Collapse regime

In another (somewhat different) approach, a probability density function (PDF) has been proposed (Moholkar and Pandit, 1997). This is used to map the cavity dynamics in the reaction medium covering all three phases of a cavity s lifetime growth, oscillation, and collapse. An ultrasonic reactor is considered highly efficient if the PDF shows peaks in the collapse regime at all of the locations in the cavitation field. This is an indication that pressure pulses exist throughout the medium and are not restricted to just a few locations. In other words, the cavitational intensity is uniformly distributed. If peaks occur in the growth and collapse regimes, it is desirable to place the reactor inside the sonicated medium at a location where the maximum probability of collapse is indicated. 

Figure 6 BAM images of a monolayer of stearic acid in the collapse regime, along with the surface pressure ( r)-mean molecular area (A) isotherm at 20 °C on a pH 3 subphase. Reproduced from Ref. 33. American Chemical Society, 1996, and the figure caption reads as follows Surface pressure/area isotherm (a) of a stearic acid monolayer on acidified aqueous subphase (pH 3.0) at 20 °C and BAM images of (b) 3D nuclei (bright dots") observed at a surface pressure of 40mNm within the steep part of the isotherm (i.e., the 3D nucleation occurs at n < Ttc = 50 mNm ) and (c) overlap and growth of the 3D nuclei within the plateau region of the n/A isotherm (after the pressure spike at Jtc). The bar represents 200p,m.

SAWs with nearest-neighbor attraction (—> theta point). The transition of polymer conformation from the high-temperature (good-solvent) regime to the theta point to the collapsed regime is well modeled by the self-avoid-... 

The regime v < fiis the collapsing regime of syneresis and no stable solution of the functional integrals exists. ... 

On the other hand, it is clear that in the classical regime, T> (T i is the crossover temperature for stepwise transfer), the transition should be step-wise and occur through one of the saddle points. Therefore, there should exist another characteristic temperature. r 2> above which there exist two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points atlT = T i. The existence of the second crossover temperature, 7, 2, for two-proton transfer has been noted by Dakhnovskii and Semenov [1989]. 

Note that the quantity (1- Q D)) can be thought of as a disorder parameter distinguishing the different regimes of behavior. Numerical measurements of this disorder parameter around the transition point at. which the zigzag pattern collapses spontaneously (i.e. around etc 3.92) suggest a critical-like behavior [kaneko89a] ... 

The slip correction factors are important for particles smaller than 1 pm in diameter, which is rarely the case for pharmaceutical aerosols. Slip correction is required for the Stokes equation to remain predictive of particle behavior for these small particles. Therefore, assuming the absence of shape effects for particles in the Stokes regime of flow, Eq. (1) collapses into the following expression ... 

The uncertainty in the age of pre main sequence stars is therefore of the order of the thermal timescale at the luminosity of D-burning smaller than a few times 105 yr for normal T Tauri, and larger than 106 yr for very low mass stars and brown dwarfs (BD). In fact, comparing observations spanning a wide range of masses we could even constrain the models, for example we can ascertain whether the Stahler et al. (1986) picture of collapse is valid also in the BD regime, or... 

P.-G. de Gennes later also considered the multisegment attraction regime. He suggested the so-called p-cluster model [11] in order to explain certain anomalies in behavior observed in many polymer species such as polyethyle-neoxide (PEO) see also [12]. The scenario of coil-globule transition with dominating multisegment interaction first considered by I.M. Lifshitz has been recently studied in [13]. The authors used a computer simulation of chains in a cubic spatial lattice to show that collapse of the polymer can be due to crystallization within the random coil. 

At high-particle number densities or low coefficients of normal restitution e, the collisions will lead to a dramatical decrease in kinetic energy. This is the so-called inelastic collapse (McNamara and Young, 1992), in which regime the collision frequencies diverge as relative velocities vanish. Clearly in that case, the hard-sphere method becomes useless. 

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