Critical expansion

In 1954, Mullins, in a modification to the Overton hypothesis, proposed that besides the membrane concentration of the anesthetic, its volume, expressed as its volume fraction (mole fraction X partial molal volume), is important. This reasoning implied that the anesthetic, due to its solubility properties, expands the cell membrane, and that anesthesia occurs when a critical expansion value is reached, at about 0.3-0.5% of the original volume. 

Additional comment deserve magnetostriction measurements near the ordering temperature 7c reflecting critical phenomena. Few data for critical expansion is available, such as have been reported by Dolejsi and Swenson (1981) for the case of Gd metal. The thermal expansion coefficient in the critical region should assume the form 1(7 — Tc)/Tc °-The critical exponent or should be the same as for the specific heat and depend only on the universality class (dimensionality, No. of degrees of freedom) of the system. For Gd metal this universality class has been determined by Frey et al. (1997). 

Other theories give fragment size as a function of a hypothesized critical expansion velocity (Rinehard and Pearson, Ref 13), or based upon a critical strain rate, a mechanism related to radial, not tangential, stress gradient across the cylinder wall (Garg and Siekmann, Ref 14, and Taylor, Ref 15). Mott, however, gives a more tractable mathematical treatment suited to first-order engineering, as compared to the others. Mott s equations are... 

Plastic piping runs should not be located near high ambient temperature sources, including other piping, ductwork, or conductors. Plastic piping supports should be closer together than for a metal pipe to compensate for the more critical expansion allowance. Formation of hot spots by the... 

The expansion is done around the principal axes so only tliree tenns occur in the simnnation. The nature of the critical pomt is detennined by the signs of the a. If > 0 for all n, then the critical point corresponds to a local minimum. If < 0 for all n, then the critical point corresponds to a local maximum. Otherwise, the critical points correspond to saddle points. 

Iiifomiation about the behaviour of the 3D Ising ferromagnet near the critical point was first obtained from high- and low-temperatnre expansions. The expansion parameter in the high-temperatnre series is tanli K, and the corresponding parameter in the low-temperatnre expansion is exp(-2A ). A 2D square lattice is self-dual in the sense that the bisectors of the line joining the lattice points also fomi a square lattice and the coefficients of the two expansions, for the 2D square lattice system, are identical to within a factor of two. The singularity occurs when... 

Figure A2.3.29 Calculation of the critical temperature and the critical exponent y for the magnetic susceptibility of Ising lattices in different dimensions from high-temperature expansions.

At the critical pohit (and anywhere in the two-phase region because of the horizontal tie-line) the compressibility is infinite. However the compressibility of each conjugate phase can be obtained as a series expansion by evaluating the derivative (as a fiuictioii of p. ) for a particular value of T, and then substituting the values of p. for the ends of the coexistence curve. The final result is... 

In 1972 Wegner [25] derived a power-series expansion for the free energy of a spin system represented by a Flamiltonian roughly equivalent to the scaled equation (A2.5.28). and from this he obtained power-series expansions of various themiodynamic quantities around the critical point. For example the compressibility... 

Easier V, Schweiss P, Meingast C, Obst B, Wuhl H, Rykov A I and Tajima S 1998 3D-XY critical fluctuations of the thermal expansivity in detwinned YBa2Cu30y g single crystals near optimal doping Phys. Rev. Lett. 81 1094-7... 

In the vicinity of the critical point (i.e. t < i) the interfacial width is much larger than the microscopic length scale / and the Landau-Ginzburg expansion is applicable. 

With disk diameters above 5.25 in., all parameters, eg, water absorption and thermal expansion, become more critical which aggravates the expansion or warp of disks. If in the future disk rotation speeds have to be increased significantly to boost data transfer rates, higher demands will be placed on warp (tilt angle) and modulus to avoid creeping (ie, irreversible elongation in radial direction). A survey of the requirement profile for the substrate material of optical disks is given in Table 5 (182,186,187,189). 

The critical property for conformal coatings is resistance to chemicals, moisture, and abrasion. Other properties, such as the coefficient of thermal expansion, thermal conductivity, flexibiHty, and modulus of elasticity, are significant only in particular appHcations. The dielectric constant and loss tangent of the conformal coating are important for high speed appHcations. 

Fig. 15. Sihca solubihty diagram, where (- x -) is the turbine expansion line and ( ) is the critical point (10). Numbers represent pressure in MPa. To...

Supercriticalfluid solvents are those formed by operating a system above the critical conditions of the solvent. SolubiHties of many solutes ia such fluids often is much greater than those found for the same solutes but with the fluid at sub atmospheric conditions. Recently, there has been considerable iaterest ia usiag supercritical fluids as solvents ia the production of certain crystalline materials because of the special properties of the product crystals. Rapid expansion of a supercritical system rapidly reduces the solubiHty of a solute throughout the entire mixture. The resulting high supersaturation produces fine crystals of relatively uniform size. Moreover, the solvent poses no purification problems because it simply becomes a gas as the system conditions are reduced below critical. 

The equations given predict vapor behavior to high degrees of accuracy but tend to give poor results near and within the Hquid region. The compressibihty factor can be used to accurately determine gas volumes when used in conjunction with a virial expansion or an equation such as equation 53 (77). However, the prediction of saturated Hquid volume and density requires another technique. A correlation was found in 1958 between the critical compressibihty factor and reduced density, based on inert gases. From this correlation an equation for normal and polar substances was developed (78) ... 

The tables given under this subject are reprinted by permission from the Smithsonian Tables. For more detadea data on thermal expansion, see International Critical Tables tabular index, vol. 3, p. 1 abrasives, vol. 2, p. 87 alloys, vol. 2, p. 463 building stones, vol. 2, p. 54 carbons, vol. 2, p. 303 elements, vol. 1, p. 102 enamels, vol. 2, p. 115 glass, vol. 

For supercritical temperatures, it is satisfactory to ever-higher pressures as the temperature increases. For pressures above the range where Eq. (4-190) is useful, but below the critical pressure, the virial expansion in density truncated to three terms is usually suitable ... 

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