Films fluctuations, random

If the balance of van der Waals attraction, electric double layer repulsion, capillary pressure, structure propagation, etc., favours an equilibrium film thickness, random fluctuations in film thickness will, in any case, tend to be neutralised. 

The existence of asperity contacts in mixed lubrication causes great many local events and significant consequences. For example, the parameters describing lubrication and contact conditions, such as film thickness, pressure, subsurface stress, and surface temperature, fluctuate violently and frequently over time and space domain. It is expected that these local events would have significant effects on the service life of machine elements, but experimental measurements are difficult because of the highly random and time-dependent nature of the signals. Only a few successes were reported so far in experimental studies of mixed lubrication, mostly limited to the artificially manufactured... 

When the film thickness is of the order of roughness heights, the effects of roughness become significant which have to be taken into account in a profound model of mixed lubrication. The difficulty is that the stochastic nature of surface roughness results in randomly fluctuating solutions that the numerical techniques in the 1970s are unable to handle. As... 

If we select the z axis to be normal to the plane of the clay film, the angle 0 can be imagined to fluctuate rapidly as the probe tumbles in the interlayer. This time-averaged system is symmetric about the z axis because of the random orientation of the a and b axes of the clay platelets in the x y plane. 

We begin in Section II with a review of the fundamental concepts of hydrodynamics and boundary conditions. In Section III, we present some common descriptions of coupling, followed in Section IV by a discussion of viscoelastic adsorbate films and the so-called inner slip. In Section V, we consider with the concept of stochastic boundary conditions, which we believe will be an important topic in situations where random fluctuations are strong. Finally, in Section VI, we present our concluding ideas and discuss some areas for future study. 

This result is interesting, since it gives the slip length as a function of parameters that can be measured experimentally or a priori, for simple systems in a linear approximation. The bulk shear viscosity can be approximated from the literature, and the monolayer density can be determined from optical techniques. To a first approximation, for rigidly adsorbed layers, the sliptime is related to the autocorrelation function of random momentum fluctuations in the film, given by [40]... 

It is most remarkable that the entropy production in a nonequilibrium steady state is directly related to the time asymmetry in the dynamical randomness of nonequilibrium fluctuations. The entropy production turns out to be the difference in the amounts of temporal disorder between the backward and forward paths or histories. In nonequilibrium steady states, the temporal disorder of the time reversals is larger than the temporal disorder h of the paths themselves. This is expressed by the principle of temporal ordering, according to which the typical paths are more ordered than their corresponding time reversals in nonequilibrium steady states. This principle is proved with nonequilibrium statistical mechanics and is a corollary of the second law of thermodynamics. Temporal ordering is possible out of equilibrium because of the increase of spatial disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Contrary to Boltzmann s interpretation, which deals with disorder in space at a fixed time, the principle of temporal ordering is concerned by order or disorder along the time axis, in the sequence of pictures of the nonequilibrium process filmed as a movie. The emphasis of the dynamical aspects is a recent trend that finds its roots in Shannon s information theory and modem dynamical systems theory. This can explain why we had to wait the last decade before these dynamical aspects of the second law were discovered. 

An important element of the kinetics of film rupture is the random character of the process. Here the question is about the correct description of the effect of fluctuations on the evolution of single waves. A formula derived in [68] seams suitable from experimental point of view... 

The z(C) dependence has been investigated with the help of microscopic foam bilayers of both ionics and nonionics [419,420]. Due to the fluctuation character of the film rupture, the film lifetime is a random parameter. Experimentally, the film mean lifetime r has been determined by averaging from a great number of measurements. Because of the assumption that the monomer and the total surfactant concentrations are practically equal, in all t(C) dependences given below, C refers to the total concentration. Using Eq. (3.120) to analyse the experimentally obtained time dependence of the probability P(t) of film rupture it was found... 

When the minimum dimensions of a resist pattern are <100 nm, their dimensional tolerances approach the scale of the molecular components of the film. At this level, LER, random fluctuation in the width of a resist feature, may limit the advancement of lithography. Possible contribution to LER include polymer molecular weight, molecular weight distribution, molecular structure of resist components, inhomogeneity in component distribution within the film, statistical effects influencing film dissolution, intrinsic properties of the imaging, chemistry, image contrasts of irradiation, etc. 

In film-based mammography systems, another important noise source is the random fluctuation contributed by the granularity of the film itself The film emulsion is composed of grains of silver halide and their random structure increases the total noise. The overall image noise can be estimated by treating noise sources as being statistically independent, giving ... 

By varying several parameters such as the W/O ratio, one can induce an inversion from an O/W to a W/O microemulsion and vice versa. The type of structure in the inversion domain depends essentially on the bending constant a characteristic of the elasticity of the surfactant layer [7]. If Ke is on the order of kT (where k is the Boltzmann constant and T absolute temperature), the persistence length of the film (i.e., the distance over which the film is locally flat) is microscopically small. The interfacial film is flexible and is easily deformed under thermal fluctuations. The phase inversion occurs through a bicontinuous structure formed of water and oil domains randomly interconnected [8,9]. The system is characterized by an average curvature around zero, and the solubilization capacity is maximum. When K kT, is large and the layers are flat over macroscopic distances. The transition occurs through a lamellar phase. 

STM images (Fig. 16) of the surface evolution of a vapor deposited Au film on smooth glass with the film thickness, S, (or time) clearly reveal the development of random fluctuations as S increases due to the competition among small growing columns (20nm size). 

Fig. 5.25. (a) Stable multimode operation of a HeNe laser (exposure time 1 s) (b) two short-time exposures of the multimode spectrum of an argon laser superimposed on the same film to demonstrate the randomly fluctuating mode distribution... 

We consider two spheres of fixed radii to have collided to form a film between them of thickness, say, /z- (less than some value to be defined presently). We further assume a random force such as that arising due to turbulent pressure fluctuations that produces a random film drainage process. A positive force is assumed to drain the film while a negative force causes it to thicken by inflow. Although the process is strictly three-dimensional, we shall assume a one-dimensional model, letting the force be always normal to the film. Further, we stipulate that if the film drains to some critical thickness, say h, the film snaps to allow aggregation between the particles. We shall see later how such a model can be formulated mathematically. The instantaneous film thickness H will serve to describe the position of one of the particles relative to the other. 

If the short-range repulsive disjoining pressure is large enough, the black foam films are stable. There are two types of black foam films common and Newtonian. While the common black films are the thicker type of black films (from about 5 to 20 nm in thickness), the Newtonian black (NB) films are bimolecular thin films (less than 5 mn in thickness). A mechanism of rupture of NB films is considered as a process of new phase nucleation in a two-dimensional system [105 108]. There exist in the film elementary vacancies (unoccupied positions of surfactant molecules) moving randomly, which associate to form clusters of vacancies called holes. A hole can grow up by fluctuations to a critical size and become a nucleus of a hypothetical two-dimensional phase of vacancies. Further spontaneous growth of the nucleus leads irreversibly to the rupture of the film. When the rupture of NB film is due to formation of holes in it by a nucleation mechanism, it has been shown that the mean film lifetime r depends on the monomer surfactant concentration C as ... 

---