Single component phase diagrams
2D phase diagram
The picture shown above is a typical phase diagram. The dotted line gives the anomalous behaviour of water. The green lines mark the freezing point and the blue line the boiling point, showing how they vary with pressure.The markings on the phase diagram show the points where the free energy is non-analytic. The open spaces, where the free energy is analytic, correspond to the phases. The phases are separated by lines of non-analyticity, where phase transitions occur, which are called phase boundaries.
In the phase diagram, the phase boundary between liquid and gas does not continue indefinitely. Instead, it terminates at a point on the phase diagram called the critical point. This reflects the fact that, at extremely high temperatures and pressures, the liquid and gaseous phases become indistinguishable, in what is known as a supercritical fluid. In water, the critical point occurs at around Tc=647.096 K (1,164.773 °R), pc=22.064 MPa (3,200.1 psi) and ρc=356 kg/m³.
The existence of the liquid-gas critical point reveals a slight ambiguity in the above definitions. When going from the liquid to the gaseous phase, one usually crosses the phase boundary, but it is possible to choose a path that never crosses the boundary by going to the right of the critical point. Thus, the liquid and gaseous phases can blend continuously into each other. However, the solid-liquid phase boundary can only end in a critical point this way if the solid and liquid phases have the same symmetry group.
Noteworthy is that the solid-liquid phase boundary in the phase diagram of most substances has a positive slope. This is due to the solid phase having a higher density than the liquid, so that increasing the pressure increases the melting point; the temperature at which a substance melts. In some parts of the phase diagram for water the solid-liquid phase boundary has a negative slope (especially the portion corresponding to standard pressure). This reflects the fact that ice has a lower density than water, which is an unusual property for a material.
In addition to just temperature or pressure, other thermodynamic properties may be graphed in phase diagrams. Examples of such thermodynamic properties include specific volume, specific enthalpy, or specific entropy. For example, single-component graphs of Temperature vs. specific entropy (T vs. s) for water/steam or for a refrigerant are commonly used to illustrate thermodynamic cycles such as a Carnot cycle, Rankine cycle, or vapor-compression refrigeration cycle.
In a two-dimensional graph, two of the thermodynamic quantities may be shown on the horizontal and vertical axes. Additional thermodymic quantities may each be illustrated in increments as a series of lines - curved, straight, or a combination of curved and straight. Each of these iso-lines represents the thermodynamic quantity at a certain constant value.
Rankine Cycle Example
The Rankine cycle is the ideal model used to describe the operation of steam heat engines. It was named after William John Macquorn Rankine, a Scottish engineer and physicist shown below:
He developed the complete theory of the steam engine and placed a huge emphasis on practical applications of his work. The Rankine cycle deals with heat being added at a constant pressure leading to isentropic expansion and heat extraction at a constant pressure leading to isentropic compression. It is used to determine the efficiencies of systems like heat-engines and heat-pumps. One of the significant advantages of this cycle is that the working fluid is found in the liquid phase (as you'll see below) during the compression stage, resulting in very little work required to actually drive the pump.
The Rankine cycle consists of four processes, each of which deal with a phase change. Below is a Ts diagram of the typical Rankine cycle:
In an ideal situation, the work output would be maximized through reduce entropy generation. You would see the processes 1-2 and 3-4 represented by the vertical lines on the graph. Process 1-2 refers to when the fluid is a liquid and is being pumped from low to high pressure. 2-3 demonstrates the period that the liquid is in the boiler and the phase change to vapor. 3-4 shows that vapor going through the turbine to decrease the pressure and temperature of the vapor and finally 4-1 shows the phase transition back to liquid as it goes through the condenser.
The actual vapor cycle differs from the ideal shown here due mainly to irrevesibilities (like friction and heat loss to the surroundings) and resulting pressure drops. As a result, you will never see the efficiency of the Carnot cycle achieved through the Rankine cycle no matter how many regeneration cycles are utilized. The inefficiencies are shown in the following graph:
In practice, the Rankine cycle follows a closed loop and is re-used constantly. One of the biggest examples you'll see of this is in power generation plants where power is generated by alternately vaporizing and condensing the working fluid. Water is most commonly used as the working fluid but other refrigerants like ammonia can be seen.
Dislocation-mediated Melting in 2 Dimensions
In a 1979 paper, Nelson and Halperin developed a mathematical theory for a crystalline solid melting on different substrates in two dimensions. Depending on the smoothness of the structure, two different kinds of transitions may occur. For a smooth substrate, an intermediate liquid crystal phase will occur between the solid and liquid phases. For a course periodic substrate, this phase will not occur. The figure below is a speculative phase diagram they put forth. More information: "Dislocation-mediated melting in two dimensions", Phase transitions in liquid crystals