Influence of topographically patterned angled guidelines on directed self-assembly of block copolymers

Single chain in mean-field Monte Carlo simulations were employed to study the self-assembly of block copolymers (BCP) in thin films that use trapezoidal guidelines to direct the orientation and alignment of lamellar patterns. The present study explored the influence of sidewall interactions and geometry of the trapezoidal guidelines on the self-assembly of perpendicularly oriented lamellar morphologies. When both the sidewall and the top surface exhibit preferential interactions to the same block of the BCP, trapezoidal guidelines with intermediate taper angles were found to result in less defective perpendicularly orientated morphologies. Similarly, when the sidewall and top surface are preferential to distinct blocks of the BCP, intermediate tapering angles were found to be optimal in promoting defect free structures. Such results are rationalized based on the energetics arising in the formation of perpendicularly oriented lamella on patterned substrates.

Figure 1

Schematic of an assembled BCP system to scale with (a) rectangular guideline and (b) trapezoidal (tapered) guideline. Grafted neutral copolymer brushes line substrate (dark blue) between the guideline. By modulating the interaction of the copolymer with the substrate side and top, effective directed self-assembly can be achieved.

Figure 2

The following parameters characterize our model: film dimensions ($L_x, L_y, L_z$), guideline height ($H$), width ($W$), taper angle ($\theta$), and the relative chemical affinities of the guideline top surface (ST), thin film top (FT), and sidewall (SW) for each block.

Figure 3

Displayed is a simplified schematic for a model without a guideline along with 2D maps of sample configurations with different ${\mathrm{\Lambda }}_{\text{FT}}^A$ and ${\mathrm{\Lambda }}_{\text{FT}}^B$ initial conditions. Only the interaction parameters ${\mathrm{\Lambda }}_{\text{FT}}^A$ and ${\mathrm{\Lambda }}_{\text{FT}}^B$ play a role in BCP self assembly when the guideline is absent.

Figure 4

Two dimensional cross section of the density of A segments with varying bottom guideline widths ($W$) [(a)–(c)], but possessing the same height ($[H=4.06R_g$ ($1.0L_o$)]. (c) and (d) have the same geometry but different interaction affinities. The scores for the different morphologies are as follows: (a) 0.95, (b) 0.84, (c) 0.88, and (d) $<0.75$.

Figure 5

Representative morphologies (corresponding to the volume fraction profiles of A segments) from 2D and 3D simulations. The respective scores are: (a) 0.81 (2D) and 0.78 (3D); (b) 0.86 (2D) and 0.75 (3D); (c) 0.96 (2D) and 0.94 (3D).

Figure 6

Tuning the top surface affinity for each BCP component when the sidewall attracts component $A$ produced four distinct morphologies: (red square) vertical orientation with minimal to no defects and a thin layer over the top surface; (blue square) vertical orientation with minimal to no defects without this thin layer; (red circle) vertical orientation with defects and a thin layer; (blue circle) vertical orientation with defects without a thin layer. $W=8.13R_g$ (2.0 $L_o$), $H=4.06R_g$ (1.0 $L_o$), taper is $68.2^{\circ }$, and ${\mathrm{\Lambda }}_{\text{SW}}^A=4.0$ and ${\mathrm{\Lambda }}_{\text{SW}}^B=6.0$.

Figure 7

2D density profiles. (a), (b) Sidewall and top surface of the guideline attract the same component; (c), (d) Sidewall and top surface of the guideline attract different components; Strong disparity between ${\mathrm{\Lambda }}_{\text{ST}}^A$ and ${\mathrm{\Lambda }}_{\text{ST}}^B$ for (a) and (c); and weak disparity between ${\mathrm{\Lambda }}_{\text{ST}}^A$ and ${\mathrm{\Lambda }}_{\text{ST}}^B$ for (b) and (d). $\{{\mathrm{\Lambda }}_{\text{ST}}^A,{\mathrm{\Lambda }}_{\text{ST}}^B\}$ = (a) {$-3,2$}; (b) {$-1,-2$}; (c) {$1,-1$}; (d) {$-0.5,-0.25$}.

Figure 8

Tuning the top surface affinity for each BCP component when the sidewall attracts component $B$ produced four distinct morphologies: (red square) vertical orientation with minimal to no defects and a thin layer over the top surface; (blue square) vertical orientation with minimal to no defects without this thin layer; (red circle) vertical orientation with defects and a thin layer; (blue circle) vertical orientation with defects without a thin layer. $W=8.13R_g$ (2.0 $L_o$), $H=4.06 R_g$ (1.0 $L_o$), taper is $68.2^{\circ }$, and ${\mathrm{\Lambda }}_{\text{SW}}^A=0.75$ and ${\mathrm{\Lambda }}_{\text{SW}}^B=0.5$.

Figure 9

(a) Effect of $\chi N$ and guideline width on vertical lamellar formation when the top surface and sidewall attract the same blocks of the BCP; (b) 2D color map depiction for scores as a function of $\chi N$ for $H=L_o$ and taper angle $\theta =68.2^{\circ }$.

Figure 10

(a) Effect of taper angle and guideline width on vertical lamellar formation when the top surface and sidewall attract the same component, with $H=L_0$; (b) 2D color maps for scores as a function of taper angle.

Figure 11

Schematic representation of the competing energetics for the case where A segments are attracted by both top and by side (for clarity, only half planes for the lamellae are shown in the figures). (a) Steep taper: Due to steeper incline, A segments in Region 2 are unable to take advantage of the favorable interactions with the sidewalls of the guidelines; (b) Shallow taper: Shallow taper exposes the side wall area to the B segments in Region 4.

Figure 12

Schematic representations of chain conformations and the resulting density maps. (a) Steep taper: Due to the presence of steep taper there is less area available for conformational rearrangement of random copolymer brush and the overlaying polymer segments; (b) Intermediate taper: An inclined sidewall is seen to present more conformational freedom for the brush and the overlaying polymer.

Figure 13

Schematic representation of the competing energetics for the case where A segments are attracted by both top and by side (for clarity, only half planes for the lamellae are shown in the figures). (a) Smaller bottom width: Due to smaller widths, A segments in Region 2 are unable to take advantage of the favorable interactions with the sidewalls of the guidelines; (b) Large bottom width: Larger bottom width exposes the side wall area to the B segments in Region 4.

Figure 14

Morphological scores as a function of the variation in guideline width for different guideline heights at taper angle corresponding to (a) $68.2^{\circ }$ (left); and ${80}^{\circ }$ (right). 2D color maps for scores as a function of (c) Height with ${68}^{\circ }$ taper angle and; (d) Height with ${80}^{\circ }$ taper angle.

Figure 15

Schematic representation of the competing energetics for the case where A segments are attracted by both top and by side (for clarity, only half planes for the lamellae are shown in the figures). (a) Small heights: Due to smaller heights, A segments in Region 2 are unable to take advantage of the favorable interactions with the sidewalls of the guidelines; (b) Large heights: Larger height exposes the side wall area to the B segments in Region 4.

Figure 16

Effect of $\chi N$ and guideline width on vertical lamellar formation when the top surface and sidewall attract different components with $H=4.06 R_g$ and taper angle to $68.2^{\circ }$ using (a) normal plot; (b) 2D representation.

Figure 17

Effect of taper angle and guideline width on vertical lamellar formation when the top surface and sidewall are attracted to different components, with $H=4.06 R_g$ using (a) 2D plot; (b) 2D color map.

Figure 18

Schematic representation of the competing energetics for the case where A segments are attracted by both top and B segments are attracted by the sidewall (for clarity, only half planes for the lamellae are shown in the figures). (a) Shallow taper: Shallow taper exposes the side wall area to the A segments in Region 2; (b) Steep taper: Due to steeper incline, B segments in Region 4 are unable to take advantage of the favorable interactions with the sidewalls of the guidelines.

Figure 19

Variation of guideline height ($H$) versus width ($W$) at (a)/(c) $68.2^{\circ }$ (left) and; (b)/(d) ${80}^{\circ }$ (right) when the top surface and sidewall attract different components.

Figure 20

Best lamellar structure (highest score) for (a) when surface and sidewall attract same component; (b) when surface and sidewall attract different components.