Asymmetrical phase diagram

I he diagram of Fig. 9-1 is typical of mixtures of two small molecules, or of two polymers of comparable molecular weight and comparable viscosity. Polymer solutions, or blends of two polymers with very different molecular weights, have asymmetric phase diagrams, reflecting the asymmetry of the molecular sizes (see Fig. 9-2). 

Fig. 40. Phase diagrams for normal and anomalous structures, (a) Symmetrical phase diagram and coupled zone associated with a normal structure, (b) Asymmetrical phase diagram and coupled zone associated with an anomalous structure. After Elliot [23]. Reproduced with permission of the ASM International, Metals Park.

Figure 31.7 A polymer phase diagram. Xi" on the y-axis is proportional to temperature. Mixtures of polymers and small molecules have asymmetric phase diagrams, in which the critical volume fraction of polymer < ) < 1/2 (see Equation (31.28)). Equation (31.29) shows that as iV oo, the critical exchange parameter Xc 1/2. 

Fig. 5 Mean-field phase diagrams for asymmetric ABA triblocks spanning between diblock (r = 0.0) and symmetric triblock (r = 0.5) limits at segregations of a xN = 20, b /N = 30 and c xN = 40 calculated with SCFT. Dotted curves critical asymmetries rc predicted by SST beyond which short A blocks are extracted from their domains. From [32]. Copyright 2000 American Institute of Physics...

Fig. 8 Phase diagram for PI-fc-PEO system. Only equilibrium phases are shown, which are obtained on cooling from high temperatures. ODT and OOT temperatures were identified by SAXS and rheology. Values of /AT were obtained using /AT = 65/T + 0.125. Dashed line spinodal line in mean-field prediction. Note the pronounced asymmetry of phase diagram with ordered phases shifted parallel to composition axis. Asymmetric appearance can be accounted for by conformational asymmetry of segments. Adopted from [53]...

Fig.66 Phase diagrams of a symmetric (peo = 0.51, Mn = 2700, Mw/Mn = 1.10) and b asymmetric (0peo = 0.32, Mn = 2100, Mw/Mn = 1.14) PEO-fc-PEP block copolymers blended with epoxy resin. Phase transitions which originate from swelling of PEO chains with epoxy and/or curing agent are drawn as single lines, without implication that there are no coexistence regions. From [197]. Copyright 2001 Wiley...

Figure 7.6. Comparison of (a) experimental phase diagram for the Cu-Au system (Hansen 1958) with (b) predictions for the Cu-Au system calculated using the tetrahedron approximation but including asymmetric four-body interactions (de Fontaine and Kikuchi 1978).

Fig. 2.38 Phase diagram computed using the strong segregation limit theory of Helfand and Wasserman (1982) for the poly(ethylene oxide)-poly(butylene oxide) (PEO-PBO) diblock system. Because the ratio of statistical segment lengths aPB0/ 1, the phase diagram is asymmetric about/= 0.5 (Hamley 1997).

The resulting phase diagram for diblock copolymers is shown in Fig. 2.40.The theory predicts that microphase separation occurs to a body-centred cubic structure for all compositions except where a direct second-order transition to a lamellar structure is predicted. First-order transitions to hex and lam phases are expected on further lowering the temperature for asymmetric diblocks. 

Matsen (1995ft) also calculated sections through the phase diagram in the weak segregation limit for a fixed value for blends with given values of/ . Because asymmetric copolymers were considered, ordered phases other than lamellae... 

Fig. 6.40 A phase diagram calculated using SCFT for a mixture containing equal amounts of two homopolymers and a symmetric diblock, all with equal chain length (Janert and Schick 1997a). A-rich and B-rich swollen lamellar bilayer phases are denoted LA and LH respectively whilst the corresponding disordered phases are denoted A and B. The con-solute line of asymmetric bilayer phases LA and Lu, shown dotted, is schematic.The dashed line is the unbinding line. The arrows indicate the locations of the unbinding transition X jN and multicritical Lifshitz point, cMiV " 6.0.

In conclusion, we have found the ratchet current for strong and weak asymmetric potentials. It exhibits a set of universal power dependencies on the voltage and can grow as the voltage decreases. In Ref. [25] our analysis was extended to include the electron spin. This leads to a complicated phase diagram with several qualitatively different transport regimes for different interaction strengths. 

As described earlier for classical resolutions (Figure 7.1), asymmetric transformations can also be described by phase diagrams. However, because the overall composition of the ternary mixture (2 diastereoisomers + solvent) is not constant, only starting and end composition can be visualized (Figure 7.10). 

FIGURE 7.10 Description of a crystallization-induced asymmetric transformation by a ternary phase diagram. 

This chapter describes in a step-by-step manner a generic strategy that we have been using for the development of various industrial processes. The chapter begins with a discussion of the overall workflow for process development, followed by a description of the way in which a process is synthesized, which relies heavily on the use of phase diagrams. The deviation from equilibrium behavior is accounted for using a model-based approach. This is illustrated with an example on the asymmetric transformation of an enantiomer. 

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