Bending elastic forces

In summary, the different pressures are shown schematically in Fig. 2. The attractive van der Waals forces act across the film and favor thinner film thicknesses, the bending elastic forces depend on the fourth-order derivatives of the height variations and essentially penalizes curvature in the height, the contact pressure is included for numerical stability and ensures that the solid capping layers do not overlap, and finally the stretching elastic forces describe the pressure on the fluid due to in-plane displacements in the solid capping layers. The behavior of the model depends upon the interactions of all these pressures, as well as the kinetics of fluid transport, in response to these pressures. 

Figure 4. Plots of (a) van der Waals forces, (b) bending elastic forces, and (c) stretching elastic forces in Pascals for the system shown in Fig. 1 after 5 minutes. The variations in these forces are responsible for fluid flow and the evolution of the system.

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-... 

In the later stage of fusion, the neck expansion velocity slows down by more than two orders of magnitude. Here the dynamics is mainly governed by the displacement of the volume of fluid around the fusion neck between the fused vesicles. The restoring force is related to the bending elasticity of the lipid bilayer [36, 37]. 

Fig. 1 shows a formation scheme and a tube obtained. The elastic forces Fi and F2 give rise to a moment M of forces which tends to bend the bilayer. The initial structure consists of AlAs sacrificial layer, InGaAs compressed layer and a conplex layer with i-GaAs quantum well and AlGaAs doped barrier. The tubes were rolled up by means of selective etching of an AlAs sacrificial layer with HF-based etchant... 

Using the analogy that can be drawn between the system of end-grafted polymer chains and the chain part of a surfactant monolayer, Milner and Witten [41] obtained simple analytical expressions for the bending elastic quantities. For the case of a melt brush," i.e., one in which the density is forced to be uniform, they obtain... 

Another model frequently used to describe the polymer chain is WLC. In WLC model, a polymer is treated as a homogenous string of constant bending elasticity. Both entropic and enthalpic contributions are combined in this model [6,61]. Scheme 30.3b shows the WLC model and the relationship between force and the extension of a WLC is shown as follows ... 

Many authors have described structural transitions in dilute lamellar phases under the influence of shear. For example, Roux and co-workers (19) studied different dilute lamellar phases which were stabilized by undulation forces and contained flat bilayers with defects, at rest. With increasing shear rates, these bilayers undergo a transition into relatively monodisperse multilamellar vesicles above a characteristic shear rate. The size of the formed vesicles is indirectly proportional to the shear rate. Beyond a second characteristic shear rate, the vesicles are again transformed into flat oriented bilayers. These results were explained in terms of a balance between shear stress and elastic forces which come from the bending and the Gaussian moduli of the bilayers. The same authors observed a similar sequence with increasing shear rate for other lamellar phases. It was found... 

In Section 9.1.3, we introduced a filament made of superparamagnetic colloidal particles that are linked to each other by double-stranded DNA. We have modeled this filament as a bead-spring configuration with bending elasticity as described in Section 9.2.3. In addition, we have to include now the forces on the beads due to the dipole-dipole interaction induced by the external magnetic field. Our model very well describes the constructed artificial swimmer [4] and allows to explore the filament s capacity for transporting fluid. 

Here Ki, K2 and K3 are the splay, the twist, and the bend elasticity constants, respectively, while n is a local director of the phase. The term k2 introduces the chirality and forces the phase to have a left-handed (k2 < 0) or right-handed (k2 > 0) twist. This twist reaches an equilibrium because of the counteraction of the torques connected with the first (k2-term) and the third term (K2-tom). De Gennes [13] has shown that the reciprocal pitch is proportional to the number of chiral points, i.e. the concentration c of the chiral molecules. The pseudoscalar proportionality ccmstant P(k2) determines size and sense of the helix ... 

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