Small Separations

The dispersion energy between two spheres 1 and 2 is inversely proportional to their separation for small values of d and to the sixth power of their separation for large d. 

When applying Eq. (2.32) to interactions obeying power laws m 7, we have to keep in mind that (n—1) fdrr = lnr rather than — (n —for n= 1. However, this necessary change in notation does not affect the conclusions drawn from Eq. (2.32). 

An exact integration of pair interactions is restricted to a few simple structures like spheres or half-spaces. The case of half-spaces was treated by de Boer in 1936 [90], the treatment of spheres was reported one year later by Hamaker [91], 

If other structures are considered we have to introduce approximations before carrying out integrations of the type considered in Eq. (2.27). It is often convenient to use different approximations for small and large separations. At small separations d it is appropriate to count the number of possibilities P r) in which a fixed distance r = Vij can be placed between atoms i of particle 1 and atoms j of particle 2. Between spheres with radii i i, i 2 we find the weight function 

The two-dimensional weight function P(r) between cylinders with radii Pi, R2 is calculated in Section 2.8. We obtain 

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At small separations or for molecules with more complicated charge distributions, the more general form... 

Often the van der Waals attraction is balanced by electric double-layer repulsion. An important example occurs in the flocculation of aqueous colloids. A suspension of charged particles experiences both the double-layer repulsion and dispersion attraction, and the balance between these determines the ease and hence the rate with which particles aggregate. Verwey and Overbeek [44, 45] considered the case of two colloidal spheres and calculated the net potential energy versus distance curves of the type illustrated in Fig. VI-5 for the case of 0 = 25.6 mV (i.e., 0 = k.T/e at 25°C). At low ionic strength, as measured by K (see Section V-2), the double-layer repulsion is overwhelming except at very small separations, but as k is increased, a net attraction at all distances... 

Fig. VI-6. The force between two crossed cylinders coated with mica and carrying adsorbed bilayers of phosphatidylcholine lipids at 22°C. The solid symbols are for 1.2 mM salt while the open circles are for 10.9 roM salt. The solid curves are the DLVO theoretical calculations. The inset shows the effect of the van der Waals force at small separations the Hamaker constant is estimated from this to be 7 1 x 10 erg. In the absence of salt there is no double-layer force and the adhesive force is -1.0 mN/m. (From Ref. 66.)...

This shows that the dielectric constant e of a polar solvent is related to the cavity fimction for two ions at large separations. One could extend this concept to define a local dielectric constant z(r) for the interaction between two ions at small separations. 

For the separation of immiscible liquids a small separating funnel of the conventional type should be used whenever practicable, a pear-shaped funnel (Fig. 16, p. 35) of 5-10 ml. capacity being particularly... 

This is done so that a small separating-funnel of about 100 ml. capacity may be used for the subsequent washing of the nitrobenzene. 

The mixture was then cooled to 30-40°C and the dissolved acetylene was sucked Over" in a water-pump vacuum (10-20 mmHg) and condensed in a receiver cooled at -78°C. The contents of the receiver were shaken three times with 20-ml portions of ice-water in a small separating funnel or dropping funnel in order to remove small amounts of tert.-butyl alcohol. After drying over a very small amount of... 

Zone refining is one of a class of techniques known as fractional solidification in which a separation is brought about by crystallization of a melt without solvent being added (see also Crystallization) (1 8). SoHd—Hquid phase equiUbria are utilized, but the phenomena are much more complex than in separation processes utilizing vapor—Hquid equiHbria. In most of the fractional-solidification techniques described in the article on crystallization, small separate crystals are formed rapidly in a relatively isothermal melt. In zone refining, on the other hand, a massive soHd is formed slowly and a sizable temperature gradient is imposed at the soHd—Hquid interface. 

Airless spray uses hydrauHc pressure to deUver the paint. Paint is brought to the spray gun under 7—40 mPa (1000—6000 psi), where it is divided into small separate streams and forced through a very small orifice to produce the spray. Airless spray is faster, cleaner, and less wasteful than air atomization, but demands good technique because it deUvers paint very quickly. 

It is seen from equation (22) that there will, indeed, be a temperature at which the separation ratio of the two solutes will be independent of the solvent composition. The temperature is determined by the relative values of the standard free enthalpies of the two solutes between each solvent and the stationary phase, together with their standard free entropies. If the separation ratio is very large, there will be a considerable difference between the respective standard enthalpies and entropies of the two solutes. As a consequence, the temperature at which the separation ratio becomes independent of solvent composition may well be outside the practical chromatography range. However, if the solutes are similar in nature and are eluted with relatively small separation ratios (for example in the separation of enantiomers) then the standard enthalpies and entropies will be comparable, and the temperature/solvent-composition independence is likely be in a range that can be experimentally observed. 

The contents of the separator are collected in a special small separation vessel. The emulsified explosive oil is separated and delivered... 

The effect of temperature on column efficiency, however, is frequently exploited, particularly in size exclusion chromatography (SEC). As has already been discussed, high efficiencies are essential in SEC due to the limited peak capacity of the column and consequently, the very small separation ratios. However the effect of temperature on column efficiency is not well understood by many analysts and consequently, will be discussed in some detail. It was shown on page... 

The colloid probe technique was first applied to the investigation of surfactant adsorption by Rutland and Senden [83]. They investigated the effect of a nonionic surfactant petakis(oxyethylene) dodecyl ether at various concentrations for a silica-silica system. In the absence of surfactant they observed a repulsive interaction at small separation, which inhibited adhesive contact. For a concentration of 2 X 10 M they found a normalized adhesive force of 19 mN/m, which is small compared to similar measurements with SEA and is probably caused by sufactant adsorption s disrupting the hydration force. The adhesive force decreased with time, suggesting that the hydrophobic attraction was being screened by further surfactant adsorption. Thus the authors concluded that adsorption occurs through... 

So far in our revision of the Debye-Hiickel theory we have focused our attention on the truncation of Coulomb integrals due to hard sphere holes formed around the ions. The corresponding corrections have redefined the inverse Debye length k but not altered the exponential form of the charge density. Now we shall take note of the fact that the exponential form of the charge density cannot be maintained at high /c-values, since this would imply a negative coion density for small separations. Recall that in the linear theory for symmetrical primitive electrolyte models we have... 

This potential is similar to one derived by Stratt [44] using different techniques. It is problematic though for simulations of molecular fluids due to the middle term on the right side [12], The square of the repulsive component of the force leads to nonphysical behavior in the effective potential for small separations. 

For relatively small separations, Figure 3.10 indicates that there are four electrostatic interactions two positive ones, and two negative ones, so the potential energy of the system is ... 

The velocity profile is uniform across the entire width of the channel if the channel is open at the electrodes, as is most often the case. However, if the electric field is applied across a closed channel (or a backpressure exists that just counters that produced by the pump), a recirculation pattern forms in which fluid along the center of the channel moves in a direction opposite to that at the walls further, the velocity along the centerline of the channel is 50% of that at the walls (Fig. 11.32a, see Plate 12 for color version). Figure 11.32b (see Plate 12 for color version) illustrates an electric field generating a net force on the fluid near the interface of the fluid/solid boundary, where a small separation of charge occurs due to the equilibrium between adsorption and desorption of ions. The charge region from excess cations localized near the interface by coulombic... 

Lund et al. (1999) reported finite element simulations of conduction between two contacting spheres without fluid flow. They modeled a small separation between the spheres to allow for interparticle micro-asperity gaps, and then the... 

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