Sequential Self-Assembly of Polystyrene-block-Polydimethylsiloxane for 3D Nanopatterning via Solvent Annealing

This study aims to develop a strategy for the fabrication of multilayer nanopatterns through sequential self-assembly of lamella-forming polystyrene-block-polydimethylsiloxane (PS-b-PDMS) block copolymer (BCP) from solvent annealing. By simply tuning the solvent selectivity, a variety of self-assembled BCP thin-film morphologies, including hexagonal perforated lamellae (HPL), parallel cylinders, and spheres, can be obtained from single-composition PS-b-PDMS. By taking advantage of reactive ion etching (RIE), topographic SiO2 monoliths with well-ordered arrays of hexagonally packed holes, parallel lines, and hexagonally packed dots can be formed. Subsequently, hole-on-dot and line-on-hole hierarchical textures can be created through a layer-by-layer process with RIE treatment as evidenced experimentally and confirmed theoretically. The results demonstrated the feasibility of creating three-dimensional (3D) nanopatterning from the sequential self-assembly of single-composition PS-b-PDMS via solvent annealing, providing an appealing process for nano-MEMS manufacturing based on BCP lithography.


Theoretical Framework: Self-Consistent Field Theory (SCFT)
The feasibility of achieving the hole-on-dot and line-on-hole patterns was investigated using Self-Consistent Field Theory (SCFT).In this section, we present a brief description of the SCFT.Here, we consider a system of  AB diblock copolymers confined in a thin-film geometry with a total volume  .Each diblock copolymer, modeled as a continuous Gaussian chain, contains a total of N segments, where  A ( = ) are segments of type A and  B ( = (1 -)) are segments of type B. For simplicity, we assume a uniform segment density ρ 0 so that the total volume of the copolymer melt is  =   0 .In our calculations, A is the minority block corresponding to the PDMS block while B is the majority block corresponding to the PS block.The confinement effect is incorporated by using the mask method [1][2][3] .In the mask method, an extra component is introduced into the system to act as walls that confine the system.The wall component is represented by a preassigned and fixed wall density,  W ().
After the standard mean-field treatment with the incorporation of the wall density, the Helmholtz free energy per chain has the form: Minimization of the above free energy with respect to the copolymer segment density and the conjugate fields results in a set of SCFT equations: In Eqs. 2, the forward and backward propagators ((,) and  † (,)) describe the statistical mechanics of the polymer chains and satisfy the following modified diffusion equations: where (,) =  A () when  ∈ [0,] and (,) =  B () when  ∈ [,1].We solve Eqs. 3 and 4 by using the pseudo-spectral method [4] with the initial conditions (0,) = 1 and  † (1,) = 1.
To describe the interface between the polymeric and wall materials, a common choice of the form for the wall density is the hyperbolic tangent function [5,6] .To reflect the geometries of the PS-coated PS-b-PDMS thin film in the first layer, wall densities with desired morphologies need to be constructed.To achieve this, we first set up a  W () = 0.5 surface (wall surface) that separates the polymer-rich region (Ω P ) and wall-rich region (Ω W ) in the 3dimensional space.The desired geometry is depicted by the shape of the wall surface.We then assign values to  W () at all spatial points in the computational box, which contains one unit cell of the periodic nanostructure.Specifically, for each spatial point r, we determine the shortest distance between that point and the wall surface, denoted as   , and assign  W () , if  ∈ Ω P and   <  0, if  ∈ Ω P and   ≥ Here  describes the steepness of the polymer-wall interface and σ is a cut-off distance.
Throughout this work, we keep  = 0.1  and  = 0.25  constant, where   is the unperturbed radius of gyration of the AB diblock copolymer.Additionally, we set  W () = 1 wherever  W () > 0.95 to facilitate the convergence of the SCFT equations.The resulting  W () as a function of   is plotted in Figure S1.
With a preassigned wall density, we initialize the polymer density and numerically solve the SCFT equations.We solve Eqs.S3 and S4 pseudospectrally using the second-order operator-splitting method and iterate Eq.S2 by simple mixing. [7]The chain contour is discretized into 100 pieces and 64 grid points are used to discretize each spatial dimension.
After solving the SCFT equations for different nanostructures, we can determine their relative stability by comparing their free energy per chain (Eq.S1).To confirm that the spatial resolution used in our calculations is sufficient, we repeated the calculations for a small number of selected phases using twice the number of grid points (128) along each spatial dimension.
We observed that the relative free energies only shifted marginally, indicating that 64 grid points lead to sufficiently accurate results.

Wall-Profile Construction
To investigate the thermodynamic stability of the experimentally observed lines-on-holes and holes-on-dots patterns, we design two different wall profiles to mimic the holes and dots geometries of the first (bottom) layer.Figures S2a and b illustrate the wall geometries used to mimic holes and dots, respectively.
In our calculations, we use  AB  = 30.The bottom walls represent the PS coating layer, while the top walls represent the PS blocks of the wetting layer, making both the bottom and top walls selective to the PDMS (B) blocks.This selectivity is achieved by setting  AW  = 30 and  BW  = 0.The A-block compositions  A used for the cases of the holes and dots geometries are 0.34 and 0.41, respectively, matching those of the samples used in experiments.
To estimate the length scales between the dots or holes in the first layer, we first performed SCFT calculations confined between two flat walls to obtain the dots (hexagonally packed spheres) and holes (hexagonally perforated lamellae) structures (Figure S3a), in which the free energies of the different structures are also minimized with respect to   and   .This allows us to estimate the   and   to be used in the calculations for the second layer.We found that the optimized   for holes is ∼5  and that for dots is ∼4.5  (Figure S3b).The   for both cases is related to   via   = 3  .We note that although the optimized periodicities for holes and dots change slightly with variations in  A and , these values provide reasonable S5 estimations for   to be used in the top-layer calculations.For the other geometric parameters, i.e.,  and ℎ, we set {, ℎ} = {/3, } for the holes shown in Figure S2a and {, ℎ} = { /3, /2} for the dots shown in Figure S2b.For both wall geometries, we perform calculations for all candidate structures spanning a range of .The procedure to generate the candidate phases for each wall geometry, as well as the morphologies of the candidate phases (Figures 9 and 10), can be found in the main text.

Figure S1 .
Figure S1.The wall density  W () as a function of   in the (a) polymer-rich and (b) wall-

Figure S2 .
Figure S2.The schematics of the two wall geometries used in our SCFT calculation: (a) holes

Figure S3 .
Figure S3.Schematics for the nanostructures in the first layer: (a) holes; (b) dots.The

Figure S4 .
Figure S4.More detailed visualizations of the H0, H1, and D0 structures.The top, middle, and

Figure S6 .
Figure S6.Top-view FESEM images of as-cast PS-b-PDMS thin films with initial thicknesses

Figure S7 .
Figure S7.Cross-section view FESEM image after RIE treatments at which the sample was