Two Different Spheres

The results obtained above can be directly applied to calculate the force between two different spheres. In the general case, the two interacting spheres can have different contact angles, 1 and 2, and different radii, l i and R2. The height coordinate h describes in this case the distance between the surface of sphere 1 and sphere 2. Rather than considering the shape of each sphere explicitly, we transform the geometry and consider the equivalent case of plane interacting with a sphere of 

For tv o identical spheres of radius Rs, the effective radius is R = Rs/2. For a sphere of radius interacting vith a plane, we have R = Ri. 

Both equations are valid at the same time. Depending on the boundary condition, one or the other is more suitable to use. If the vapor phase is in equilibrium vith the condensed liquid all the time, the radius r is constant and Eq. (5.31) is convenient to use. For nonvolatile liquids and fast processes, the volume is likely to be constant and Eq. (5.32) is appropriate. The volume of the liquid meniscus is [484, 522, 523[ 

The adhesion force, that is, the force required to separate tv o spherical surfaces from each other, is in both cases [511[ 

It only depends on the radii of the particles and the surface tension of the liquid. Neither does it depend on the actual radius of curvature of the meniscus nor does it depend on the vapor pressure. This at first sight surprising result is due to the fact that vith increasing vapor pressure the cross section of the meniscus I increases. At the same time, the capillary pressure decreases because r increases. The product of cross-sectional area and pressure difference, nfAP, remains constant and both effects compensate each other. 

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Initially, the spheres are positioned randomly in the box, periodic boundary conditions are used in the x- and /-direction and no-slip on the -direction. The spheres move according to the flow field and the viscosity is calculated for several time steps, and for each configuration an average suspension viscosity is obtained. The box is divided into 216 elements with 650 nodes, and each sphere into 96 elements with 290 nodes. The computational time depends, as for any particulate simulation, on the number of spheres. Two different sphere radii were used in the simulations 0.05 length units and 0.07 length units. In the same way, the box dimensions were set to lxlxl (length units)3 and 0.8 x 0.8 x 0.8 (length units)3. Each case was simulated with 10,20, 30 and 40 spheres. 

In case of two different spheres with radii a and h we have ... 

Two patterns of packing two different spheres are shown here. For each structure (a) draw the two-dimensional unit cell (b) determine the angle between the lattice vectors, 7, and determine whether the lattice vectors are of the same length or of different lengths (c) determine the type of two-dimensional lattice (from Figure 12.4). 

Figure 36 A plot of the ADH//NLDH fraction of surface charge neutralized by a bulk 1 1 electrolyte according to Eq. [319] for a charged sphere as a function of the surface charge Qa for two values of the Del ye constant kd (0.01 and 0.1 A" ) and for two different sphere radii <3(10 and 20 A) as listed at right.

Unlike the cylindrical case of Eq. [265], this expression is not a function of Koa alone. We have plotted this fraction in Figure 36 for two values of the Debye constant (0.01 and 0.1 and for two different sphere radii (10 and 20 A). For micelles of radius 20 A, there is a noticeable difference between Kd = 0.01 and 0.1 A in the rate at which the limiting fraction plateau is reached. The condensation radius determined from Eq. [269] and the corresponding potential as a function of the scaled charge density are displayed in Figures 29 and 30, respectively, and compared with those for the plane and cylinder. Comments follow those given earlier for a charged cylinder. 

To display properties on molecular surfaces, two different approaches are applied. One method assigns color codes to each grid point of the surface. The grid points are connected to lines chicken-wire) or to surfaces (solid sphere) and then the color values are interpolated onto a color gradient [200]. The second method projects colored textures onto the surface [202, 203] and is mostly used to display such properties as electrostatic potentials, polarizability, hydrophobidty, and spin density. 

We shall see in Sec. 9.10 that sedimentation and diffusion data yield experimental friction factors which may also be described-by the ratio of the experimental f to fQ, the friction factor of a sphere of the same mass-as contours in solvation-ellipticity plots. The two different kinds of contours differ in detailed shape, as illustrated in Fig. 9.4b, so the location at which they cross provides the desired characterization. For the hypothetical system shown in Fig. 9.4b, the axial ratio is about 2.5 and the protein is hydrated to the extent of about 1.0 g water (g polymer)". ... 

Two modifications of the duidized-bed reactor technology have been developed. In the first, two gas-phase duidized-bed reactors coimected to one another have been used by Mobil Chemical Co. and Union Carbide to manufacture HDPE resins with broad MWD (74,75). In the second development, a combination of two different reactor types, a small slurry loop reactor followed by one or two gas-phase duidized-bed reactors (Sphetilene process), was used by Montedision to accommodate a Ziegler catalyst with a special particle morphology (76,77). This catalyst is able to produce PE resins in the form of dense spheres with a diameter of up to 4—5 mm such resins are ready for shipping without pelletization. 

Figure 10 Models of complexes between BLBP and two different fatty acids. The fatty acid ligand IS shown in the CPK representation. The small spheres in the ligand-bmdmg cavity are water molecules, (a) Model of the BLBP-oleic acid complex, in which the cavity is not filled, (b) Model of the BLBP-docosahexaenoic acid complex, m which the cavity is filled. The figure was prepared using the program MOLSCRIPT [236].

Colloidal crystals . At the end of Section 2.1.4, there is a brief account of regular, crystal-like structures formed spontaneously by two differently sized populations of hard (polymeric) spheres, typically near 0.5 nm in diameter, depositing out of a colloidal solution. Binary superlattices of composition AB2 and ABn are found. Experiment has allowed phase diagrams to be constructed, showing the crystal structures formed for a fixed radius ratio of the two populations but for variable volume fractions in solution of the two populations, and a computer simulation (Eldridge et al. 1995) has been used to examine how nearly theory and experiment match up. The agreement is not bad, but there are some unexpected differences from which lessons were learned. 

In the case of computer simulations of fluids with directional associative forces a less intuitive but computationally more convenient potential model has been used [14,16,106]. According to that model the attraction sites a and j3 on two different particles form a bond if the centers of reacting particles are within a given cut-off radius a and if the orientations of two spheres are constrained as follows i < 6 i and [tt - 2 < The interaction potential is... 

Fig. 21.21 The oxygen atoms coordination of the two different Ba sites in Ba3BP30- 2 (Ba atoms, black spheres O atoms, grey spheres).

FIG. 6 Successive coupling of two different biotinylated compounds with the DNA-STV conjugates 2 [33]. In a first step, a macromolecular functional component (FC, represented by the shaded ellipse), such as a biotinylated enzyme or oligonucleotide, is coupled. In a second step, a biotinylated low-molecular-weight modulator M, represented by the shaded sphere) is coupled to the remaining free biotin-binding sites. The modulator is used to modify the conjugate s hybridization properties or to supplement its functionality. 

Now consider adding a third layer of close-packed spheres. This new layer can be placed in two different ways because there are two sets of dimples in the second layer. Notice in Figure ll-29b that the view through one set of dimples reveals the maroon spheres of the first layer. If spheres in the third layer lie in these dimples, the third... 

The formation of nanopattemed functional surfaces is a recent topic in nanotechnology. As is widely known, diblock copolymers, which consist of two different types of polymer chains cormected by a chemical bond, have a wide variety of microphase separation structures, such as spheres, cylinders, and lamellae, on the nanoscale, and are expected to be new functional materials with nanostructures. Further modification of the nanostructures is also useful for obtaining new functional materials. In addition, utilization of nanopartides of an organic dye is also a topic of interest in nanotechnology. 

In order to obtain the velocity v, we consider two concentric spheres, the first representing the drop surface with radius r, and the second one having a Active radius r2 as the liquid is not compressible, we can choose r2 (although altering) in such a way that the volume difference AV of the spheres is constant, i.e. AV = V2 - Vi = jn(r - rj), so that for a very slight difference r2 - r, = x we find... 

The CsCl type offers the simplest way to combine atoms of two different elements in the same arrangement as in body-centered cubic packing the atom in the center of the unit cell is surrounded by eight atoms of the other element in the vertices of the unit cell. In this way each atom only has adjacent atoms of the other element. This is a condition that cannot be fulfilled in a closest-packing of spheres (cf. preceding section). 

The number of octahedral holes in the unit cell can be deduced from Fig. 17.1(c) two differently oriented octahedra alternate in direction c, i.e. it takes two octahedra until the pattern is repeated. Flence there are two octahedral interstices per unit cell. Fig. 17.1(b) shows the presence of two spheres in the unit cell, one each in the layers A and B. The number of spheres and of octahedral interstices are thus the same, i.e. there is exactly one octahedral interstice per sphere. 

Packings of spheres having occupied tetrahedral and octahedral interstices usually occur if atoms of two different elements are present, one of which prefers tetrahedral coordination, and the other octahedral coordination. This is a common feature among silicates (cf. Section 16.7). Another important structure type of this kind is the spinel type. Spinel is the mineral MgAl204, and generally spinels have the composition AM2X4. Most of them are oxides in addition, there exist sulfides, selenides, halides and pseudohalides. 

The specific structure of [(H20)5Ni(py)]2+ was observed in the complexes with the second-sphere coordination of calix[4]arene sulfonate.715 There are two different [(H20)5Ni(py)]2+ cations in the complex assembly. In one the hydrophobic pyridine ring is buried in the hydrophobic cavity of the calixarene with the depth of penetration into the calixarene cavity being 4.3 A (Figure 9). The second independent [(H20)5Ni(py)]2+ cation is intercalated into the calixarene bilayer. 

Electron transfer between metal ions contained in complexes can occur in two different ways, depending on the nature of the metal complexes that are present. If the complexes are inert, electron transfer occurring faster than the substitution processes must occur without breaking the bond between the metal and ligand. Such electron transfers are said to take place by an outer sphere mechanism. Thus, each metal ion remains attached to its original ligands and the electron is transferred through the coordination spheres of the metal ions. 

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