Order / Disorder equilibriums

The last example demonstrates how sensitive the order/disorder equilibrium depends on the molecular architecture. Dielectric measuements can help us to understand such complicated relations which are also very important in the life sciences. 

Nearly all experimental eoexistenee eurves, whether from liquid-gas equilibrium, liquid mixtures, order-disorder in alloys, or in ferromagnetie materials, are far from parabolie, and more nearly eubie, even far below the eritieal temperature. This was known for fluid systems, at least to some experimentalists, more than one hundred years ago. Versehaflfelt (1900), from a eareflil analysis of data (pressure-volume and densities) on isopentane, eoneluded that the best fit was with p = 0.34 and 8 = 4.26, far from the elassieal values. Van Laar apparently rejeeted this eonelusion, believing that, at least very elose to the eritieal temperature, the eoexistenee eurve must beeome parabolie. Even earlier, van der Waals, who had derived a elassieal theory of eapillarity with a surfaee-tension exponent of 3/2, found (1893)... 

Whereas only a few atomie jumps may be neeessary to enable ehanges in the equilibrium degree of SRO, without atomie movement over long distanees ehanges in LRO may not be suffieient to reaeh equilibrium. This ean lead to a eompetition between inerease of SRO in the matrix and formation of the new LR0-phase when lowering the temperature below an order/disorder phase boundary. In those cases, thermal and/or meehanieal pretreatment of the sample is of erueial importanee for what is observed in the sample. 

For example, 0 describes the temperature dependence of composition near the upper critical solution temperature for binary (liquid + liquid) equilibrium, of the susceptibility in some magnetic phase transitions, and of the order parameter in (order + disorder) phase transitions. 

Fig. 8.8 The principle of the Dyson model. Each point in the phase diagram represents a possible composition of a molecular population. The horizontal axis is a, where (a+ 1) is the number of monomer types. On the vertical axis, b represents the quality factor of the polymeric catalysis. The transition region consists of populations which can have both an ordered and a disordered equilibrium state. In the death region there are only disordered states, while in the immortal region (in the Garden of Eden ), there is no disordered state (Dyson, 1988)...

Intuitively we might associate the low shear limit with the order-disorder transition at literature data for the packing fraction in this limit is more widely scattered. We must remember that the approach to equilibrium in these systems can take a while to progress. So it is feasible that some systems have been measured away from the equilibrium state when the samples have been transferred and placed in the measuring geometry on an instrument. We could... 

Earlier work on systems such as Ni-Al-Cr reported in Sanchez et al. (1984b) used FP methods to obtain information on phases for which there was no experimental information. In the case of Ni-base alloys, the results correctly reproduced the main qualitative features of the 7 — 7 equilibrium but cannot be considered accurate enough to be used for quantitative alloy development. A closely related example is the work of (Enomoto and Harada 1991) who made CVM predictions for order/disorder (7 — 7 ) transformation in Ni-based superalloys utilising Lennard-Jones pair potentials. 

Figure 1-19 Schematic diagram showing how Ku for the Fe-Mg order-disorder reaction varies during cooling. The arrow indicates the progression of time. The thin dashed curve shows how the equilibrium iCo varies with temperature as the system cools. The solid curve shows how iCo varies with temperature during rapid quench in a volcanic rock. The thick and long dashed curve shows how Kj) varies during slow cooling in a plutonic rock.

Box 2.2 Calculation of the equilibrium species concentrations of the Fe-Mg order-disorder reaction in orthopyroxene... 

This section focuses on how the Fe-Mg order-disorder reaction (Section 2.1.4) is applied as a geospeedometer. The equilibrium and kinetics of the reaction are discussed in Section 2.1.4 and only a brief review is provided here. Although there is some complexity in the kinetics of this reaction (e.g.. Figure 2-5), it is minor, and is hence usually ignored so that the forward and backward reactions are treated as elementary reactions. The rate coefficient for the forward reaction of this reaction (Reaction 2-55)... 

Stimpfl M., Ganguly J., and Molin G. (1999) Fe +-Mg order-disorder in orthopyroxene equilibrium fractionation between the octahedral sites and thermodynamic analysis. Contrib. Mineral. Petrol. 136, 297-309. 

Fig. 2.51 Effect of reciprocating shear (strain amplitude, k = 200%) on the ODT of an /pep = 0.55 PEP-PEE diblock (Koppi etal. 1993). Here y denotes the shear rate.The equilibrium order-disorder transition (, A) and disordered state stability limit (A.O) are shown. The upper curve is a fit to the scaling relation Tom y2- The lower curve represents the. scaling rs(A) A-i,3Todt> where A = y/y, with y an adjustable, parameter. Points given by and O were obtained at fixed temperature by varying y, while those represented by A and A were determined by varying the temperature at fixed y.

Because of the potential importance for industrial-scale catalysis, we decided to check (i) whether an influence of a semiconductor support on a metal catalyst was present also if the metal is not spread as a thin layer on the semiconductor surface but rather exists in form of small particles mixed intimately with a powder of the semiconductor, and (ii) whether a doping effect was present even then. To this end the nitrates of nickel, zinc (zinc oxide is a well-characterized n-type semiconductor) and of the doping element gallium (for increased n-type doping) or lithium (for decreased n-type character) were dissolved in water, mixed, heated to dryness, and decomposed at 250°-300°C. The oxide mixtures were then pelleted and sintered 4 hr at 800° in order to establish the disorder equilibrium of the doped zinc oxide. The ratio Ni/ZnO was 1 8 and the eventual doping amounted to 0.2 at % (75). 

Figure 8.3. Equilibrium constants (log K) for the metastable coexistence of completely ordered and disordered CaMg(C03>2 with dolomite in its stable order/disorder state as a function of temperature at constant pressure. SAT refers to the vapor-liquid curve for pure H2O. (After Bowers et al., 1984.)...

Fig. XVII-8.—Phase equilibrium diagram for the system Cu-Zn, in which a number of phases of variable composition arc formed, mixtures of the phases being stable between the regions of stability of the pure phases. The phase a is face centered cubic, as Cu is, j8 is body centered, 7 is a complicated structure, e and r) are hexagonal. The transition between and /S is an order-disorder transition, /3 being disordered, and /8 ordered, as discussed in the following chapter.

---