Phase transitions diagrams

Equilibrium (bridging - 3-fold/atop) phase transition diagram for CO / Pd(111)... 

Figure 26.3 Sol-to-gel phase transition diagram gelator in C4mim/BF4 and (Q) gelator ...

The impact of crystallinity, however, goes beyond surface area and porosity properties. Phase transition temperatures and thermal stability of a product also change with the size of the primary crystals. This is demonstrated in figure 8, showing a typical phase transition diagram for a material of small and large crystallites, representing pseudoboehmite and boehmite alumina. 

To gain a balanced general view of the many ways in which man has crystallized diamond, the thermodynamically stable and metastable phases of elemental carbon and reaction dynamics between them, over obtainable pressures and temperatures, should be considered. The most up to date phase and transformation diagrams are to be found in a review by Bundy et al. [9]. Figure 7 is an adaptation of the pressure-temperature phase transition diagram taken from this review article, together with further information gleaned from the extensive diamond- and carbon-related scientific literature. 

Figure /. Equilibrium volume phase transition diagram for the gel The polymer volume fraction (f> is represented as a function of Flory s interaction parameterX-...

Sol-gel-sol phase transition diagrams of composite hydrogel were recorded using test tube-inverting method [18, 36]. The sol-gel-sol transition was visually observed by inverting the vials, and conditions of sol and gel were defined as flow liquid sol and no flow solid gel in one minute respectively. 

Z. Li, and G. Demopoulos, Model-based Construction of Calcium Sulfate Phase Transition Diagrams in the HCl-CaCl2-H20 System between 0 and 100°C, Industrial Engineering and Chemistry Research, 45 (2006), 4517-4524. 

To account for this, Rory suggested introducing the van Laar term of the energy of mixing in the form of in the cited free energy equation. This method allows satisfactory considmtion of the isotropic interactions between the components and construction of the complete phase transition diagram in composition-interaction parameter % coordinates. It has usually been assumed that parameter % is correlated with the reciprocal of the temperature by the equation x = a + P/T (where a and p are coefficients). Although this direct correlation is not valid in all cases, it nevertheless holds for most systems. Special cases of the complex dependence of % on T, for example, for systems... 

Douzals JP, Perrier-Comet JM, CoquiUe JC, Gervais P. 2001. Pressure-temperature phase transition diagram for wheat starch. J Agric Food Chem 49 873-876. 

Phase transitions in binary systems, nomially measured at constant pressure and composition, usually do not take place entirely at a single temperature, but rather extend over a finite but nonzero temperature range. Figure A2.5.3 shows a temperature-mole fraction T, x) phase diagram for one of the simplest of such examples, vaporization of an ideal liquid mixture to an ideal gas mixture, all at a fixed pressure, (e.g. 1 atm). Because there is an additional composition variable, the sample path shown in tlie figure is not only at constant pressure, but also at a constant total mole fraction, here chosen to be v = 1/2. 

Figure A2.5.30. Left-hand side Eight hypothetical phase diagrams (A through H) for ternary mixtures of d-and /-enantiomers with an optically inactive third component. Note the syimnetry about a line corresponding to a racemic mixture. Right-hand side Four T, x diagrams ((a) tlirough (d)) for pseudobinary mixtures of a racemic mixture of enantiomers with an optically inactive third component. Reproduced from [37] 1984 Phase Transitions and Critical Phenomena ed C Domb and J Lebowitz, vol 9, eh 2, Knobler C M and Scott R L Multicritical points in fluid mixtures. Experimental studies pp 213-14, (Copyright 1984) by pennission of the publisher Academic Press.

Flere we discuss the exploration of phase diagrams, and the location of phase transitions. See also [128. 129. 130. 131] and [22, chapters 8-14]. Very roughly we classify phase transitions into two types first-order and continuous. The fact that we are dealing with a finite-sized system must be borne in mind, in either case. 

We will focus on one experimental study here. Monovoukas and Cast studied polystyrene particles witli a = 61 nm in potassium chloride solutions [86]. They obtained a very good agreement between tlieir observations and tire predicted Yukawa phase diagram (see figure C2.6.9). In order to make tire comparison tliey rescaled the particle charges according to Alexander et al [43] (see also [82]). At high electrolyte concentrations, tire particle interactions tend to hard-sphere behaviour (see section C2.6.4) and tire phase transition shifts to volume fractions around 0.5 [88]. 

Fig. 3. PF diagram for a pure fluid (not to scale) point c is the gas—liquid critical state, is the constant pressure at which phase transition occurs at...

It was estabhshed ia 1945 that monolayers of saturated fatty acids have quite compHcated phase diagrams (13). However, the observation of the different phases has become possible only much more recendy owiag to improvements ia experimental optical techniques such as duorescence, polarized duorescence, and Brewster angle microscopies, and x-ray methods usiag synchrotron radiation, etc. Thus, it has become well accepted that Hpid monolayer stmctures are not merely soHd, Hquid expanded, Hquid condensed, etc, but that a faidy large number of phases and mesophases exist, as a variety of phase transitions between them (14,15). 

Fig. 5. The phase and transition diagram for carbon (simplified from [62]).

Phase transitions in two-dimensional layers often have very interesting and surprising features. The phase diagram of the multicomponent Widom-Rowhnson model with purely repulsive interactions contains a nontrivial phase where only one of the sublattices is preferentially occupied. Fluids and molecules adsorbed on substrate surfaces often have phase transitions at low temperatures where quantum effects have to be considered. Examples are molecular layers of H2, D2, N2 and CO molecules on graphite substrates. We review the path integral Monte Carlo (PIMC) approach to such phenomena, clarify certain experimentally observed anomalies in H2 and D2 layers, and give predictions for the order of the N2 herringbone transition. Dynamical quantum phenomena in fluids are analyzed via PIMC as well. Comparisons with the results of approximate analytical theories demonstrate the importance of the PIMC approach to phase transitions where quantum effects play a role. 

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... 

Phase transitions in adsorbed layers often take place at low temperatures where quantum effects are important. A method suitable for the study of phase transitions in such systems is PIMC (see Sec. IV D). Next we study the gas-liquid transition of a model fluid with internal quantum states. The model [193,293-300] is intended to mimic an adsorbate in the limit of strong binding and small corrugation. No attempt is made to model any real adsorbate realistically. Despite the crudeness of the model, it has been shown by various previous investigations [193,297-300] that it captures the essential features also observed in real adsorbates. For example, the quite complex phase diagram of the model is in qualitative agreement with that of real substances. The Hamiltonian is given by... 

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.)... 

FIG. 4 Qualitative phase diagram close to a first-order irreversible phase transition. The solid line shows the dependence of the coverage of A species ( a) on the partial pressure (Ta). Just at the critical point F2a one has a discontinuity in (dashed line) which indicates coexistence between a reactive state with no large A clusters and an A rich phase (hkely a large A cluster). The dotted fine shows a metastability loop where Fas and F s are the upper and lower spinodal points, respectively. Between F2A and Fas the reactive state is unstable and is displaced by the A rich phase. In contrast, between F s and F2A the reactive state displaces the A rich phase. 

Another interesting class of phase transitions is that of internal transitions within amphiphilic monolayers or bilayers. In particular, monolayers of amphiphiles at the air/water interface (Langmuir monolayers) have been intensively studied in the past as experimentally fairly accessible model systems [16,17]. A schematic phase diagram for long chain fatty acids, alcohols, or lipids is shown in Fig. 4. On increasing the area per molecule, one observes two distinct coexistence regions between fluid phases a transition from a highly diluted, gas -like phase into a more condensed liquid expanded phase, and a second transition into an even denser... 

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