Curvature-induced motion

This idea can be illustrated in schematic form through the example of the motion of a single curved interface in two dimensions under the action of curvature-induced forces. For convenience, we will represent the interface as a function (rather than via a discrete set of parameters as in the discussion of the variational principle above) of a parameter s as r(x). In this case, the rate of change of Gibbs free energy is given by... 

Not surprisingly, this equation of motion implies the curvature-induced shrinkage of the particle. 

Curvature-Induced Dielectrophoresis, Fig. 1 Mechanism of curvature-induced dielectrophoresis (C-iDEP) in a microchannel turn of angle, 0, where the dielectro-phoretic particle motion, Udep. competes with the elec-trokinetic particle motion, Uek, leading to a cross-stream particle deflection. The background color represent the contour of electric field, E (short arrows indicate the direction) (The figure is adapted from Ref. [2] with permission)... 

Fig. 61. Schematics of pressure-induced and applied-potential-induced BLM deformations. Application of hydrostatic pressure (by lowering a piston into the aqueous solution bathing the cis side of the BLM) displaces the BLM from position 1 to position 2. The displacement involves both translational (lateral) motion (Ft) and curvature increase (Fc). As indicated, deformation of the BLM is accompanied by a change in its torus (Plateau-Gibbs border). 2R and 2Rm represent the diameters of the aperture of the pinhole in the Tefzel film and that of the membrane (excluding the torus). The object laser beam, incident upon the trans side of the BLM and reflected by it at 45° at a shortened wavelength produces concentric optical interference fringes with the reference laser beam. Ag/AgCl electrodes, placed in the cis and trans sides of the BLM, allow for continuous electrical measurements [413]...

Analysis is simplified if 7 is isotropic—i.e., independent of geometrical attributes such as interfacial inclination n and, for internal interfaces in crystalline materials, the crystallographic misorientation across the interface. All interfacial energy reduction then results from a reduction of interfacial area through interface motion. The rate of interfacial area reduction per volume transferred across the interface is the local geometric mean curvature. Thus, local driving forces derived from variations in mean curvature allow tractable models for the capillarity-induced morphological evolution of isotropic interfaces. 

A somewhat unexpected feature of this representation is the increasing rather than decreasing order parameter of the interfacial segments (C2—C4) when going from the L to the Hi phase which is probably a result of the motional restriction of the phospholipid chains in the normal hexagonal phase. Thus, a neutral or pivotal plane, approximately at the level of C5, may be assumed about which the molecules turn in the lamellar-hexagonal phase transition [8]. A similar relation between order parameters and spontaneous curvature has been observed for liquid crystalline sodium laurate where an L -Hi phase transition can be induced by changing the water content... 

After putting kc(t) = Go t)/j into Equation (38) of the quasi-steady approximation and linearizing this equation, one can find variations of the curvature ko at the free end of the curve which are induced by the modulation of Gq. Substitution of these variations into the linearized equation (39) yields the time dependence of the angle cco- The motion of the tip of the spiral wave can then be calculated from (40) and (41). We find that in the linear... 

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