Lateral Confinement of Block CopolymerThin Films in Numerical SCFT

August Bosse a

Scott Sides b

Carlos Garcia-Cervera c

Glenn Fredrickson d

CFDC Annual Meeting

2 February 2005

aDepartment of Physics, University of California, Santa Barbara, CAbTech-X Corporation, Boulder, COcDepartment of Mathematics, University of California, Santa Barbara, CAdDepartments of Chemical Engineering and Materials, University of California, Santa Barbara, CA

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Topics of Discussion

1. Motivation• Masking methods and sub-micron fabrication• Artificially assisted self-assembly and long-range order• ...

2. Introduction to the Method—Lateral Confinement via Matsen-type“Masking” Fields

3. Qualitative “Results”

4. Future Work• Improved annealing techniques• Improved numerical techniques• Analysis• More sophisticated confinement modeling methods

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Motivation

Why block copolymer thin films?• Sub-micron scale fabrication and lithography using microphase

separated block copolymers (e.g., Cheng, et al., 2002; Hamley, 2003).

Figure: Park, Yoon, & Thomas, 2003CFDC Annual Meeting, 2 February 2005 – p.3/28

• Sub-micron masks in semiconductor and magnetic media fabrication(e.g., Katsuyuki Naito, et al. from Toshiba).

• “Further study is necessary to obtain uniform dot diameter and regulardot position” (Naito, et al., 2002).

• Order in 2D crystals (2D array of microphase domains).• Artificially assisted self-assembly (e.g., wall interactions, defects, 2D

order, etc.) (Segalman, Hexemer, & Kramer, 2003a).• KTHNY theory of 2D melting (Segalman, et al., 2003):

Crystal ⇔ Hexatic ⇔ Liquid.• Artificially assisted self-assembly & KTHNY (Segalman, Hexemer, &

Kramer, 2003b).

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Introduction to the Method

For an incompressible melt of n AB diblock copolymers in a three-dimensionalvolume V , the canonical partition function is

Z =

Z

nY

α=1

δrα(s)

!

δ(ρ̂A + ρ̂B − ρ0) exp(−U0 − U1),

where

U0 =1

4R2g0

nX

α=1

Z 1

0

ds

„

drα(s)

ds

«2

,

and

U1 =χAB

ρ0

Z

dr ρ̂Aρ̂B.

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U0 is the “Gaussian thread” free energy term (from harmonic stretching). U1

models A-B interactions:• ρ0 = nN/V is the total monomer density,• χAB is the Flory-Huggins interaction parameter,• Rg0 is the unperturbed radius of gyration for the Gaussian chain,• and ρ̂A and ρ̂B are the microscopic monomer densities:

ρ̂A(r) = N

nX

α=1

Z f

0

ds δ(r− rα(s)),

ρ̂B(r) = N

nX

α=1

Z 1

f

ds δ(r− rα(s)).

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This model (and its variations) has successfully handled:• Numerous different polymer architectures (AB, ABC, star, etc.),• Both unit cell and large cell simulations,• With periodic boundary conditions,• And specific confined geometries (e.g., confined slit).• However, for arbitrary confinement and/or arbitrary wall-polymer

interactions, we need something new...

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How about if we modify the current model (cf., Matsen, 1997):

δ(ρ̂A + ρ̂B − ρ0) → δ(ρ̂A + ρ̂B − ρ0(r))

U1 → U1 +χWA

ρ0

Z

dr h(r)ρ̂A +χWB

ρ0

Z

dr h(r)ρ̂B.

Here:• h(r) is an external “wall interaction” field,

h(r) ∈ [0, ρ0],

• χWA and χWA are Flory-type parameters for wall-monomer interactions,• and ρ0(r) is the total monomer density function (a.k.a., “mask” function).

Typically we set

ρ0(r) = ρ0(1 − h(r)/ρ0) = ρ0(1 − H(r)).

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• Next ...• H.S. transformation yielding WA,WB , and P fields.• Mean-field approximation: saddle point equations for WA,WB , and P .• Everything is as one would expect, except the saddle point equations

are slightly modified:

WA(r) = χABN(φB(r) − φ̄B) + P (r) + χWANH(r)

WB(r) = χABN(φA(r) − φ̄A) + P (r) + χWBNH(r)

φA(r) + φB(r) = φ0(r)

φA(r) = −V

Q

δQ

δWA

φB(r) = −V

Q

δQ

δWB

.

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General Properties of the Masking Method:• Computationally wasteful,• Useful for studying the long range effects of confinement (i.e., away from

walls),• Easy and fast to develop (existing SCFT code base by Scott Sides),• Versatile!

General Properties of Scott’s SCFT Code (Sides & Fredrickson, 2003):• Distributed memory parallel implementation,• Pseudo-spectral splitting for diffusion equations,• Saddle point equations solved via simple relaxation (forward Euler, or a

variation) or Scott’s new “spectral-filtering” algorithm.

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Constructing h(r)

• We would like the possibility for arbitrary masks.• Smooth “walls” are desirable:

w ∼ t

(Wall Width) ∼ (AB Interface Thickness) ∼ O(Rg0)

(good for the FFTs and relaxation speed). We use a hyperbolic tangentprofile for the mask (Alexander-Katz, Moreira, & Fredrickson, 2003).

• The Method:• First a mask outline is created (using inequalities, level curves, etc.),• Then the tanh profile is added via a distance function. E.g.,

tanh

„

d(x, y)

w

«

.

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Lx = Ly = 100Rg0, w ∼ O(Rg0), and nx = ny = 256. CFDC Annual Meeting, 2 February 2005 – p.13/28

Lx = Ly = 100Rg0, w ∼ O(Rg0), and nx = ny = 256. CFDC Annual Meeting, 2 February 2005 – p.14/28

Lx = Ly = 100Rg0, w ∼ O(Rg0), and nx = ny = 256. CFDC Annual Meeting, 2 February 2005 – p.15/28

Qualitative “Results”

How do walls affect the ordering of microdomains? Let’s run some simulations!

(A-type monomer in white)f = 0.50, χABN = 25.1, χWAN = −10.0, and χWBN = 10.0.Lx = Ly = 25.6Rg0, w ∼ O(10−1Rg0), and nx = ny = 256.

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Larger confined systems have more defects...

(A-type monomer in white)f = 0.50, χABN = 25.1, χWAN = −10.0, and χWBN = 10.0.Lx = Ly = 51.2Rg0, w ∼ O(10−1Rg0), and nx = ny = 512.

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We can look at how the shape of the boundary affects microdomain ordering:

(A-type monomer in white)f = 0.75, χABN = 25.1, χWAN = −10.0, and χWBN = 10.0.Lx = Ly = 25.6Rg0, w ∼ O(10−1Rg0), and nx = ny = 256.

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(A-type monomer in white)f = 0.75, χABN = 25.1, χWAN = −10.0, and χWBN = 10.0.Lx = Ly = 76.8Rg0, w ∼ O(10−1Rg0), and nx = ny = 768.

Voronoi Diagram: Courtesy Gila Stein

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(A-type monomer in white)f = 0.75, χABN = 25.1, χWAN = −10.0, and χWBN = 10.0.Lx = Ly = 25.6Rg0, w ∼ O(10−1Rg0), and nx = ny = 256.

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(A-type monomer in white)f = 0.75, χABN = 25.1, χWAN = −10.0, and χWBN = 10.0.Lx = Ly = 76.8Rg0, w ∼ O(10−1Rg0), and nx = ny = 768.

Voronoi Diagram: Courtesy Gila Stein

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We can also examine how the walls distort microdomains and create defects(an extreme example):

(A-type monomer in white)f = 0.75, χABN = 25.1, χWAN = −10.0, and χWBN = 10.0.Lx = Ly = 12.8Rg0, w ∼ O(10−1Rg0), and nx = ny = 128.

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Not surprisingly, we can extend these methods beyond 2D. Here we have anABA triblock copolymer in 3D. There is a wall at z = 0 and z = Lz:

F̃ = −0.425 F̃ = −0.433

fA, fB, fA = 0.35, 0.30, 0.35; χABN = 32; χWAN = −0.2; and χWBN = −0.4.Lx = Ly = Lz ≈ 11Rg0, w ∼ O(10−1Rg0), and nx = ny = nz = 48.

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Future Work

• More sophisticated annealing techniques (remove metastable defects):• Spectral-filtering• Simulated annealing:

• Relax W+ = (WA + WB)/2• Monte Carlo on W

−= (WA − WB)/2

• Speed up relaxation and reduce error: semi-implicit-Seidel (SIS)(Ceniceros & Fredrickson, 2004)

• Analysis techniques:• Local orientational order parameters• Correlation functions• Voronoi diagrams and defect counting• Microdomain (cylinder) spacing• FT of densities

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• More sophisticated methods for modeling confined systems:• Special boundary conditions (e.g., Dirichlet, ...) and specialized basis

functions• Multigrid• ...

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Acknowledgments

• Scott Sides• Carlos Garcia-Cervera• Glenn Fredrickson• Gila Stein• Alexander Hexemer• and everyone in the Fredrickson group!

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References

• Alexander-Katz, A.; Moreira, A.G.; Fredrickson, G.H. J. of Chem. Phys.,2003, 118, 9030.

• Ceniceros, H.D.; Fredrickson, G.H. Multiscale Model Simul. 2004, 2, 452.• Cheng, J.Y.; Ross, C.A.; Thomas, E.L.; Smith, H.I.; Vancso, G.J. Applied

Physics Letters 2002, 81, 3658.• Hamley, I.W. Nanotechnology 2003, 14, R39.• Matsen, M.W. J. Chem. Phys., 1997, 106, 7781.• Naito, K.; Hieda, H.; Sakurai, M.; Kamata, Y.; Asakawa, K. IEEE

Transactions on Magnetics, 2002, 38, 1949.• Park, C.; Yoon, J.; Thomas, E.L. Polymer 2003, 44, 6725.• Segalman, R.A.; Hexemer, A.; Hayward, R.C.; Kramer, E.J.

Macromolecules 2003, 36, 3272.

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• Segalman, R.A.; Hexemer, A.; Kramer, E.J. Macromolecules 2003a, 36,6831.

• Segalman, R.A.; Hexemer, A.; Kramer, E.J. Phys. Rev. Lett. 2003b, 91,No. 19.

• Sides, S.W.; Fredrickson, G.H. Polymer 2003, 44, 5859.

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