Two spheres

Consider next two interacting spherical colloidal particles 1 and 2 of radii ai and 02 having surface potentials ij/oi and ij/oi, respectively, at separation R = H+ai+a2 between their centers in a general electrolyte solution. According to the method of Brenner and Parsegian [6], the asymptotic expression for the interaction energy V (R) is given by 

Here Q ta ( = 1, 2) is the hypothetical point charge that would produce the same asymptotic potential as produced by sphere i and is the asymptotic form of the unperturbed potential at a large distance R from the center of sphere i in the absence of interaction. The hypothetical point charge geffi gives rise to the potential 

The asymptotic expression for the unperturbed potential of sphere i (/ = 1, 2) at a large distance R from the center of sphere i may be expressed as 

Note that Bell et al. [7] derived Eq. (11.73) by the usual method, that is, by integrating the osmotic pressure and the Maxwell tensor over an arbitrary closed surface enclosing either sphere. Equation (11.73) agrees with the leading term of the exact 

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A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... 

The integration in Eq. VI-21 may be carried out for various macroscopic shapes. An important situation in colloid science, two spheres of radius a yields... 

Fig. VI-5. The effect of electrolyte concentration on the interaction potential energy between two spheres where K is k in cm". (From Ref. 44.)...

For hard spheres, the coefficients are independent of temperature because the Mayer/-fiinctions, in tenns of which they can be expressed, are temperature independent. The calculation of the leading temiy fy) is simple, but the detennination of the remaining tenns increases in complexify for larger n. Recalling that the Mayer /-fiinction for hard spheres of diameter a is -1 when r < a, and zero otherwise, it follows thaty/r, 7) is zero for r > 2a. For r < 2a, it is just the overlap volume of two spheres of radii 2a and a sunple calculation shows tliat... 

The first case is relevant in the discussion of colloid stability of section C2.6.5. It uses the potential around a single sphere in the case of a double layer that is thin compared to the particle, Ka 1. Furthennore, it is assumed that the surface separation is fairly large, such that exp(-K/f) 1, so the potential between two spheres can be calculated from the sum of single-sphere potentials. Under these conditions, is approximated by [42] ... 

It is convenient to begin by backtracking to a discussion of AS for an athermal mixture. We shall consider a dilute solution containing N2 solute molecules, each of which has an excluded volume u. The excluded volume of a particle is that volume for which the center of mass of a second particle is excluded from entering. Although we assume no specific geometry for the molecules at this time, Fig. 8.10 shows how the excluded volume is defined for two spheres of radius a. The two spheres are in surface contact when their centers are separated by a distance 2a. The excluded volume for the pair has the volume (4/3)7r(2a), or eight times the volume of one sphere. This volume is indicated by the broken line in Fig. 8.10. Since this volume is associated with the interaction of two spheres, the excluded volume per sphere is... 

Figure 8.10 Excluded volume for two spheres (dotted surface) as determined by the distance of closest approach.

Fig. 3. The two-sphere model illustratiag material transport paths I—VII duriag sintering where (a) represents coarsening (b), the two spheres before...

Hertz [27] solved the problem of the contact between two elastic elliptical bodies by modeling each body as an infinite half plane which is loaded over a contact area that is small in comparison to the body itself. The requirement of small areas of contact further allowed Hertz to use a parabola to represent the shape of the profile of the ellipses. In essence. Hertz modeled the interaction of elliptical asperities in contact. Fundamental in his solution is the assumption that, when two elliptical objects are compressed against one another, the shape of the deformed mating surface lies between the shape of the two undeformed surfaces but more closely resembles the shape of the surface with the higher elastic modulus. This means the deformed shape after two spheres are pressed against one another is a spherical shape. 

This allows for the equivalence between crossed cylinders and the particle on a plane problem. Likewise, the mechanics of two spheres can be described by an equivalently radiused particle-on-a-plane problem. The combination of moduli and the use of an effective radius greatly simplifies the computational representation and allows all the cases to be represented by the same formula. On the other hand, it opens the possibility of factors of two errors if the formula are used without realizing that such combinations have been made. Readers are cautioned to be aware of these issues in the formulae that follow. 

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... 

Pietersen (1988) describes the San Juan Ixhuatepec disaster. The storage site consisted of four spheres of LPG with a volume of 16(X) m (56,500 ft ) and two spheres with a volume of 2400 m (85,000 ft ). An additional 48 horizontal cylindrical tanks of various dimensions were present (Figure 2.24). At the time of the disaster, the total site inventory may have been approximately 11,000-12,000 m (390,000-420,000 ft ) of LPG. 

Figure 5-7 shows a simple electrometer. It consists of two spheres of very light weight, each coated with a thin film of metal. The spheres are suspended near each other by fine metal threads in a closed box to exclude air draffs. Each suspending thread is connected to a brass terminal. Next to the box is a battery —a collection of electrochemical cells. There are two terminal posts on the batteiy. We shall call these posts Pi and Pi. If post Pi is connected by a copper wire... 

This time the two spheres move apart—they repel each other When both spheres are given electric charge from the battery post labeled P-, they repel instead of attract. In Figure 5-7 we... 

Now let us reverse the wires so that both spheres are charged from battery post P. This time, Pi is connected to the base of the electrometer. Again we observe that the spheres move apart. Whenever both spheres are connected to the same battery post, the two spheres repel each other. 

It is known that electric charges attract or repel each other with a force that is inversely proportional to the square of the distance between them. If two spheres like those in the electrometer (Figure 5-7) are negatively charged, what would be the change in the force of repulsion if the distance between them were increased to four times the original distance ... 

Absolute values of 5 and px are represented in the xz plane in Figs. 1 and 2. s is spherically symmetrical, with the value 1 in all directions. px consists of two spheres as shown (the x axis is an infinite symmetry axis), with the maximum value /3 along the x axis. py and />, are similar, with maximum values of /3 along the y and z axis, respectively. From Rule 5 we conclude that p electrons will form stronger bonds than s electrons, and that the bonds formed by p electrons in an atom tend to be oriented at right angles to one another. 

The simplest molecules contain just two atoms. For example, a molecule of hydrogen is made up of two hydrogen atoms. A molecule that contains two atoms is classified as a diatomic molecule. Figure FA represents a diatomic hydrogen molecule as two spheres connected together. 

A hydrogen molecule can be represented by connecting two spheres together, with each sphere representing one hydrogen atom. 

Let us now examine why this mecheuiism might be true. In the case of sintering of spheres, we can define two cases as before, that of no shrinkage and that of shrinkage, both as a function of volume. If we have two spheres in... 

Since the gravitational constant is extremely small, it is natural to expect that the force of interaction is also very small too. For illustration, consider two spheres with radius Im and mass 31.4x10 kg made from galena, with the distance between their centers 10 m. Then, the force of interaction, Equation (1.1), is... 

The author assumed that the Born radii of atoms can be estimated from the solvent exposure factors for sampling spheres around the atoms. Two spheres were used in a five-parameter equahon to calculate the Born radii. The parameters of the equahon were eshmated using numerical calculahons from X-ray protein structures for dihydrofolate reductase. In addition to AGol the author also considered the AGJ term accounting for cavity formahon and dispersion of the solute-solvent interactions as ... 

Multiparticle collision dynamics provides an ideal way to simulate the motion of small self-propelled objects since the interaction between the solvent and the motor can be specified and hydrodynamic effects are taken into account automatically. It has been used to investigate the self-propelled motion of swimmers composed of linked beads that undergo non-time-reversible cyclic motion [116] and chemically powered nanodimers [117]. The chemically powered nanodimers can serve as models for the motions of the bimetallic nanodimers discussed earlier. The nanodimers are made from two spheres separated by a fixed distance R dissolved in a solvent of A and B molecules. One dimer sphere (C) catalyzes the irreversible reaction A + C B I C, while nonreactive interactions occur with the noncatalytic sphere (N). The nanodimer and reactive events are shown in Fig. 22. The A and B species interact with the nanodimer spheres through repulsive Lennard-Jones (LJ) potentials in Eq. (76). The MPC simulations assume that the potentials satisfy Vca = Vcb = Vna, with c.,t and Vnb with 3- The A molecules react to form B molecules when they approach the catalytic sphere within the interaction distance r < rc. The B molecules produced in the reaction interact differently with the catalytic and noncatalytic spheres. 

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. 

In this category, among the molecular, electrostatic and magnetic interparticle bonds, interest is primarily centered on the van der Waals-type attractive forces that may predominate in the absence of liquid and solid bonds. The force of the van der Waals attraction between two spheres of equal size is (R4)... 

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