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Numerical Simulation for the Self-assembly of Polymer Blends with
Nano-scaled Features
BYYINGRUI SHANG
PhD DissertationUNIVERSITY OF MASSACHUSETTS LOWELL
2008
Dissertation Supervisors: David O. Kazmer, Ph.D., Carol F. Barry, D. Eng., and Joey Mead, Ph.D.
November 25, 2008
Outline
• Introduction– Thermodynamics
– Numerical assumptions
– Theoretical fundamentals
– Numerical methods
• Mechanisms of Phase Separation
• Validation of Modeling
• Conclusions and Future Work
• Acknowledgement
Nanomanufacturing Through High-rate/High-volume Templates for Guided Assembly of Nanoelements
Surface functionalization
Templates directed phase separation
Introduction
• Spinodal decomposition– Phase separation can
be induced by small composition fluctuations
– The spinodal decomposition can be directed by substrate functionalization
Local Free Energy in Ternary Blend
Ternary phase diagram
Spinodal line
Starting point of phase separation
Free energy of ternary mixture
Introduction to Numerical Simulation
Template Resulting concentration:
• Modelling assumptions– Random distribution initial situation– Incompressible fluid– Isothermal– Bulk-diffusion-controlled coarsening– Evaporation rate: h=h0exp(t)
Fundamentals
• The total free energy of the ternary (Cahn-Hilliard equation),
– F : total free energy– f : local free energy
– : the composition gradient energy coefficient
– Ci : the composition of component i
Mass Flux
• Ji, net
: the net mass flux
• Mi : mobility of component i
• i: chemical potential of component i
• Ci : composition of component i
Flory-Huggins Free Energy• The bulk free energy
– R : gas constant– T : absolute temperature
– mi : degree of polymerization of i
– Greater D. P., higher energy barrier for mixing
– ij : immiscibility parameter of i and j
– vsite: Molar volume of reference site
Fundamentals
Cahn-Hilliard Equation
C1+C2+C3=1
Mass flux: ,
– i,j : represent component 1 and component 2.– Mij : mobility
Determination of Controlling Factors• Flory-Huggins interaction parameter,
– 12,c
: critical interaction parameter. >12,c
for spinodal
decomposition to occur.
– Determines the miscibility of the polymer pair
–
–
– i: solubility parameter of component i
– The difficulties to obtain accurate solubility parameters.
– =0.221, when m1=174,m2=22
– 12,c
=0.0417
Determination of Controlling Factors
• Gradient energy coefficient,
– a : Monomer size, the affecting radius of van de
Waals force– Determines the domain size and interface thickness– – D: Diffusivity– Determines the kinetics of the phase transaction.
The values of and D are estimated by benchmarking with the experimental results, as shown later.
Numerical Method
• Discrete cosine transform method for PDEs
– and are the DCT of and – is the transformed discrete Laplacian,
Outline
• Introduction
• Mechanisms of Phase Separation− Linear relation of log(R)~log(t)
− Pattern size should match the value of R
− Effects of solvent and evaporation
• Validation of Modeling
• Conclusions and Future Work
• Acknowledgement
Constant Solvent Concentration
Polymer 1 Polymer 2 Solvent Polymer 1 Polymer 2 Solvent
t*=1024
t*=4096
t*=2048
(a) (b)
(a) Csolvent=60% (b) Csolvent=30%
Constant Solvent Concentration
• Measurement of the characteristic length, R
– Index numbers are shown on the figure
– The evolution of the domain size, R(t)~t, fits the rule that R(t)∝t1/3
0.3690.341
Results in a Binary (Annealing) System:With Patterns
64Characteristic length
Phase Separation with Solvent Evaporation
Lz=L0exp(-a*t), where t is the time, a is a constant, and Lz is the thicknessof the film at time t, and L0 is the thickness at t=0
Polymer 1 Polymer 2 Solvent
Time
Effects of Solvent and Evaporation
The compatibility, Cs, on the solution-substrate interface evolution with time.
Outline
• Introduction
• Mechanisms of Phase Separation
• Validation of Modeling− Summary
− Determination of parameters
− Effects of processing conditions
• Conclusions and Future Work
• Acknowledgement
Validation Experiments• Chemically heterogeneous substrate on Au surface
– Ebeam lithography followed by self-assembly of alkanethiol monolayer
– Hydrophylic strips covered by 11-Amino-1-undecanthiol (NH2)
– Hydrophobic strips covered by 1-octadecanethiol (ODT)
• Ternary system of polymers used– Polyacrylic acid (PAA): Negative static electrical force
– Polystyrene (PS): Hydrophobic
– Dimethylformamide (DMF): Solvent, on the order of 98% weight
• Experimental procedure– Polymer solution placed on substrate by pipette
– 6 minutes quiescence at room temperature and low humidity
– Polymer solution spin coated at varying RPM for in 30 seconds
Validation Experiments• Investigated factors
– Spin coating speed: 1000rpm, 3000rpm, 7000rpm
– Pattern substrate strip width: 667nm, 1000nm, 133nm
– Polymer composition ratio PS/PAA: 30/70, 50/50, 70/30
– Molecular weight of PAA: 2k, 50k, 450k
• Image acquisition– Field emission scanning electron microscopy (JEOL 7401)
– Atomic force microscopy (non-contact mode, Veeco NanoScopella)
– Fourier transform analysis (PSIA, v. 1.5)
• Model parameters then tuned by inspection of experimental and simulation results
Determination of M and
After comparison of the simulation and the experimental results
M=3.63·10-21 m5/(J*s)=1.82·10-7J/m
Experimental condition:• Spin coating speed: 3000 rpm• Time: 30 seconds• Solvent w%: 99%• PS/PAA (weight) : 7:3
Characteristic length, R=0.829m
Experiment
Experiment
The Effects of the Rotation Speed
The initial and final thickness of the film is measured experimentally. The evaporation constant , in h=hexp(t) can then be determined. The faster the rotation speed, the faster the evaporation, the smaller the The faster rotation speed results in a smaller R value, due to the effects of the faster solvent evaporationIncrease in the mobility, M, or in the value of result in larger domain size. Higher mobility will amplify the effects of the rotation speed.
Validation with the Experiments-- with the Patterned Substrate
Measure of the compatibility parameter, Cs
Experiment: SEM images are compared with the template patterns
, and the greater the better match of the morphologyto the pattern substrate.
Simulation: Comparison of result pattern and substrate template are compared element by element
s1(k) - the parameter in the surface energy expression for polymer oneSk - the quantitative representation of the substrate attraction.
Different Pattern Strip Widths
The pattern size has to match the intrinsic R value The simulation results generally matches the experimental value
Different PS:PAA Weight Ratios
The volume ratio of PS/PAA has to match the functionalized pattern area ratio
Effects of PAA Mw
The molecular weight of PAA will affect the shape of the Flory-Huggins local free energy Smaller molecular weight results in a more compatible pattern
Self-assembly in Thick Film
Initial thickness: 1mm, final thickness 8 m
Thickness dimension scaled by 2:1
The phase separation in the bulk domain will affect the morphology in the surface in a thick film
4m
8m
2m
128
64
64
More Complicated Substrate Pattern
Substrate pattern directed phase separation with different attraction forces
The substrate pattern
12m
12m
Graphic User Interface Program in MATLAB and C
Conclusion
The 3D numerical model for ternary system is established
The evolution mechanism is investigated. The R(t)∝t1/3 rule is fitted.
The model is fully tested and the numerical results are validated with the experimental results
The parameters are benchmarked, such as the mobility the gradient energy coefficient, and the surface energy M=3.63E-22m5/(J*s), =1.82E-7J/m, |fs|=4.82E3J/m2
Effects of different parameters are investigated. Recommendations for processing parameters
A GUI program is developed and tested, which can be used to assist the experiment and theoretical work.
Acknowledgement• Advisor, Professor David O. Kazmer
• Professor Joey Mead and Professor Carol Barry
• Liang Fang and Dr. Ming Wei assisted in the experimental results
• Center of High rate Nano-manufacture at UMass Lowell
• National Science Foundation funds (#NSF-0425826)
• All the people helped in this work
• Professor Jan Huang
• Ms. Lois Heath, and Ms. Adrianna Morris
• Ms. Ying Zeng
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