Phase disordered

In both cases the late stages of kinetics show power law domain growth, the nature of which does not depend on the mitial state it depends on the nature of the fluctuating variable(s) which is (are) driving the phase separation process. Such a fluctuating variable is called the order parameter for a binary mixture, tlie order parameter o(r,0 is tlie relative concentration of one of the two species and its fluctuation around the mean value is 5e(/,t) = c(r,t) - c. In the disordered phase, the system s concentration is homogeneous and the order... 

The Ag (100) surface is of special scientific interest, since it reveals an order-disorder phase transition which is predicted to be second order, similar to tire two dimensional Ising model in magnetism [37]. In fact, tire steep intensity increase observed for potentials positive to - 0.76 V against Ag/AgCl for tire (1,0) reflection, which is forbidden by symmetry for tire clean Ag(lOO) surface, can be associated witli tire development of an ordered (V2 x V2)R45°-Br lattice, where tire bromine is located in tire fourfold hollow sites of tire underlying fee (100) surface tills stmcture is depicted in tlie lower right inset in figure C2.10.1 [15]. 

M. Schoen, D. J. Diestler, J. H. Cushman. Stratification-induced order-disorder phase transitions in molecular confined films. J Chem Phys 707 6865-6872, 1994. 

In this section we review several studies of phase transitions in adsorbed layers. Phase transitions in adsorbed (2D) fluids and in adsorbed layers of molecules are studied with a combination of path integral Monte Carlo, Gibbs ensemble Monte Carlo (GEMC), and finite size scaling techniques. Phase diagrams of fluids with internal quantum states are analyzed. Adsorbed layers of H2 molecules at a full monolayer coverage in the /3 X /3 structure have a higher transition temperature to the disordered phase compared to the system with the heavier D2 molecules this effect is... 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

Bicontinuous disordered phase (Bicontinuous microemulsion Sponge phase)... 

Lattice models for bulk mixtures have mostly been designed to describe features which are characteristic of systems with low amphiphile content. In particular, models for ternary oil/water/amphiphile systems are challenged to reproduce the reduction of the interfacial tension between water and oil in the presence of amphiphiles, and the existence of a structured disordered phase (a microemulsion) which coexists with an oil-rich and a water-rich phase. We recall that a structured phase is one in which correlation functions show oscillating behavior. Ordered lamellar phases have also been studied, but they are much more influenced by lattice artefacts here than in the case of the chain models. 

The example illustrates how Monte Carlo studies of lattice models can deal with questions which reach far beyond the sheer calculation of phase diagrams. The reason why our particular problem could be studied with such success Hes of course in the fact that it touches a rather fundamental aspect of the physics of amphiphilic systems—the interplay between structure and wetting behavior. In fact, the results should be universal and apply to all systems where structured, disordered phases coexist with non-struc-tured phases. It is this universal character of many issues in surfactant physics which makes these systems so attractive for theoretical physicists. 

FIG. 13 Phase diagram of a vector lattice model for a balanced ternary amphiphilic system in the temperature vs surfactant concentration plane. W -I- O denotes a region of coexistence between oil- and water-rich phases, D a disordered phase, Lj an ordered phase which consists of alternating oil, amphiphile, water, and again amphi-phile sheets, and L/r an incommensurate lamellar phase (not present in mean field calculations). The data points are based on simulations at various system sizes on an fee lattice. (From Matsen and Sullivan [182]. Copyright 1994 APS.)... 

Langevin simulations of time-dependent Ginzburg-Landau models have also been performed to study other dynamical aspects of amphiphilic systems [223,224]. An attractive alternative approach is that of the Lattice-Boltzmann models, which take proper account of the hydrodynamics of the system. They have been used recently to study quenches from a disordered phase in a lamellar phase [225,226]. 

Figure 5 Free energy surface at l l(Fig. 5a) [22, 24, 28] and 1 3 (Fig. 5b) [23, 24, 33] stoichiometries in the vicinity of disordered state ( f=0.0) at T—. 7Q and 1.6, respectively. The solid line in left-hand (right-hand) figure indicates the kinetic path evolving towards the L q LI2 ordered phase when the system is quenched from T—2.5 (3.0) down to 1.70 (1.60), while the broken lines are devolving towards disordered phase. The open arrows on the contour surface designate the direction of the decrease of free energy, and the arrows on the kinetic path indicate the direction of time evolution or devolution.

When the bulk transition is of first order, the above mentioned arguments based on dimensionality do not apply and the would be roughening transition temperature T j may be larger than the bulk transition temperature T, in which case there is simply no roughening transition. The situation is further complicated by the wetting phenomena. When we approach T from below, the disordered phase becomes metastable and may wet the interface a large layer of disordered phase develops in between the two ordered domains. 

We know that another interesting phenomenon occurs when the temperature increases up to the bulk transition Tj. Previous studies have shown that the APB is wetted by the disordered phase a large layer of disordered phase develops in between the two ordered domains. In other words, the APB is splitted into two order-disorder interfaces, whose separation diverges as In(T), - T). We display in Fig. 5 the 2-dlmensional maps for T=1687 K, i.e. very close to the first-order transition. As expected, we see that the APB separates into two order-disorder interfaces. Moreover, the width of the penetrating disordered layer varies along the APB. This means that each order-disorder interface develops its own transverse fluctuations and that the APB begins to behave as two separate objects. 

T. Hashimoto, T. Miyoshi and H. Ohtsuka, Investigation of the relaxation process in the Cu3Au-alloy order-disorder phase transition near the transition point, Phys. Rev. B 13.1119 (1976). 

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. 

The equilibrium state of the system corresponds to that set of values of the cf that minimizes A at a given temperature and pressure. From our definitions the disordered phase is characterized by cf = 0. In... 

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. 

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