Interaction between two hard spheres

For the interaction between two hard spheres 1 and 2 having radii a and 02. and constant surface potentials i/ oi and ij/o2, respectively, at separation H between the two spheres, the interaction energy (R) is given by [3,6,8,10]... 

Fig. 1.8 Sketch of the depletion interaction between two hard spheres...

For more general interactions - i.e. not necessarily between two hard-spheres - we introduce a differential scattering cross section a v,8), defined by b db d(j> a u,8)dU. Boltzman s equation (equation 9.32) becomes... 

To conclude this section, I would like to add two sets of results on the pair correlation function (PCF) between two simple solutes in an L/ solvent. These results may or may not be relevant to the problem of Hhard-sphere (particles labeled A) solutes in an LJ solvent (particles labeled B) with varying strength of the energy parameter sbb- All the calculations for this and the subsequent demonstration were done by solving the Percus-Yevick integral equations with the following molecular... 

We now proceed to reduce (292) for the case of hard-core interaction. The dynamical description of a collision between two hard spheres of diameter cr is depicted in Fig. 5. The initial relative separation and momentum are r and p. It is evident that there will be no collision unless r p < 0 and where b is the... 

In Fig. 1.8 the AOV interaction potential WdepQi) between two hard spheres in a solution containing free polymers is plotted. The minimum value of the potential Wdep is achieved when the particles touch h = 0). 

FIGURE 7.7 Schematic Baxter (—), Van der Waals attraction (—), and steric stabilization (.) potential curves describing the interaction between two DMDBTDM A aggregates. Rc is the radius of the polar core, Rhs is the hard-sphere radius, and (8-Rhs) represents the distance of the effective attractive interaction. (From L. Martinet, Organisation Supramoleculaire des Phases Qrganiques de Malonamides du Precede d Extraction DIAMEX. PhD thesis. Rapport CEA-R-6105, 2005. With permission.)... 

The simplest models view the interacting bodies as hard spheres (e.g., billiard balls). Mathematically, if r is the separation between the center of two molecules, we write the potential energy of interaction between them as ... 

FIGURE 7.1 Collisions between two particles that interact through a hard-sphere potential (left) can be described by conservation of momentum and energy. A potential that is more realistic for atoms and molecules (right) changes the trajectories. 

Several classical trajectories may result from such a collision process, as sketched in the figure. What makes the manifold of trajectories possible are the internal states i and j of the colliding molecules. To make that evident, let us first consider a situation where there are no internal states of the molecules and where the interaction potential only depends on the distance between the molecules, like for two hard spheres. Then there will only be one trajectory possible for a given b, , v, because the initial conditions for the deterministic classical equations of motion are completely specified. This will not be the case when the molecules have internal degrees of freedom, even if... 

FIGURE 14.3 Interaction between two charged hard spheres 1 and 2 of radii ai and U2 at a separation R between their centers. H = R — — 02) is the closest distance between their... 

Equations (15.49) and (15.50), respectively, agrees with the expression for the electrostatic interaction energy between two parallel hard plates at constant surface charge density and that for two hard spheres at constant surface charge density [4] (Eqs. (10.54) and (10.55)). 

It is well known [4,5] that in the case of hard spheres di = d.2 = 0), the electrostatic force between two dissimilar spheres with charges of unlike sign is attractive for large kH but becomes repulsive at small kH, that is, there is a minimum in the interaction energy except when a lox =1. The case of nonzero Kd and xd2,... 

Isolated atoms show spherical symmetry, and it is natural to model atoms by spheres of some suitably defined radii. The potential energy of interaction between two atoms rises very sharply at short internuclear distances during atomic collisions, not unlike the potential energy increase in the collisions of hard, macroscopic bodies. In a somewhat crude, approximate sense, atoms behave as hard balls. This analogy can be used for a simple molecular model where atoms are represented by hard spheres. Once a choice of atomic radii is made, the approximate atomic surfaces can be defined as the surfaces of these spheres. 

The quantity Gaa (or G ) is often referred to as representing the solute-solute interaction. In this book, we reserve the term interaction for the direct intermodular interaction operating between two particles. For instance, two hard-sphere solutes of diameter a do not interact with each other at a distance R> a, yet the solute-solute affinity conveyed by Gaa may be different from zero. Therefore, care must be exercised in identifying DI solutions as arising from the absence of solute-solute interactions. 

Returning to the energy of interaction between two molecules as a function of distance, we see that the energy at large distances decreases as 1/r until r reaches the value a. At r < cr, the energy becomes infinitely positive this is shown by the vertical line in Fig. 26.6(a). This form of the interaction energy results from the supposition that the molecules are hard spheres of diameter cr. 

As already mentioned, it is hard to obtain a quantitative measure of PMF between two hydrophobic solutes, say for example, between two phenylalanine residues in a protein. Theoretical studies have often modeled this process by studying the interaction between two spheres as a fimction of the distance between them, as shown in Figure 15.4. 

Two specific interaction schemes are considered a) the particles interact by predominantly hard-sphere repulsive forces b) a short range attractive interaction between particles exists, such that a weak tendency for self association results. Likely candidates for the attractive potential between PSM primary aggregates are hydrophobic and/or hydrogen-bonding interactions of the carbohydrate side chains S. 

The constants k, k",... are related to the interaction between two particles, three particles, and so on. For hard spheres, for which k = 2.5, a value of 6.2 has been calculated for k. The similarity between Equation 17.8 and Equation 3.105 may be striking. However, k, k",... are partly of dynamic nature (influenced by particle interactions due to motion in the flow field), whereas B, C,... pertain to thermodynamic quantities, such as osmotic pressure. 

The polymer density profile of ideal chains next to a hard sphere for arbitrary size ratio q was first ealeulated by Taniguchi et al. [125] and later independently by Eisenriegler et al. [126]. Eisenriegler also considered the pair interaction between two colloids for Rg< R [127] and for Rg R [128], as well as the interaction between a sphere and a flat wall due to ideal chains [129]. Depletion of excluded volume polymer chains at a wall and near a sphere was considered by Hanke et al. [130]. One of their results is that the ratio /Rg at a flat plate, which is 1.13 for ideal chains [118, 119], is slightly smaller (1.07) for excluded-volume chains. 

In this chapter we consider the depletion interaction between two flat plates and between two spherical colloidal particles for different depletants (polymers, small colloidal spheres, rods and plates). First of all we focus on the depletion interaction due to a somewhat hypothetical model depletant, the penetrable hard sphere (phs), to mimic a (ideal) polymer molecule. This model, implicitly introduced by Asakura and Oosawa [1] and considered in detail by Vrij [2], is characterized by the fact that the spheres freely overlap each other but act as hard spheres with diameter a when interacting with a wall or a colloidal particle. The thermodynamic properties of a system of hard spheres plus added penetrable hard spheres have been considered by Widom and Rowlinson [3] and provided much of the inspiration for the theory of phase behavior developed in Chap. 3. 

For the calculation of the depletion interaction due to hard spheres we need the concentration profile between two confining walls. This problem was treated analytically by Glandt [45] and by Antonchenko et al. [46] using Monte Carlo computer simulations. Like for a single waU we present the calculation of the concentration profile between two confining walls to order n. For hdepletion zone of a sphere overlaps with the depletion zones of both walls (see Fig. 2.22) and we can write... 

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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