Divergence thickness

Figure B3.6.3. Sketch of the coarse-grained description of a binary blend in contact with a wall, (a) Composition profile at the wall, (b) Effective interaction g(l) between the interface and the wall. The different potentials correspond to complete wettmg, a first-order wetting transition and the non-wet state (from above to below). In case of a second-order transition there is no double-well structure close to the transition, but g(l) exhibits a single minimum which moves to larger distances as the wetting transition temperature is approached from below, (c) Temperature dependence of the thickness / of the enriclnnent layer at the wall. The jump of the layer thickness indicates a first-order wetting transition. In the case of a conthuious transition the layer thickness would diverge continuously upon approaching from below.

Typically specimens for reflectivity measurements are prepared on flat, smooth, rigid substrates. For example, these substrates can be polished fused silica, quartz, or silicon. It is important, however, that the substrates be thick to avoid distonions of the specimen when mounted in the reflectometer. Any curvature or bowing will increase the divergence of the incident beam and result in a deterioration of the resolution. 

When the temperature lies in between the wetting temperatures of both substrates, then one substrate is wet and the film thickness on that part diverges when the distance from the boundary between different substrates increases. This divergence is found to be represented by a power law x for... 

The adsorption transition also shows up in the behavior of the chain linear dimension. Fig. 6(a) shows the mean-square gyration radii parallel, i gl, and perpendicular, to the adsorbing plate. While these components do not differ from each other for e for e > ej i g strongly increases whereas Rh decreases. In the first case (non-adsorbed chain) oc oc N as a dilute solution in a good solvent in the bulk. For adsorbed chains R /N 0 for oo because the thickness is finite it is controlled by the distance e- e from the adsorption threshold, but does not diverge as N oo. The adsorbed chain follows in fact a... 

Complete wetting of the substrate is not exhibited in either case. This is only satisfied by a system for which the film thickness diverges continuously as In the system with W=2, the thickness reaches a limit of... 

Here is seen the most obvious contrast with the kinematical theory. Both curves are plotted as a function of thickness in Figure 4.23. It is seen that the formulae give similar results for small thicknesses, that the kinematic theory diverges drastically at larger thicknesses, and that the dynamic intensity shows oscillations about a saturated level after a thickness of about. ... 

One of the successes of the EH theory is its ability to provide an exphcit result for the lamellar thickness, based on kinetic considerations of secondary nucleation. In the EH theory, 8L is a natural result of the steady state and not due to any subsequent thickening. The form of Eq. (1.94) is exactly the same as Eq. (1.1), observed experimentally. However, one of the serious difficulties posed by Eq. (1.94) is that it predicts a divergence in 5L at... 

Since A i is proportional to the supercooling, the LH theory predicts that as the crystallization temperature is lowered to a value when Eq. (1.95) is satisfied, the lamellar thickness would diverge. This is referred to as the 8L catastrophe. By taking reasonable experimental values for the various parameters and assuming t / = 1 (no barriers for attachment of each stem), the necessary supercooling for the appearance of this catastrophe is 55 K for polyethylene. This is not observed experimentally. To fix this discrepancy, vf/ is taken to be zero so that... 

The transverse voltage is the drop of potential across the double layer and is of the order of 0-01 to 0 05 volt. It will be noted that the amount of liquid transported is dependent on the nature of the liquid and on the current and is independent of the diameters or lengths of the tubes of the diaphragm. Somewhat divergent views are held as to the actual thickness of the double layer (Lamb, Phil. Mag. xxv. 52, 1888) a point which we have 7 referred to. 

Among the first practical applications, research studies have been interested to fathom possible potentials and limitations of LAPS-based measurements. One distinguishing feature of the LAPS is its spatial resolution. The resolution of a LAPS system is limited by several factors such as the definition of the light source (e.g., wavelength, beam diameter, beam divergence, beam intensity, etc.) and the LAPS geometry of the build-up (layer scheme, layer thickness, material properties,... 

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