Transition line

Sigaud G, Flardouin F and Aohard M F 1979 A possible polar smeotio A-non-polar smeotio A transition line in a binary system Phys.Lett. A 72 24... 

A natural question is just how big does Mq have to be to see this ordered phase for M > Mq. It was shown in Ref 189 that Mq <27, a very large upper bound. A direct computation on the Bethe lattice (see Fig. 2) with q neighbors [190,191] gives Mq = [q/ q — 2)f, which would suggest Mq 4 for the square lattice. By transfer matrix methods and by Pirogov-Sinai theory asymptotically (M 1) exact formulas were derived [190,191] for the transition lines between the gas and the crystal phase (M 3.1962/z)... 

FIG. 2 Phase diagram in the M-z plane for a square lattice (MC) and for a Bethe lattice q = A). Dashed lines Exact results for the Bethe lattice for the transition lines from the gas phase to the crystal phase, from the gas to the demixed phase and from the crystal to the demixed phase full lines asymptotic expansions. Symbols for MC transition points from the gas phase to the crystal phase (circles), from the gas to the demixed phase (triangles) and from the crystal to the demixed phase (squares). (Reprinted with permission from Ref. 190, Fig. 7. 1995, American Physical Society.)... 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

The second difference is related to the structure of the lamellar phase. The Euler characteristic has been assumed zero in the whole lamellar phase by Gompper and Kraus [47], whereas we show that it fluctuates strongly in the lamellar phase between the transition line and the topological disorder fine. The notion of the topological disorder line has not appeared in their paper. We think that the topological disorder line is much closer to the transition... 

FIG. 12 The behavior of the internal energy U (per site), heat capacity Cy (per site), the average Euler characteristic (x) and its variance (x") — (x) close to the transition line and at the transition to the lamellar phase for/o = 0. The changes are small at the transition and the transition is very weakly first-order. The weakness of the transition is related to the proliferation of the wormhole passages, which make the lamellar phase locally very similar to the microemulsion phase (Fig. 13). Note also that the values of the energy and heat capacity are not very much different from their values (i.e., 0.5 per site) in the Gaussian approximation of the model [47]. (After Ref. 49.)... 

Fig. 11. Intensity of the single mode of Ar+ 514.5 nm as mode is swept through gain curve. The single mode will match the iodine transition line and will be absorbed ( 1).

If point F in Fig. 2 is reached without physical burn-out occurring, then, as shown by Nukiyama, a further increase in heat flux will raise the surface temperature in the direction of E until physical burn-out does occur. If, however, the heat flux at point F is decreased, the surface temperature does not revert to the value at C, but moves along the curve towards D. On reaching D, it was observed by Nukiyama that the surface temperature undergoes another jump discontinuity along the dotted line DG, and stabilizes at G in the nucleate-boiling region. Both the transition lines CF and DG can be passed only in the direction shown by the arrows in Fig. 2. 

Fig. 2.31a,b Flow pattern maps with experimental transition lines for R-134a, d = 0.5 mm, 70 mm, Ts = 30 °C and AFsub = 3 K. Reprinted from Revellin et al. (2006) with permission... 

Fig. 5.15 Comparison between the experimental flow patterns obtained by Triplett et al. (1999a) and the experimental flow regime transition lines of Damianides and Westwater (1988) representing their 1 mm diameter cireular test section. Reprinted from Triplett et al. (1999a) with permission...

Figure 5.16c indicates that as the channel size was reduced to Jh = 0.866 mm, the dispersed bubbly flow pattern vanished from the flow regime map. Figure 5.16a-c indicates that the slug-churn flow transition line shifted to the right, as the channel size was reduced. Similar trends were also found in small circular tubes by the... 

There has been relatively little work done on the development of two-phase flow regime maps for micro-channels. The general trends of how the transition lines are shifted as the diameter is decreased are unclear. Figure 5.19 shows a flow pattern map obtained for air-water two-phase flow in a 20 pm i.d. silica tube by Serizawa et al. (2002) at nearly atmospheric pressure. 

The smectic phases Ai, A2 and A have the same macroscopic symmetry, differing from each other in the wavelength of spacing. Hence it is possible to go from Ai to Aa or from Aa to A2 by varying only the layer periodicity in a continuous or discontinuous way(with the jump in the layer spacing d). Smectic-smectic transition lines of first order may terminate at a critical point, where the differences between the periodicities of the smectic A phases vanish, providing a continuous evolution from an Aa to bilayer A2 phase [12]. 

The denominator in Equation 24.19 is evaluated along the glass transition line, Tg P), so that... 

The sol-gel transition has been determined visually, with calcium and copper, for different pectins under different external conditions. As shown in Figure 5 for sample C44 the homogeneous gel phase is situated between the two transition lines. The extension of this phase was found to depend mainly on the DE, temperature and nature of the cation. With calcium the amount of cation required to get a gel increased with the degree of esterification and above 50% it became impossible to get a gel [8]. 

The gel/sol transitions cannot be distinguished for pectins extracted by after acid treatment and water-soluble pectins after extrusion. A minimal pectin concentration of 0.2% is required for gelation and no gels can be obtained below a sucrose concentration of 45 %. Commercial pectin (Hercules) with a dm 73 % has a lower phase transition line with a minimal pectin concentration of 0.1 % and sucrose concentration of 40 %. 

Fig. 1. Quantification of framework Ti sites and unit cell expansion versus Ti contents using XPS 2p transition lines ( O ). XANES profiles at Ti K-edge (A) and Rietveld refinement of the XRD powder patterns ( ). Solid data points for TS-1 prepared as in ref [9].

Figure 5.6 Flow pattern map for a gas/liquid flow regime in micro channels. Annular flow wavy annular flow (WA) wavy annular-dry flow (WAD) slug flow bubbly flow annular-dry flow (AD). Transition lines for nitrogen/acetonitrile flows in a triangular channel (224 pm) (solid line). Transition lines for air/water flows in triangular channels (1.097 mm) (dashed lines). Region 2 presents flow conditions in the dual-channel reactor ( ), with the acetonitrile/nitrogen system between the limits of channeling (I) and partially dried walls (III). Flow conditions in rectangular channels for a 32-channel reactor (150 pm) (T) and singlechannel reactor (500 pm) (A) [13].

Recoil Energy Loss in Free Atoms and Thermal Broadening of Transition Lines... 

Fig. 7.3 Effect of magnetic dipole interaction (7/m), electric quadmpole interaction (Hq), and combined interaction// = Hu + //q, Em> q on the Mossbauernuclear levels of Ni. The larger spacings between the sublevels of the ground state are due to the somewhat larger magnetic dipole moment of the nuclear ground state as compared to the excited state. The relative transition probabilities for a powder sample as well as the relative positions of the transition lines are indicated by the stick spectra below...

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