Phase Separation of a Binary Mixture
Brian Espino
M.S. Defense
Oct. 18, 2006
Examples of binary mixtures
Mixtures may be solid or liquid
Brass (85% copper-15% zinc) in solid form is an
example of a solid binary mixture. When raised to above
the respective melting temperatures, a liquid binary
mixture is produced. It is in this phase that the
components can be mixed.
Example of a liquid mixture at room temperature.
Gasohol (10% ethanol-90% gasoline)
What is Phase Separation?
The process of phase separation is the unmixing of a
thermodynamically unstable solution.
Essentially the mixture self segregates based on
thermodynamic principles.
States of a mixture
• Homogeneous state
In the homogeneous state, the respective concentration
of the components is the same at all locations.
• Heterogeneous state
In the heterogeneous state, there are local domains
richer in one of the components
Phase Diagrams
A phase diagram is a plot with variables:
temperature (T) and concentration (F)
Composition, F, is defined as F(r) = Fa(r) - Fb(r).
Fi(r) is the relative concentration of each component at
location r
State of mixture is described by its location on the phase
diagram and defined by (F, T)
States of a mixture related to the phase diagram
• Which state the system is in is determined by its
location on the phase diagram.
• If the state of the mixture is above the
coexistence curve, it is homogeneous.
• If the state of the mixture is below the
coexistence cure, the system is globally unstable
in the homogeneous state.
Phase Diagram
Coexistence Curve
Spinodal Curve
Tc
T
-1
fc
+1
Features of phase diagram
Coexistence curve separate regions of homogeneous and
heterogeneous states. Along this curve the mixed and
unmixed states are in equilibrium.
When quenched below the coexistence curve, the system
is locally stable in the homogeneous state. Between the
coexistence and spinodal curves, the homogeneous state
is said to be metastable.
Beneath the spinodal curve, the system is unstable in the
homogeneous state.
Two methods of phase separation
• Nucleation
• Spinodal Decomposition
Methods of phase separation
•Nucleation
–
–
–
–
–
–
Metastable process
Must overcome free energy barrier
Occurs between coexistence and spinodal curves
Involves large fluctuations in composition
Local process
Produces round domains (drops for 3-D, circular
domains for 2-D systems)
•Spinodal Decomposition
Free energy barrier for nucleation
DF
Stable
Metastable
I
II
III
DF
FI
FIII
composition
Phase Diagram
Coexistence Curve
Spinodal Curve
Tc
T
-1
Nucleation
Nucleation
Metastable
Metastable
fc
+1
Nucleation of a droplet
DF = free energy corresponding to presence of a droplet
For a 3-D droplet DF = 4sR2-4eR3/3
s = surface energy per area compared to surrounding medium
e = bulk energy per volume
Critical radius, Rc = 2s/e
In 2-D droplets are replaced by circles
DF = 2sR-eR2
s = surface energy per length
e = bulk energy per area
Critical radius, Rc = s/e
Nucleation of a droplet
Only droplets/circular regions with DF < 0 will grow.
Since the critical radius maximizes DF, only regions with radius
R > Rc will grow. Those with R < Rc will shrink away.
DF
radius
R<Rc
Rc
R>Rc
Methods of phase separation
• Nucleation
• Spinodal decomposition
–
–
–
–
–
Unstable process
Occurs when state is below spinodal curve
Involves small fluctuations in composition
Long range process
Produces elongated domains
• Key in producing a percolation
Unstable free energy
DF
Unstable
I
II
DF
composition
III
Phase Diagram
Coexistence Curve
Spinodal Curve
Tc
T
Spinodal Decomposition
Unstable
-1
fc
+1
Goals
• To observe that the elongated domains grow as
a function of time: size(t) ~ta
• Compute a, by calculating the Fourier transform
(structure factor) as a function of time, simulating
an optical study of the system.
• Compute a, by using the correlation function to
calculate time dependence of the correlation
length.
• Calculate the critical exponent and compare to
known value of a = 1/3
Mathematical Background
Mixture of elements A and B.
System is defined by order parameter F(r, t)
F(r, t) = Fa(r, t) - Fb(r, t)
Fa(r, t) = concentration of element A as a function of (r, t)
Fb(r, t) = concentration of element B as a function of (r, t)
F(r, t) runs from (-1, 1)
When F(r, t) is positive, A-rich region
When F(r, t) is negative, B-rich region
Mathematical Background
• The kinetics of the systems is described by the
continuity equation:
F(r , t )
t
 j  0
j   Mm
• The diffusive current is:
• M represents the mobility of the particles
• Chemical potential, m, is defined as the functional
derivative of the Free Energy with respect to particle
number:
m
F
F(r , t )
Mathematical Background
• Free Energy is described by the functional:
F   f (r , t )d 3 r
• f(r, t) is the Ginzburg-Landau function:
 F(r , t ) 2 F(r , t ) 4 K
f (r , t ) 

 (F(r , t )) 2
2
4
2
• The free energy becomes:
F   f F(r , t ), F(r , t ) )d r
3
Mathematical Background
• Using the functional derivative obtain:
 f
 3
f
F   
F 
F d r
F(r , t )
 F(r , t )

• Minimizing the free energy and integrating by parts yields:
F
F
F


F(r , t ) F(r , t )
(F(r , t ))
• Which is defined as the chemical potential:
F
F
m

F(r , t )
 (F (r , t ))
Mathematical Background
• Combining the chemical potential with the definition of
the diffusive current we obtain:
 F

F

j  M

(F(r , t )) 
 F(r , t )
• When replaced in the continuity equation with use of the
following partial derivatives:
f
 F ( r , t )  F ( r , t ) 3
F(r , t )
f
 KF(r , t )
F(r , t )
• Yields the Cahn-Hilliard Equation:
F(r , t )
 M 2  F(r , t )  F (r , t ) 3  K 2 F (r , t )  0
t


Cahn-Hilliard Equation
The Cahn-Hilliard equation, when integrated with
respect to time, determines the time evolution of
the composition of the system.
F(r , t )
 M 2  F(r , t )  F (r , t ) 3  K 2 F (r , t )  0
t


Euler’s method will be used to solve this equation
numerically giving us the composition as a function
of position and time. F(r, t)
Length Scaling
Of key interest is the size or length of the elongated
domains as a function of time. The length is defined as:
L(t ) ~
 A(t ) 
 P(t ) 
<A(t)> = average area of domains as a function of time
A(t) is defined as the number of points making up the domain.
<P(t)> = average perimeter of domains as a function of time
P(t) is defined as the number of points on a domain that border
a region made up of the opposite particle type.
For a circular domain: A = R2 and P = 2R, where R is the
radius of the domain.
The length parameter is then: L =R/2 or L(t) ~R(t)
Scaling law for droplets, non-rigorous approach
Rewrite the continuity equation in terms of chemical
potential:
F (r , t )
2
 M m (r , t )
t
Using the Gibbs-Thompson relation for liquid droplets, the
chemical potential is defined as:
m
s
DF
with

d  1)

R
s the surface tension, d is the dimensions of the system, and DF
is the change in order parameter at the droplet’s surface)
Scaling law for droplets, non-rigorous approach
The continuity equation becomes:
F(r , t )
2
2 s ( d  1)
 M m (r , t )  M
t
DFR
Which yields:
F(r , t )
1
 3
t
R
After integrating the equation, dimensional analysis gives
us a scaling law: R ~ tn , with n=1/3. Note that the critical
exponent in the growth law is independent of the system’s
dimensions.
Modeling a binary mixture undergoing spinodal
decomposition (2-D study)
2-D square lattice is created with each lattice site receiving
a random real number in the range (-1, 1).
This is the initial value of F at each site, F(r, 0).
Random number generator is weighted so that the overall
particle concentrations are A = 52.5% and B = 47.5%.
Goal is to find F(r, t)
Finding F(r, t)
F(r, t) is found in the following manner:
At each time step the Cahn-Hilliard equation is determined
using the process described in the previous section.
After being solved numerically, the Cahn-Hilliard equation
produces a new value of F(r, t).
These steps are repeated a set number of times for each
time increment.
Qualitative description of particle dynamics
Particle motion at each lattice site is determined by the
interaction energy from 4 nearest neighbors.
If adjacent sites are rich with the same particle type, energy
is attractive.
If adjacent sites do not share predominant particle type,
energy is positive and repulsive.
Constraint on the system
Continuity equation acts a restraint.
F(r , t )
 j  0
t
Each particle type has a GLOBALLY conserved quantity.
This conservation law is not enforced at each lattice site.
The total energy of the system must be lowered for particle
motion to be allowed.
Energy may increase locally between neighbors if the total
energy of the system is reduced.
Goals of the system
(what the particles want to do)
To reduce the total energy, the sum of the nearest
neighbor energies, to a minimum value.
This would occur if all the A-rich domains are in
one continuous region, while all the B-rich domains
are in a second continuous region.
Examples include
A-rich B-rich
Methods of observing the system
• Visual observation
Images of the system
• Structure Factor
Similar to conducting a light scattering
experiment.
• Correlation length
Using the Correlation function
Observing the system
The program records a snapshot of the system after every
time step. Each of these snapshots is a surface plot
mapping of F(r) = Fa(r) - Fb(r).
The time of each snapshot is determined by:
Time = (snapshot #)(# of updates per scan)(delta time)
The example below is the 10th snapshot, with 5000 updates
per scan, and dt = 0.01 units.
Time = 10*5000*0.01=500 units.
Snapshot of system at time = 500 units
Structure Factor
The program calculates the 2-D Fourier Transform
of each surface plot of F(r) = Fa(r) - Fb(r).
F (k x , k y )   f ( x, y)e
x
2i ( k x x  k y y )
y
With the radial wave number being:
k  k x2  k y2
Structure Factor
K , is obtained from its respective wave number
using the relation K = 2k/D, where D is the size of
the system. In our system, D is equal to 256.
Kmax corresponds to the inverse of the most
common length scale of the contiguous regions
Of key interest is the time dependence of the
maximum wave number/vector.
kmax(t) ~ t-a or
Kmax(t) ~ t -a
Example of Structure factor
10
Structure factor with order parameter =0.05 at time =20 units
Intensity
8
6
4
2
0
0
10
20
30
Wave num ber k
40
50
Correlation length
The correlation length is defined as the spatial range over which
fluctuations in one region are correlated to fluctuations in another
region.
As domains increase in size, the number of adjacent lattice sites
correlating with each other also will increase.
The scaling law for the correlation length is: L(t) ~ ta
The correlation length is also defined as the first zero of the
correlation function :
G (r , R ) 
 F ( R )F ( R  r )
pairs
Example of Correlation function
Correlation Function at time = 20 units
with initial order parameter = 0.05
0.25
0.2
0.15
0.1
0.05
0
-0.05 0
5
10
15
-0.1
-0.15
-0.2
length (lattice sites)
20
25
Results
Plots were made of the structure factor for time ranging up to 900 units
Structure factor for times 20-120 units
Structure factor for times 140-240 units
40
70
40
30
60
180
100
20
160
50
80
25
140
60
120
15
Intensity
Intensity
20
35
200
220
40
240
30
20
10
10
5
0
0
0
10
20
Wave number k
30
0
40
20
30
Wave number k
Structure factor for times 260-450 units
100
500
280
140
550
300
120
600
70
350
60
400
50
450
40
30
Intensity
80
Structure factor for times 500-900 units
160
260
90
Intensity
10
700
100
800
80
900
60
40
20
20
10
0
0
0
10
20
Wave number k
30
0
10
20
Wave number k
30
Time dependence of maximum wave number
Maximum wave number vs. time (log-log)
Maximum wave number vs. time
35
100
30
-0.3217
y = 82.006x
20
Kmax
Kmax
25
15
10
10
5
0
1
0
200
400
600
time
800
1000
10
100
1000
time
The power law found from the measurement resulted in a
critical exponent of a = 0.3217. This compares favorably to
the accepted value of a = 1/3.
The correlation function is shown for a range
of times extending to 900 units
Correlation function for times 180-350 units
Correlation function for times 20-160 units
0.7
0.8
20
0.6
80
0.4
Correlation
Correlation
0.5
100
0.3
120
0.2
140
160
0.1
0.6
200
220
0.5
240
0.4
260
280
0.3
300
0.2
350
0.1
0
-0.1
180
0.7
40
60
0
0
5
10
15
20
25
-0.1 0
5
10
Correlation function for times 400-900 units
0.3
0.7
400
0.6
450
500
0.2
0.5
550
0.15
0.4
600
0.3
700
800
0.2
900
0.1
0
5
10
15
20
Correlation function for times 20 -160 units zoomed
20
40
60
80
100
120
140
160
0.25
Correlation
Correlation
25
length
length
-0.1 0
20
-0.2
-0.2
0.8
15
25
0.1
0.05
0
-0.05 4
4.5
5
5.5
6
-0.1
-0.15
-0.2
-0.2
length
length
6.5
7
7.5
8
Time dependence of correlation length
Correlation length vs. time (log-log)
Correlation length vs. time
100
12
Correlation length
Correlation length
10
y = 1.8681x0.2695
8
6
4
2
0
10
1
0
200
400
600
time
800
1000
1
10
100
time
The power law found from the measurement resulted in a
critical exponent of a = 0.2695. This is lower than the
critical exponent for the structure factor plots by ~16%.
1000
• One point of interest is that by observing the
snapshots up to a time value of 900 units, the
domains are still predominantly elongated.
• The simulation was extended to 4950 units to
see if the large circular domains became
present.
• Check to see if the scaling law for the
correlation length versus time changed over a
longer time scale.
Snapshots for time = 900 and 4950 units.
The correlation function is shown for a range
of times extending to 4950 units
Correlation function for times 2250-3500 units
Correlation function for times 100-2000 units
0.9
0.9
500
0.7
2250
2500
0.5
2750
0.5
1250
0.3
1500
1750
0.1
2000
-0.1 0
-0.3
5
10
15
20
correlation
1000
3000
0.3
3250
0.1
3500
-0.1 0
25
5
10
15
-0.3
-0.5
-0.5
length
length
Correlation function for times 3750-4950 units
1
0.8
3750
4000
4250
4500
4750
4950
0.6
correlation
correlation
0.7
100
0.4
0.2
0
-0.2
0
5
10
15
-0.4
-0.6
length
20
25
20
25
Time dependence of correlation length
Correlation length vs. time (times up to 4950 units log-log)
Correlation length vs. time (times up to 4950 units)
100
20
15
Correlation length
Correlation length
25
y = 1.5676x0.3091
10
5
0
10
1
0
1000
2000
3000
time
4000
5000
1
10
100
1000
10000
time
With the longer range of times, the critical exponent a, was
found to be 0.3091. This was in better agreement with the
value for the critical exponent found from the structure
factor data.
Conclusions
• Data from structure factor plots produced critical
exponent of a = 0.3217, which was the closest calculate
value of a to the accepted value of a = 1/3
• Initially the correlation length data yielded a = 0.2695
• When the time was extended by a factor of 5 the
correlation length yielded a = 0.3091.
• Elongated domains remained present
Acknowledgements
• My Committee
– Larry Weaver
– Amit Chakrabarti
– Chris Sorensen