Curvature Force

Still another force will be present if a dislocation is curved. In such cases, the dislocation can reduce the energy of the system by moving to decrease its length. An effective force therefore tends to induce this type of motion. Consider, for example, the simple case of a circular prismatic dislocation loop of radius, R. The energy of such a loop is 

The vector form of Eq. 11.9 is readily obtained. If r is the position vector tracing out the dislocation line in space and ds is the increment of arc length traversed along the dislocation when r increased by dr,4 

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In die operation of die first source, the driving force for sintering is the difference in curvature between the neck and the surface of the sphere. The curvature force A l, is given by... 

Solution. The source will be able to become active if the driving osmotic climb force is large enough to overcome the restraining curvature force that reaches a maximum when the dislocation segment has bowed out to the minimum radius of curvature corresponding to R = L/2. Setting /M = fK, we then have the critical condition... 

Now let us assume that both parabolas have the same curvature (force constant /). The reactants correspond to the parabola with the minimum at qp (without losing generality, we adopt a convention that at q = qp, the energy equals zero) ... 

FIGURE 5.9 Amphiphilic block copolymers form different structures with increasing length of the hydrophobic segment. The resulting curvature forces the formation of micelles, polymersomes, or wormlike structures. 

The contemporary theory of the electron transfer reaction was proposed by Rudolph Marcus. The theory is based to a large extent on the harmonic approximation for the diabatic potentials involved, i.e. the diabatic curves represent parabolas. One of the parabolas corresponds to the reactants VR(q), the other to the products Vp(q) of the electron transfer reaction (Fig. 14.22). Now, lei us assume that both parabolas have the same curvature (force constant f). The reactants correspond to the parabola with the minimum at qp (without loosing generality we adopt a convention that at = 0 the energy is equal zero)... 

If accurate tunneling corrections are required, the situation rapidly becomes more complicated. Issues related to separability of motion along the reaction coordinate, curvature of the reaction coordinate, and multidimensional tunneling arise and must be dealt with. Marcus and Coltrin [51] found that reaction path curvature forces the reaction to cut... 

This process is similar to GTA welding but a filler metal is used. Takasu and Toguri [47] have shown that there are four forces affecting the fluid motion (Figure 25) in the pool when the molten filler metal drop hits the molten pool, namely (i) a stirring force due to the momentum of the drop (ii) a buoyancy force related to the density difference between the drop and the pool (iii) a curvature force related to the surface tension normal to the surface and (iv) the Marangoni force related to the difference in surface tension of the drop and pool. Takasu and Toguri showed that when (a) ydrop > ypooi the droplet penetrated into the pool and (b) ydrop < ypooi the drop will spread out over the surface. 

Changes in the curvature (force constant and vibrational frequency) of one potential relative to another affect the transition energies via the o and " values in Equation 10.61b, and they can also affect the Franck-Condon overlap. So, intensities of the bands depend on the overlap between the initial and final vibrational states, which in turn depend on the nature of the two potential curves. 

One of the difficulties in theoretical studies of the hydrodynamic effects on vesicle dynamics is the no-slip boundary condition for the embedding fluid on the vesicle surface, which changes its shape dynamically under the effect of flow and curvature forces. In early studies, a fluid vesicle was therefore modeled as an ellipsoid with fixed shape [194]. This simplified model is still very useful as a reference for the interpretation of simulation results. 

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... 

Bikerman [179] has argued that the Kelvin equation should not apply to crystals, that is, in terms of increased vapor pressure or solubility of small crystals. The reasoning is that perfect crystals of whatever size will consist of plane facets whose radius of curvature is therefore infinite. On a molecular scale, it is argued that local condensation-evaporation equilibrium on a crystal plane should not be affected by the extent of the plane, that is, the crystal size, since molecular forces are short range. This conclusion is contrary to that in Section VII-2C. Discuss the situation. The derivation of the Kelvin equation in Ref. 180 is helpful. 

The interest in vesicles as models for cell biomembranes has led to much work on the interactions within and between lipid layers. The primary contributions to vesicle stability and curvature include those familiar to us already, the electrostatic interactions between charged head groups (Chapter V) and the van der Waals interaction between layers (Chapter VI). An additional force due to thermal fluctuations in membranes produces a steric repulsion between membranes known as the Helfrich or undulation interaction. This force has been quantified by Sackmann and co-workers using reflection interference contrast microscopy to monitor vesicles weakly adhering to a solid substrate [78]. Membrane fluctuation forces may influence the interactions between proteins embedded in them [79]. Finally, in balance with these forces, bending elasticity helps determine shape transitions [80], interactions between inclusions [81], aggregation of membrane junctions [82], and unbinding of pinched membranes [83]. Specific interactions between membrane embedded receptors add an additional complication to biomembrane behavior. These have been stud-... 

Fig. 3.10 Contributions to the lowering of chemical potential of the condensed liquid in a capillary, arising from adsorption forces (c) and meniscus curvature (Ap). The chemical potential of the free liquid is , and that of the capillary condensed liquid is (= ) z is the distance from the capillary wall. (After Everett. )...

We note that if the crack opening is zero on F,, i.e. [%] = 0, the value of the objective functional Js u) is zero. We also assume that near F, the punch does not interact with the shell. It turns out that in this case the solution X = (IF, w) of problem (2.188) is infinitely differentiable in a neighbourhood of points of the crack. This property is local, so that a zero opening of the crack near the fixed point guarantees infinite differentiability of the solution in some neighbourhood of this point. Here it is undoubtedly necessary to require appropriate regularity of the curvatures % and the external forces u. The aim of the following discussion is to justify this fact. At this point the external force u is taken to be fixed. 

Fig. 3. Two-dimensional schematic illustrating the distribution of Hquid between the Plateau borders and the films separating three adjacent gas bubbles. The radius of curvature r of the interface at the Plateau border depends on the Hquid content and the competition between surface tension and interfacial forces, (a) Flat films and highly curved borders occur for dry foams with strong interfacial forces, (b) Nearly spherical bubbles occur for wet foams where...

AH distortions of the nematic phase may be decomposed into three basic curvatures of the director, as depicted in Figure 6. Liquid crystals are unusual fluids in that such elastic curvatures may be sustained. Molecules of a tme Hquid would immediately reorient to flow out of an imposed mechanical shear. The force constants characterizing these distortions are very weak, making the material exceedingly sensitive and easy to perturb. 

Fig. 6. The three basic curvature deformations of a nematic Hquid crystal bend, twist, and splay. The force constants opposing each of these strains are...

The primary driviag force for material transport comes from the chemical potential difference that exists between surfaces of dissimilar curvature within the system. The greater the curvature, ie, the finer the particle size, the greater the driving force for material transport and sintering. 

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