Single hysteresis loop

The stationary-state response curves, or bifurcation diagrams shown in Figs 1.13(b) and 1.12(f), represent two of the simplest possible patterns monotonic variation and a single hysteresis loop respectively. These are the only qualitatively different responses possible for the cubic autocatalytic step on its own. They are also found for a first-order exothermic reaction in an adiabatic flow reactor (see chapter 6). With only slightly more complex chemical mechanisms a whole array of extra exotic patterns can be found, such as those displayed in Fig. 1.14. The origins of these shapes will be determined in chapter 4. 

Fig. 6.19. The five stationary-state loci patterns found when the uncatalysed conversion of A to B is included in the model (a) unique, (b) isola (c) mushroom (d) single hysteresis loop (breaking wave) (e) hysteresis loop + isola.

With the exponential approximation (y 0) and the assumption that the inflow and ambient temperatures are equal, we have a stationary-state equation which links ass to tres and which involves two other unfolding parameters, 0ad and tn. Depending on the particular values of the last two parameters the (1 — ass) versus rres locus has one of five possible qualitative forms. These different patterns are shown in Fig. 7.4 as unique, single hysteresis loop, isola, mushroom, and hysteresis loop plus isola. The five corresponding regions in the 0ad-rN parameter plane are shown in Fig. 7.5. This parameter plane is divided into these regions by a straight line and a cusp, which cut each other at two points. 

FONI model (a) unique (b) single hysteresis loop or breaking wave (c) isola (d) isola + hysteresis loop (e) mushroom. With the full Arrhenius rate law and the provision pre-heating or cooling, two additional patterns are found (f) reversed hysteresis loop and (g) reversed hysteresis loop + isola. Also shown are various degenerate loci corresponding to parameter values on the boundaries or special points in the parameter plane (see Fig. 7.5). 

This corresponds to an ordinary cusp, with an unfolding to a unique locus or a single hysteresis loop. 

Fig. 7.8. The full unfolding of the stationary-state locus (a) corresponding to a winged cusp singularity, (b) unique (c) isola (d) mushroom (e) single hysteresis loop (f) hysteresis loop + isola (g) reverse hysteresis loop (h) reverse hysteresis loop + isola.

FIG. 9.6. The five qualitative forms for the stationary-state locus ass(0)-D for cubic autocatalysis with decay in a reaction-diffusion cell (a) unique (b) single hysteresis loop (c) mushroom (d) isola (e) isola + hysteresis loop, (f) The division of the Pcx-k2 parameter plane giving the five regions corresponding to the stationary-state forms in (a)-(e) note that the region for response (e), shown inset, is particularly small and has not yet been successfully located. 

Fig. 12.5. (a) The region of multiple stationary-state behaviour for the Takoudis-Schmidt-Aris model of surface reaction, with = 10 3 and k2 = 2 x 10-3 (b), (c), and (d) show how the stationary-state reaction rate varies with the gas-phase pressure of reactant R for different values of p, giving isola, mushroom, and single hysteresis loop respectively. (Adapted and reprinted with permission from McKarnin, M. A. et al. (1988). Proc. R. Soc., A415, 363-87.)... 

Fig. 4.5-96a,b MHPOBC. D — E hysteresis loops obtained from switching-current measurements. D is the electric displacement. Sample ceU thickness = 3 jjim. (a) Double hysteresis loop at 2 Hz. (b) Single hysteresis loop at 100 Hz... 

Hysteresis loops M—H have been determined for single atomic monolayers of Fe and Ni that were prepared and measured in ultrahigh vacuum. 

TbPc2 SIMM (single-ion molecular magnet) and Ni magnetization on out-of-plane Ni/Cu( 100) films, (b) Element-resolved hysteresis loops of Ni (top) and Tb (bottom) for TbPc2/Ni/Cu(100) measured at out-of-plane (8 = 0°) incidence at T = 8K. The units of... 

After several cycles of the compression and expansion, the dynamic jc-A curve becomes a single closed loop, somewhat distorted from a genuine ellipsoid. In order to analyze the forms of the hysteresis loop under stationary conditions, we have measured the time trace of the dynamic surface pressure after five cycles of the compression and expansion, and then Fourier-transformed it to the frequency domain. The Fourier-transformation was adapted to evaluate the nonlinear viscoelasticity in a quantitative manner. The detailed theoretical consideration for the use of the Fourier transformation to evaluate the nonlinearity, are contained in the published articles [8,43]. 

As discussed above, hysteresis loops can appear in sorption isotherms as result of different adsorption and desorption mechanisms arising in single pores. A porous material is usually built up of interconnected pores of irregular size and geometry. Even if the adsorption mechanism is reversible, hysteresis can still occur because of network effects which are now widely accepted as being a percolation problem [21, 81] associated with specific pore connectivities. Percolation theory for the description of connectivity-related phenomena was first introduced by Broad-bent et al. [88]. Following this approach, Seaton [89] has proposed a method for the determination of connectivity parameters from nitrogen sorption measurements. 

Fig. 7.12 Left Hysteresis loops of MD (top) and SD magnetite (bottom) (modified from Dunlop, 1990, with permission). Right The ratio Jrs/Js as a function of the ratio of Hcr/Hc. The fields for single domain (SD), pseudo-single domain (PSD), and multi domain (MD) particles are given for magnetite.

Figure 8 Partial magnetization hysteresis loop at 5 K for a superconducting YE Cu O g single crystal. The solid triangles represent the response to increasing fields while the open triangles are for decreasing fields.

Figure 10 Comparison of partial magnetization hysteresis loops at 30 and 77 K for a superconducting T Ca j Ba2Cu20o single crystal (Tc is 105 K). The lack of hysteresis at 77 K is due to weak flux pinning.

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