Particle radius, effective

STM and AFM profiles distort the shape of a particle because the side of the tip rides up on the particle. This effect can be corrected for. Consider, say, a spherical gold particle on a smooth surface. The sphere may be truncated, that is, the center may be a distance q above the surface, where q < r, the radius of the sphere. Assume the tip to be a cone of cone angle a. The observed profile in the vertical plane containing the center of the sphere will be a rounded hump of base width 2d and height h. Calculate q and r for the case where a - 32° and d and h are 275 nm and 300 nm, respectively. Note Chapter XVI, Ref. 133a. Can you show how to obtain the relevent equation ... 

Limitations It is desirable to have an estimate for the smallest particle size that can be effectively influenced by DEP. To do this, we consider the force on a particle due to DEP and also due to the osmotic pressure. This latter diffusional force will randomize the particles and tend to destroy the control by DEP Figure 22-32 shows a plot of these two forces, calciilated for practical and representative conditions, as a func tion of particle radius. As we can see, the smallest particles that can be effec tively handled by DEP appear to be in range of 0.01 to 0.1 piTidOO to 1000 A). 

FIG. 22-40 Normalized free-energy difference between distributed (II) and nondistributed (I) states of tbe solid particles versus tbree-pbase contact angle (collection at tbe interface is not considered). A negative free-energy difference implies tbat tbe distributed state is preferred over tbe nondistributed state. Note especially tbe significant effect of n, tbe ratio of tbe liquid droplet to solid-particle radius. [From Jacques, Ho-oaron ura, and Hemy, Am. Inst. Cbem. Eng. J., 25 1), 160 (1979).]... 

Figure 6.10 Effect of particle radius on colloidal interaction...

In equation (2) Rq is the equivalent capillary radius calculated from the bed hydraulic radius (l7), Rp is the particle radius, and the exponential, fxinction contains, in addition the Boltzman constant and temperature, the total energy of interaction between the particle and capillary wall force fields. The particle streamline velocity Vp(r) contains a correction for the wall effect (l8). A similar expression for results with the exception that for the marker the van der Waals attraction and Born repulsion terms as well as the wall effect are considered to be negligible (3 ). 

Fig. 8.1. Effect of particle radius on the boundary layer thickness of digoxin and oxazepam. Graph constructed from data in Bisrat et al. [19],...

For powders and granular solids, there are two types of antistatic agents surface- and volume-active additives. Surface-active agents, which increase the surface conductivity of individual particles, are effective because triboelectric charge is always situated on the surfaces of individual particles. Most if not all surface-active agents are hygroscopic and thus attract a thin film of water to the surface it is this moisture that is responsible forthe increased surface conduction (van Drumpt, 1991). The effective bulk resistivity of the particles % -may be estimated by assuming that the particles are spherical and of radius R (Jones, 1995). 

Comparison of our theory with experimental data shows excellent agreement, both with respect to the molecular weight dependency and to the effect of particle radius and particle concentration. 

The polymer radius has to be larger than 80% of the particle radius to avoid adsorption limitation under orthokinetic conditions. As a rule of thumb a particle diameter of about 1 pm marks the transition between perikinetic and orthokinetic coagulation (and flocculation). The effective size of a polymeric flocculant must clearly be very large to avoid adsorption limitation. However, if the polymer is sufficiently small, the Brownian diffusion rate may be fast enough to prevent adsorption limitation. For example, if the particle radius is 0.535 pm and the shear rate is 1800 s-, then tAp due to Brownian motion will be shorter than t 0 for r < 0.001, i.e., for a polymer with a... 

Figure 3.23 The tertiary electroviscous effect observed for particles of polystyrene latex with a copolymer of polyacrylic acid at the outer surface. The experimental points were obtained at pH 3 and 10. The dry particle radius was 75 nm and Ka 25...

To account for their data (Fig. 2.7), Mondain-Monval et al. hypothesized that these two forces simply add and that the repulsion between micelles and droplets increases the effective diameter of the droplets (or micelles) [22]. This force is derived by integrating the osmotic pressure Posm over the accessible zone for micelles of diameter 2r (r = 2.35 nm) from 6 = n to 9 = 7t -Oi, with 9i defined in Fig. 2.6. The distance at which the small micelles are excluded from the gap between the droplets is evidently influenced by the electrostatic micelle-droplet repulsion. To account for this repulsion, droplets (or micelles) may be considered as particles of effective radius (a + S) [or micelles of radius (r + 5)]. From... 

A probabilistic kinetic model describing the rapid coagulation or aggregation of small spheres that make contact with each other as a consequence of Brownian motion. Smoluchowski recognized that the likelihood of a particle (radius = ri) hitting another particle (radius = T2 concentration = C2) within a time interval (dt) equals the diffusional flux (dC2ldp)p=R into a sphere of radius i i2, equal to (ri + r2). The effective diffusion coefficient Di2 was taken to be the sum of the diffusion coefficients... 

The second approach uses multiple detectors (Figure 3.13), allowing a double extrapolation to zero concentration and zero angle with the data forming what is called a Zimm plot (Figure 3.14). The extrapolation to zero angle corrects for finite particle size effects. The radius of gyration, related to polymer shape and size, can also be determined from this plot. The second extrapolation to zero concentration corrects for concentration factors. The intercepts of both plots are equal to /M . 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Colloidal CdS particles 2-7 nm in diameter exhibit a blue shift in their absorption and luminescence characteristics due to quantum confinement effects [45,46]. It is known that particle size has a pronounced effect on semiconductor spectral properties when their size becomes comparable with that of an exciton. This so called quantum size effect occurs when R < as (R = particle radius, ub = Bohr radius see Chapter 4, coinciding with a gradual change in the energy bands of a semiconductor into a set of discrete electronic levels. The observation of a discrete excitonic transition in the absorption and luminescence spectra of such particles, so called Q-particles, requires samples of very narrow size distribution and well-defined crystal structure [47,48]. Semiconductor nanocrystals, or... 

For the other extreme of the free molecular regime where Kn - oc, the particle radius is small compared to the mean free path. In this case, the thermal velocity distribution of the gas is not distorted by uptake at the surface. In effect, the gas molecules do not see the small particles. For this case, Fuchs and Sutugin (1970, 1971) show that for diffusion to a spherical particle of radius a... 

It is well known that swelling of resins increases the particle radius. When resin beads are immersed in the solution, water uptake takes place almost immediately, resulting in a new swollen radius. The swollen radius is inversely proportional to the initial solution concentration (Hellferich, 1962). This effect has not been taken into account in the original model of Levenspiel (1972). However, the actual swollen radius of the resin should be used in the model equations, and so measurements should be performed in order to estimate this radius. 

Other dimensionless groups that compare the thickness of the adsorbed polymer layer to the radius of the particle or the radius of gyration of the polymer to the particle radius in polymer/colloid mixtures can also be easily defined. We are mostly concerned with the volume fraction and the Peclet number Pe in our discussions in this chapter. However, the other dimensionless groups may appear in the equations for intrinsic viscosity of dispersions when the dominant effects are electroviscous or sterically induced. 

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