Comparing the backfilling of mesoporous titania thin films with hole conductors of different sizes sharing the same mass density

Mesoporous titania thin films backfilled with the conjugated polymer PTB7-Th or the small-molecule PhenTe-BPinPh have been studied as novel materials for hybrid photovoltaics. Together with observed structural changes due to backfilling, volumetric mass density can be excluded in determining factors that influence backfilling efficiency for solar cell applications.

IUCrJ (2020). 7, doi:10.1107/S2052252520000913 Supporting information, sup-2 Hochtemperaturöfen GmbH) to remove the polymer template and to induce anatase-type crystallization. Figure S1 Profiles of X-ray scattering length density as function of the distance from the substrate surface as extracted from XRR measurements; (a) PTB7-Th spin-coated at 1000 rpm on glass, (b) PTB7-Th spin-coated at 1500 rpm on silicon, (c) PhenTe-BPinPh spin-coated at 1000 rpm on silicon with subsequent solvent vapor annealing (SVA) and (d) without SVA.

S4. Footprint correction for XRR measurements with the Empyrean (PANanalytical) X-ray diffractometer
The footprint correction corrects the intensity ( Figure S2) for very small angles in the case that the Xray beam width is larger than the sample length as follows where is the corrected intensity, 1 is the uncorrected intensity, is the corrected sample length in beam direction, 1 is the real uncorrected sample length, is the beam width in beam direction, and is the incident angle. With the geometric relation = ℎ 1 = IUCrJ (2020). 7, doi:10.1107/S2052252520000913 Supporting information, sup-3 where ℎ is the fraction of illuminating the sample, the footprint can be corrected with knowledge of the fixed beam width = 0.08 mm which is determined by the used slit and the sample length of 1 = 20 mm. The correction is done for small angles where 1 < .

Figure S2
Geometrical depiction of parameters used for XRR footprint correction. Incident X-ray beam with width represented in orange, impinging under an angle on the sample with length 1 represented in black. With the beam fraction ℎ illuminating the sample, the corrected beam length can be calculated.

S5. Calculation of density and neutron scattering length density
The Abeles matrix method implemented in Motofit was applied to XRR data and an X-ray SLD was obtained. From this, the material volumetric density and the neutron SLD was calculated. In good approximation the SLD is defined as (Roe, 2000): = 2 2 and the refractive index for X-rays is defined as (Attwood, 1999) = 1 − + with δ = 2 2 1 and = 2 2 2 for one atom, where is the classical electron radius, = is the number density, 1 and 2 are atomic scattering factors, is the volumetric density, is the Avogadro constant and is the molar mass (g/mol). The scattering length for X-rays is defined as = 1 .
With the relation (Roe, 2000) = √2 and plugging the definition of δ into the equation for , using the definition for , and solving for yields: The form factors for single atoms were obtained from literature and the total form factors for PTB7-Th and PhenTe-BPinPh were calculated as follows (Chantler et al., 2005): PhenTe-BPinPh (C28H25BO2Te): = 266.63 The scattering lengths were different for neutrons than for X-rays and could also be obtained from literature (Rauch & Waschkowski, 2003). The relation between volumetric density and SLD was used to extract the neutron SLD by plugging in neutron scattering lengths and the calculated volumetric density. modeling. For as-prepared, mesoporous titania, the structure diameter is determined by cylindrical titania structures, arranged such that porous areas are formed and thereby defining the structure center-to-center distance. When material is infiltrated and adheres to pore walls, the average cylinder size grows, while the center-to-center distance remains constant.

S9. Gravitational correction in ToF-GISANS measurements
In GISANS and NR measurements, the nominal incident angle is usually determined by the slope of the sample surface (goniometer plane), while the incident beam is collimated and directed along the horizon. Here, we used = 0.62°. However, (i) since REFSANS has a horizontal scattering geometry; (ii) because of the ToF-mode and of the wide wavelength range used; (iii) due to the vertical fall of neutrons caused by the presence of the gravitational field, the nominal value does not correspond to the real incident angle of the neutrons, , which will actually depend on the wavelength. The determination of the real incident angle is, of course, very important because ignoring it would result in a significant systematic error in the calculation of the q-values. Figure S6 shows a schematic view of the scattering process in which the motion of neutrons with a certain wavelength is analyzed. The frame delimited by the dotted line on the right side represents a zoom of the scattering process occurring at the sample position. Starting from the last slit, the incident beam is directed towards the sample forming an incident angle of which is larger than the nominal angle because of the parabolic "trajectory" indicated in blue. In the absence of the gravitational field, the neutron motion would be a straight line, which is indicated in black. The transmitted beam continues its travel and reaches the detector plane at the position 0 , which is lower than the hypothetical position of an undisturbed beam, indicated with 0 .
To correctly evaluate the real incident angle, it is necessary to apply a robust method, which is independent of the instrumental settings. What is done is to divide the neutrons recorded by the detector in wavelength slices each of which has a predefined width, which can be selected after the experimental investigation, since all the events are recorded in list-mode. For each slice, a ROI containing the specular signal on the detector is selected: This can be easily done though a vertical line cut performed along the horizontal position corresponding to the incident beam. The specular signal is easily detectable because it usually corresponds to the most intense one and even when this is not the case, it may be detected because it must be located around the nominal value , at least for the shortest wavelengths (with least gravitational effect). The vertical position (in pixels) of the specular signal is detected for each slice, fitting a Gaussian to the experimental data belonging to the appropriate chosen ROI. In figure S6, the trajectory of the specular reflected beam is also indicated, along with the hypothetical trajectory of the beam not experiencing the presence of the gravitational IUCrJ (2020). 7, doi:10.1107/S2052252520000913 Supporting information, sup-7 field, which reaches the detector at the position . A similar procedure is applied to determine 0 : In this case the sample and the beam-stop are removed and the incident beam is directly measured on the detector, eventually using an attenuator to avoid saturating or damaging the detector. For a typical value of , even for neutrons with a wavelength of 20 Å, the difference between and over a distance of ≅ 10 m is of the order of few micrometers. Thus, it is negligible with respect to the detector pixel sizes (some millimeters).
Provided that the vertical shift is identical for both the direct and the specular beam, the real incident angle is readily obtained as where is the vertical pixel size, which converts the distance measured in pixels in a real value. The presence of the absolute value suppresses the necessity to know the orientation of the vertical axis.

S10. ToF-GISANS real incidence angle and penetration depth
The wavelength dependent penetration depth of neutrons is determined by the ratio of incident angle and critical angle. Neutrons either deeply penetrate the film when the incident angle is larger than the critical angle, or are surface sensitive when the incident angle is smaller than the critical angle (Müller-Buschbaum, 2013). As shown in Figure S7, the real incident angles in the ToF-GISANS experiments ranged from 0.62° to 0.68° and thus were always larger than the critical angles of our investigated materials. Therefore, it can be concluded that all shown measurements are bulk sensitive. Supporting information, sup-10

S12. Calculation of porosity and backfilling efficiency
The following equation states the relation between the critical angle , the neutron wavelength , and the material specific scattering length density (SLD) (Kaune et al., 2010): Through plotting of vs. and extraction of the slope via linear regressions, ( ) 1 2 is obtained and accordingly the SLD of the measured sample.
The porosity of the mesoporous titania is expressed by the following equation (Rawolle et al., 2013): where is the measured neutron SLD of the mesoporous titania film and is the theoretical neutron SLD of a solid titania film ( = 3.78 g/cm 3 ) with a value of = 2.349 × 10 −6 Å −2 (Fu et al., 2018).
The backfilling efficiency of material infiltrated into the mesoporous titania film is expressed as follows: , where is the measured neutron SLD of a composite film of mesoporous titania and infiltrated material and is the theoretical neutron SLD of the infiltrated polymer or small molecule (Rawolle et al., 2013).

S13. Remarks to the determination of SLDs
Due to the respective two Yoneda peaks (as-prepared titania and Si substrate or backfilled titania and Si substrate) lying close together, they cannot be distinguished by eye and their peak position is therefore determined by fitting a double Gaussian function into the vertical line cuts. By fitting one Gaussian for the silicon Yoneda Peak and one Gaussian for the as-prepared/backfilled titania into the vertical cuts, the material specific Yoneda peaks are determined as the peak centers of the single Gauss curves plus the angle corresponding to one detector pixel. This way, the determination of a Yoneda peak below the critical angle due to the maximum resolution of one pixel is avoided. The peak positions were determined with an accuracy of one pixel, represented by the error bars in angle.
The wavelength dependent critical angles of the Si substrate were calculated from the literature Si density (Greenwood & Earnshaw, 1988) of = 2.336 g/cm 3 , resulting in a neutron SLD value of SLD Si = 2.079 × 10 −6 Å −2 and were fixed as one wavelength dependent peak position during fitting of the double Gaussian into the vertical cuts.