Spherical shells

The steady-state solution for diffusion through spherical shells with boundary conditions dependent only on r may be obtained by integrating twice and determining the two constants of integration by fitting the solution to the boundary conditions. 

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While most vesicles are formed from double-tail amphiphiles such as lipids, they can also be made from some single chain fatty acids [73], surfactant-cosurfactant mixtures [71], and bola (two-headed) amphiphiles [74]. In addition to the more common spherical shells, tubular vesicles have been observed in DMPC-alcohol mixtures [70]. Polymerizable lipids allow photo- or chemical polymerization that can sometimes stabilize the vesicle [65] however, the structural change in the bilayer on polymerization can cause giant vesicles to bud into smaller shells [76]. Multivesicular liposomes are collections of hundreds of bilayer enclosed water-filled compartments that are suitable for localized drug delivery [77]. The structures of these water-in-water vesicles resemble those of foams (see Section XIV-7) with the polyhedral structure persisting down to molecular dimensions as shown in Fig. XV-11. 

The spherical shell model can only account for tire major shell closings. For open shell clusters, ellipsoidal distortions occur [47], leading to subshell closings which account for the fine stmctures in figure C1.1.2(a ). The electron shell model is one of tire most successful models emerging from cluster physics. The electron shell effects are observed in many physical properties of tire simple metal clusters, including tlieir ionization potentials, electron affinities, polarizabilities and collective excitations [34]. 

In addition, the volume element of interest is not the box dx dy dz shown in Fig. 1.6a but, rather, a spherical shell of radius r and thickness dr as shown in Fig. 1.6b. The result of expressing the volume element in spherical coordinates and integrating over all angles is the replacement... 

Figure 1.6 A flexible coil attached at the origin at one end and (a) in a volume element dx dy dz at the other end and (b) in a spherical shell of volume 47rr dr. (Reprinted from Ref. 4, p. 116.)...

The factor containing r increases with increasing r and reflects the fact that there are more locations to place the loose chain end within larger spherical shells, but. ... 

Only a fraction of the chain segments will be present in this spherical shell, but whatever their number is, it will increase with the degree of polymerization n. Therefore, in the volume element associated with the expansion of the coil, the volume fraction of chain segments 0 is proportional to n/dV, or 0 n/a ro dro ... 

Plot 4nr Rl against p (or r), as shown in Figure 1.7(c). The quantity 4nr Rl is called the radial charge density and is the probability of finding the electron in a volume element consisting of a thin spherical shell of thickness dr, radius r, and volume 4 ir dr. 

Internal-pressure design rules and formulas are given for cylindrical and spherical shells and for ellipsoidal, torispherical (often called ASME heads), hemispherical, and conical heads. The formulas given assume membrane-stress failure, although the rules for heads include consideration for buckling failure in the transition area from cylinder to head (knuckle area). 

There are four commonly occurring states of stress, shown in Fig. 3.2. The simplest is that of simple tension or compression (as in a tension member loaded by pin joints at its ends or in a pillar supporting a structure in compression). The stress is, of course, the force divided by the section area of the member or pillar. The second common state of stress is that of biaxial tension. If a spherical shell (like a balloon) contains an internal pressure, then the skin of the shell is loaded in two directions, not one, as shown in Fig. 3.2. This state of stress is called biaxial tension (unequal biaxial tension is obviously the state in which the two tensile stresses are unequal). The third common state of stress is that of hydrostatic pressure. This occurs deep in the earth s crust, or deep in the ocean, when a solid is subjected to equal compression on all sides. There is a convention that stresses are positive when they pull, as we have drawn them in earlier figures. Pressure,... 

The impressed current, 7, flows from the spherical anode radially in a symmetrical field (i.e., the equipotential lines represent spherical shells). It follows from Eq. (24-1)... 

A nucleic acid can never code for a single protein molecule that is big enough to enclose and protect it. Therefore, the protein shell of viruses is built up from many copies of one or a few polypeptide chains. The simplest viruses have just one type of capsid polypeptide chain, which forms either a rod-shaped or a roughly spherical shell around the nucleic acid. The simplest such viruses whose three-dimensional structures are known are plant and insect viruses the rod-shaped tobacco mosaic virus, the spherical satellite tobacco necrosis virus, tomato bushy stunt virus, southern bean mosaic vims. 

E = Joint efficiency in cylindrical or spherical shells or ligaments between openings (see ASME Code Par.lJW-12 or UG-53) e = natural logarithm base, e = 2.718 e, = TNT equivalent (explosion) (see Table 7-26)... 

Mampel extended the treatment to include due allowance for three-dimensional growth of product into the particles by considering the latter to consist of a series of concentric thin spherical shells. The fractional reaction within each such shell was calculated and the total reaction found by integration to include all such shells. This analysis, which includes the effects of overlap, ingestion and also particle size of the reactant, is not amenable to general solution, but the following special cases are of interest. 

The filled polymer is considered as a collection of repesentative volume elements (RVE) of many spherical or cylindrical composites of various sizes. Each of these contains a filler particle and two concentric spherical shells, a thin one corresponding to the mesophase, and another thicker, representing the matrix respectively. The volume fraction of the filler in each composite is the same, as the total volume fraction of the filler in the filled polymer. 

Figure 9.8. Conduction through thick-walled tube or spherical shell The heat flow at any radius r is given by ...

The basic differential equation for mass transfer accompanied by an nth order chemical reaction in a spherical particle is obtained by taking a material balance over a spherical shell of inner radius r and outer radius r + Sr, as shown in Figure 10.12. 

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