Chapter 1
1.1 Introduction to phase separation
The study of phase separation in alloys and liquid mixtures has been of prime
interest over the last couple of decades as the processes involved are direct consequences
of non-equilibrium thermodynamics. Examples include but are not limited to binary and
ternary metal alloys, liquid mixtures, and systems of magnetic spins. Metal alloys are
produced when the constituent metals are heated to their liquid phases. The liquids are
then free to form a molten mixture. When the temperature is then lowered below all of
the respective melting points, a solid mixture of metals is produced. One familiar
example is an alloy composed of copper and zinc, which combine to form brass. Binary
liquids are produced in the same manner, except the change to the solid phase is not
necessary. Common binary liquids involve components such as methanol and ethanol.
One example of recent interest is gasohol which is a mixture of gasoline (90%) and
ethanol (10%). While systems can consist of four or more different species, these higher
order systems are difficult to study. Phase diagrams can be constructed for higher order
systems, yet mixtures containing two or three elements are intrinsically easier to study.
In our study we will look at mixtures involving two components. Bulk metals and fluids
are naturally three dimensional objects. However, we will limit our work to two
dimensional cases for the sake of our computer simulations. This method will then be
suitable for modeling thin films.
Before phase separation can take place, the components must be mixed. As
previously described, metallic mixtures are generally produced at high temperatures in
the molten phase. Liquids however, can be mixed at lower temperatures, generally by
1
mechanical means, for example shaking. When a mixture is present, the components
form either a homogeneous or heterogeneous state. In the homogeneous state, the
respective concentration of the components is the same at all locations. For the
heterogeneous case, there are local domains richer in one of the components.
1.2 Phase diagram
We consider a binary mixture whose components are called A and B. The phase
diagram we use is a plot with the variables being the temperature, T, and the
concentration,
, which is defined as
(r) =
a(r)
-
b(r).
The state of the mixture is
described by its location on the mixture’s phase diagram (fig. 1) and is defined by its
composition and temperature ( , T). The composition describes the relative
concentrations of the constituent liquids.
Coexistence Curve
Spinodal Curve
Tc
T
Nucleation
Metastable
Spinodal Decomposition
Unstable
-1
c
Fig. 1
2
Nucleation
Metastable
+1
The coexistence curve divides the phase diagram into the homogeneous and
heterogeneous regions. On points along this curve the mixed and unmixed states are in
equilibrium with each other. When the state ( , T) of a binary mixture is located above
the coexistence curve on the phase diagram then the state is present as a homogeneous
mixture. When quenched below the coexistence curve, the system is now unstable as a
homogeneous mixture and will begin to separate into regions where only one component
of the mixture is present. In the region between the coexistence and spinodal curves, the
2
free energy is stable,
2
F
0 , locally in the homogeneous state. Figure 2 shows the
barrier, F, needed to be overcome in order for the free energy to be further reduced.
F
is the energy gain involved with having a phase separation compared to not having a
phase separation. The mixture is stable to small changes in composition over local
domains in region I and metastable in region III, however, a large enough fluctuation in
composition may produce a phase separation. This process is called nucleation.
F
Stable
Metastable
II
I
III
F
composition
Fig. 2
The process of phase separation is the unmixing of a thermodynamically unstable
solution. For a mixture made up of components A and B, there will be A-rich regions
3
and B-rich regions. The forces at work in this study are the A-A attraction, B-B
attraction, and A-B repulsion. B type particles in an A-rich region will migrate over to
the B-rich domain. Likewise, A type particles in the B-rich region may drift over to the
locally A-rich region. In doing so, areas that were well mixed, split into A-rich and Brich domains. Thus we have the presence of the heterogeneous state in which
varies as
a function of position.
There is a second curve of importance located in the heterogeneous region called
2
the spinodal curve. Within the spinodal, the homogeneous state is unstable,
F
2
0.
Figure 3 shows the change in free energy, F as a function of composition in this
unstable region. In region II, any small fluctuation in composition will result in a
decrease in free energy. Due to the presence of fluctuations in the mixture, this will
result in spontaneous growth during phase separation. This process is known as spinodal
decomposition.
F
I
Unstable
II
III
composition
Fig. 3
1.3 Nucleation
For nucleation to occur, when the state is between the coexistence and spinodal
curves, the system must be able to lower its free energy. A reduction in energy occurs
4
when the composition changes from a value corresponding to region III,
composition occurring in region I,
I.
III,
to a
Since the change in composition varies over a
range of values included in region II, nucleation occurs when the composition undergoes
a significant variation. The rate that this variation can occur is the nucleation rate.
The change in free energy produced by the presence of a droplet is F. Since the
system wants to reduce its total free energy, only droplets that have F < 0 associated
with them will grow. This F is equal to the difference in energy with a drop present
compared to no drop present. In three dimensions, F = 4
R2-4 R3/3 [1], where
and
are the surface free energy per unit area and the bulk free energy per unit volume
respectively of the drop, compared to the energies of the surrounding medium. Droplets
of a critical radius, Rc, will maximize F. This happens when R = Rc = 2 / . In our
study we are limited to two dimensions so spherical drops are replaced by circular
regions. In this case the change in free energy is F = 2
dimensional system,
R- R2. For the two
has units of energy per length and has units of energy per area.
Minimizing the change in free energy in this case results in a critical radius of Rc = / .
Figure 4 shows F as a function of radius with the location of the critical radius shown.
F
R<Rc
Rc
R>Rc
Fig. 4
5
radius
For small droplets, R < Rc, the energy needed to form the interface is greater than
the energy held in the bulk, so F > 0. The only way for a small droplet to reduce the
energy of the system is to reduce in size. Therefore, droplets with a radius less than Rc
will proceed to shrink away. When the system contains larger droplets, R > Rc, and it
will because of fluctuations in
, there is a reduction of free energy due to the droplet’s
presence, F < 0. Larger droplets will then grow in size to further reduce the free energy.
Of particular interest is the rate of nucleation. This will be determined by location
of the system on the phase diagram and the dynamics. The further below the coexistence
line, the higher the nucleation rate. For systems that are quenched to states just above the
spinodal, the nucleation rate may be so large that droplets are formed before the quench is
over [2].
1.4 Spinodal decomposition
Within the spinodal curve, the homogeneous phase is unstable towards
infinitesimal fluctuations in composition. Here any changes in the local concentration of
the constituent liquids will cause the two components to separate from each other. The
unmixing of a solution due to long range, infinitesimal changes in composition is called
spinodal decomposition. The separation process can occur while the composition
undergoes a variation of
as shown in figure 3. The key difference between spinodal
decomposition and nucleation is that since spinodal decomposition is an unstable process,
no additional energy is needed to start the process. While spinodal decomposition begins
to separate the mixture spontaneously, the rate that the separation occurs will be
determined by the variations in composition as a function of location and time.
6
Recall that for nucleation to occur, enough energy must be added to get over the
adjacent local maxima. The other main distinction is that nucleation is a local
phenomenon while spinodal decomposition can be a long range affect. Because spinodal
decomposition can act over long ranges, it produces elongated domains while nucleation
formed droplets. When a mixture containing components A and B experiences spinodal
decomposition, the two constituent liquids spontaneously group themselves into
intertwined domains. This initial formation is said to be spontaneous since there is no
nucleation barrier to cross. Some of these elongated domains are A-rich while the others
are B-rich. The domain of the majority component will form a percolate across the
sample when its concentration is greater than a critical density needed to do so. The
minority component will be grouped into smaller continuous domains which are
intertwined with the dominant species. For a three dimensional simple cubic system, the
critical probability for a percolate being present is approximately 1/3 [3, 4]. This allows
both components to form backbones, setting up a bicontinuous network. When these
elongated domains grow, they too, want to reduce the interface area, in order to reduce
the energy involved with their presence. In order to reduce surface area, the domains
coalesce and eliminate any interpenetration that may be present. The behavior of binary
liquids undergoing spinodal decomposition has been widely studied [1-11]. Of key
concern are the growth laws involved and the transition from the bicontinuous network
and elongated domains to smaller droplet shaped domains. This study will examine the
growth laws of continuous domains and compare the critical exponents to previous
studies.
7
Chapter 2
2.1 Mathematical background of spinodal decomposition
Spinodal decomposition is a diffusive process dictated by the thermodynamic
flow of energy. Energy is transferred by the two types of particles that have different
chemical potentials. For a binary mixture consisting of elements A and B, their
respective concentrations at each point are
a(r,
t) and
b(r,
t). These values, which run
from zero to one, determine whether a position in the system is A-rich or B-rich at a
given time. The order parameter is defined as
plus one to minus one. Values of
(r, t) =
a(r,
t) -
b(r,
t) and runs from
(r, t) greater than zero correspond to A-rich locations,
while values below zero are for B-rich positions. The chemical potential is defined as the
functional derivative of the Free Energy with respect to particle number;
F
.
(r , t )
The kinetics of the systems is described by the continuity equation:
(r , t )
t
j
(eqn. 2.1)
0
where the diffusive current is j
M
. M represents the mobility of the particles to be
determined by the systems components. The Free Energy is described by the functional,
F
f (r , t )
f (r, t )d 3 r , where we use the Ginzburg Landau function:
(r , t ) 2
2
(r , t ) 4
4
The free energy becomes F
we obtain: F
f
(r , t )
K
(
2
f
(r , t )) 2 .
(eqn. 2.2)
(r, t ) d 3 r . Using the functional derivative
(r, t ),
f
(r , t )
d 3 r . Minimizing the free energy and
8
integrating by parts yields
F
(r , t )
F
(r , t )
chemical potential:
F
(r , t )
F
, which is defined as the
(r , t ))
(
F
.
(r , t ))
(
(eqn. 2.3)
Combining equation 2.3 and the definition of the diffusive current j
j
F
(r , t )
M
M
gives us:
F
(r , t ))
(
(eqn. 2.4)
Combining equation 2.1 and equation 2.4 with the following partial derivatives:
f
(r , t )
equation:
(r , t )
(r , t )
t
(r , t ) 3 and
M
2
(r , t )
f
(r , t )
K
(r , t ) 3
(r , t ) , we obtain the Cahn-Hilliard
K
2
(r , t )
This equation describes how the order parameter,
0
(eqn. 2.5)
changes as a function of
position and time. The Cahn-Hilliard equation can be integrated with respect to time to
determine how the composition changes with the time evolution of the system. In this
study we use Euler’s method to solve the Cahn-Hilliard equation numerically.
2.2 Length scaling
The characteristic of the elongated domains that is most closely investigated is the
size or length. The length is defined as L(t ) ~
A(t )
P(t )
where <A(t)> is the average area
and <P(t)> is the average perimeter of the domains as a function of time. <P(t)> is the
average number of points of a domain that border a region made up of the opposite
particle type. A circular domain for example would have an area of A = R2 and a
perimeter of P=2 R, where R is the radius of the domain. The length parameter is then
L =R/2 or L(t) ~R.
9
A non-rigorous approach to find the scaling law pertaining to droplets is to
(r , t )
t
rewrite equation 2.1 in terms of the chemical potential:
M
2
(r , t ) .
Using the Gibbs-Thompson relation for liquid droplets, the chemical potential is defined
as
, with
d 1
,
R
the surface tension, d is the dimensions of the system,
is the change in order parameter at the droplet’s surface [10].
and
continuity equation will yield:
(r , t )
t
(r , t )
t
M
2
1
.
R3
(r , t )
M
2
Rewriting the
(d 1)
this gives us:
R
(eqn. 2.6)
After integrating both sides of equation 2.6, dimensional analysis gives us a scaling law
of: r
t n with n = 1/3. This growth law is independent of the system’s dimensions.
10
Chapter 3
3.1 Modeling a binary mixture undergoing spinodal decomposition
A two dimensional square lattice is initially created with each lattice site receiving
random real numbers varying between plus one and minus one. This is the value of
on
the lattice site. The random number generator is weighted so that the particle
concentrations are A = 52.5% and B = 47.5% respectively. Our goal is to determine
(r,
t) as time elapses.
The change in composition is computed in the following manner. At each time,
the energy of the system is computed by calculating the Ginzburg-Landau free energy:
f (r , t )
(r , t ) 2
2
(r , t ) 4
4
K
(
2
(r , t )) 2 . The functional derivative
F
is
(r , t )
F
(r , t )
F
.
(r , t ))
computed, which gives the chemical potential;
The diffusive current, j
F
(r , t )
(
, is found by taking the gradient of the chemical
M
potential with a suitable value of the mobility constant. We have scaled the length and
time so that K = M = 1. The continuity equation
Hilliard equation:
(r , t )
t
2
(r , t )
(r , t ) 3
(r , t )
t
2
j
0 , gives us the Cahn-
(r , t ) . This equation is then
solved numerically to find the new composition. These steps are repeated a set number
of times for each time increment.
Qualitatively, the dynamics of the local particle concentrations will be decided by
the four nearest neighbor values which will determine the energy. If two adjacent sites
are both predominantly the same type of particle, the energy is attractive. If not, the
11
energy is positive and repulsive. The constraint that acts on the particle motion is the
continuity equation,
(r , t )
t
j
0 . The continuity equation requires that each
particle type has a globally conserved quantity over the system. The conservation law,
while global, is not required at each lattice site. This allows the number of particles of
each constituent to change in local regions. The particles simply shift to an adjacent
region as members of the other particle type from the adjacent region take their place. As
the local concentration changes, the attractive energy between lattice sites will increase
and the repulsive interactions will be reduced. This occurs because the net free energy of
the entire system must be lowered in order for particle motion to be allowed. The
lowering of the total free energy is a requirement in addition to the continuity equation
that must be followed when changing the composition. Notice that while movement may
increase the energy between a set of two adjacent sites, the move is allowed if the total
energy of the system is lowered.
3.2 Structure factor
The program scans through the lattice a set number of updates for every time step
before taking a snapshot of the system, and going to the next time. The snapshot is a
three dimensional surface plot where the height of the plot represents
(r) =
a(r)
-
b(r).
Where the plot is positive, particle A represents the majority, where the plot is negative,
B dominates. This surface plot is then represented by a two dimensional color map where
the color of a local region represents which particle dominates the region. The time value
corresponding to each snapshot is the product of the snapshot number, the number of
updates per scan, and a small delta time value which was fixed to be 0.01 units. After
each time step, the program simulates a light scattering experiment by taking the Fourier
12
transform of the surface plot. This Fourier transform is proportional to the structure
factor obtained in the light scattering measurement. The Fourier transform of the
snapshot is calculated to measure the size distribution of the continuous regions in k-
space. The one dimensional Fourier transform is defined as F (k )
This can be generalized to two dimensions to F (k x , k y )
f ( x)e
f ( x, y)e
ikx
dx .
2i ( k x x k y y )
dxdy .
However, the integrals are replaced by summations since we are concerned with a
discrete function for f(x,y):
F (k x , k y )
f ( x, y)e
x
For our application, f(x, y) is equal to
2i ( k x x k y y )
.
y
(x, y), which was color mapped in order
to visually survey the domains and the phase transitions. In order to calculate the Fourier
transform in two dimensions, first the transform is applied to one of the dimensions to
produce F(kx, y) or F(x, ky). Next the transform is applied to the remaining dimension to
result in F(kx, ky). By taking the two dimensional Fourier transform at each moment of
time, we are able to observe the structure factor of the mixture as time evolves. One
example is the plot shown below of the structure factor for an initial order parameter of
0.05 at a time of 20 units (fig. 5).
This plot shows that the majority of the domains are described by wave numbers
ranging from 24 to 33. The horizontal axis is the radial wave number, k
k x2
k y2 .
The wave vector, K, which maximizes the structure factor, is the inverse of the most
common length scale of the continuous regions. To obtain the length of a wave vector,
13
K, from its respective wave number use the relation K = 2
k
where D is the size of the
D
system. In our system, D is equal to 256. Our analysis will be of the location and shape
of the peaks as a function of time. Of particular interest is the scaling of the time
dependence of the maximum wave number.
10
Structure factor with order parameter =0.05 at time =20 units
Intensity
8
6
4
2
0
0
10
20
30
40
50
Wave num ber k
Fig. 5
3.3 Correlation length
Our study also involved the use of the correlation function to determine the
growth rate of the correlation length. This gives us another indication of the rate that the
domains grow. 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 correlation length can be obtained by finding the first zero of the
correlation function. The correlation function is defined as G(r, R )
(R ) (R
r) ,
pairs
where R is the location of a chosen lattice site and r is the distance from the initial lattice
site to all sites a distance r away. This function is oscillatory in nature, and an example
is given below (figure 6). The program obtains the correlation function by scanning all
14
locations on a spherical shell centered on a selected site. Starting with the smallest shell
possible, a shell with the radius of nearest neighbor separation distance, the program
computes G (r , R ) . The radius of the shell is increased and the process is repeated. The
program repeats this process for all lattice points and averages over the number of sites.
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
20
25
-0.1
-0.15
-0.2
length (lattice sites)
Fig. 6
The first zero is located at approximately 4.5 units. On the average, for a pair of
points within this distance, the order parameter correlates positively. Within this region
one of the particle types is dominant. For distances just beyond the first zero the order
parameter anti-correlates. In this region, the other particle type is dominant and a phase
separation is present between the two domains. The correlation is partial and varies
continuously. If there were absolute correlation between lattice sites, the correlation
function would consist of regions with a slope of zero. Also, there would be
discontinuities at the roots of the correlation function. This would only be the case if the
particle densities varied discretely across a phase boundary.
15
Finding the scaling laws to determine how the peak wave vector and correlation
length scale with time is of key interest. The location of the first root of G (r , R ) migrates
to the right as time increases. The correlation length has been shown in previous research
to follow a power law L ~ t ,[1,5-10] where the critical exponent, , determines the
growth rate. The exponent
has been shown to be 1/3 for all dimensions, when the order
parameter is conserved [7]. Likewise, the wave vector corresponding to the peak of the
structure factor graph, also will follow a power law Kmax ~ t - . This corresponds to the
location Kmax moving to the left. Using these power laws we can predict the growth rate
of the coarsening domains.
3.3 Results
Plots were made of the structure factor for times ranging up to 900 units (figures 7a-d).
Structure factor for times 20-120 units
40
Intensity
20
35
40
30
60
80
25
100
20
120
15
10
5
0
0
10
20
Wave number k
Fig. 7a
16
30
40
Structure factor for times 140-240 units
70
140
160
60
180
Intensity
50
200
220
40
240
30
20
10
0
0
10
20
30
Wave number k
Fig. 7b
Structure factor for times 260-450 units
100
260
90
280
Intensity
80
300
70
350
60
400
50
450
40
30
20
10
0
0
10
20
Wave number k
Fig. 7c
17
30
Structure factor for times 500-900 units
Intensity
160
500
140
550
120
600
700
100
800
80
900
60
40
20
0
0
10
20
30
Wave number k
Fig. 7d
It is observed that as time increases, the location of the peak of the structure factor
moves to smaller wave numbers. In addition to the location of the peak moving with
time, the height of the peak increases with time. This shows that as time evolves, more
wave vectors of shorter length are being scattered from the specimen, corresponding to
larger domain sizes. The location of the peaks were recorded and plotted as a function of
time and fitted with a function of the form: f ( x)
ct
from the measurement resulted in a critical exponent of
(fig. 8a). The power law found
= 0.3217. The data is also
plotted in a log-log plot, (fig 8b), where the slope of a linear fit line would correspond to
the power law. The slope of the line was found to -0.3217. The calculated value of
compares favorably to the previously known value of 1/3 [2, 5-10]. The percent error is
calculated to be 3.5%.
18
Maximum wave number vs. time
35
30
-0.3217
y = 82.006x
20
15
10
5
0
0
200
400
600
800
1000
time
Fig. 8a
Maximum wave number vs. time (log-log)
100
Kmax
Kmax
25
10
1
10
100
time
Fig. 8b
19
1000
The similar graphing procedure was repeated with plots of the correlation
function at different times in order to determine the dependence of the correlation length
as a function of time. The correlation function is shown for a range of times extending to
900 time units (figures 9 a-c).
Correlation function for times 20-160 units
0.7
20
0.6
40
60
Correlation
0.5
80
0.4
100
0.3
120
0.2
140
160
0.1
0
-0.1
0
5
10
15
20
25
-0.2
length
Fig. 9a
Correlation function for times 180-350 units
0.8
180
Correlation
0.7
0.6
200
220
0.5
240
0.4
260
280
0.3
300
0.2
350
0.1
0
-0.1 0
5
10
15
-0.2
length
Fig. 9b
20
20
25
Correlation function for times 400-900 units
Correlation
0.8
0.7
400
0.6
450
500
0.5
550
0.4
600
0.3
700
800
0.2
900
0.1
0
-0.1 0
5
10
15
20
25
-0.2
length
Fig. 9c
The first zero of the correlation function, the correlation length, was found for each time
by fitting the data with a smoothed line and zooming in on the graph to obtain a value
accurate to within one-tenth of a unit. An example is shown below in figure 10.
0.3
Correlation function for times 20 -160 units zoomed
20
40
60
80
100
120
140
160
0.25
0.2
Correlation
0.15
0.1
0.05
0
-0.05 4
4.5
5
5.5
6
-0.1
-0.15
-0.2
length
Fig. 10
21
6.5
7
7.5
8
Once the correlation length was obtained, it was plotted as a function of time. In figures
11 a-b, this is shown in a linear and log-log plot.
Correlation length vs. time
12
Correlation length
10
y = 1.8681x0.2695
8
6
4
2
0
0
200
400
600
800
1000
time
Fig. 11a
Correlation length vs. time (log-log)
Correlation length
100
10
1
1
10
100
time
Fig. 11b
22
1000
The correlation length versus time graphs were fitted with a power law and the
critical exponent was found to be 0.2695. This is lower than the critical exponent for the
structure factor plots by ~16%. One point of interest is that by observing the snapshots
up to a time value of 900 units, the domains are still predominantly elongated, and the
large droplets observed by Demyanchuk [11] were just becoming present. This can be
observed in the image below (fig. 12) of the morphology at time equals 900 units.
The study was then extended by carrying out the simulation to a time of 4950 units to see
if larger spherical domains became present. Also, it would be possible to observe if the
scaling law for the correlation length versus time varied over different time scales. The
domains after time of 4950 units can be observed in figure 13. From the image we can
observe some larger drops present. However, the presence of the remaining elongated
domains reveals that even at time 4950 units, the percolation to droplet transition is not
yet complete.
Mixture at time = 900 units
Mixture at time = 4950 units
Fig. 12
Fig. 13
23
With the time drawn out to 4950 units, correlation functions were again plotted
and the time dependence of the correlation length was found. These correlation functions
are shown in figures 14 a-c. One key observation is that at these later times, time
1000
units, the oscillations in the correlation function occur slowly. When using the same
horizontal scale as in the earlier correlation function graphs, the second zero is not seen.
Correlation function for times 100-2000 units
0.9
100
500
correlation
0.7
1000
0.5
1250
0.3
1500
1750
0.1
2000
-0.1 0
-0.3
5
10
15
20
25
-0.5
length
Fig. 14a
Correlation function for times 2250-3500 units
correlation
0.9
0.7
2250
2500
0.5
2750
3000
0.3
3250
0.1
-0.1 0
3500
5
10
15
-0.3
-0.5
length
Fig. 14b
24
20
25
Correlation function for times 3750-4950 units
1
0.8
3750
4000
4250
4500
4750
4950
0.6
correlation
0.4
0.2
0
-0.2
0
5
10
15
20
25
-0.4
-0.6
length
Fig. 14c
The method of zooming in was used to obtain the correlation length, and it was plotted as
a function of time on both linear and log-log scales (figures 15 a-b). Again the slope of
the line of best fit on the log-log plot (fig. 15b) corresponds to the exponent in the power
law fit on the linear scale graph (fig. 15a).
Correlation length vs. time (times up to 4950 units)
Correlation length
25
20
15
y = 1.5676x0.3091
10
5
0
0
1000
2000
3000
time
Fig. 15a
25
4000
5000
Correlation length vs. time (times up to 4950 units log-log)
Correlation length
100
10
1
1
10
100
1000
10000
time
Fig. 15b
Using the larger range of times, the critical exponent , was found to be 0.3091.
This was in better agreement with the value for the critical exponent found from the
structure factor data. One possible explanation is that during the later times, the
correlation function does not grow as fast. This means that the values of the correlation
length taken during the earlier times may have more easily deviated from the trend during
the time of more rapid growth.
3.4 Conclusion
Critical exponents were found to be in general agreement with the accepted value
of 1/3. The data from the structure factor plots resulted in a critical exponent with a value
of
= 0.3217, which agreed with the accepted value better than the value of
obtained from the correlation length data. The value of
= 0.2695
from the correlation length
data, increased to 0.3091 when the simulated time was increased by a factor of five.
26
Of key note is the double peak that was observed in the study by Demyanchuk et
al [11] during the percolation to droplet transition, was not observed. While the
snapshots of the system that I studied showed that the percolation to droplet process was
never completed, the double peak structure from the light scattering simulation was not
observed. One possible explanation is that Demyanchuk’s system was a physical three
dimensional mixture, where my calculations were performed for a two dimensional
system. Of key relevance is that for two dimensions the critical probability for obtaining
a site percolation is 0.5927 [3]. For a three dimensional cubic lattice, the critical
probability is 0.325 +/- 0.023 [4]. Therefore, it is statistically less likely for a percolation
to be present in two dimensions. In fact, for a three dimensional mixture, since the
critical probability for forming a percolation is noticeably lower, there is a much better
chance for forming a bicontinuous network. Setting up the computer program in three
dimensions, while possible, would have significantly increased the processing time.
27
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