DIBLOCK COPOLYMERS UNDER NANO-CONFINEMENT Dong Meng, Christopher Driscoll, Yuhua Yin, Xinghua Zhang, and Qiang (David) Wang Department of Chemical and Biological Engineering, Colorado State University Laboratory of Computational Soft Materials At Large : Structure of Concentric Cylinders Concentric cylinders A B B A d 1 d 2 d 3 d 4 At Small : Undulation of A-B Interfaces in Slab Morphology Due to the surface preference, A-B interfaces in the slab morphology are undulated so that more A segments can be in contact with the pore surface. Self-Assembly of Diblock Copolymers (DBC) L 0 : Bulk structure period; f A : Volume fraction of A block; : Flory-Huggins parameter between A and B segments; N: Copolymer chain length. Self-Consistent Field (SCF) Calculations 0 1 2 Hamiltonian: Canonical ensemble Incompressibility constraint: A (r) B (r) (r) 0 entropic contribution of copolymer chains; 1 A-B repulsion described by the parameter; 2 surface-copolymer interactions. 0 0 ( ) : normalized (by ) density field of ( A,B) segments; ( ) : total segmental density at spatial position . j j r r r A B Performed in real space without a priori knowledge about possible morphologies. 6 g R N a A and B segments are assumed to have the same statistical segment length a and bulk density 0 . Bulk phase diagram from Cochran et al., Macromolecules, 39, 2449 (2006). f A Part I. Symmetric DBC (f A 0.5) Thin Films Three effects of confining surfaces: 1. The surface preference for one of the blocks favors parallel lamellae. 2. The surface confinement favors perpendicular lamellae when the surface separation D is incommensurate with the bulk lamellar period L 0 . 3. The hard-surface effect favors perpendicular lamellae. This is due to the presence of impenetrable (hard) walls, at which the overall polymer segmental density 0 is reduced to 0 (from 1 in the interior of the film). Chains also lose conformational entropy near hard walls. (D. Meng and Q. Wang, J. Chem. Phys., 126, 234902 (2007)) Perpendicular Lamellae Parallel Lamellae : surface preference – 0 prefers A, 0 prefers B, 0 is neutral. A B l ≥0 u D N15 L 0 3.71R g The transitions among ||, T 0 and 2T are of the second order. Mixed morphologies (T 1 (2D), T 0 and 2T) form due to their subtle balance among chain entropy, A-B repulsion, and surface energies. T 1 (3D) can be distinguished from T 1 (2D) and || by their top-views. A novel 3D mixed morphology T 1 (3D) is found, which has smaller A-B repulsion than T 1 (2D). 3D Phase Diagrams at l 15 (without Hard-Surface Effect, 0 (r)1) 2D Phase Diagram at DL 0 (with Hard-Surface Effect) 3D Phase Diagram at DL 0 (without Hard-Surface Effect, 0 (r)1) PVP-dPS, D1.05L 0 Koneripalli et al., Langmuir, 12, 6681 (1996) PS-PBMA (both surfaces prefer PBMA) DL 0 DL 0 Fasolka et al., Macro- molecules, 33, 5702 (2000) Possible Experimental Evidence of T 1 (3D) Dissimilar surfaces In the left, the stable region of mixed morphologies (T 1 (3D) and T 1 (2D)) shrinks as D increases from L 0 , and vanishes for D1.2L 0 . As D further increases towards 2L 0 , T 1 (2D) appears as the only stable mixed morphology and has prolonged perpendicular A-B interfaces compared to T 1 (2D) at DL 0 , which help alleviate the frustration at the T-junctions. Similar surfaces at D1.5L 0 In the above, T 2 (3D) is similar to T 1 (3D) but has two parallel A-B interfaces near the lower surface and an inverted A-B pattern near the upper surface. T 1 (2D) between similar surfaces has different A-B interfacial curvatures near the upper surface than those in T 1 (2D) between dissimilar surfaces. 0 2 4 6 8 (1.5) (2) u T 2 (3D) T 1 (2D) Part II. Symmetric DBC (f A 0.5) on Topographically Patterned Substrates The area enclosed by the red dashed lines is the unit cell used in SCF calculations. We performed 2D calculations in accordance to the experimental setup. All input parameters are taken directly from experiments, where both the sidewalls and substrate are neutral, leading to perpendicular lamellae. N25, L 0 31nm, D 1 100nm, D 2 200nm. D 2 D 1 L x y L x L y Lamellae Tilting vs. Bending 1. Lamellae tilt at 90º to alleviate the bending of lamellae. As increases from 70º to 110º, reducing the lamellae bending at the cost of increasing lamellae tilting explains the decrease of . 3. At 130 o and larger, the lamellae break and form -kinks with highly localized entropic penalty. This helps release the entropic penalty in other regions. therefore suddenly increases to about 90 o and levels off in this region. 2. At 70 o , as the elbow corner is approached, the available space for copolymers become smaller and arc structures with smaller form instead of bend structures with high entropic penalty at the bending vertices. The same reasoning applies when is decreased from 70 o , explaining the decrease of in this region. Segmental Density and Chain Elastic Free Energy Distributions from SCFT The elastic free energy (k B T/chain) due to chain conformational entropy is high at bending vertices and -kinks. Top-View SEM Images from Experiments S. M. Park, D. Meng, C. T. Rettner, D. S. Dandy, Q. Wang, and H. C. Kim, Macromolecules, in press We have performed real-space self-consistent field (SCF) calculations with high accuracy to study the self-assembled morphology of diblock copolymers (DBC) under nano- confinement for several systems, including 1D lamellae- forming DBC confined between two homogeneous and parallel surfaces (thin films), in nano-pores, and on topographically patterned substrates; 2D cylinder-forming DBC on chemically stripe-patterned substrates; and 3D sphere- and gyroid-forming DBC thin films. The stable phases are identified through free-energy comparisons, and our SCF results are compared with available experiments and Monte Carlo simulations. The surface preference, geometry, confinement, and pattern all have significant influence on the self-assembled morphology of DBC under nano-confinement. Much richer phase behaviors are found in all of these systems, with complex morphologies that are very different from those in the bulk. Understanding and predicting the self-assembled morphology of DBC under nano-confinement will help us obtain the desirable morphology for targeted applications. Real-space SCF calculation is a powerful tool for this purpose. Summary Part V. Cylinder-Forming DBC (f A 0.3) on Chemically Nano-Patterned Substrates Top-View SEM Images Self-Consistent Field Calculations The green stripes on the substrate prefer A segments (5) forming the cylinders, while the yellow has no preference (0). Input parameters are taken directly from experiments. N L 0 4.03R g D L s wL 0 /2 (a) L s L 0 45nm, D40nm On a commensurate substrate pattern, two layers of paralle half-cylinders form and are registered with the stripe pattern on the substrate. 1.0 0.0 0.8 0.6 0.4 0.2 A 1.0 0.5 0.9 0.8 0.7 0.6 S. O. Kim, B. H. Kim, D. Meng, D. O. Shin, C. M. Koo, H. H. Solak, and Q. Wang, Adv. Mater., 19, 3271 (2007) (b) L s 2.3L 0 100nm, D40nm On an incommensurate substrate pattern, mixed morphology forms with two layers of parallel half-cylinders on preferential stripes and perpendicular cylinders on neutral stripes. (c) L s 2.3L 0 100nm, D20nm As D is reduced by half from (b), well ordered mixed morphology forms with alternating parallel half-cylinders on preferential stripes and perpendicular cylinders on neutral stripes. PMMA-PS (dark-bright) Scale bar – 100nm Scale bar – 200nm Part IV. Preliminary Results of Asymmetric DBC Thin-Films Between Identical Surfaces Between strongly A-preferential surfaces (10) 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 DL 0 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 DL 0 Between strongly B-preferential surfaces (10) Between weakly A-preferential surfaces (2) Sphere-Forming DBC Gyroid-Forming DBC N20 f A 0.357 L 0 9.50R g N23.5 f A 0.2 L 0 4.49R g Preliminary SCF calculations are performed without the hard-surface effect, i.e., 0 (r)1. Various morphologies (cylinders, lamellae, and perforated lamellae) different from the bulk morphology, deformed spheres, and no gyroid structure are found in thin films. More calculations are needed and on-going. (1) (2) (1.5) T 1 (2D) T 1 (2D) (2.5) T 1 (3D) u T 1 (2D) D. Meng and Q. Wang, Soft Matter, submitted Part III. Symmetric DBC (f A 0.5) in Nano-Pores B A D2R Pore surface prefers A segments (>0) N24, L 0 4.28R g : radial distance from the pore center; () A () B (): order parameter. The mixed morphology has been reported in several lattice MC studies. Our SCF calculations indicate that this morphology is unstable due to its large entropic penalty (chain-stretching). At Intermediate : Morphology Stability via Free-Energy Comparison The parallel morphology was also reported in lattice MC simulations. Our SCF calculations show that this morphology is unstable due to its large A-B repulsion. D2L 0 Free Energy (k B Tchain) D1.65L 0 Free Energy (k B Tchain) Lattice Monte Carlo (MC) simulation data are taken from Q. Wang, J. Chem. Phys., 126, 024903 (2007). Good agreement between SCF and MC results are found for concentric cylinders. D. Meng, X. Zhang, and Q. Wang, Macromolecules, to be submitted Our SCF calculations are in good agreement with experimental observations, and further provide the 3D structures in the films and their formation mechanisms. DL 0 0.9 1.0 1.1 1.2 1.3 1.5 1.4 1.6 1.7 1.8 1.9 2.0 2.1 2.2 1/ 2 1/2 // // C C 1/2 1/2 ~ ~ S S S 1/ 2 1/ 2 // // ~ ~ C S C 1/ 2 1/ 2 // // C S C 1/2 2 1/2 ~ ~ s S S S 1/2 1/ 2 // // d C S C 1/2 1/2 // // ~ ~ d C S C 1/2 2 1/ 2 ~ ~ S S S (SQR) (SCL) (SQR) (HEX) (HEX) (SQR) (SQR) 2 ~ ~ PL S PL 2 ~ ~ s PL S PL (HEX) (SQR) R0.79L 0 n: number of A-B interfaces
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DIBLOCK COPOLYMERS UNDER NANO-CONFINEMENTDong Meng, Christopher Driscoll, Yuhua Yin, Xinghua Zhang, and Qiang (David) Wang
Department of Chemical and Biological Engineering, Colorado State University
Laboratory of Computational Soft Materials
Laboratory of Computational Soft Materials
At Large : Structure of Concentric Cylinders
Concentric cylinders
ABBAd1d2d3d4
At Small : Undulation of A-B Interfaces in Slab Morphology
Due to the surface preference, A-B interfaces in the slab morphology are undulated so that more A segments can be in contact with the pore surface.
Self-Assembly of Diblock Copolymers (DBC)
L0: Bulk structure period;fA: Volume fraction of A block; : Flory-Huggins parameter
between A and B segments;N: Copolymer chain length.
Self-Consistent Field (SCF) Calculations
0 1 2Hamiltonian: Canonical ensemble
Incompressibility constraint: A(r) B(r) (r)
0 entropic contribution of copolymer chains;1 A-B repulsion described by the parameter;2 surface-copolymer interactions.
0
0
( ) : normalized (by ) density field of ( A,B) segments;
( ) : total segmental density at spatial position .j j
r
r r
A B
Performed in real space without a prioriknowledge about possible morphologies.
6gR N a
A and B segments are assumed to have the same statistical segment length a and bulk density 0.
Bulk phase diagram from Cochran et al., Macromolecules, 39, 2449 (2006).fA
Part I. Symmetric DBC (fA0.5) Thin FilmsThree effects of confining surfaces:1. The surface preference for one of the blocks favors parallel lamellae.2. The surface confinement favors perpendicular lamellae when the surface separation D
is incommensurate with the bulk lamellar period L0. 3. The hard-surface effect favors perpendicular lamellae. This is due to the presence of
impenetrable (hard) walls, at which the overall polymer segmental density 0 is reduced to 0 (from 1 in the interior of the film). Chains also lose conformational entropy near hard walls. (D. Meng and Q. Wang, J. Chem. Phys., 126, 234902 (2007))
Perpendicular LamellaeParallel Lamellae
: surface preference – 0 prefers A, 0 prefers B, 0 is neutral.
A B
l≥0
u
DN15L03.71Rg
The transitions among ||, T0 and 2T are of the second order. Mixed morphologies (T1(2D), T0 and 2T) form due to their subtle
balance among chain entropy, A-B repulsion, and surface energies.
T1(3D) can be distinguished from T1(2D) and || by their top-views.
A novel 3D mixed morphology T1(3D) is found, which has smaller A-B repulsion than T1(2D).
3D Phase Diagrams at l15 (without Hard-Surface Effect, 0(r)1)
2D Phase Diagram at DL0 (with Hard-Surface Effect) 3D Phase Diagram at DL0 (without Hard-Surface Effect, 0(r)1)
PVP-dPS, D1.05L0
Koneripalli et al., Langmuir, 12, 6681 (1996)
PS-PBMA (both surfaces prefer PBMA)
DL0 DL0
Fasolka et al., Macro-molecules, 33, 5702 (2000)
Possible Experimental Evidence of T1(3D)
Dissimilar surfaces
In the left, the stable region of mixed morphologies (T1(3D) and T1(2D)) shrinks as D increases from L0, and vanishes for D1.2L0. As D further increases towards 2L0, T1(2D) appears as the only stable mixed morphology and has prolonged perpendicular A-B interfaces compared to T1(2D) at DL0, which help alleviate the frustration at the T-junctions.
Similar surfaces at D1.5L0
In the above, T2(3D) is similar to T1(3D) but has two parallel A-B interfaces near the lower surface and an inverted A-B pattern near the upper surface. T1(2D) between similar surfaces has different A-B interfacial curvatures near the upper surface than those in T1(2D) between dissimilar surfaces.
02468
(1.5)(2)
u
T2(3D)T1(2D)
Part II. Symmetric DBC (fA0.5) on Topographically Patterned Substrates
The area enclosed by the red dashed lines is the unit cell used in SCF calculations. We performed 2D calculations in accordance to the experimental setup. All input parameters are taken directly from experiments, where both the sidewalls and substrate are neutral, leading to perpendicular lamellae.
N25, L031nm,D1100nm, D2200nm.
D2 D1 L
x
y
Lx
Ly
Lamellae Tilting vs. Bending1. Lamellae tilt at 90º to
alleviate the bending of lamellae. As increases from 70º to 110º, reducing the lamellae bending at the cost of increasing lamellae tilting explains the decrease of .
3. At 130o and larger, the lamellae break and form -kinks with highly localized entropic penalty. This helps release the entropic penalty in other regions. therefore suddenly increases to about 90o and levels off in this region.
2. At 70o, as the elbow corner is approached, the available space for copolymers become smaller and arc structures with smaller form instead of bend structures with high entropic penalty at the bending vertices. The same reasoning applies when is decreased from 70o, explaining the decrease of in this region.
Segmental Density and Chain Elastic Free Energy Distributions from SCFT
The elastic free energy (kBT/chain) due to chain conformational entropy is high at bending vertices and -kinks.
Top-View SEM Images from Experiments
S. M. Park, D. Meng, C. T. Rettner, D. S. Dandy, Q. Wang, and H. C. Kim, Macromolecules, in press
We have performed real-space self-consistent field (SCF) calculations with high accuracy to study the self-assembled morphology of diblock copolymers (DBC) under nano-confinement for several systems, including 1D lamellae-forming DBC confined between two homogeneous and parallel surfaces (thin films), in nano-pores, and on topographically patterned substrates; 2D cylinder-forming DBC on chemically stripe-patterned substrates; and 3D sphere- and gyroid-forming DBC thin films. The stable phases are identified through free-energy comparisons, and our SCF results are compared with available experiments and Monte Carlo simulations.
The surface preference, geometry, confinement, and pattern all have significant influence on the self-assembled morphology of DBC under nano-confinement. Much richer phase behaviors are found in all of these systems, with complex morphologies that are very different from those in the bulk.
Understanding and predicting the self-assembled morphology of DBC under nano-confinement will help us obtain the desirable morphology for targeted applications. Real-space SCF calculation is a powerful tool for this purpose.
Summary
Part V. Cylinder-Forming DBC (fA0.3) on Chemically Nano-Patterned Substrates
Top-View SEM ImagesSelf-Consistent Field Calculations
The green stripes on the substrate prefer A segments (5) forming the cylinders, while the yellow has no preference (0). Input parameters are taken directly from experiments.
NL04.03Rg
D
Ls wL0/2(a) LsL045nm, D40nm
On a commensurate substrate pattern, two layers of paralle half-cylinders form and are registered with the stripe pattern on the substrate.
1.0
0.0
0.8
0.6
0.4
0.2
A
1.0
0.5
0.9
0.8
0.7
0.6
S. O. Kim, B. H. Kim, D. Meng, D. O. Shin, C. M. Koo, H. H. Solak, and Q. Wang,Adv. Mater., 19, 3271 (2007)
(b) Ls2.3L0100nm, D40nmOn an incommensurate substrate pattern, mixed morphology forms with two layers of parallel half-cylinders on preferential stripes and perpendicular cylinders on neutral stripes.
(c) Ls2.3L0100nm, D20nmAs D is reduced by half from (b), well ordered mixed morphology forms with alternating parallel half-cylinders on preferential stripes and perpendicular cylinders on neutral stripes.
PMMA-PS (dark-bright)
Scale bar – 100nm
Scale bar – 200nm
Part IV. Preliminary Results of Asymmetric DBC Thin-Films Between Identical Surfaces
Between strongly A-preferential surfaces (10)0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
DL0
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6DL0
Between strongly B-preferential surfaces (10)
Between weakly A-preferential surfaces (2)Sphere-Forming DBC
Gyroid-Forming DBCN20 fA0.357 L09.50Rg
N23.5 fA0.2 L04.49Rg
Preliminary SCF calculations are performed without the hard-surface effect, i.e., 0(r)1.
Various morphologies (cylinders, lamellae, and perforated lamellae) different from the bulk morphology, deformed spheres, and no gyroid structure are found in thin films.
More calculations are needed and on-going.
(1) (2)(1.5)
T1(2D)
T1(2D)
(2.5)
T1(3D)u
T1(2D)
D. Meng and Q. Wang, Soft Matter, submitted
Part III. Symmetric DBC (fA0.5) in Nano-Pores
BA D2R
Pore surface prefers A segments (>0)
N24, L04.28Rg
: radial distance from the pore center;()A()B(): order parameter.
The mixed morphologyhas been reported in several lattice MC studies. Our SCF calculations indicate that this morphology is unstabledue to its large entropic penalty (chain-stretching).
At Intermediate : Morphology Stability via Free-Energy Comparison
The parallel morphologywas also reported in lattice MC simulations. Our SCF calculations show that this morphology is unstable due to its large A-B repulsion. D2L0
Free
Ene
rgy
(kBT
chai
n)
D1.65L0
Free
Ene
rgy
(kBT
chai
n)
Lattice Monte Carlo (MC) simulation data are taken from Q. Wang, J. Chem. Phys., 126, 024903 (2007).
Good agreement between SCF and MC results are found for concentric cylinders.
D. Meng, X. Zhang, and Q. Wang, Macromolecules, to be submitted
Our SCF calculations are in good agreement with experimental observations, and further provide the 3D structures in the films and their formation mechanisms.
DL0
0.9 1.0 1.1 1.2 1.3 1.51.4 1.6 1.7 1.8 1.9 2.0 2.1 2.2
1/2 1/2/ / / /C C
1/2 1/2~ ~S S S 1/2 1/2/ / //~ ~C S C
1/2 1/2/ / / /C S C
1/2 2 1/2~ ~sS S S 1/2 1/2/ / / /
dC S C
1/2 1/2/ / / /~ ~dC S C
1/2 2 1/2~ ~S S S(SQR)(SCL)
(SQR)
(HEX)
(HEX)(SQR)
(SQR)2~ ~PL S PL
2~ ~sPL S PL(HEX)(SQR)
R0.79L0
n: number of A-B interfaces
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