Surface-to-Volume Ratios

The difference between a positively curved surface and a negatively curved surface is nothing more than the surface-to-volume ratio, y and the slight difference in the coordinating environment of the undercoordinated atoms. Considering a sphere of K radius with n + 1/2) spherical cavities of Lj radius lined along the K radius, as illustrated in Fig. 11.7b, the total number of voids is An n+ 1/2) I2 inside the K sphere. For the hollow sphere ( = 0) with only one void in the center, this expression needs slight revision. We can estimate the volume Vo occupied by atoms as the entire volume of the sphere less the voids. The sum of the volume of skins of the voids and the sphere surface is Vi. The Vo and the V, are calculated as follows  

The parameters of n, L, and K are constrained by the relation 2(L+l)(n+l/2) K. — 2 because of the allowed maximum number of cavities aligned along the radius K. 2(L +1) represents the diameter of the void including the one layer of surface skin and K—2 the radius of the sphere excluding the surface skin. This expression covers situations of a solid sphere, a hollow sphere, and a sphere with uniformly distributed cavities of the same size. This relation also applies to a solid rod, a hollow tube, and a porous nanowtre as well. 

With the derived r, (n, L, K) relation and the given expressions for the q zt, dj(t), Ei(t)), one can readily predict the size, cavity density, and temperature dependence of the 2 of a system with large fraction of undercoordinated atoms without involving hypothetic parameters. 

The following sections will focus on the size dependency of the known bulk properties and the emerging properties of nanostmctures toward consistent insight into the nature of the size- and shape-induced property change in nanosolids. 

Physical quantities of a bulk solid can normally be categorized as follows  

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Attempts have been made to perform thermal retorting ia a gas barrier flexible pouch or tray. The retort pouch, under development for many years, has a higher surface-to-volume ratio than a can and employs a heat seal rather than a mechanical closure. Similarly, plastic retort trays have higher surface-to-volume ratios and are usually heat seal closed. Plastic cans iatended for microwave reheating are composed of bodies fabricated from multilayer plastic including a high oxygen barrier material, plus double-seam aluminum closures. 

PhenoHc MicrobaUoons appHcations in plastics take advantage of low density, porosity, and surface-to-volume ratio to produce lightweight parts. Probably the most notable example is the syntactic foam. 

A spray comprises a cloud of liquid droplets randomly dispersed ia a gas phase. Depending on the appHcation, sprays may be produced ia many different ways. The purposes of most sprays are (/) creation of a spectmm of droplet sizes to iacrease the Hquid surface-to-volume ratio, (2) metering or control of the hquid throughput, (J) dispersion of the Hquid ia a certain pattern, or (4) generation of droplet velocity and momentum. 

The properties of thin films generally differ from the values for the material in bulk form (see Formation techniques. In many cases, the growth and properties of thin films are affected by the properties of the underlying substrate material. The properties of the film can also be affected by the high surface-to-volume ratio of the film. 

Electrical Properties. Generally, deposited thin films have an electrical resistivity that is higher than that of the bulk material. This is often the result of the lower density and high surface-to-volume ratio in the film. In semiconductor films, the electron mobiHty and lifetime can be affected by the point defect concentration, which also affects electromigration. These effects are eliminated by depositing the film at low rates, high temperatures, and under very controUed conditions, such as are found in molecular beam epitaxy and vapor-phase epitaxy. 

Increasing surface-to-volume ratio increases susceptibiUty to oxidation. Thin film and fiber are much more sensitive to oxidation than thick specimens (26). The effectiveness of an antioxidant for products with high surface-to-volume ratios is deterrnined not only by its inherent activity in a particular polymer, but also by the rate of loss by volatilization. 

Capillary Electrophoresis. Capillaries were first appHed as a support medium for electrophoresis in the early 1980s (44,45). The glass capillaries used are typically 20 to 200 p.m in diameter (46), may be filled with buffer or gel, and are frequendy coated on the inside. Capillaries are used because of the high surface-to-volume ratio which allows high voltages without heating effects. The only limitations associated with capillaries are limits of detection and clearance of sample components. 

Surface Area Determination The surface-to-volume ratio is an important powder property since it governs the rate at which a powder interacts with its surroundings. Surface area may be determined from size-distribution data or measured directly by flow through a powder bed or the adsorption of gas molecules on the powder surface. Other methods such as gas diffusion, dye adsorption from solution, and heats of adsorption have also been used. It is emphasized that a powder does not have a unique surface, unless the surface is considered to be absolutely smooth, and the magnitude of the measured surface depends upon the level of scrutiny (e.g., the smaller the gas molecules used for gas adsorption measurement the larger the measured surface). 

Cf, C y, and Cq are the concentrations of the substance in question (which may be a colligend or a surfactant) in the feed stream, bottoms stream, and foamate (collapsed foam) respectively. G, F, and Q are the volumetric flow rates of gas, feed, and foamate respectively, is the surface excess in equilibrium with C y. S is the surface-to-volume ratio for a bubble. For a spherical bubble, S = 6/d, where d is the bubble diameter. For variation in bubble sizes, d should be taken as YLnid fLnidj, where n is the number of bubbles with diameter dj in a representative region of foam. 

Fluidized bed catalytic reactors seem to have so many advantageous features that they were considered for many processes. One of the advantages is their excellent heat transfer characteristics, due to the large catalyst surface to volume ratio, so very little temperature difference is needed for heat transfer. This would make temperature control problem-free. The second is the uniformity of reaction conditions in the bed. 

For a flame to be quenched the flame arrester passageways must be small enough to extract heat from the flame faster than it can be generated by chemical reactions. The surface to volume ratio of flame arresters is impor-... 

When the polypeptide chains of protein molecules bend and fold in order to assume a more compact three-dimensional shape, a tertiary (3°) level of structure is generated (Figure 5.9). It is by virtue of their tertiary structure that proteins adopt a globular shape. A globular conformation gives the lowest surface-to-volume ratio, minimizing interaction of the protein with the surrounding environment. 

One general benefit of subunit association is a favorable reduction of the protein s surface-to-volume ratio. The surface-to-volume ratio becomes smaller as the radius of any particle or object becomes larger. (This is because surface area is a function of the radius squared and volume is a function of the radius cubed.) Because interactions within the protein usually tend to stabilize the protein energetically and because the interaction of the protein surface with... 

N.B. Short-path distillation and a high surface-to-volume ratio is necessary in order to prevent dimerization of the product. 

The other important factor is a, the geometric surface area exposed to gas per volume of reactor, which depends on the void fraction and the dimension of the packing. The product of the transfer coefficient and the surface-to-volume ratio governs the rate of heat and mass transfer per... 

The mercury film electrode has a higher surface-to-volume ratio than the hanging mercury drop electrode and consequently offers a more efficient preconcentration and higher sensitivity (equations 3-22 through 3-25). hi addition, the total exhaustion of thin mercury films results in sharper peaks and hence unproved peak resolution in multicomponent analysis (Figure 3-14). 

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