Thickness fluctuations

The model has been applied successfully to predicting the performances of bearings, gears, seals, and engines [10-12]. A fundamental limitation of the statistic models is their inability to provide detailed information about local pressure distribution, film thickness fluctuation, and asperity deformation, which are crucial for understanding the mechanisms of lubrication, friction, and surface failure. As an alternative, researchers paid a great interest to the deterministic ML model. 

An alternative description of membrane stability has been based on hydrodynamic models, originally developed for liquid films in various environments [54-56]. Rupture of the film was rationalized by the instability of symmetrical squeezing modes (SQM) related to the thickness fluctuations. In the simplest form it can be described by a condition [54] d Vdis/dh < where is the interaction contribution related to the dis-... 

Membrane thickness fluctuations were initially discussed in the local approach by Hladky and Gruen (HG) [102] in conjunction with their possible effect on membrane capacitance. They are directly related to the spectrum of 5-modes ... 

In Ref 53 these calculations were reproduced using the nonlocal approach. Following HG, the results were contrasted for two X cutoffs, = 1 nm and = 10 nm. The nonlocal results for RMS thickness fluctuations A/z = h — are comparable in... 

Further progress in understanding membrane instability and nonlocality requires development of microscopic theory and modeling. Analysis of membrane thickness fluctuations derived from molecular dynamics simulations can serve such a purpose. A possible difficulty with such analysis must be mentioned. In a natural environment isolated membranes assume a stressless state. However, MD modeling requires imposition of special boundary conditions corresponding to a stressed state of the membrane (see Refs. 84,87,112). This stress can interfere with the fluctuations of membrane shape and thickness, an effect that must be accounted for in analyzing data extracted from computer experiments. 

Lindahl, E. and Edholm, O. (2000). Mesoscopic undulations and thickness fluctuations in lipid bilayers from molecular dynamics simulations, Biophys. J.,19,426-433. 

FIGURE 12.4 Apparent LNAPL thickness fluctuations with time reflecting installation of additional recovery wells. 

The occurrence of yam breaks was reported early and is connected with thickness fluctuations in extruded ribbons or films, as described collectively by the phenomenon of draw resonance , which is characterized by oscillations of the fiber diameter, which ultimately lead to yarn break. The latter is defined as brittle fracture and is thus related to melt temperature, molecular weight, quenching conditions, and particularly to the role of viscoelasticity, as described in the following section. 

In the wall thickness fluctuations up to 5 % may occur. As a result of the uneven temperature in the molten polymer during rotation, and also by the not always exactly reproducible rate of cooling, deviations in the dimensions of the finished product may amount to 5 %. Requirements are, that the materials can be molten completely, that the melt is sufficiently low-viscous, and that the molten polymer does not degrade too rapidly. Besides plasticised PVC, HDPE and LDPE are often used, as well as copolymers of PE such as EVA (ethylene - vinyl acetate copolymerj.Because the shear stresses in this process are extremely low, a narrow molar mass distribution is to be recommended, as discussed in 5.4. Cycle times vary between 3 and 40 minutes, dependent on the wall thickness. Cycle times can be reduced considerably by using machines with multiple moulds, since the cycle time... 

A major source of real-structure inhomogenities are wire-thickness fluctuations and polycrystallinity, and geometrical features at wire ends [54, 55, 114, 158]. As expected from 2.3.4, the localization of the modes is accompanied by a coercivity reduction [54, 55, 114]. 

At constant external pressure the enthalpy is the relevant thermodynamic potential for the Boltzmann distribution. A large area A implies that the distribution of the thicknesses is confined near the minimum of the enthalpy, while a small value of A corresponds to large thickness fluctuations. The first case corresponds to a low enthalpy, but a large entropy, whereas the second to a large enthalpy, but a low entropy. The real distribution is provided by the minimization of the Gibbs free energy with respect to A at constant external pressure II. 

For these reasons, operation of a nearly dry second doctor blade may lead to tape cast thickness fluctuations. Much more stable operation can be obtained from operating with only one doctor blade with a full doctor box and control of tape speed and applied pressure. 

For Bingham fluids, these thickness fluctuations will self-level to a height, A8 , equivalent to the height of fluid supported by the yield stress or... 

When two emulsion droplets come into close contact in a floe or creamed layer, or during Brownian diffusion, thinning and disruption of the liquid film may occur that results in eventual rupture. On close approach of the droplets, film thickness fluctuations may occur alternatively, the liquid surfaces may undergo some fluctuations to form surface waves, as illustrated in Figure 10.31. These surface waves may grow in amplitude and the apices may join as a result of the strong van der Waals attractions (at the apex, the film thickness is the smallest). The same applies if the film thins to a small value (critical thickness for coalescence)... 

The rupture mechanisms of thin liquid films were considered by de Vries [15] and by Vrij and Overbeek [16]. It was assumed that thermal and mechanical disturbances (having a wavelike nature) cause film thickness fluctuations (in thin films), leading to the rupture or coalescence of bubbles at a critical thickness. Vrij and Overbeek [16] carried out a theoretical analysis of the hydrodynamic interfacial force balance, and expressed the critical thickness of rupture in terms of the attractive van der Waals interaction (characterised by the Hamaker constant A), the surface or interfacial tension y, and the disjoining pressure. The critical wavelength, for the perturbation to grow (assuming that the disjoining pressure just exceeds the... 

From the theory of waves on single surfaces (for a review, see Ref. 85) it was known that for small values of surface wavelength A (i.e., large values of K), the relaxation of the waves was dominated by the action of surface tension. In the case of thin films, also, the effect of interaction forces was expected to influence the relaxation of the film to its mean thickness. Especially, the squeezing mode relaxation, where local thickness fluctuations appear, should show up the influence of colloidal interactions. 

Fig. 7.3. Measured autocorrelation function ( + ) of the photocurrent fluctuations. Wavelength of the thickness fluctuation (squeezing mode) A =4.90 ftm. The curve indicates nonlinear least-squares fit to the experimental data according to (7.13). Film was drawn from solution with an ionic strength of 1.9 X 10" mol/dm .

It is worthwhile formulating a model in terms of material properties that are measurable (or at least, potentially so). For instance, the short-range membrane thickness fluctuations at a point r depend on the averaged values of the elastic constants in its vicinity ... 

Figure 19.11 Representation of a system of two phases and their impact in the function Yiir)-(a) Periodic system of two phases, (h) effect of variations in the period L, (c) effect of additional thickness fluctuations, (d) effect of the introduction of diffuse frontiers (interface thickness). Source Reproduced with permission from Strohl GR, Schneider M. J Polym Sci Polym Phys Ed 1980 18 1343 [36], Copyright 1980 Wiley Periodicals, Inc.

Brochard and Dalliant have indicated that liquid films on non-wettable substrates are spinodally unstable against thickness fluctuation if the thickness e is thinner than lOOnm (7). Their model of linearized capillary wave instability predicts that the thickness fluctuation with a characteristic wave vector qu is amplified most rapidly to nucleate initial dry spots separated by 2it/qtt fix>m each odiw. Their result is... 

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