Michael P. Tate, Hugh W. Hillhouse

General method for simulation of 2D GISAXS intensities for any nanostructured film using discrete Fourier transforms

Journal of Physical Chemistry C 111 (2007) 7645–7654

A fast, flexible 2D GISAXS simulation method based on the distorted-wave Born approximation (DWBA) has been developed for nanostructured thin films using discrete Fourier transforms of a N x N x N matrix that represents the electron density. By expressing the electron density in matrix form, various models of electron density distributions can be easily simulated and compared to experimental data. In addition to modeling the effects of overall symmetry and orientation on the relative intensities of the Bragg peaks, this approach can be used to evaluate specific details of the nanostructure such as pore connectivity, domain size, domain shape, positional disorder, orientation disorder, and polydispersity. These effects are included in a natural way without making a decoupling approximation between structure factor and form factor. The range of reciprocal space simulated is set by the size of the matrix N and the scale factor beta that sets the real-space length of each matrix element, given by 1/(2N beta) < vertical bar s vertical bar < 1/(2 beta). However, the computation time of the 3D transform scales with N as N(3)log(N-3). A matrix with N = 200 and a real-space resolution of beta = 1 nm was sufficient to model the relevant features of self-assembled nanomaterials while remaining computationally inexpensive. Here, we describe the methodology, show simulations for several examples, and compare simulation to experimental 2D GISAXS patterns. Specific examples include simulated 2D GISAXS patterns for 2D hexagonal nanostructures (p6mm) where the pores are perpendicular to the substrate and for (110)-oriented body-centered cubic nanostructures (Im(3) over bar m) based on the level surface approximation of the I-WP surface. For the latter, results show that systematic suppression of Bragg peaks occurs for specific values of the contour level and may be used to identify accessible phases. In addition, we compare simulated patterns to experimental 2D GISAXS synchrotron data for (111)-oriented rhombohedral (R(3) over bar m) nanostructured films. Curved arcs in the experimental data are identified by simulations to result from domain shape effects. Simulations show that the domains in the film are rhombus shaped, where the edges of the domain are co-aligned with the (100) faces of the R(3) over bar m unit cell. The simulation code, entitled NANODIFT, is written in Mathematica and is available upon request.

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