Strategy to simulate and fit 2D grazing-incidence small-angle X-ray scattering patterns of nanostructured thin films

A strategy to simulate and fit 2D grazing-incidence small-angle X-ray scattering patterns of supported, nanostructured soft-matter thin films using the distorted-wave Born approximation is introduced. The different scattering contributions of the nanostructure, surface roughness and background scattering are treated separately and are adjusted step by step. To minimize calculation efforts, 1D line cuts are chosen, in which the scattering is predominantly attributed to one of the contributions, and the parameters found are used in the subsequent steps. Hence, separate measurements of the bare substrate are beneficial.

The beam is chosen to impinge on the surface of the film under an incident angle of 0.2°, which, at the wavelength given (0.15 nm), is slightly higher than the critical angles of the film (αc,film  0.115°) and of the substrate (αc,sub  0.162°).Thus, the X-ray beam penetrates the film fully and is partially reflected by the film-substrate interface.This angle has often been used, because it results in higher scattering intensities than higher incident angles, facilitating time-resolved measurements.The 2D GISAXS pattern of the model sample was simulated using BornAgain v1.19.0 (Figure S1b).At this, random noise is introduced by transforming each pixel I(q) according to a Gaussian distribution with a standard deviation of σnoise(q)  γnoise I(q) 1/2 , where γnoise = 0.5 is a scale factor.For the sake of clarity, no additional parasitic scattering background was included in the model.The 2D GISAXS pattern of the model sample features high intensity in the region qz = 0.2-0.3nm -1 , which is extended along the qy axis and comprises a few intensity oscillations along qz.This is the socalled Yoneda band, i.e. the region between the critical angles of the substrate (αc,sub  0.162°) and the polymer film (αc,film  0.115°), i.e. qz = 0.265 and 1.230 nm -1 .The reflected beam is at qz = 0.292 nm -1 and qy = 0 and is masked in the simulation.A high-intensity streak centred at qy = 0, that extends along qz, is observed and features intensity oscillations as well.It is due to scattering and reflections from the film/vacuum and the film/substrate interfaces, and its width in qy is related to their roughnesses.Such streaks, albeit of weaker intensity, are also present at qy = ±0.3nm -1 , indicating a certain periodicity in the film plane.Generally speaking, the length of the streak is related to the height of the features which are at its origin.Finally, a halo of scattering extends around the reflected beam, which seems to contain information about the structure within the film.At qz values below the critical angle of the film, the halo-like scattering quickly decays, while significant scattering of the vertical streaks remains.
These features are reflected in the 5 linecuts, that are described in the main text (see Figure 1).They are shown in Figure S2 along with the contributions calculated in the 4 steps.Linecut I is the intensity profile along qy in the Yoneda band.Linecut II describes the enhanced intensity in the Yoneda band as well as the intensity oscillations within.Linecut III shows both, the intensity decay around qy = 0 and the position of the streaks at qy = ±0.3nm -1 .Linecut IV is the intensity profile along qz at qy = 0 and contains the same oscillations as linecut II.In addition, it features intensity oscillations above the Yoneda band.Linecut V is constant, has a weak maximum at qy = 0.3 nm -1 , presumably due to the streaks due to the protrusions, and a strong decay above.

S3. Simulations of substrates for different roughness parameters
Simulations of bare substrates are performed to illustrate the dependence of the scattering patterns on the surface roughness parameters.At this, the parameters in Table S1 were used (excluding the film and its surface and inner structure) and varying either the root-mean-square roughness, σrms,sub, the Hurst parameter Hsub or the lateral correlation length ξsub.
The effect of σrms,sub on the 2D GISAXS patterns of the bare substrate and the corresponding linecuts      Table S3 Parameters used in the simulation of the example film in Figure 10.

Figure
Figure S1 (a) Sketch of the model sample.(b) Simulated 2D GISAXS pattern of the model sample characterized by the parameters given in Table S1.(c) Close-up of (b) in the Yoneda band region.The arrows indicate (from top to bottom) the calculated positions of the reflected beam (covered by a mask) and the critical angles of the substrate and the film.

Figure
Figure S2 Linecuts I at qz  0.26 nm −1 (a), II at qy  0.17 nm −1 (b), II at qy  0.29 nm −1 (c) III at qz  1.01 nm −1 (d), IV (e) and V at qz  0.13 nm −1 (f) of the 2D GISAXS pattern of the model sample (open symbols) and of the fits in steps 1 (grey lines), step 2 (blue lines), step 3 (green lines) and step 4 (red lines).Insets in (b), (c) and (e) are close-ups of the Yoneda band region between 0.2 and 0.4 nm −1 .

Figure
Figure S3 Linecuts I (a), III (b), IV (c) and V (d) of GISAXS patterns of the sample (black dotted line) and the background contributions IDB (red dashed line) and Isur (green dashed line) given in Figure 4.The sum of the three linecuts is shown as a blue full line.
and IV is shown for for ξsub  200 nm and Hsub = 1.0 in FigureS4.With increasing σrms,sub, the intensity of the vertical scattering streak is reduced at high qz values (FigureS4c and e).At the same time, an overall increase of intensity is observed.The shape of the horizontal linecut III is, apart from the increase in intensity, unaffected by a change of σrms (FigureS4b and d).

Figure
Figure S4 (a) Simulated 2D GISAXS patterns of bare substrates having root-mean-square roughnesses, σrms, of 0.1 nm, 0.5 nm, 1.0 nm and 2.0 nm (left to right).All other parameters in the simulations are the same as in Table S1, i.e., H  1.0 and ξ  200 nm.(b, c) Linecuts III (b) and IV (c) of the patterns in (a).(d, e) Same linecuts as in (b) and (c), but normalized to the highest intensity of the cuts at σrms  0.1 nm.Note that the normalization factors in (d) and (e) are different to ensure maximum overlap of the respective curves.The effect of ξsub on the 2D GISAXS patterns of the bare substrate and the corresponding linecuts III and IV is shown for σrms,sub  0.1 nm and Hsub = 1.0 in FigureS5.With decreasing ξsub, the vertical streak broadens along qy, and its overall intensity decreases.This goes along with a shift of the onset of the intensity decay in the horizontal linecut III towards larger qy (FigureS5b and d).The vertical linecut features a plateau which extends towards larger qz values, when ξ is decreased.

Figure
Figure S5 (a) Simulated 2D GISAXS patterns of bare substrates having lateral correlation lengths ξ of 200 nm, 100 nm, 50 nm and 10 nm (left to right).All other parameters in the simulations are the same as in Table S1, i.e., σrms  0.1 nm and H  1.0.(b, c) Linecuts III (b) and IV (c) of the patterns in (a).(d, e) Same linecuts as in (b) and (c), but normalized to the highest intensity of the linecuts at ξ  200 nm.Note that the normalization factors in (d) and (e) are different to ensure maximum overlap of the respective curves.The effect of Hsub on the 2D GISAXS patterns of the bare substrate and the corresponding linecuts III and IV is shown for σrms,sub  0.1 nm and ξsub  200 nm in Figure S6.The value of Hsub affects the 2D GISAXS patterns overall only weakly, but it defines the steepness of the intensity decay at large qvalues: With decreasing Hsub, the decays in the horizontal linecut III (Figure S6b and d) and the vertical linecut IV (Figure S6c and e) become less steep.

Figure
Figure S6 (a) Simulated 2D GISAXS patterns of bare substrates having Hurst parameters, H, of 1.0, 0.75, 0.50 and 0.25 (left to right).All other parameters in the simulations are the same as in Table S1, i.e., σrms  0.1 nm and ξ  200 nm.(b, c) Linecuts III (b) and IV (c) of the patterns in (a).(d, e) Same linecuts as in (b) and (c), but normalized to the highest intensity of the cuts at H  1.00.Note that the normalization factors in (d) and (e) are different to ensure maximum overlap of the respective curves.

Figure S8
Figure S8 Residual plots (see equation 3 in the main text) of experimental and simulated patterns of the example film at step 1 (a), step 2 (b), step 3 (c) and step 4 (d).

Table S1
Parameters used in the simulation of the model film

Table S2
Parameters used in the simulation of the bare substrate in Figure9and FigureS7.