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Numerical Simulation of the Phase Separation of a Ternary
Systems on a Heterogeneously Functionalized Substrate
Yingrui Shang, Liang Fang, David Kazmer, Ming Wei, Joey Mead, and Carol Barry
University of Massachusetts Lowell Center for High-rate Nanomanufacturing
ABSTRACT
MATERIALS AND METHODS
• A numerical model for a polymerpolymer-solvent ternary system has been
established.
The Cahn-Hilliard equation for a ternary
system is established as:
• The functionalization of the template is
implemented numerically, and the
relation of the domain size and the time
are investigated.
Polymer 1 Polymer 2
Elements
• The free energy profile of the domain is
described by the Cahn-Hilliard equation.
• The discrete cosine transform method is
used to to solve the evolution equation
with numerical stability and efficiency.
RESULTS
Solvent
Polymer 1
Polymer 2 Solvent
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t*=1024
Pattern Size: Should match the intrinsic
polymer domain size value
F: total free energy
t*=2048
f: local free energy
: the composition gradient energy coefficient
Ci: the composition of component i
The system then can be described as a
function of the compositions. Considering
C1+C2+C3=1, the evolution equation can then
be written as a function of only C1 and C2,
(a) Csolvent=60%
t*=4096
(b) Csolvent=30%
The evolution of the domain size, R(t)~t, fits the
rule that R(t)∝t1/3. The influence of the solvent
concentration on the rate of morphology evolution
is significant. The
less the solvent, the
faster the
agglomeration of
the domains.
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• The numerical model can be used to
investigate the evolution mechanism of the
phase separation.
• The optimized parameters can be virtually
established from numerical & sensitivity
studies.
• Materials parameters which are difficult to
measure can also be estimated via the
simulation.
R(t)
SIGNIFICANCE
i,j: represent components 1 and component 2.
Mij: mobility of component i through j
The mobility M should is a function of the
compositions of polymer 1 and polymer 2.
The free energy of ternary system can be
plotted in a 3D view. The spinodal line is also
calculated.
• A user friendly software can be designed
to assist the experiments and practical
production.
C3=0.20
C3=0.35
Series3
Power (Series3)
Experimental system:
PS/PAA/DMF ternary
t*
solution spin coated at
3000rpm in 30 s. Patterned substrate:
ODT/NH2. The characteristic length, R, and
the compatibility parameter, Cs, are measured
from the SEM images. To determine the
mobility, M, and the gradient energy
coefficient, , the simulation is benchmarked
with the experimental results.
1
100
1000
10000
Molecular Weight: Affects the shape of
the Flory-Huggins local free energy, with
lower molecular weight resulting in a
more compatible pattern
CONCLUSIONS
• The 3D numerical model for ternary system
is established.
• The evolution mechanism is investigated,
and verifies that R(t)∝t1/3 rule as expected.
APPLICATIONS
• The simulation is validated by experiments.
The self-assembly of polymer blends
directed by a patterned substrate is a
promising method for rapid nanomanufacturing.

The numerical simulation of this process
can be used to investigate the mechanism
of the evolution and to estimate the material
properties & optimal process parameters.
MAIN EFFECTS
Free energy of ternary mixture

Simulation results
Nanoscale Science and Engineering
Center for High-rate Nanomanufacturing
EEC-0425826
• A method to benchmark the immeasurable
parameters by comparison of simulation
and the experimental results are developed.
• The patterned substrate is implemented
into the ternary system with solvent
evaporation.
• The effects of different parameters, such
as the spin coating rotation speed, polymer
weight ratios, and the PAA molecular
weight are investigated.
Starting point
of phase separation
Spinodal line
Experimental results
Volume Ratio: Should match the
functionalized pattern area ratio
Ternary phase diagram
 Rotation Speed: The faster rotation speed
results in a smaller domain size due to the effects
of the faster solvent evaporation.
• The numerical results are compatible with
the experimental results and can be used
to assist the experimental and theoretical
work.