular-weight distribution than the latter. If the molecular-weight distribution of a sample is nearly monodisperse, its r]o value is little affected by the distribution while its J° value is significantly affected. 

As the steady-state compliance J° is sensitive to the molecular-weight distribution, the experimental results of the nearly monodisperse samples are higher than the theoretical values for ideal monodispersity. Shown in Fig. 10.11 is the comparison of the experimental data of (the experimental results shown in Fig. 10.11 are consistent with those shown in Fig. 4.12) with four theoretical curves curve 1 is calculated from Eq. (9.25) curves 2 and 3 are numerically calculated from the substitution of Eq. (9.19) into... 

Fig. 17.7 Comparison of the equilibrium-simulated G(t) curve (o) for the twenty-bead Praenkel chain with Hp = 400kT and the predicted experimental curve (solid line) for an ideally monodisperse polystyrene sample with the molecular weight equivalent to N = 20 also shown are the points (+) representing the relaxation times of the 19 Rouse normal modes.

More recently, Dushkin [89] (1998) derived a model interposed between the idealized monodisperse and the more realistic polydisperse. This pseudo-polydisperse systan allowed for two types of micelles, one at each side of the bell-shaped size distribution. 

The specific surface area of an ideal monodisperse catalyst per unit of its mass is related to its diameter d as S = 6/pd (where p is the density). In the case of platinum, a particle diameter of 5 nm corresponds to a specific surface area of 56 m /g. With a particle size reduced to 2.2 nm, 5 = 127 m /g, which is the largest specific surface area attainable with platinum. At this crystallite diameter all platinum atoms of a particle are surface atoms (i.e., the particle contains no inert bulk atoms). The specific surface area of disperse catalysts can be measured quite accurately by low-temperature nitrogen or helium adsorption (the BET method) or, in the case of platinum, in terms of the amount of charge consumed for the electrochemical adsorption or desorption of a monolayer of hydrogen atoms. In subsequent sections we consider specific questions that arise in the use of different catalysts when building and operating fuel cells. 

More uniform (ideally monodisperse) particle size distribution - determined by atomizer design... 

Figure 4.4 Variation of volume Jraction with height during creaming of (a) ideal monodisperse emulsion and (b) ideal bidisperse emulsion...

In solution, nanocrystals are ideal spectroscopic samples however many of dieir most important properties can only be realized when diey are assembled into more complex stmctures. One way of building complex stmctures is to rely on die inlierent tendency for monodisperse spheres to crystallize. Figure C2.17.3 shows die hexagonal close-... 

The partitioning principle is different at high concentrations c > c . Strong repulsions between solvated polymer chains increase the osmotic pressure of the solution to a level much higher when compared to an ideal solution of the same concentration (5). The high osmotic pressure of the solution exterior to the pore drives polymer chains into the pore channels at a higher proportion (4,9). Thus K increases as c increases. For a solution of monodisperse polymer, K approaches unity at sufficiently high concentrations, but never exceeds unity. 

Various PIB architectures with aromatic finks are ideal model polymers for branching analysis, since they can be disassembled by selective link destmction (see Figure 7.7). For example, a monodisperse star would yield linear PIB arms of nearly equal MW, while polydisperse stars will yield linear arms with a polydispersity similar to the original star. Both a monodisperse and polydisperse randomly branched stmcture would yield linear PIB with the most-probable distribution of M jM = 2, provided the branches have the most-probable distribution. Indeed, this is what we found after selective link destruction of various DlBs with narrow and broad distribution. Recently we synthesized various PIB architectures for branching analysis. 

In order to make practical use of the physical properties of nanoparticles, whether individual or collective, one has to find a way to address them. If we leave out the near field techniques, this in turn requires that the particles be monodisperse and organized in two or three dimensions. It is therefore necessary to imagine techniques allowing the self-organization and even, ideally, the crystallization of nanoparticles into super-lattices. 

The scaling exponent a can be related to the particle shape. One finds a = 2,0, 0.5, and 0.8 for a thin rod, solid sphere, ideal chain, and swollen chain, respectively. For most polymers K and a have been tabulated [23]. For a monodisperse sample Equation (36) can be used for a crude determination of the molar mass ... 

After normalization to the asymptotic baseline, g2(r) decays from two to unity if measured with a perfect instrument. A real instrument always suffers from some loss of coherence, and for a monodisperse solution of ideal, non-interacting solute molecules the intensity autocorrelation function g2(r) takes the form... 

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