Nonspherical cells

Pahlow et al. [67] have also carefully examined the role of cell shape on nutrient uptake by diatoms in the presence of advection, under the assumption that nonspherical cells will rotate intermittently. In that case, the relative increase in transport capacity is small for solitary, nonspherical cells with respect to spheres [67], These authors found that advection could significantly increase transport for cell chains, even though it did not compensate for the loss in diffusive flux when calculations were made on a cell surface area basis. Therefore, although advection has a much greater effect on the supply of solute to chains than it does to solitary cells, solitary cells will always maintain an advantage over chains. 

As previously mentioned, nonspherical cells have been modeled as spheres with equivalent radius and effective complex index of refraction. This approximation can be justified by the fact that they are typically well mixed and randomly oriented in the PBRs. Then, the equivalent radius r q can be approximated such that either the volume or the surface area of the equivalent sphere is identical to that of the actual cell. The radius of the volume-equivalent sphere can be expressed as... 

This equation is a reasonable model of electrokinetic behavior, although for theoretical studies many possible corrections must be considered. Correction must always be made for electrokinetic effects at the wall of the cell, since this wall also carries a double layer. There are corrections for the motion of solvated ions through the medium, surface and bulk conductivity of the particles, nonspherical shape of the particles, etc. The parameter zeta, determined by measuring the particle velocity and substituting in the above equation, is a measure of the potential at the so-called surface of shear, ie, the surface dividing the moving particle and its adherent layer of solution from the stationary bulk of the solution. This surface of shear ties at an indeterrninate distance from the tme particle surface. Thus, the measured zeta potential can be related only semiquantitatively to the curves of Figure 3. 

Figure 2.9b is an end-on, side view of the cell rhombus. In the figure, are outlines of three 51268 cavities (labeled A, B, and C) are shown with the vertical borders of the rhombus at centroids of each 51268. The fourth 51268 of Figure 2.9a is aligned behind the middle 51268 cavity in Figure 2.9b. This view shows both the nonspherical nature of the 51268 cavities and their nonplanar, strained hexagonal faces in contrast to the almost planar hexagonal faces in si and sll.

The surface matching theorem makes it possible to generalize the idea of muffin-tin orbitals to a nonspherical Wigner-Seitz cell r. Each local basis orbital is represented as (p =a x + V on the cell surface a, where y and p are the auxiliary functions defined by the surface matching theorem. An atomic-cell orbital (ACO) is defined as the function — y, regular inside r. By construction, the smooth continuation of this ACO outside r is the function p. The specific functional forms are... 

In some cases, if the complex ion is not too far from spherical, the compound may assume a structure similar to the simple ones already described, hut with slight distortion. In calcium carbide, for example, the layout of positive and negative ions is the same as in sodium chloride, but the nonspherical Cl ions are lined up with their axes parallel, distending one dimension of the square units to form rectangles. In potassium nitrate, again similar to the sodium chloride structure, the unit cell has been even further distorted from a perfect cube, all faces becoming rhombuses rather than squares. 

Consider a spatially periodic suspension whose basic unit cell contains N particles, all of whose sizes, shapes, and orientations may be different. The particulate phase of the system is completely specified geometrically by the values of the 3N spatial coordinates rN, and 3N orientational coordinates eN of the (generally nonspherical) particles, as in Section II,A. A particle is identified by the pair of scalar and vector indices i (i = 1,..., N) and n,... 

Acetylene blacks are chemically very pure. Their particles significantly deviate from the other carbon blacks in being significantly nonspherical. They are utilized in dry cells, due to their higher electrical conductivity and their good absorptive capacity. 

Barker [92-94] has presented a general formulation of the cell theory and we give a brief review of his approach here. We will restrict our discussion to single-component atomic solids and discuss the application to mixtures and nonspherical molecules later. Suppose we have a system of N molecules in the canonical ensemble. The configurational partition function, Eq. (2.205), may be rewritten by breaking the volume into N identical subvolumes or cells so that... 

Obscuration is the fraction of light that is obscured by the crystals in a flow cell thus, obscuration is defined as 1 - I/Io where I is the intensity of undiffracted light that passes through the suspension of crystals and 7o is the intensity of the incident light. The obscuration can be accurately measured, and the Beer-Lambert law provides a means by which it can be modelled for comparison with experimental data. As discussed by Witkowski et al. (1990), the obscuration provides a measure of the second moment of the CSD. Preliminary experimental tests indicate that the theorem discussed above relating geometrical cross section to surface area may be helpful in extending the use of obscuration measurement to cases with nonspherical particles. 

Nonspherical units and noncubic arrays were investigated by changing the shape of the unit and the cell. Within definitive limits, spherical units in cubic arrays are shown to be more reactive than other configurations of units of the same mass. 

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