Two-dimensional phase diagram

Sometimes the third variable will be shown as contours on the two-dimensional phase diagrams. Thus, a series of isobars may be shown on a (E v) phase diagram. 

Fig. 6.45 Two-dimensional phase diagram as a function of chain composition for a blend of two diblocks (

The analogy between three- and two-dimensional phase diagrams can be carried much further. Monomolecular amphiphilic films show ordered phases similar to three-dimensional systems [579], The phases of an amphiphilic monolayer can be detected most conveniently in pressure-area (7r-versus-OA) isotherms. These may look different for different substances. The behavior of simple amphiphilic molecules, like long-chain alcohols, amines, or acids, was extensively investigated (reviews Refs. [580,581]). In monolayers so-called mesophases can occur. In a mesophase the tail groups are ordered over relatively large areas, while the order in the hydrophilic head groups is only over a much smaller distances. 

Two-dimensional phase diagrams are often displayed in the form of In [p] against 1/7 (at a constant specific amount adsorbed), which provides a convenient way of indicating the conditions for the coexistence of two phases (see Figure 4.3). Indeed, the application of the Phase Rule indicates that when two adsorbed phases coexist in equilibrium, the system has one degree of freedom therefore, at constant... 

Two-dimensional phase diagrams, i.e. r A) curves at various temperatures, are one of the basic means for studying the rich behaviour, whether the layers are gas-like, llquid-llke, solid-like, or mixed, including the structural transitions that monolayers can exhibit. Let us for the moment assume that such ti[A] isotherms have been properly measured, that hysteresis is under control, etc. By way of introduction to the types of phases, emd their succession that may be observed, consider first the schematic picture of fig. 3.6. This way of plotting is phenomenological A is the total area. However, in describing specific examples, it is more useful to use the average area per molecule (oj, in nm ), obtained as A/N°. Here, N must be inferred from the amount of surfactant spread. If so desired, the surface concentration, = n /A (in mole m" ) can also be reported. When there is only one component, the subscript i may be dropped. 

For systems with more than four independent components, two-dimensional phase diagrams can be made by fixing the amount of each additional component. An isotherm for such a system will then consist of a series of phase diagrams with varying amounts of these components. 

Two-dimensional phase diagrams and order-disorder transitions... 

The two-dimensional phase diagram of the monolayer of Xe adsorbed on graphite is displayed in Figs. 1 [6,8,14] and 2 [11,15,16] in two usital representations, coverage 0 vs. pressure and pressure vs. T. Fig. 2 extends the data of Fig. 1 to lower pressures and temperatures and allows the presentation of the second layer formation and of the bulk sublimation curve, as well. 

The two-dimensional phase diagram of system as determined using RHEED-TRAXS... 

Several studies of the two-dimensional phase diagram for Lennard-Jones atoms have been reported [26,75]. In particular, the 2D liquid-vapor coexistence curve has been determined together with the critical temperature and density (/cT cr/e , - 1.316, = 0.304) (Fig. 10). These studies are relevant to mono-... 

Phase Diagram. A graphical representation of the equilibrium relationships between phases in a system. For multicomponent systems, and considering varying temperatures, more than a simple two-dimensional phase diagram will be required. 

For a fixed value of a, the two parameters v and Ado determine the shape of vesicles which can be placed in a two-dimensional phase diagram as shown in Figure 14a (143). There are numerous shapes including spherical, tubular, oblate, and starfish vesicles. Vesicles with outwardly curved shapes and prolate vesicles are in the upper part of the diagram for Ado > 1, and inwardly curved shapes and oblate vesicles are in the lower part for Ado > 1- In this phase diagram, all the shapes are drawn to scale, have the same area, and differ only in their values of V and Ado. Lipid and block copolymers exhibit a similar multitude of shapes as shown for PB-PEO block copolymer vesicles in Figure 14b (144). 

Figure 8.3 Two-dimensional phase diagram of a Langmuir film. Reproduced from K. Kolasinski, Surface Science Foundations of Catalysis and Nanoscience, John Wiley Sons, Ltd. Chichester, 2002. Copyright 2002 by John Wiley Sons, Limited. Reproduced with permission.

Two-dimensional phase diagrams for a single-substance system can be generated as projections of a three-dimensional surface in a coordinate system with Cartesian axes p, V/n, and T. A point on the three-dimensional surface corresponds to a physically-realizable combination of values, for an equilibrium state of the system containing a total amount n of the substance, of the variables p, V/n, and T. 

We can derive a more general form of the lever rule that will be needed in Chap. 13 for phase diagrams of multicomponent systems. This general form can be applied to any two-phase area of a two-dimensional phase diagram in which a tie-line construction is valid, with the position of the system point along the tie line given by the variable... 

We can independently vary the temperature, pressure, and composition of the system as a whole. Instead of using these variables as the coordinates of a three-dimensional phase diagram, we usually draw a two-dimensional phase diagram that is either a temperature-composition diagram at a fixed pressure or a pressure-composition diagram at a fixed temperature. The position of the system point on one of these diagrams then corresponds to a definite temperature, pressure, and overall composition. The composition variable usually... 

The way in which we interpret a two-dimensional phase diagram to obtain the compositions of individual phases depends on the number of phases present in the system. 

A binary system with three phases has only one degree of freedom and eannot be represented by an area on a two-dimensional phase diagram. Instead, there is a horizontal boundary line between areas, with a speeial point along the line at the junction of several areas. The compositions of the three phases are given by the positions of this point and the points at the two ends of the line. The position of the system point on this line does not uniquely specify the relative amounts in the three phases. 

A ternary system is one with three components. We can independently vary the temperature, the pressure, and two independent composition variables for the system as a whole. A two-dimensional phase diagram for a ternary system is usually drawn for conditions of constant T and p. 

Some selected basic properties of water are given in Table 15.1. The unique characteristics of water, although not obvious, will become apparent as we learn more about it. The three-dimensional phase diagram for water, up to the pressure of 10,000 atm and for the temperature range of —50°C to - - 50°C, is shown in Fig. 15.1. The density of ice (ice I) is less than that of water up to about 2,200 atm. Above this pressure, ice exists in various different crystalline modifications. The two-dimensional phase diagram for water is shown in Fig. 15.2, where the triple point, 0.0100°C, consists of solid ice, water, and water vapor at 4.579 torr in equilibrium. Also shown is the critical point above which liquid water cannot exist in the liquid state. The negative slope of the P-T line is due to the difference in molar volumes of liquid and solid, that is, (V, — VJ < 0, and because... 

Minimizing the total energy, F = F,. + now leads to pears, budding and a multitude of starfish vesicles, some of which are shown in Figure 7.3 [7], These shapes depend not just on the reduced volume v but on one more parameter which basically is the number difference of lipid molecules in the two layers. These shapes of lowest energy can be arranged in a two-dimensional phase diagram [15,18]. As... 

A complete two-component phase diagram requires three dimensions to plot T, P, and the mole fraction of one component, as depicted in Figure 6.4. Two-dimensional phase diagrams are produced by passing planes of constant T or constant P through the diagram. 

Figire 9 Two-dimensional phase diagram for a diblock copolymer swollen with a solvent displaying selectivity for one of the blocte. The diagram displays not only a shift in the mean-field critical point but also possesses an asymmetric shape. Reprinted with permission from Huang, C.-l. Lodge, T. P. Macro mlecules 99Z. 31,3556-3565. Copyright 1998, American Chemical Society. 

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