Entropic repulsion

The well defined contact geometry and the ionic structure of the mica surface favours observation of structural and solvation forces. Besides a monotonic entropic repulsion one may observe superimposed periodic force modulations. It is commonly believed that these modulations are due to a metastable layering at surface separations below some 3-10 molecular diameters. These diflftise layers are very difficult to observe with other teclmiques [92]. The periodicity of these oscillatory forces is regularly found to correspond to the characteristic molecular diameter. Figure Bl.20.7 shows a typical measurement of solvation forces in the case of ethanol between mica. 

Pigment dispersions are stabilized by charge repulsion and entropic, ie, steric or osmotic, repulsion. Although both types of stabilization force may be present in most cases, for pigment dispersions in solvent-bome coatings entropic repulsion is usually the most important mechanism for stabilization. 

The entropic repulsion theory by Clayfield and Lumb [79], who suggest that the bulk of the protective macromolecular layer prevents other particles from approaching close enough for the attractive forces to cause agglomeration [80]. 

A similar simplifying assumption has been used by Allain et al.(90) in analyzing their experiments on a flexible fluorescent anthracene-polystyrene copolymer coil in the vicinity of a nonadsorbing wall. The analysis appears to confirm a local decrease in C(z) for small z at the solid/solution interface. Such a depletion layer is interpreted in terms of an entropic repulsion ... 

To extract a value of the step-mobility h from the grating relaxation experiments [12], we must evaluate the strength of the step-step interaction y. Computational work suggests that ydue to elastic interactions between Si(OOl) steps is 0.2 eV run [29], while, we estimate that the entropic interaction is 10 times larger. (We use a step stiffness P calculated from the geometric mean of P for Sa and Sb steps given in Ref [30] P, 0.03 eV mn-. ) Therefore, entropic repulsion should dominate, and... 

The experimental evidence for this scenario, mentioned at the outset of this section, is less compelling since it is harder to control this sort of dosage-sensitive STM experiment than a Monte Carlo simulation. (Specifically, it problematic to convert from CO dosage to the evolution time from initial instability the analysis would be better if the surface could be instantaneously de-oxidized.) From an earlier examination of the terrace-width distribution for Ag(110)2Y—>[001], Ozcomert et al. (1993) concluded that to a good approximation the step-step interactions were purely entropic repulsions (by finding a good fit to a free-fermion form (Jobs et al, 1991)). (But see also Pai et al (1994) for remarkable behavior under different conditions.) From the relationship (Bartelt el a/., 1992)... 

When a negative free energy is associated to a crossing point the network of steps tends to condense to maximize the density of crossings. On the other side, entropic repulsions (Gruber and Mullins, 1967) favor configurations where steps are far apart and tend to stabilize the network. 

Direct step-step interaction terms in the step energy ( direct interactions are entropic repulsion, strain terms, electronic structure effects etc.) do influence the step fluctuations, and they also drive the spreading of step trains, wires and bumps. Nevertheless, it is instructive to first ignore these direcf step-step repulsion, as is done in... 

The effects of improved wettability, entropic repulsion, and sterical hindrance undoubtedly play a role in stabilizing dispersed solid particles by block or graft copolymers. However, since the dispersions of titanium dioxide in toluene stabilized by carboxylated styrene-butadiene block copolymers are so much more stable than dispersions stabilized by carboxylated homopolymers under otherwise identical conditions, we must assume that an additional factor comes into play when block copolymers are used. The model in Figure 1 is an attempt to explain this additional... 

As a consequence, real surfaces will not exhibit such evenly sized terrace or evenly spaced kinks as suggested by Fig. 8.12. The terrace-ledge-kink (TLK) model [332] can provide a more realistic description of vicinal surfaces. The distribution of the terrace widths is calculated taking into account the entropic repulsion between ledges. The confinement of a ledge between two neighboring steps (that cannot be crossed) leads to a reduction of the number... 

Also noteworthy is the appreciable coalescence caused by the shear flows in the single screws, of the rheology section of the TSMEE following the mixing element section. Flow of dispersed immiscible blends involves continuous breakdown and coalescence of the dispersed domains (122). Shear flows, where droplet-to-droplet collisions are frequent—in contrast to extensional flows—favor coalescence over dispersion. The presence of compatibilizers shifts the balance toward reduced coalescence rate. Macosko et al. (123) attribute this to the entropic repulsion of the compatibilizer molecules located at the interface as they balance the van der Waals forces and reduce coalescence, as shown on Fig. 11.36. 

A. The Thermal Fluctuations of the Interfaces for Arbitrary Interactions. After the Helfrich initial theory,18 Helfrich and Servuss17 suggested an alternate derivation of the entropic repulsion due to the confinement of a membrane between rigid walls, by considering the lipid bilayer composed of many independent pieces , whose area is related to the root mean square fluctuations of the positions of the undulatingbilayer. As shown below, this representation can be extended to interfaces interacting via arbitrary potentials. 

Supplementing the entropic repulsion of the Carnahan-Starling (1969) equation with a van der Waals attraction leads to the SCF (Ploehn and... 

In non-aqueous media of low dielectric constant ionic charge stabilization is unlikely to be very important. In such cases stabilization depends on steric or entropic repulsion and polymeric agents are preferred... 

Elmendorp and van der Vegt 1986). This has been inferred from the observation that the droplet size does not increase as much with increasing droplet volume fraction as it does when the block copolymer is absent (Sundararaj and Macosko 1995 Beck Tan et al. 1996). Milner and Xi (1996) have put forth a theory that explains the suppression of coalescence in terms of an effective entropic repulsive force between droplets with block copolymers at their surfaces. 

Morphology of long alkanes is also being studied in order to evaluate the role of cilia in lamellar splay, a key factor in the formation of polymer spheru-lites.204-217 The hypothesis is that cilia exert an entropic repulsive force when crystalline lamellae approach each other closely. Earlier results seemed to have indicated that in long alkanes spherulites are... 

For the rod geometry, on the other hand, only a fraction of the backbone charges are neutralized by condensed counterions. This means that each rod still carries a net charge, so two rods will repel each other electrostatically. On the other hand, the entropic repulsion due to confinement of counterions is much weaker. 

The slope, P, that is a measure of the coalescence rate, is plotted as a function of the residence time in Figure 8.28. One can see that, even at low PP-MA content (PP-MA = 1 phr), a good stabilization mechanism was established after the reaction time, t = 10 min. This suggests sufficient coverage of the interface by the in situ formed graft copolymers, which provide the entropic repulsion between neighboring particles. When PP-MA content is higher, the mechanism is set up in shorter t. 

Another counter-argument against a strong wall-repulsion effect derives from the so-called surface anchoring effect [110] water-soluble initiators used in emulsion polymerization will lead to polar endgroups of the polymer chains. These endgroups are expected to be affixed near the svuface of the particles and counteract the entropic repulsion. 

Interparticle forces include the van der Waals attractive forces, electrostatic r ul-sive forces arising firom surface chaiges on the particles, and entropic repulsive forces due to water-soluble polymera adsoibed/anchored to the particle surface and/or due to adsorbed surfactants. These interparticle forces become important as the interparticle distance, h, becomes smaller, and are significant factors at h < 10 nm. Both smaller particle diameter, d, and higher volume fracfion, < >, lead to decreasing values of h, as shown in Equation (13.8). As already seen in Section 13.2.4, the thickness of adsorbed surfactant layers can be an appreciable fraction of the interparticle distance. Adsorbed polymer layers can be of the order of 10 nm in thickness, and in systons with low ionic strength, electrical charge effects can extend much further. 

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