Repulsive potentials

The presence of the large repulsive potential barrier between the secondary minimum and contact prevents flocculation. One can thus see why increasing ionic strength of a solution promotes flocculation. The net potential per unit area between two planar surfaces is given approximately by the combination of Eqs. V-31 and VI-22 ... 

For purely repulsive potentials (u(r) > 0 ),f(r) is negative is positive. For purely attractive potentials,... 

Hyperspherical coordinates have the properties that q motion is always bound since q = 0 and q = P correspond to cases where two of the three atoms are on top of one another, yielding a very repulsive potential. Also, p —> 0 is a repulsive part of the potential, while large p takes us to the reagent and product valleys. 

The final scattering angle 0 is defined rising 0 = 0(t = oo). There will be a correspondence between b and 0 that will tend to look like what is shown in figure A3.11.5 for a repulsive potential (liere given for the special case of a hard sphere potential). 

Figure B2.3.10. Potential energy eiirves [42] of the ground X and exeited A eleetronie states of the hydroxyl radieal. Several vibrational levels are explieitly drawn in eaeh eleetronie state. One vibrational transition is explieitly indieated, and the upper and lower vibrational wavefiinetions are plotted. The upper and lower state vibrational quantum numbers are denoted V and v", respeetively. Also shown is one of the three repulsive potential energy eurves whieh eorrelate with the ground 0( P) + H dissoeiation asymptote. These eause predissoeiation of the higher rotational and vibrational levels of the A state.

Fig. 3. Attraction—repulsion potentials as a function of distance between particle centers. Curve 1 represents the attractive potential caused by van der Waals forces, curve 2 is the repulsive potential caused by double-layer forces, and curve 3 is the resultant force experienced by the two particles.

The relative value of the two potentials reveals the destabdization action of salts added to the emulsion. Addition of an electrolyte to the continuous phase causes a reduction of the electric double-layer repulsion potential, whereas the van der Waals potential remains essentially unchanged. Hence, the reduced electric double-layer potential causes a corresponding reduction of the maximum in the total potential, and at a certain concentration of electrolyte the maximum barrier height is reduced to a level at which the stabdity is lost. 

Fig. 2. Sketch of the interaction potential between segments m and n. The potential can be decomposed into a hard core repulsive potential Unm (hard) and a weak attractive potential Unn, (attr)...

Since we are interested in this chapter in analyzing the T- and P-dependences of polymer viscoelasticity, our emphasis is on dielectric relaxation results. We focus on the means to extrapolate data measured at low strain rates and ambient pressures to higher rates and pressures. The usual practice is to invoke the time-temperature superposition principle with a similar approach for extrapolation to elevated pressures [22]. The limitations of conventional t-T superpositioning will be discussed. A newly developed thermodynamic scaling procedure, based on consideration of the intermolecular repulsive potential, is presented. Applications and limitations of this scaling procedure are described. 

Dropping the attractive term in Equation 24.5 gives the inverse power-law (IPL) repulsive potential [69-72]... 

Generally, the values of the scaling exponent are smaller for polymers than for molecular liquids, for which 3.2 < y < 8.5. A larger y, or steeper repulsive potential, implies greater influence of jamming on the dynamics. The smaller exponent found for polymers in comparison with small-molecule liquids means that volume effects are weaker for polymers, which is ironic given their central role in the historical development of free-volume models. The reason why y is smaller... 

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