Elastic repulsion

Equilibrium is established when the attractive surface forces are balanced by elastic repulsion forces between the materials. The DMT model states that the elastic repulsion force is related to the attractive force within the contact region Fs by... 

Jones developed an equation of the Griin-eisen type, based on the Einstein model of a solid, of the form p=Ae av— B+fRT, where a, A, B and f are constants. Lutzky, however, preferred to use an equation based on the discussion by Zel dovich Kompaneets (Ref 21) of the equation derived by Landau Stanyukovich (Ref 2). In their view, the comparatively stable molecules of the detonation products are in a highly compressed state, being at a density over twice that of the liquid gases. The predominant part of the pressure is due to elastic repulsion. 

The adsorbed layers between the particles may interpenetrate and so give a local increase in the concentration of polymer segments. Depending on the balance between polymer-polymer and polymer-dispersion medium interactions, this may lead to either repulsion or attraction by an osmotic mechanism. Enthalpic and entropic changes will be involved. If interpenetration takes place to a significant extent, elastic repulsion will also operate. 

Steric stabilisers are usually block copolymer molecules (e.g. poly (ethylene oxide) surfactants), with a lyophobic part (the anchor group) which attaches strongly to the particle surface, and a lyophilic chain which trails freely in the dispersion medium. The conditions for stabilisation are similar to those for polymer solubility outlined in the previous section. If the dispersion medium is a good solvent for the lyophilic moieties of the adsorbed polymer, interpenetration is not favoured and interparticle repulsion results but if, on the other hand, the dispersion medium is a poor solvent, interpenetration of the polymer chains is favoured and attraction results. In the latter case, the polymer chains will interpenetrate to the point where further interpenetration is prevented by elastic repulsion. 

From this discussion, the mechanism of dependence of pH on relaxation properties is as follows. Introduced side chains are ionized with an increase in pH, so that wood swells increasingly because of electrostatic repulsion of main chains of wood components. The swelling pressure increases until the balancing by elastic repulsive force due to the ultrastructure of wood is reached... 

Another approach is the model proposed by Johnson, Kendall, and Roberts (JKR), which considers the elastic deformation and takes into account of the adhesion contribution in contact mechanics (Hugel and Seitz 2001 Janshoff et al. 2000). This model considers the influence of van der Waals forces within the contact zone, and the diminished force of elastic repulsion caused by the attraction. A general equation relating contact area and load is described as follows ... 

An added atom from columns 3 or 5 neutralizes the three atoms below it Just as added atoms from columns 1 and 7 neutralize the single atoms below them. This would leave the alternate inward-outward reconstruction between such added titoms. We would expect the three back atoms to relax in order to reduce the angular discrepancy. It is possible that they move into the surface, tending to raise the atoms around them. The covertige of such atoms would certainly be limited to one added atom for each three surface atoms, and might well be limited further hy the elastic repulsion between the extended distortion patterns. 

Let us now consider again picture a in Fig. 12. If is increased to values ranging from 2-3 V/pm, we observe that the quadrupoles come into contact and can coalesce despite the elastic repulsion evidenced above. Indeed, for these field values, the attractive electrostatic force completely overwhelms the elastic repulsion force which eventually leads to a one-dimensional coalescence of the drops as shown in Fig. 14. Thus, the system is again unstable in the presence of high electric fields. Depending on the field intensity, we can then control the droplet coalescence and thereby their size. 

Fig. 14. For o>l V/nm,the electric field dominates the elastic repulsion and the quadrupoles are forced to coalesce within the chains. Scale bar 14.1 pm...

The elastic repulsion hypothesis has arisen from the studies using partially hydrolysed poly(vinyl alcohol) as stabilizer. In this instance, no interactional groups are apparently necessary on the surface of the colloidal particles, presumably because the blocks of poly(vinyl acetate) segments, which are nominally insoluble in water, anchor the poly(vinyl alcohol) stabilizing moieties to the surface. Of course, poly(vinyl alcohol) is a decidedly atypical... 

It is possible that an elastic contribution to steric stabilization could have a kinetic, rather than a thermodynamic, origin. The polymer segments in the stabilizing moieties may relax too slowly to allow interpenetration to proceed in the time span of a Brownian collision. A denting mechanism would then be operative in order to transmit the repulsive stress to the particles. This kinetic mechanism for the introduction of an elastic repulsion would predict that the enhancement should be strongly dependent upon the particle size which determines the duration of a Brownian encounter. No experimental evidence for such a particle size dependence has apparently been published. 

Finally, we note that the maximum in the UCFT observed with poly(vinyl alcohol) is readily explicable without the elastic repulsion hypothesis. It merely requires the adsorbed, partially hydrolysed polymer to change from a relatively flat conformation at lower concentrations to a conformation more extended normal to the interface, and thus exhibiting reduced multipoint anchoring, at higher concentrations. This explanation is closely analogous to that proposed by Dobbie et al. (1973) for poly(oxyethylene) attached to polystyrene latices containing surface carboxylic acid groups. 

The interpenetrational-plus-compressional domain is also important in heterostericdly stabilized dispersions where the different stabilizing sheaths are composed of compatible polymers. Elastic repulsion can give rise to heterosteric stabilization. 

Again a Derjaguin integration can be used to evaluate these expressions for the elastic repulsion for spheres. As demanded by physical considerations. 

It is apparent that the experimentally measured repulsion extends much further out from the surface than would be expected for isolated chains. The theoretical curve presented in Fig. 12.2 ignores the elastic repulsion that must... 

The elastic repulsion. Once the interpenetrational-plus-compressional domain is entered, not only must the mixing contribution to the steric repulsion be considered but so too must the elastic contribution. As the second particle approaches closer than the span of the stabilizing chains, the chains are compressed and so must lose configurational entropy. This is the origin of the elastic repulsion. The elastic repulsion is relatively insensitive to the solvency of the dispersion medium, being influenced by its nature only insofar as the solvency affects the chain conformation. 

Evans el al. (1977) have shown how the loss of configurational entropy can be calculated assiuning, somewhat unrealistically, that the segments behave as disconnected gas molecules with the appropriate number distribution function over the available space. Results obtained in this way for several distributions are summarized in Table 12.4. Also included in this table are the corresponding expressions for spheres obtained by a Deijaguin integration. These allow the elastic repulsion for spheres to be calculated from... 

The elastic repulsion calculated in this fashion is a rather sensitive function of the assumed form of the segment density distribution function. It is likely to be in error if only because of the unphysical nature of the model employed. Since the elastic term rises so steeply, however, its precise functional form is rarely of great moment. 

The distance dependent functions for the elastic repulsion of spheres and flat plates... 

Comparison of theory with experiment. It will be shown in Section 13.3.2.1 that the flat plate potentials can be used to calculate the osmotic disjoining pressures in concentrated monodisperse sterically stabilized dispersions. Evans and Napper (1977) have compared the theoretical predictions using the above equations with those measured by Homola and Robertson (1976) for polystyrene latex particles stabilized by poly(oxyethylene) of molecular weight ca 2 000 in aqueous dispersion media. The elastic repulsion in the interpenetrational-plus-compressional domain was estimated from the following expression for the constant segment density model... 

It is apparent from Fig. 12.5 that the measured distance dependence of the osmotic pressure is well described by the theoretical equations in the interpenetrational domain (/fo 9nm). On closer approach, however, the predicted repulsion rose too steeply, due presumably to an overestimation of the elastic repulsion. Better agreement between theory and experiment could be achieved by arbitrarily softening the elastic repulsion this was accomplished by replacing in equation (12.55) by (5o(3- o)V4. 

The simplest interpretation of the dramatic increase in the measured force when D Rg is that it is the elastic repulsion arising from the compression of the attached polystyrene chains by the close approach of the opposite plate. Since the first sign of steric interaction occurs at ca 3/J, this suggests that the steric layer thickness is 1 -SRg. Consequently, elastic compression of the chains would certainly be expected when D Rg. 

Notwithstanding these criticisms, it remains an experimental fact that the theories of steric stabilization which consider the sole origin of repulsion in worse than 0-solvents to be the elastic repulsion, predict a rise in the repulsion that is seemingly nowhere near sufficiently steep to explain Klein s results. This strongly suggests the operation of another source of repulsive interaction, of which that suggested by Flory is the most soundly based. 

In elastic steric stabilization, the elastic repulsion must exceed the attractive van der Waals interaction for stabiUty to be observed. Dolan and Edwards showed that two types of aggregation occur. The first, corresponding to close adhesion of the particles, occurs if... 

In the contact of smooth particles with a smooth surface, the contact zone is deformed. As a result of the deformation of the contact surfaces, forces of repulsion are created. Hence the adhesive interaction (if we assume that no other components participate in the formation of adhesion) will be made up of the molecular forces of the contact surfaces after deduction of the force created as a result of elastic repulsion of the bodies that have been subjected to deformation. The deformation tends to increase the contact area between the particle and surface, and this causes an increase in adhesion. 

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