Interaction potential, between spheres

Fig. 31. Interaction potential between spheres of radius a with depletion layers of thickness L from Eq. (132) (Gast et

The limitation of the Deijaguin approximation is that it only provides reliable results for R > Rg. To obtain results for the interaction potential between spheres for arbitrary q = Rg/R we use the extended Gibbs adsorption equation. Taniguchi et al. [33] and, independently, Eisenriegler et al. [34] found the concentration profile of Gaussian ideal polymer chains around a single hard sphere with radius R which reads... 

In Fig. 2.26 we present the interaction potential between spheres (valid up to or, equivalently, up to cfp ). In [47] results are presented for the interaction valid up to nl including a comparison with the computer simulation results of Biben et al. [54]. 

This dissociation is in effect an extension of the diameter d2 of the second coordination sphere and the subsequent decrease in the intrinsic interaction potential of the outer sphere. Therewith, the inter-spherical interaction potential between the central atom and the first coordination sphere increases, leading to shortening of the distance di, which in turn leads to an increase in the frequency of the Ta-F bond vibration. 

Figure 7.13 Left interaction potential and force between an atom at the apex of the tip and an atom in the surface. Tip-surface interactions can be described by a summation of these potentials over all combinations of atoms from the tip and the surface. Right interaction potential between the tip, approximated as a sphere, and a plane surface, valid in the non-contact mode of force microscopy. To stress the long-range character of the non-contact potential, the Lennard-Jones interaction potential between two atoms has been included as well (dotted line).

An alternative to the hard-sphere collision rate constant in Eq. 10.155 is used for the case of a Lennard-Jones interaction potential between the excited molecule (1) and the collision partner (2) characterized by a cross section a 2 and well depth en... 

The simple formula derived for viscosity in Eq. 12.49 predicts that /z should be independent of pressure and should increase as the square root of temperature. It is typically found that the viscosity of a gas is independent of pressure except at high and low pressure extremes. At very high pressure, molecular interactions become more important and the rigid-sphere approximation becomes inappropriate, leading to a breakdown in Eq. 12.49. At very low pressures, the gas no longer behaves like a continuum fluid, and the steady-state flow picture of Fig. 12.1 is no longer valid. Viscosity is usually found experimentally to increase with T faster than the n = 1 /2 power. Consideration of the interaction potential between molecules, as is discussed in the next section, is needed to more closely match the observed temperature dependence of /x. 

In the real world, however, the interaction potential between molecules cannot be described by the hard-sphere potential. It is continuous in nature. This makes the calculations difficult, and even an exact calculation of the binary collision term for a continuous potential is numerically formidable [46]. Sjogren and Sjolander have developed a repeated ring kinetic theory for a one-component system where the interaction is described by a continuous potential [9]. They have also included the effect of the full many-body propagators in describing the intermediate propagation. 

In the condensed phase the AC permanently interacts with its neighbors, therefore a change in the local phase composition (as were demonstrated on Figs. 8.1 and 8.2) affects the activation barrier level (Fig. 8.6). Historically the first model used for surface processes is the analogy of the collision model (CM) [23,48,57]. This model uses the molecular-kinetic gas theory [54]. It will be necessary to count the number of the active collisions between the reagents on the assumption that the molecules represent solid spheres with no interaction potential between them. Then the rate constant can be written down as follows (instead of Eq. (6)) ... 

In Model 2 the ratio 3a/2y may be considered approximately to represent the ratio of the dispersion interaction potential between an adsorbate molecule and a solid surface for a polarized as against a rigid, unpolarized adsorbate molecule, assuming in both cases that the potential may be represented by the 3-9 Lennard-Jones (surface) function. This approximation is based additionally on the assumption that the adsorbate is effectively hard sphere in the multilayer region. This ratio turns out to be 3.3 and 3.5 for 02 and N2 on anatase, respectively. Furthermore, the adsorbate-adsorbent interactions in the adsorbate-polarization case must evidently amount to 1.8 EL and 2.5 EL for 02 and N2 on anatase, re-... 

The interaction energy between spheres at constant surface potential involves only the function G (0 (which depends only on the sphere radius a,), while the interaction at constant surface charge density is characterized by the function H (i) (which depends on both sphere radius u, and relative permittivity Ep,). The interaction energy in the mixed case involves both G (i) and // (/) ... 

Electrostatic interactions in solutions containing charged particles and ions can be described using the Poisson-Boltzmann equation. A charged surface attracts counterions into a double layer of thickness defined by the Debye length, which depends on counterion concentration and solvent dielectric constant. From simplified theories, expressions can be derived for the attractive interaction potential between charged spheres. 

Several models have been developed to describe these phenomena quantitatively, the main difference being the interaction potential between the particles. There are two major approaches the hard sphere and the soft sphere. The hard sphere assumes that the only interaction between particles is a strong repulsion at the point of contact. The soft sphere is more realistic and assumes a potential with a barrier and a primary minimum like in DLVO theory (Figure 11.8). 

If the encounters between A B complexes and X molecules are not assumed to be those between elastic hard spheres, the functional form of ABx( r) appropriate to the assumed interaction potential between A B and X must be known before Eq. (2-38) can be integrated. The arbitrary cutoffs introduced by defining an A B complex as an A and a B molecule with internuclear separations in the range rf B b reduces the sub-... 

Equation 15 can be utilized to derive the EDL interaction potential between a sphere of radius a and a flat plate by setting ai = a, U2 oo, to obtain... 

The first positive term on the right-hand side represents the osmotic repulsion between the brushes and the second negative term originates from the elastic energy gain upon retraction of chains (less stretching). The repulsion dominates the interaction for hpressure yields the interaction potential between two plates from which also the interaction between two spheres can be derived. 

Fig. 1.9 Sketch of the total interaction potential between two spheres covered with polymer brushes in a good solvent in a solution containing non-adsorbing polymer chains...

Interaction Potential Between Two Spheres Using the Force Method... 

Some of the above results also follow directly from the so called Deijaguin approximation. Deijaguin [10] showed that there exists a simple (approximate) relation for the foree between curved objects and the interaction potential between two flat plates. In the Deijaguin approximation the spherical surface is replaced by a collection of flat rings. Consider two spheres with radius 1 at a center-to-center... 

Here W h)is the interaction potential between two flat plates at distance h. Clearly this approximate relation between the force for spheres and the interaction potential for plates is more accurate the larger the radius of the spheres compared to the range of the interaction. In this chapter we shall frequently use this Derjaguin approximation. It is a useful tool which, under the right conditions (see above), is very accurate but one has to be careful and be aware of its limitations. 

This is an important relation as it allows one to obtain the interaction potential between two parallel plates from the measured force between a sphere and a wall (see Sect. 2.6)... 

Fig. 2.14 Interaction potential between two big hard spheres as a function of the closest distance between the surfaces of the spheres...

Using the Derjaguin approximation (2.27) for the force between two spheres the interaction potential between the spheres can be obtained from... 

Fig. 2.18 Interaction potential between two spheres for RjRg = 100. Dashed curve (2.64) using (2.63). Solid curve (2.58)...

Fig. 2.25 Interaction potential between two hard plates due to small hard spheres (< = 0.1)...

The effective pair interactions measured with these techniques are the direct pair interactions between two colloidal particles plus the interactions mediated by the depletants. In practice depletants are poly disperse, for which there are sometimes theoretical results available. For the interaction potential between hard spheres we quote references for the depletion interaction in the presence of polydisperse penetrable hard spheres [74], poly disperse ideal chains [75], poly-disperse hard spheres [76] and polydisperse thin rods [77]. 

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