Critical separation distance

In studies of steric stabilizers too little attention is generally paid to the dispersion force attractions between particles and the critical separation distance (H ) needed to keep particles from flocculating. Adsorbed steric stabilizers can provide a certain film thickness on each particle but if the separation distance between colliding particles is less than H the particles will flocculate. The calculation of H is not cr difficult and measurements to prove or disprove such calculations are not difficult either. For equal-sized spheres of substance 1 with radius or in medium 2 the Hamaker equation for the dispersion force attractive energy (Uj2i) at close approach is (7) ... 

Figure 3. Critical separation distance (H ) of spherical particles of radius (a) to prevent floccuSation at 25°C. Carbon-in-oil dispersions (C/0), polystyrene in water latexes (PS/W), and oil-in-water emulsions (0/W).

The critical separation distance calculated from the quenching data was found to be 13 A which is of the same order as the van der Waals separation. The critical separation distance remained unchanged when halogen substituted naphthalenes were used. The halo-substitution is expected to increase T1A SoA transition probability in naphthalenes. Since oscillator strengths / (naphthalenes /(iodonaphthalenes) is as 1 1000, no increase in transfer efficiency is clear indication of the lack of dependence on the oscillator strength. 

Parallel phosphorescence studies indicated that the O—O band for cinnamic acid occurred at 20,000 cm-1. Tsuda 43> reported a value of 17,600 cm-1. In the same study Kikuchi and Nakamura 42> measured the quenching of the phosphorescence of cinnamic acid with a group of typical sensitizers, among which were 2-nitro-fluorene and -nitroaniline. The critical separation distance between cinnamic acid and sensitizers for effective quenching was found to be 10A. 

In practice, as kET is inversely proportional to r6, it is often more convenient to measure the efficiency of energy transfer (ET) rather than kE v. Determination of the critical separation distance, Ra, provides a means of quantifying, experimentally, the ET efficiency. 

Consequently, Eq. (2.22) predicts that oc y and dtc/d0d < 0 - both confirmed by experiments. The coalescence involves collision between two drops, which flattens them, entrapping the matrix Drainage of the entrapped liquid depends on A and K [123]. The film might break when the separation between drops is smaller than the critical separation distance, hc = 5nm [124, 125]. 

K critical separation distance between coalescing drops... 

Briefly, the experiments coinsisted of measuring the critical separation distance between three subcritical clusters ol 2.3S wt% 11 enriched UC>2 rods in water. The experiments were of two basic types (a).those in which only water separated the fuel clusters, and (b) those in which selected neutron-absorbing materials were positioned m e water between the fuel clusters. 

The shotgun test involves shooting a cylinder of explosive or rocket propellant from a shotgun at a target. The sympathetic detonation test involves two cubes of explosive or propellant, one of which is detonated. The extent of reaction in the acceptor cube is determined by its effect on a lead witness cylinder in contact with the acceptor cube. The critical separation distance at which no detonation is observed is determined. These tests were modeled using the 2DE code with Forest Fire burn in reference 27. The gap test, shotgun test, and sympathetic detonation test were modeled. They are shock initiation experiments with a substained shock pulse. A detonation will occur when the shock wave is of sufficient strength and duration to build up to detonation. 

The basic charging of solid particles in a gas stream with a confining wall is assumed to depend on the contact potential and the critical separation distance Zq (Baum, Cole, and Mobbs, 1970). The theory follows that of a parallel plate capacitor relating the charge to the potential by the capacitance ... 

For a line spark source, the flame volume is initially cylindrical with the cylinder length equal to the separation distance between the electrodes. Thus, for a cylindrical flame, = e, and the critical ignition volumes are equation 7 for a spherical flame and equation 8 for a cylindrical flame where = critical ignition volume, m /kg e = thickness of flame front, m and d = flame height, m. 

Given that the atoms form a stable molecule at a separation of 0.3 nm with an energy of -4 eV, calculate A and B. Also find the force required to break the molecule, and the critical separation at which the molecule breaks. You should sketch an energy/distance curve for the atom, and sketch beneath this curve the appropriate/orce/distance curve. 

As a replacement for bulk oil circuit breakers for distribution purposes, vacuum is a rival to SFg. The contact interruption chamber is a sealed unit with a small contact-separation distances (typically, 16mm for 24kV). Choice of contact material is critical since the material affects ... 

The transition of the compressibility, and other properties of the polyelectrolyte brnshes, is most likely accounted for in terms of the transition in the binding mode of the connterion to the polyelectrolytes, from the loosely bonnd state to the tightly bound one, which rednces inter- and intrachain repulsive interactions. The following snpports this ac-connt (1) At the critical density, = 0.20 chain/nm, the separation distance between polyelectrolyte chains, d, is 2.4 nm. This distance is close to the snm, 2.6 nm, of the chain diameter, 1.3 nm, and the size of two hydrated connterions, 1.32 nm, indicating that the abrupt... 

In polyelectrolyte solutions, the counterion condensation on linear polyelectrolyte chains is known to occur when the charge density along the chain exceeds the critical value [40]. Our work indicates the existence of a critical value for the separation distance between chains, where the interchain interaction changes drastically, most likely due to the transition in the binding mode of the counterions (see Fig. 13). Many peculiar forms of behavior, which are often interpreted by the cluster formation or the interchain organization of polyelectrolytes, have been reported for high concentrations of aqueous polyelectrolytes... 

If the critical separation is determined for a large number of relative geometries of the electron and molecule it is possible to obtain a three-dimensional picture of the probability of ionization as a function of the orientation of the molecule. Effectively, the idea of an ionization cross section, the area the target molecule presents to the electron, is extended to a three-dimensional object defined by the critical distances, with ionization occurring when the electron penetrates the surface enclosing this volume. The volume enclosed by the electron impact ionization surface may be used to obtain an estimate for the cross section (volume averaged cross section) ... 

For the potential given by (5.3), it is easy to show that when b > bc the distance of closest approach is bc /21/2, whereas for b < b, the only thing preventing interpenetration is a repulsive core potential, which is not explicitly considered here. Equation (5.4) is actually the classical collision cross section for the problem. To translate this into a reaction cross section, we may assume that there is another critical separation r0 such that when r < rg chemical forces complete the reaction and no reaction takes place if r > rg. If rg is less than b /2m, then Eq. (5.4) is also the reaction cross section, since reaction definitely takes place if b < b. and it definitely does not take place if b > b.. According to this modification, the high-energy limit of the reaction cross section is nr2 rather than zero as given by (5.4). One therefore has... 

Table I lists some characteristic wave lengths from the work of Gregory (9). The calculations of f shown in Figure 2 are taken from the work of Clayfield and Lumb.(lO) By using these calculations one can determine the attractive energy per pair of particles at various separation distances, and determine for any particular value gf Aj2i> Xj, and radius (a) the critical value of H that makes U j 21= kT, where k is the Boltzmann constant and kT is the average vibrational energy of a pair of particles flocculated at separation distance H. If Uj2l is greater than -kT the particles will nearly always bounce apart on collision, but if it is less than -kT the particles tend to flocculate.

Solution When the two solid surfaces are far apart (d = oo), the free end of each adsorbed rod has access to sites, with = 2itL2 and L the length of the rod. When the separation is such that the second surface cuts off access to some of these sites, the number of accessible sites becomes The subscript here indicates a separation less than some critical distance that is the threshold for interaction. The exact form of and the critical separation at which it begins to apply depends on whether one or both surfaces carry the adsorbed rods. The fraction of the area of the hemisphere that remains accessible to the free ends of the rods could be calculated from geometrical considerations. Using these Q values as substitutions in Equation (3.46), we obtain ASR = kB In (fia/fic). Since ASR is negative. This gives the effect per... 

In Figure 7 we present the free energy for an asymmetric Gaussian distribution (a = 1.4) as a function of distance for various values of the Hamaker constant (with all the other parameters unchanged). For H > 3.825 x 10-21 J, a stable minimum is obtained at a finite distance. For H < 3.825 x 10 21 J, the stable minimum is at infinite distance however, for 3.825 x 10-21c7 > H > 3.45 x 10-21 J, a local (unstable) minimum is still obtained at finite distance. For H = 3.825 x 10-21 J, a critical unbinding transition occurs, since the minima at finite and infinite distances become equal. However, these two minima are separated by a potential barrier, with a maximum height of 1.68 x 10 7 J/m2, located at a separation distance of 90 A. The results remained qualitatively the same for any combination of the interaction parameters. 

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