# Difference between revisions of "Phase separation"

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<center><math>\epsilon = zN[V_{AB} - {1 \over 2}(V_{BB}+V_{BB})]</math> </center> | <center><math>\epsilon = zN[V_{AB} - {1 \over 2}(V_{BB}+V_{BB})]</math> </center> | ||

− | <math>V</math> represents bond energies, <math>z</math> is the number of neighbors, <math>N</math> is the total number of atoms, and <math>x</math> is the mole fraction. The sign of <math>\epsilon</math> is essential in determining the solution's behavior as a function of temperature. If <math>\epsilon</math> < 0 then the change in free energy upon mixing is always negative, so the atoms will always want to be fully mixed. For <math>\epsilon</math> > 0, there is a temperature dependance, and either mixing or phase separation can occur. Figure 1 shows a plot of <math>G_{mix}</math>/<math>RT</math> versus composition for various values of <math>\epsilon</math>/<math>RT</math>. For <math>\epsilon</math>/<math>RT</math> values < 2, we can see that the curve only has a single minimum. This means that, for any concentration, the system is at a lower free energy when fully mixed than it would be if it separated into two different compositions. Thus full mixing will occur and we will only have a single phase. However, for <math>\epsilon</math>/<math>RT</math> values | + | <math>V</math> represents bond energies, <math>z</math> is the number of neighbors, <math>N</math> is the total number of atoms, and <math>x</math> is the mole fraction. The sign of <math>\epsilon</math> is essential in determining the solution's behavior as a function of temperature. If <math>\epsilon</math> < 0 then the change in free energy upon mixing is always negative, so the atoms will always want to be fully mixed. For <math>\epsilon</math> > 0, there is a temperature dependance, and either mixing or phase separation can occur. Figure 1 shows a plot of <math>G_{mix}</math>/<math>RT</math> versus composition for various values of <math>\epsilon</math>/<math>RT</math>. For <math>\epsilon</math>/<math>RT</math> values < 2, we can see that the curve only has a single minimum. This means that, for any concentration, the system is at a lower free energy when fully mixed than it would be if it separated into two different compositions. Thus full mixing will occur and we will only have a single phase. However, for <math>\epsilon</math>/<math>RT</math> values ≥ 2, something quite different occurs because the curve has two minima. The system can lower its free energy by separating into compositions at the two minima rather than saying as a homogenous mixture. Thus any composition in this region will spontaneously separate into two phases --- <b>phase separation</b>! |

− | + | the curve has two minima. Consider graph (b), It shows that combinations of the two low energy concentrations, | |

+ | |||

+ | \varphi _{1}\text{ and }\varphi _{2} will have a lower free energy than the homogeneous mixture, \varphi _{0}. Any composition in this region will spontaneously separate into two phases, a phase separation. | ||

[[Image:Phasesep_fig1-2.png |thumb| 600px | center| Figure 1 Free energy versus composition for various values of <math>\epsilon</math>/<math>RT</math> (made by [[Emily Redston]])]] | [[Image:Phasesep_fig1-2.png |thumb| 600px | center| Figure 1 Free energy versus composition for various values of <math>\epsilon</math>/<math>RT</math> (made by [[Emily Redston]])]] | ||

− | + | We can define this more explicitly by saying that phase separation occurs when | |

<center><math>{G_{mix} \over x_A} = 0</math></center> | <center><math>{G_{mix} \over x_A} = 0</math></center> | ||

By taking this derivative and solving for <math>T_{P}</math> (below which phase separation occurs), we will define a [[phase boundary]] on our temperature versus composition [[phase diagram]] (Figure 2). | By taking this derivative and solving for <math>T_{P}</math> (below which phase separation occurs), we will define a [[phase boundary]] on our temperature versus composition [[phase diagram]] (Figure 2). |

## Revision as of 22:35, 9 December 2011

Entry by Emily Redston, AP 225, Fall 2011

We typically talk about phase separation in terms of the regular solution model of liquids. The free energy of mixing can be written as,

<math>V</math> represents bond energies, <math>z</math> is the number of neighbors, <math>N</math> is the total number of atoms, and <math>x</math> is the mole fraction. The sign of <math>\epsilon</math> is essential in determining the solution's behavior as a function of temperature. If <math>\epsilon</math> < 0 then the change in free energy upon mixing is always negative, so the atoms will always want to be fully mixed. For <math>\epsilon</math> > 0, there is a temperature dependance, and either mixing or phase separation can occur. Figure 1 shows a plot of <math>G_{mix}</math>/<math>RT</math> versus composition for various values of <math>\epsilon</math>/<math>RT</math>. For <math>\epsilon</math>/<math>RT</math> values < 2, we can see that the curve only has a single minimum. This means that, for any concentration, the system is at a lower free energy when fully mixed than it would be if it separated into two different compositions. Thus full mixing will occur and we will only have a single phase. However, for <math>\epsilon</math>/<math>RT</math> values ≥ 2, something quite different occurs because the curve has two minima. The system can lower its free energy by separating into compositions at the two minima rather than saying as a homogenous mixture. Thus any composition in this region will spontaneously separate into two phases --- **phase separation**!

the curve has two minima. Consider graph (b), It shows that combinations of the two low energy concentrations,

\varphi _{1}\text{ and }\varphi _{2} will have a lower free energy than the homogeneous mixture, \varphi _{0}. Any composition in this region will spontaneously separate into two phases, a phase separation.

We can define this more explicitly by saying that phase separation occurs when

By taking this derivative and solving for <math>T_{P}</math> (below which phase separation occurs), we will define a phase boundary on our temperature versus composition phase diagram (Figure 2).

We can also define a critical temperature <math>T_c</math>, which is the maximum temperature below which phase separation will occur (the maximum of the <math>T_P</math> curve). This will occur at <math>x_A = 0.5</math>.

From here, we can go on to consider the spinodal lines.

See also:

Phase separation in Phases and Phase Diagrams from Lectures for AP225.

## References

The Role of Polymer Polydispersity in Phase Separation and Gelation in Colloid−Polymer Mixtures