Bulking force

Solid surfaces of single crystals provide to some extent the realization of a well-defined two-dimensional (2D) periodic array of atoms. However, the loss of vertical translational invariance at the surface changes the local force held with respect to the bulk forces. As seen in section 3 the charge redistribution is... 

Alcohols The magnitude of y changes by 23.7 - 22.1 = 1.6 mN/m per -CH2-group. This is based upon the y data of ethanol (22.1 mN/m) and propanol (23.7 mN/m). These observations indicate the molecular correlation between bulk forces and surface forces (surface tension y) for homologous series of substances. 

The interaction between two spherical colloids can be transformed by the Derjaguin approximation [29] to the interaction between two flat surfaces (see Appendix A). The net osmotic pressure in an electric double layer is the difference between the internal force, F n, and the external or bulk force, Fex, and is related to the force between two colloids Posm = F n — Fex/a, where a is the area. 

A dispersion Is a system made of discrete objects separated by a homogeneous medium In colloidal dispersions the objects are very small In at least one dimension. Colloidal sizes range from 1 to 100 nm however these limits are somewhat arbitrary, and It Is more useful to define colloids as dispersions where surface forces are large compared to bulk forces. Here we are concerned with systems where the dispersion medium Is a liquid examples are droplets In emulsions or mlcroemulslons (oll/water or water/oll), aggregates of amphiphilic molecules (surfactant micelles), foams, and all the dispersions of solid particles which are used as Intermediates In the manufacture of ceramics. At this stage we are not too concerned with the nature of the constituents, but rather with the structures which they form this Is a geometrical problem, where the system Is characterized by Its surface area A, by the shapes of Its Interfaces (curvatures - b ), and by the distances between opposing surfaces (d — concentration parameter). 

Such a relative motion can be induced by external electric fields or by pressure gradients or bulk forces (e.g. gravity). It is possible that particles move in a quiescent solvent or that the solvent flows through a fixed bed of particles. A detailed description on electrokinetic phenomena is e.g. given by Hunter (1988). Zeta-potential measurements on colloidal suspensions are fiequently conducted via electrophoresis or by means of electroacoustics. Besides this, there are recent techniques based on non-linear optics that are sensitive to interfacial changes. 

The external bulk force G exerted on the director by the electric field E = E -f E is ... 

Now we come to the additional equations that are specific to the processes driving the instability. In EHC the bulk force in the Navier-Stokes Equation 14 is derived from the Maxwell stress tensor, which here reduces to... 

In RBC the bulk force in the Navier-Stokes equation is f = pg. In the spirit of the Boussi-nesq approximation one has for the mass density p = Pm[ - a(T-To)], where g is the gravitational acceleration and a the thermal expansion coefficient. One needs in addition the heat conduction equation... 

The surface phonon dispersion of the clean surface, as measured by HREELS [86Wut] and confirmed by HATOF [93Ben, 94Hof], is reported in Fig. 16. The RW is stiffer than predicted by LDM using the bulk force constant. The phenomenon was ascribed to the presence of tensile surface stress. 

Hydrogen adsorption restores the surface phonon frequencies to the values predicted by LDM calculations using the bulk force constants. In particular, compared to the clean surface case, the RW is softened by 0.7 meV at X and by 1.5 meV at M. The longitudinal resonance disappears almost completely from the spectra. The data are discussed in ref. [87Ber], but no dispersion curves were reported. 

Fig. 16. Cu(lOO). Surface phonon spectrum for the bare surface [86Wut]. The circles are the experimental HREELS data points, dashed and solid lines are the result of the LDM fit with bulk force constants and with modified surface force constants, respectively.

Fig. 53a, b. Ru(OOOl) (1x1) H. (a) Surface phonon dispersion [97Bra]. The lines are the result of a LDM calculation with bulk force constants, (b) Surface phonon dispersion curve in presence of the (1x1) H overlayer as determined by HATOF. The lines are the result for surface phonon and resonances of a LDM fitting the bulk phonon dispersion [97Bra]. LR denotes the longitudinal resonance. The SBZ is reported in the inset in (a). 

The phenomenological theory of rubberlike elasticity is based on continuum mechanics. It provides a mathematical structure from which, in principle, the deformation produced within a vulcanized elastomer by applied surface and bulk forces can be calculated. In the theory, the material is idealized by the assumption that it is perfectly elastic, isotropic in the undeformed state and incompressible. The most general form of the strain energy function (which vanishes at... 

The new surface modes appear because of the broken translational symmetry at the surface. However, at real surfaces, the force constants within the first layer and between the very first layers will be modified with respect to the bulk force constants. This originates from the modified electronic structure at the surface and the surface relaxation. Furthermore, we considered here for simplicity only central nearest-neighbor forces. For a better description, the range of interaction has to be extended to next-nearest neighbors and even longer distances. Additional noncentral forces as, for example, bond-bending forces have to be included for a proper description of the phonon dispersion in most materials. 

As the experimentally determined optical surface phonons of BaFj] ) have about half the energies of corresponding bulk branches and could be well described with the bulk force constant between nearest neighbors and with the bulk bonding angles, surface relaxation was considered rather unlikely for BaFj] ). 

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