Dynamic phase diagrams

The topological transformations in an incompatible blend can be described by the dynamic phase diagram that is usually determined experimentally at a constant shear rate. For equal viscosities, a bicontinuous morphology is observed within a broad interval of the volume fractions. When the viscosity ratio increases, the bicontinuous region of the phase diagram shrinks. At large viscosity ratios, the droplets of a more viscous component in a continuous matrix of a less viscous component are observed practically for all allowed geometrically volume fractions. 

Fig. 6. Dynamical phase diagram of the ascorbic acid/copper(II)/oxygen system in a CSTR in the kf — [Cu2+]0 plane. Fixed reactor concentrations [H2Asc]0 = 5.0x10 4M [H2SO4]0 = 6.0 x 10-5 M [Na2SO4]0 = 0.04M. Symbols O, steady state , oscillations , bistability. The asterisk ( ) marks the Takens-Bogdanov point. Strizhak, P. E. Basylchuk, A. B. Demjanchyk, I. Fecher, F. Shcneider, F. W. Munster, A. F. Phys. Chem. Chem. Phys. 2000, 2, 4721. Reproduced by permission of The Royal Society of Chemistry on behalf of the PCCP Owner Societies.

A very rich scenario was, on the other hand, observed with Pt(110) if perturbed under conditions for which regular autonomous oscillations with well-defined frequencies existed (91-93). Typically, these autonomous oscillations were established at fixed external parameters and then one of the partial pressures was periodically modulated by use of a feedback-regulated gas inlet system with frequencies up to 0.5 s l and relative amplitudes around 1% (31, 33). Following the pioneering mathematical treatment of forced nonlinear oscillations by Kai and Tomita (SO), the results can be rationalized in terms of a dynamic phase diagram characterizing the response of the system as a function of the amplitude A and of the period of the pressure modulation Tcx with respect to that of the... 

Fig. 17. Experimental dynamic phase diagram for forced oscillations in the CO/O, reaction on Pt(l 10). Periodic modulation of the 02 pressure with amplitude A (as a percentage of the constant base pressure) and period length, Tex, with respect to that of the autonomous oscillations, T . (From Ref. 91.) Fixed parameters, for the subharmonic (superharmonic) range p(h = 3.0 (4.15) x 10 5 torr, p , = 1.6 (2.1) x 10 5 torr, T = 525 (530) K.

Fig. 21. Theoretical dynamic phase diagram for forced oscillations on Pt(l 10), evaluated from Eqs. (4), (5a), and (6). The calculations have been performed up to larger forcing amplitudes than were experimentally accessible (compare with Fig. 17). For small A, the bands characterizing entrainment and quasiperiodicity are discernible, whereas for larger A more complex bifurcations result. Types of bifurcations ns, Neimark-Sacker pd, period doubling snp, saddle-node of periodic orbits. (From Ref. 69.)...

The behavior of such a system may be characterized by a dynamic phase diagram, in which the regions of entrainment and quasi-periodicity are identified as a function of T /Tq and A. The structure of such a diagram does not depend on the specific properties of the reacting system, but rather on its type of bifurcation around which the perturbation is applied [32/33]. 

FIGURE 7.12. Experimental dynamic phase diagram for periodically forced oscillations in the CO oxidation on a Pt(l 10) surface [34]. Existence range for entrained and quasi-periodic (shaded areas) oscillations as a function of the ratio of the periods of modulation Tex and autonomous oscillations To and amplitude A of the modulated O2 pressure. 

FIGURE 7.13. Theoretical dynamic phase diagram resulting from solution of the quoted ODEs [35]. Types of bifurcations Ns = Neimark-Sacker, pd = period doubling, and snp = saddle node. 

Figure 4.34. A "dynamic phase diagram superimposed on a conventional equilibrium diagram for the quinary system Na, K, Mg, Cl, SO4 in water at 25 °C...

Tycko, R. Molecular dynamics, phase diagrams and electronic properties of fuller-enes and alkali fullerides Insights from sohd-state nuclear magnetic resonance spectroscopy.. Solid State Nud. Masn. Reson. 1994 3 303-314. 

If normalized shear stress c/Go and shear rate yxR are introduced, it is possible to summarize the overall nonlinear rheological behavior measured at different concentrations and temperatures on a master dynamic phase diagram as shown in Fig. 12 [137]. The flow curve at 21 wt. %, a concentration close to the I-N transition, makes the link with the concentrated regime. As concentration decreases, stress... 

Figure 9.3 shows the dynamical phase diagram for the slip avalanche statistics in granular materials, obtained analytically from this model [73]. The y-axis effectively corresponds to rescaled packing fraction v = O/O, such that the upper limit on the y-axis y = 1 corresponds to the densest possible packing, d) =... 

Dynamical phase diagram (tuning packing fraction and frictional weakening)... 

Figure 9.3 Dynamical phase diagram for sheared granular materials according to [73]. The red arrow indicates lowering the packing fraction starting from random close-packed materials. The granular volume fraction v is defined asv = < /Phic, where is the packing fraction and <1) is the maximum packing fraction (like random closed packed). The black line indicates a critical volume fraction v that separates fluid-like dynamics from solidlike dynamics with larger slip avalanches (see also Table 9.1).

Fig. 28 Dynamical phase diagram as a function of viscosity contrast = Tjin/rjo, for r = 0 and various reduced volumes V, calculated from (105), (106) without thermal noise. The tank-treading phase is located on the left-hand side of the dashed lines. The solid lines represent the tumbling-to-swinging transitions. From [205]...

Fig. 29 Dynamical phase diagram of a vesicle in shear flow for reduced volume V = 0.59. Symbols correspond to simulated parameter values, and indicate tank-treading discocyte and tank-treading prolate (circles), tank-treading prolate and unstable discocyte (triangles), tank-treading discocyte and tumbling (transient) prolate (open squares), tumbling with shape oscillation (diamonds), unstable stomatocyte (pluses), stable stomatocyte (crosses), and near transition (filled squares). The dashed lines are guides to the eye. From [180]...

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