Energy model of single component phase diagram

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van der Waals, Nobel prize in Physics, 1910.

<math>\left( p+\frac{a}{V^{2}} \right)\left( V-b \right)=RT</math>

The van der Waals equation shows the transition from an ideal gas at high temperature to a two-phase system at temperaures below the critial temperature. The van der Waals equation does not describe the liquid-solid transition so lacks a triple point.
/ Transitions/vdw.pdf
The van der Waals equation is usually plotted as pressure versus volume per mole, but for our purposes, a plot of pressure versus temperature will be more illuminating. The ideas are the same except that the liquid/solid phase transition is included with the resulting triple point.
In fact the phase diagram can be plotted in three dimensions. It is thought that Maxwell constructed models such as these and sent them to Gibbs (without much approval on Gibbs' part. '(Check out this story!!)' To extend the phase diagram to the condensed phases, liquids and solids, the phase diagram is often plotted as the temperature versus density.
We can compare the pressure versus temperature with the temperature versus density representation. This accepts the features at higher density, that is the liquid and solid phases.
OneComponentPhaseDiagram Two Views.png
Geometry and entropic derivations – useful because many “particles” have quite short-range interactions and so behave as hard spheres. The free energy is simply: because the work is always zero; no change in heat. <math>A=-TS</math>
The densest packing is face-centered cubic, a volume fraction of <math>\varphi =\frac{\pi }{\sqrt{18}}\simeq 0.74</math>. There is another limit – random close packing

<math>\varphi _{RCC}\simeq 0.63</math>
We can consider what the phase diagram might look like as the strength of interaction decreases. Therefore stating with:
Manoharan 2006
we find the progression:
Manoharan 2006

Phase diagrams for globular proteins have been showing up in the literature en masse over the last few years, mostly due to the troubles of protein crystallization. MIT has been a big contributor of articles explaining modeling via hard spheres (above diagram) and the experimental creation of phase diagrams. In a review by Neer Asherie (he worked with George Benedek and Aleksey Lomakin, whom I would consider the giants of protein phase diagrams), the general process and analysis of protein specific phase diagrams is shown. Note that this was accepted in 2004, I believe really showing the youth of the field.

"The problems associated with producing protein crystals have stimulated fundamental research on protein crystallization. An important tool in this work is the phase diagram. A complete phase diagram shows the state of a material as a function of all of the relevant variables of the system. For a protein solution, these variables are the concentration of the protein, the temperature and the characteristics of the solvent (e.g., pH, ionic strength and the concentration and identity of the buffer and any additives). The most common form of the phase diagram for proteins is two-dimensional and usually displays the concentration of protein as a function of one parameter, with all other parameters held constant. Three-dimensional diagrams (two dependent parameters) have also been reported and a few more complex ones have been determined as well."

Asherie, Neer. Protein crystallization and phase diagrams. Methods 34 (2004) 266–272.[1]

One of the interesting things about liquid liquid phase separations in protein solutions is that they mimic to an extent the phase separation of a gas into a liquid (like cloud formation) except for one key difference: the phase separated states are always metastable, and will not remain phase separated indefinitely. This is a good thing for crystallographers; the protein rich state of the solution will be likely to form a distinct phase: crystals.

This is a typical schematic of a liquid-liquid phase separated protein solution from the Asherie paper: LLPS diagram.gif

One model of phase transitions is the Ising Model. The Ising model consists of a lattice of spins that can be either up or down that are coupled to each other through the coupling energy J. In addition a magnetic field can be added to the system that couples to each spin individually. The Hamiltonian of the system is:

<math>H= - \frac{1}{2} \sum_{\langle i,j\rangle} J_{ij} S_i S_j - \sum_i h_i S_i \,</math>

In 2D the Ising model causes a phase transition in spins from an unordered state to an ordered state as temperature drops below T_c. T_c depends on the lattice configuration, for a square lattice <math>k_B T_c = \frac{2J}{\ln{(1+\sqrt{2})}}</math>.

A lattice gas can be mapped onto the Ising model, with a spin up corresponding to an atom being present and a spin down corresponding to no atom. The magnetic field term includes the chemical potential.

(Source: Pathria, Statistical Mechanics.

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