Particle radius

Here iC is the intrinsic anisotropy constant due to the crystalline anisotropy. After the demagnetization in the longest direction, k is the shape-dependent constant (for an infinite cylinder k = 1.38), M is the exchange constant, and R the particle radius. An infinite cylinder with only shape anisotropy gives... 

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).]... 

The so-called JKR equation relates the adhesion-induced contact radius a to the particle radius R and the applied load P by... 

The JKR model predicts that the contact radius varies with the reciprocal of the cube root of the Young s modulus. As previously discussed, the 2/3 and — 1/3 power-law dependencies of the zero-load contact radius on particle radius and Young s modulus are characteristics of adhesion theories that assume elastic behavior. 

Upon comparison of Eqs. 29 and 36, it is readily apparent that both theories predict the same power law dependence of the contact radius on particle radius and elastic moduli. However, the actual value of the contact radius predicted by the JKR theory is that predicted by the DMT model. This implies that, for a given contact radius, the work of adhesion would have to be six times as great in the DMT theory than in the JKR model. It should be apparent that it is both necessary and important to establish which theory correctly describes a system. 

Fig. 1. Log-log plot of the contact radius as a function of particle radius for soda-lime glass particles on polyurethane (from ref. [56]).

There have been several theories proposed to explain the anomalous 3/4 power-law dependence of the contact radius on particle radius in what should be simple JKR systems. Maugis [60], proposed that the problem with using the JKR model, per se, is that the JKR model assumes small deformations in order to approximate the shape of the contact as a parabola. In his model, Maugis re-solved the JKR problem using the exact shape of the contact. According to his calculations, o should vary as / , where 2/3 < y < 1, depending on the ratio a/R. 

As is evident, there are several distinctive characteristics of adhesion-induced plastic deformations, compared to elastic ones. Perhaps the most obvious distinction is the power-law dependence of the contact radius on particle radius. Specifically, the MP model predicts an exponent of 1/2, compared to the 2/3 predicted by either the JKR or DMT models. 

An example of a Maugis-Pollock system is polystyrene particles having radii between about 1 and 6 p.m on a polished silicon substrate, as studied by Rimai et al. [64]. As shown in Fig. 4, the contact radius was found to vary as the square root of the particle radius. Similar results were reported for crosslinked polystyrene spheres on Si02/silicon substrates [65] and micrometer-size glass particles on silicon substrates [66]. 

Fig. 4. The contact radius as a function of the square root of the particle radius for polystyrene spheres on a silicon substrate (from ref. [64]).

By comparing Eq. 17 with Eq. 84, it is clear that, whereas van der Waals forces vary linearly with the particle radius, the electrostatic forces vary quadratically. 

Another manifestation of a time dependence to particle adhesion involves the phenomenon of total engulfment of the particle by the substrate. It is recognized that both the JKR and MP theories of adhesion assume that the contact radius a is small compared to the particle radius R. Realistically, however, that may not be the case. Rather, the contact radius depends on the work of adhesion between the two materials, as well as their mechanical properties such as the Young s modulus E or yield strength Y. Accordingly, there is no fundamental reason why the contact radius cannot be the same size as the particle radius. For the sake of the present discussion, let us ignore some mathematical complexities and simply assume that both the JKR and MP theories can be simply expanded to include large contact radii. Let us further assume that, under conditions of no externally applied load, the contact and particle radii are equal, that is a(0) = R. Under these conditions, Eq. 29 reduces to... 

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

This equation may be used for the estimation of the swelling capacity of the activated seed particles with the monomer. A typical graph sketched based on Eq. (11) is given in Fig. 18. This graph shows the variation of the swelling capacity of the seed polymer particles VmIVp) with the ratio of interfacial tension-initial particle radius... 

Gas constant (J/mol K) coarse particle radius Specific weight of feed Degree of mixing Time scale (s)... 

Particle sizes are measured in microns, p, A micron is 1/1000 millimeter or 1/25,400 inch. A millimicron, m,U, is 1/1000 of a micron, or 1/1,000,000 millimeter. Usually particle size is designated as the average diameter in microns, although some literature reports particle radius. Particle concentration is often expressed as grains/cubic feet of gas volume. One grain is 1/7000 of a pound. 

The rate expression for reaction of a cylindrical particle (radius r) is [469]... 

A diffusion-limited reaction proceeding in spherical particles (radius r) obeys a rate expression obtained by combining eqn. (10) with the contracting volume relation [eqn. (7), n = 3], viz. 

Mcllvried and Massoth [484] applied essentially the same approach as Hutchinson et al. [483] to both the contracting volume and diffusion-controlled models with normal and log—normal particle size distributions. They produced generalized plots of a against reduced time r (defined by t = kt/p) for various values of the standard deviation of the distribution, a (log—normal distribution) or the dispersion ratio, a/p (normal distribution with mean particle radius, p). 

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