Critical density criterion

Mott first gave an estimate (125) of this critical density at the NM-M transition in terms of the screening properties of an itinerant electron gas. The "Mott criterion is... 

Figure 6. Normalized critical stress versus crack orientation - Experimental results and theoretical predictions based on the minimum strain energy density criterion.

The field-density concept is especially usefiil in recognizing the parallelism of path in different physical situations. The criterion is the number of densities held constant the number of fields is irrelevant. A path to the critical point that holds only fields constant produces a strong divergence a path with one density held constant yields a weak divergence a path with two or more densities held constant is nondivergent. Thus the compressibility Kj,oi a one-component fluid shows a strong divergence, while Cj in the one-component fluid is comparable to (constant pressure and composition) in the two-component fluid and shows a weak... 

Measurement Requirements. Any analysis of measurement requirements must begin with consideration of the particular accuracy, repeatabihty, and range needed. Depending on the appHcation, other measurement considerations might be the speed of system response and the pressure drop across the flow meter. For control appHcations repeatabihty may be the principal criterion conversely for critical measurements, the total installed system accuracy should be considered. This latter includes the accuracy of the flow meter and associated readout devices as well as the effects of piping, temperature, pressure, and fluid density. The accuracy of the system may also relate to the required measurement range. 

Thickener control philosophies are usually based on the idea that the Iindertlow density obtained is the most important performance criterion. The o ertlow clarity is also a consideration, but this is generally not as critical. Additional factors which must be considered are optimization of tlocciilant usage and protection of the raking mechanism. 

Flow of the liquid past the electrode is found in electrochemical cells where a liquid electrolyte is agitated with a stirrer or by pumping. The character of liquid flow near a solid wall depends on the flow velocity v, on the characteristic length L of the solid, and on the kinematic viscosity (which is the ratio of the usual rheological viscosity q and the liquid s density p). A convenient criterion is the dimensionless parameter Re = vLN, called the Reynolds number. The flow is laminar when this number is smaller than some critical value (which is about 10 for rough surfaces and about 10 for smooth surfaces) in this case the liquid moves in the form of layers parallel to the surface. At high Reynolds numbers (high flow velocities) the motion becomes turbulent and eddies develop at random in the flow. We shall only be concerned with laminar flow of the liquid. 

Figure 10 Critical current criteria as described in Ref. 29. The electric field vs. current density is shown as a function of magnetic field. The electric field criterion is set and the tangent at that intersection to the field vs current curve is extrapolated to obtain the offset Jc.

As expected from the general discussion in Section III. A, the criterion (57) can also be derived from the exact free energy an alternative form involving the spinodal determinant Y is given in Appendix D. Equation (57) shows that the location of critical points depend only on the moment densities p[t py, and pijk [11, 46]. For a system with an excess free energy depending only on power-law moments up to order K - 1, the critical point condition thus involves power-law moments of the parent only up to order 3 (K — 1). 

This criterion, which is product rather than precursor-property driven, is critical to the design and synthesis of new precursors. The need for high ceramic yields arises because of the excessive volume changes associated with pyrolytic conversion to ceramic materials. Scheme 1 illustrates these changes for a SiC precursor with an 80% ceramic yield of phase pure SiC (3.2 gml-1). Most precursors densities are close to 1 gml-1, whereas most Si ceramic densities range from 2.5 to 3.5 gml-1. 

Supercritical Fluid. To be useful as a mobile phase in chromatography, a supercritical fluid must have a relatively low critical temperature and pressure, and a relatively high density/solvating power at experimentally accessible pressures and temperatures. The former criterion excludes water and most common organic solvents, whereas the latter excludes such low-boiling substances as helium, hydrogen, and methane. Commonly used fluids are listed in Table I. 

In order to achieve an overall positive energy balance the D-T plasma has to be heated to the ignition temperature of 5 keV and confined for a sufficiently long time4). Energy breakeven is usually expressed in terms of the Lawson criterion which states that the product of the plasma density, n, and the energy confinement time re has to exceed a critical value which is a function of the plasma temperature4 8). 

Many polymers exhibit neither a measurable stick-slip transition nor flow oscillation. For example, commercial polystyrene (PS), polypropylene (PP), and low density polyethylene (LDPE) usually do not undergo a flow discontinuity transition nor oscillating flow. This does not mean that their extrudate would remain smooth. The often observed spiral-like extrudate distortion of PS, LDPE and PP, among other polymer melts, normally arises from a secondary (vortex) flow in the barrel due to a sharp die entry and is unrelated to interfacial slip. Section 11 discusses this type of extrudate distortion in some detail. Here we focus on the question of why polymers such as PS often do not exhibit interfacial flow instabilities and flow discontinuity. The answer is contained in the celebrated formula Eqs. (3) or (5). For a polymer to show an observable wall slip on a length scale of 1 mm requires a viscosity ratio q/q equal to 105 or larger. In other words, there should be a sufficient level of bulk chain entanglement at the critical stress for an interfacial breakdown (i.e., disentanglement transition between adsorbed and unbound chains). The above-mentioned commercial polymers do not meet this criterion. 

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