Critical threshold

Keywords Branching process, Gelation, Cyclization, Excluded volume effects, Dimension, Concentration invariant, High concentration expansion, High dimension expansion, Gel point, Percolation threshold, Critical dimension. 

As Balberg notes in a review The electrical data were explained for many years within the framework of interparticle tunneling conduction and/or the framework of classical percolation theory. However, these two basic ingredients for the understanding of the system are not compatible with each other conceptually, and their simple combination does not provide an explanation for the diversity of experimental results [17]. He proposes a model to explain the apparent dependence of percolation threshold critical resistivity exponent on structure of various carbon black composites. This model is testable against predictions of electrical noise spectra for various formulations of CB in polymers and gives a satisfactory fit [16]. 

Lattice type Coordination number Z Packing density Percolation threshold Critical number of bonds Critkal volume liaison... 

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick. Therefore imposition of wall slip introduces a form of non-linearity into the flow model which should be handled via an iterative loop. The slip coefficient (i.e. /I in the Navier s slip condition given as Equation (3.59) is defined as... 

We noted above that the presence of monomer with a functionality greater than 2 results in branched polymer chains. This in turn produces a three-dimensional network of polymer under certain circumstances. The solubility and mechanical behavior of such materials depend critically on whether the extent of polymerization is above or below the threshold for the formation of this network. The threshold is described as the gel point, since the reaction mixture sets up or gels at this point. We have previously introduced the term thermosetting to describe these cross-linked polymeric materials. Because their mechanical properties are largely unaffected by temperature variations-in contrast to thermoplastic materials which become more fluid on heating-step-growth polymers that exceed the gel point are widely used as engineering materials. 

At low concentrations surfactant molecules adsorbed at the surface are in equilibrium with other molecules in solution. Above a threshold concentration, called the critical micelle concentration (cmc, for short), another equilibrium must be considered. This additional equilibrium is that between individual molecules in solution and clusters of emulsifier molecules known as micelles. 

In these expressions Xc is the critical (subscript c) value of x which marks the threshold at which immiscibility sets in, and 1 - 0j or 0j is the volume fraction of the solvent in the solution at this point. Rearranging Eq. (8.56), we obtain... 

B = 0 when x = 1/2, a condition we have already seen [Eq. (8.60)], corresponds to a critical value of x for a copolymer of infinite molecular weight. For finite molecular weights this condition is not quite a threshold for precipitation, but is close to it. Polymer-polymer contacts are sufficiently favored over polymer-solvent contacts that a chain of infinite length would undergo phase separation. 

Above about 250°C, the vapor-phase oxidation (VPO) of many organic substances becomes self-sustaining. Such oxidations are characterized by a lengthy induction period. During this period, peroxides accumulate until they can provide a source of new radicals to sustain a chain reaction. Once a critical threshold peroxide concentration is reached, the reaction accelerates very rapidly. 

An even wider range of wavelength, toward the infrared, can be covered with quantum well lasers. In the Al Ga As system, compressively strained wells of Ga In As are used. This ternary system is indicated in Figure 6 by the line joining GaAs and In As. In most cases the A1 fraction is quite small, X < 0.2. Such wells are under compressive strain and their thickness must be carefully controlled in order not to exceed the critical layer thickness. Lasers prepared in this way are characterized by unusually low threshold current density, as low as ca 50 A/cm (l )-... 

Stress Relaxation. Copper alloys are used extensively in appHcations where they are subjected to moderately elevated temperatures while under load. An important example is the spring member for contacts in electrical and electronic coimectors. Critical to rehable performance is the maintenance of adequate contact force, or stabiUty, while in service. Excessive decrease in this force to below a minimum threshold value because of losses in spring property can lead to premature open-circuit failure (see Electrical connectors). 

The most recent model of de Gennes and coworkers [100,101], the threshold toughness is related to the extra surface and elastic energy of the individual chains as they are pulled out. The threshold toughness and the critical crack propagation speed are given by... 

Thus, fracture occurs by first straining the chains to a critical draw ratio X and storing mechanical energy G (X — 1). The chains relax by Rouse retraction and disentangle if the energy released is sufficient to relax them to the critically connected state corresponding to the percolation threshold. Since Xc (M/Mc) /, we expect the molecular weight dependence of fracture to behave approximately as... 

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