Undulation length

Equations (7.36)-(7.37) together determine the important length scale A — the so-called undulation length (also called Odijk length after the author of the concept — Theo Odijk of Leiden University in The Netherlands) ... 

The apparatus (Fig. 82), which is constructed throughout of glass, consists of a pear-shaped bulb A (of about 5 ml. capacity) in which the solution is boiled, and which has a short length of platinum wire fused through its lowest point to assist steady boiling. The bulb A is connected near its base by a curved narrow tube B to a vertical condenser C, and from its apex by a similar tube D, undulating as shown, to the cup E. A larger outer cup F is fused to the lower neck of E as shown. 

The finite size of the box has several important consequences. One of them is that the area of the membrane piece is only of the order of 100 nm2. It is expected that the membrane is, on this length scale, roughly flat, i.e. the area is small as compared with the persistence length for the bilayer. Interestingly, however, in recent simulations the first signs of fluctuations away from the flat bilayer structure (undulations) have reportedly been found by MD simulations [33],... 

PAOP This must be lower than the PA diastolic pressure to ensure forward flow. It is drawn as an undulating waveform similar to the CVP trace. The normal value is 6-12 mmHg. The values vary with the respiratory cycle and are read at the end of expiration. In spontaneously ventilating patients, this will be the highest reading and in mechanically ventilated patients, it will be the lowest. The PAOP is found at an insertion length of around 45 cm. 

Let us examine the instability oi strained thin films. In Fig. 3, thin films of30 ML are coherently bonded to the hard substrates. The film phase has a misfit strain, e = 0.01, relative to the substrate phase, and the periodic length is equal to 200 a. The three interface energies are identical to each other = yiv = y = Y Both phases are elastically isotropic, but the shear modulus of the substrate is twice that of the film (p = 2p). On the left-hand side, an infinite-torque condition is imposed to the substrate-vapor and film-substrate interfaces, whereas torque terms are equal to zero on the right. In the absence of the coherency strain, these films are stable as their thickness is well over 16 ML. With a coherency strain, surface undulations induced by thermal fluctuations become growing waves. By the time of 2M, six waves are definitely seen to have established, and these numbers are in agreement with the continuum linear elasticity prediction [16]. 

If the misfit strain is less than a critical value, the undulations cannot mount cracktips, as demonstrated in Fig. 4, where a periodic length is equal to 100 a and film thickness is 30 ML. With the same physical parameters employed for Fig. 3, no islands are created if the misfit strain is less than 0.006. When the misfit strain is less than but close to the critical value, a permanent wave structure sets in the film as in the case ofs = 0.005. If the misfit strain is further reduced, coherency-induced undulations are swept away by thermal fluctuations. 

Figure 4. Thin films with small misfit strains under the zero-torque condition. As the misfit strains are less than the critical value, 0.006, no islands are created out of the undulations. The periodic length is equal to 100 a and the film thickness is 30 ML.

The high temporal resolution of SFM imaging uncovered elementary dynamic processes of structural rearrangements. We observed short-term interfacial undulations [111], fast repetitive transitions between distinct defect configurations [112], their spatio-temporal correlations on a length scale of several microdomains [112], and unexpected defect annihilation pathways via formation of temporal excited states [51, 111]. 

Figure 3. The experimental data for the external pressure II vs average thickness (d) are compared to the results of the present theory, which accounts for the undulation of the interfaces combined with the traditional DLVO treatment of interactions. Different bending moduli affect not only the strength of the interactions but also its slope (traditionally related to the Debye screening length).

The third issue is that, in determining the appropriate values of the Hamaker constant, it was supposed that both the exponential hydration (Eq. (1)) and the Helfrich repulsion (Eq. (3)) remain unchanged upon addition of a salt (and that the osmotic pressure due to the interlamellar salt deficit, discussed above, is negligible). While the exponential form for the hydration force has been determined at high osmotic pressures (corresponding to separations of around 10 A) it is not clear that this behavior will remain the same at distances of the order of 40 A. For a decay length of 2.2 A [14], the magnitude of the hydration forces decreases about 106 times between 10 and 40 A, and even a minuscule deviation from the exponential behavior will have drastic consequences in the evaluation of the Hamaker constant. The problems associated with a constant Helfrich repulsion (due to the thermal undulations of the bilayers) are even more complicated and will be addressed in detail in Section 3. 

The forth issue is the increase in the repulsion between bilayers at short distances. In Fig. 1, the osmotic pressure is plotted as a function of separation distance (data from Ref. [13]) for no added salt, for l M KC1 and for 1 M KBr. They reveal an increase in repulsion at short separation distances upon addition of salt. While the relatively small difference between 1 M KC1 and 1 M KBr can be attributed to the charging of the neutral lipid bilayers by the binding of Br (but not C.1-) [14], the relatively large difference between no salt and 1 M KCl is more difficult to explain. Even a zero value for the Hamaker constant (continuous line (2) in Fig. 1), in the 1 M KCl case, is not enough to explain the increase in repulsion, determined experimentally. The screening of the van der Waals interaction, at distances of the order of three Debye-Hiickel lengths (about 10 A) should lead, according to Petrache et al. calculations, to a decrease of only about 30% of the Hamaker constant (from 1.2kT to about 0.8kT, see Fig. 5C of Ref. [14]). Therefore, an additional mechanism to increase the hydration repulsion or the undulation force (or both) upon addition of salt should exist to explain the experiments. 

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