Distance, effect, interactions between

When the particle volume fraction / was increased to 0.015, the oscillations of the effective interaction between identical charged particles became larger than those for / = 0.005 at an electrolyte concentration of 10 5 M (Fig. 3). The effective interaction between identical charged particles versus the distance between particles is plotted in Fig. 3 for various values of Z (Z = 300, Z = 600, Z = 1200). As shown in Fig. 4, the colloidal dispersion has a disordered liquidlike structure for Z = 600, but a more ordered structure for Z = 1200. When the electrolyte concentration was increased to 10-4 M, the interaction between identical charged particles became completely screened. As shown in Fig. 5, no oscillations of the effective interaction potential were present for Z = 300,600, and 1200. 

Preliminary inspection of Equation 2.16 reveals that the Gibbs pair potential leads to repulsion at small interplate separations and attraction at large distances. Because it is U°n and not 17 F that is the appropriate pair potential for describing the effective interaction between the mth and nth macroions in solution under isobaric conditions, the different analytic properties of and U, have profound implications for colloid science. 

The seasoned Debye-Hiickel (D-H) theory, put forth in 1923 [33,34] takes into account the contribution of the ionic electrostatic interactions to the free energy of a solution and provides a quantitative expression for the activity coefficients. The basic concept of the D-H theory is that the long-range Coulomb interaction between two individual ions bathed in a salt solution is mediated by mobile ions from the solution. The effective charges of a certain ion are decreased as the result of charge screening by the mobile counterions it follows that, at sufficient distance, the interaction between two ions decays exponentially. We briefly outline the main considerations and assumptions of the D-H model ... 

The effective interaction between ions in aluminum. Also shown is the distribution of neighbors as a function of distance in the face-centered cubic structure. [After Harrison, 1964.]... 

It turns out that concepts of dielectric media are helpful to describe such a system so the effective interaction between a pair of opposite charges at distance r is not the bare interaction 2rr J ln(r/a), but rather it is screened by a distance-dependent dielectric constant (r) which must be calculated self-consistently. A renormalization group treatment (Kosterlilz, 1974 Young, 1978) shows that the quantity K n r/a) = J/(k Teir)) vanishes above 7kt for r - oo but behaves as... 

Note that this contribution to overall energy does not include other through-space effects such as van der Waals interactions. To take account of these effects, interactions between atoms which are separated from each other by greater than 1,4 distances are usually split into van der Waals and electrostatic components. There are many ways of describing van der Waals interactions the most common methods employ either the 6-12 (Lennard-Jones) potential or the Buckingham potential as shown below ... 

In TT-electron models the a and core electrons play the static role of screening the Coulomb interactions between the remaining degrees of freedom. In particular, they screen the nuclear-nuclear interactions, the interactions between the TT-electrons and the nuclei, and the mutual interactions between the Tr-electrons. This screening is often modelled by a static dielectric constant, and by the reduction of the effective charge of the nucleus to +Q at large distances. We now define Vp r R ) as the pseudopotential which models the effective interaction between the TT-electrons and the nuclei, while Vg g(r — r ) models the effective electron-electron interaction. 

Now let us consider two immersed particles of radii R i, R i-The free energy is then defined by the same Equation [76] with Ip = 0 at the two particle surfaces and ip = ipo far away from both particles. The free energy depends on the distance D between the particle centers W=W D). The force Fs = -dWl 3D of effective interaction between the particles can be found using the Laplace Equation [77]. The problem is analogous to the interaction of two conducting charged spheres. The result for D Rj is... 

In the above equation the effective interface potential represents the effective interaction between the two corresponding planar interfaces separated by a distance 1. The term Xyp paV fyp y fpa describes that contribution to the effective interaction between the two interfaces that stems from their corrugation. The dependence of the functions ojyp pai ) and go,( ) reflects the decay of the microscopic pair potentials governing the system. For long-ranged dispersion forces, which for large distances r decay both these functions decay for lai e . The... 

Figure B3.6.3. Sketch of the coarse-grained description of a binary blend in contact with a wall, (a) Composition profile at the wall, (b) Effective interaction g(l) between the interface and the wall. The different potentials correspond to complete wettmg, a first-order wetting transition and the non-wet state (from above to below). In case of a second-order transition there is no double-well structure close to the transition, but g(l) exhibits a single minimum which moves to larger distances as the wetting transition temperature is approached from below, (c) Temperature dependence of the thickness / of the enriclnnent layer at the wall. The jump of the layer thickness indicates a first-order wetting transition. In the case of a conthuious transition the layer thickness would diverge continuously upon approaching from below.

The angles ot, p, and x relate to the orientation of the dipole nionient vectors. The geonieti y of interaction between two bonds is given in Fig. 4-16, where r is the distance between the centers of the bonds. It is noteworthy that only the bond moments need be read in for the calculation because all geometr ic features (angles, etc.) can be calculated from the atomic coordinates. A default value of 1.0 for dielectric constant of the medium would normally be expected for calculating str uctures of isolated molecules in a vacuum, but the actual default value has been increased 1.5 to account for some intramolecular dipole moment interaction. A dielectric constant other than the default value can be entered for calculations in which the presence of solvent molecules is assumed, but it is not a simple matter to know what the effective dipole moment of the solvent molecules actually is in the immediate vicinity of the solute molecule. It is probably wrong to assume that the effective dipole moment is the same as it is in the bulk pure solvent. The molecular dipole moment (File 4-3) is the vector sum of the individual dipole moments within the molecule. 

There are two ways in which the volume occupied by a sample can influence the Gibbs free energy of the system. One of these involves the average distance of separation between the molecules and therefore influences G through the energetics of molecular interactions. The second volume effect on G arises from the contribution of free-volume considerations. In Chap. 2 we described the molecular texture of the liquid state in terms of a model which allowed for vacancies or holes. The number and size of the holes influence G through entropy considerations. Each of these volume effects varies differently with changing temperature and each behaves differently on opposite sides of Tg. We shall call free volume that volume which makes the second type of contribution to G. 

The reaction medium plays a very important role in all ionic polymerizations. Likewise, the nature of the ionic partner to the active center-called the counterion or gegenion-has a large effect also. This is true because the nature of the counterion, the polarity of the solvent, and the possibility of specific solvent-ion interactions determines the average distance of separation between the ions in solution. It is not difficult to visualize a whole spectrum of possibilities, from completely separated ions to an ion pair of partially solvated ions to an ion pair of unsolvated ions. The distance between the centers of the ions is different in... 

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