Difference between revisions of "Phase separation"

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Entry by [[Emily Redston]], AP 225, Fall 2011
 
Entry by [[Emily Redston]], AP 225, Fall 2011
  
[[Image:Phasesep_fig1.png |thumb| 400px | left| Figure 1 (from Haasen)]] [[Image:Phasesep_fig2.png |thumb| 400px | right| Figure 2 (from Haasen)]]
+
[[Image:Phasesep_fig1.png |thumb| 400px | right| Figure 1 (from Haasen)]] [[Image:Phasesep_fig2.png |thumb| 400px | right| Figure 2 (from Haasen)]]
 
We typically talk about phase separation in terms of the [[regular solution]] model of liquids. The free energy of mixing can be written as,
 
We typically talk about phase separation in terms of the [[regular solution]] model of liquids. The free energy of mixing can be written as,
  
          <font size="+2"><math>G_{mix}= RT (x _{A}\ln x _{A}+x _{B}\ln x _{B})+\epsilon x _{A} x _{B}</math></font>
+
<center><font size="+2"><math>G_{mix}= RT (x _{A}\ln x _{A}+x _{B}\ln x _{B})+\epsilon x _{A} x _{B}</math></font></center>
  
            where <math>\epsilon = zN[V_{AB} - {1 \over 2}(V_{BB}+V_{BB})]</math>  
+
<center>where <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. Phase separation occurs when  
 
<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. Phase separation occurs when  
                          <math>{G_{mix} \over x_B} = 0</math>
+
<center><math>{G_{mix} \over x_B} = 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).
 +
<center> <math>T_{P} = {-\epsilon (2 x_B -1) \over R[\ln (1-x_B) - \ln x_B]} </math></center>
 +
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).
 +
<center><math>T_C = \epsilon \over 2R</math></center>
 +
         
  
  

Revision as of 21:44, 9 December 2011

Entry by Emily Redston, AP 225, Fall 2011

Figure 1 (from Haasen)
Figure 2 (from Haasen)

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>G_{mix}= RT (x _{A}\ln x _{A}+x _{B}\ln x _{B})+\epsilon x _{A} x _{B}</math>
where <math>\epsilon = zN[V_{AB} - {1 \over 2}(V_{BB}+V_{BB})]</math>

<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. Phase separation occurs when

<math>{G_{mix} \over x_B} = 0</math>

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).

<math>T_{P} = {-\epsilon (2 x_B -1) \over R[\ln (1-x_B) - \ln x_B]} </math>

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).

<math>T_C = \epsilon \over 2R</math>





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