Nonspherical Particle Diffusion

Next, it will be helpful to anticipate a description of experimental procedures and consider the magnitude of measured diffusion coefficients. The self-diffusion coefficients for ordinary liquids with small molecules are of the order of magnitude 10 9 m2 s for colloidal substances, they are typically of the order 10"11 m2 s l. In the next section, we see that for spherical particles the diffusion coefficient is inversely proportional to the radius of the sphere. Therefore, every increase by a factor of 10 in size decreases the diffusion coefficient by the same factor. Qualitatively, this same inverse relationship applies to nonspherical particles as well. Once again, we see that diffusion decreases in importance with increasing particle size, precisely those conditions for which sedimentation increases in importance. For larger particles, for which D is very small, the diffusion coefficient also becomes harder to measure. For... 

For nonspherical particles, Muller (1928) postulated that since the diffusion equation applicable to aerosol problems is the same (except for definition of terms) as the general equation for electric fields (Laplace s equation), there should be analogs among the electrostatic terms for various properties of coagulation. For example, the potential should be analogous to particle number concentration, and field strength to particle agglomeration rate. Zebel (1966) pointed out that... 

This rotational diffusion coefficient will be used in Chapter 13 to aid in determining if nonspherical particles will orient in shear flow as they are cast to make a ceramic green body. 

For solutions of nonspherical particles the situation is more complicated and the physical picture can be described qualitatively as follows for a system of particles in a fluid one can define a distribution function, F (Peterlin, 1938), which specifies the relative number of particles with their axes pointed in a particular direction. Under the influence of an applied shearing stress a gradient of the distribution function, dFfdt, is set up and the particles tend to rotate at rates which depend upon their orientation, so that they remain longer with their major axes in position parallel to the flow than perpendicular to it. This preferred orientation is however opposed by the rotary Brownian motion of the particles which tends to level out the distribution or orientations and lead the particles back toward a more random distribution. The intensity of the Brownian motion can be characterized by a rotary diffusion coefficient 0. Mathematically one can write for a laminar, steady-state flow ... 

We can estimate a, the radius of the particle, from a knowledge of the diffusion coefficient. If the particle is not spherical, the frictional coefficient is larger than that given by the Stokes s law expression the nonspherical particles exert a frictional effect larger than that exerted by an equivalent spherical particle. 

Grashof number for mass transfer L is a characteristic dimension, i.e., the diameter of a spherical particle, or the equivalent diameter of a nonspherical particle, etc. v is the kinematic viscosity D is the binary diffusion coefficient U is the linear velocity of the gas stream flowing past the particle (measured outside the boundary layer surrounding the particle) g is the acceleration due to gravity is a characteristic concentration difference, and... 

The Debye-Stokes-Einstein relation assumes a particle to be spherical. For a nonspherical particle an alternate equation must be chosen that takes into consideration the effect of molecular shape on the diffusion properties of the particle. 

Equation (13.4.4) gives the correct order of magnitude for D. However, the diffusivity of nonspherical particles such as ellipsoids may be different in different directions. 

For dissolution of solid particles, the Hixson-Crowell cube-root law (Eq. 5.3) assumes that the thickness of the diffusion layer h is constant during dissolution. However, this is not necessarily true. In addition, most drug particles are nonspherical and nonuniform in size. Therefore, very often the dissolution mechanism of solid drug particles is actually much more complicated. Nevertheless, the Hixson-Crowell cube-root law provides the first approximation to model powder dissolution. 

The spherical shape is thermodynamically more stable than other shapes of particles which depend on the nature of materials used and the mode of preparation, such as inorganic colloids to yield ellipsoidal, rod-like, cubic, platelet, or needle-like shapes. In the case of organic particles, it is also feasible to make colloids with organic particles having a nonspherical shape, which leads to the formation of metastable thermodynamic state with their diffusion capability. 

Figures 1 and 2 show the data for the A-14 plotted according to Eq. (4) for grain boundary and volume diffusion, respectively. It is readily observed that the grain boundary plot exhibits straight lines for most temperatures after a brief curved portion, while the volume plots are all curved. There is apparently a small amount of abnormally rapid shrinkage initially, resulting in an effective Lq about 0.003 in. less than the measured Lq. This shrinkage may be due to particle nonsphericity and size distribution. The lowest temperature runs are curved even on the grain boundary plot, indicating that surface diffusion is probably significant at these temperatures ( ). The fact that...

Schcuch G, Heyder J. Dynamie shape factor of nonspherical aerosol particles in the diffusion regime. Aerosol Sei Teehnol 1990 12 270-277. 

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