The intensity modulations along the Bragg rod can be used for determination of the specific stacking sequence by fitting a simple model for a finite number of hexagonally close-packed layers. That enables direct access to structural information of finite-sized crystals on the basis of coherent diffraction measurements, but without the challenging phase retrieval data analysis typical for CXDI. This approach was developed and demonstrated by J.-M. Meijer et. al. [160], where the Bragg rod intensity profiles, extracted form the 3D dataset presented in Figure 5.3(a), were analyzed.
According to the equation (5.3) the contribution of the structure factorS(l)is only depen-dent on the stacking sequence. Therefore, groups ofhklrods with the same contribution of the form factor given by equation (4.2) can be classified into a general family, for example all rods 10l,01l,10l,01l,11l and11lbelong to the10lfamily.
Using this model, authors of work [160] performed analysis of the intensity profiles of 3 Bragg rodes extracted from the three-dimensional dataset in reciprocal space. The l values were determined by locating the middle of the rod with respect to the center of the incident beam and scaled with the q value of the (001) Bragg peak, as this is located at l = 1. The intensity profiles for the selected Bragg rod families10l,20land21lare plotted in Figure 5.8, where the intensity modulations can clearly be seen. For each profile, the peak positions are similar while the peak amplitudes differ significantly. This is expected, because for each hk combination of indices the structure factor S(l)is the same, while the form factors Phk(l)are
Figure 5.8: Normalized experimental Bragg rod profiles of the10l,20land21l families (lines + sym-bols), showing distinct intensity modulations alongl. The peak positions of the families correspond well, while their amplitudes differ because of the different form-factor and structure factor contributions at the specifichkindices.
different.
The total number of layersN in the studied crystal grain determined by fitting the width of the Bragg peaks along the11l rod was found to be12. The first layer was defined as A. After careful examination of all possible combinations of A, B, C layers, two sequences that yielded the best match to the experimental profiles were found. Figure 5.9 shows the experimental Bragg rod profiles for the10l,20land21lBragg rod families, as well as the best fits calculated for the two found sequences. The first stacking sequence, ABCBABCBABCB, is a perfect DHCP structure. The second sequence,ABCBABCBCACB, is similar to the first one with the exception of the two underlined layers. These two layers cause a stacking defect in the perfect DHCP sequence, and change the repeatingBlayer to a repeatingClayer. The calculated I(l)profiles for the two considered DHCP sequences are very similar in terms of peak positions, shapes and amplitudes and describe all three experimental Bragg rod profiles very well. Small deviations between the experimental data and the model could be explained by our simplified approach to determining I(l). The overall good agreement obtained using the simple model for I(l) shows that the DHCP structure is the dominant packing arrangement of the colloidal spheres present in the studied crystal grain.
These results are very similar to the DHCP stacking that were determined from the 3D reconstruction. The small difference in the number of layers can be explained by the finite size of the grain. The Bragg rod model assumes equal contribution from each layer, while the reconstructed grain clearly shows that the layers at the top and bottom of the grain are much smaller than the those in the center.
5.5 Conclusions
Coherent X-ray diffractive imaging experiment with a single colloidal crystal grain was per-formed. Coherent X-ray diffraction data which included several Bragg reflections together with
Figure 5.9: Normalized experimental Bragg rod profiles (black lines + symbols) for (a) the10l, (b) the 20land (c) the21lfamilies, and modeled profiles for two DHCP structures, a perfect sequence (orange lines) and a sequence with a single stacking fault (blue dashed lines). The specific layer sequence is represented by the A, B andC sequences (given along the top), where each layer can be in either an HCP environment (red) or an FCC environment (green). Arrows indicate the region where the two DHCP model profiles mismatch the most and have a single or double peak, respectively.
surrounding speckles and intensities between them was measured by a rotation series of 2D far-field diffraction patterns. The obtained 3D dataset in reciprocal space was inverted into the electron density distribution in real space using the phase retrieval approach. In the obtained reconstruction positions of individual colloidal particles were resolved in three dimensions. The crystalline structure of the sample was characterized in terms of close-packing of perfect hexag-onal layers. The determined stacking sequence revealed14layers with almost perfect double hexagonal close-packed structure. This is a remarkable result because the DHCP structure has not been observed for colloidal crystals before. The stacking defect in the lattice was visualized in the projection of the reconstructed electron density on the [1¯10] crystallographic direction.
The reconstruction results were compared with an independent analysis of the reciprocal space data, based on the theoretical model for a scattered intensity along the Bragg rod for an exact stacking sequence of a finite number of hexagonally close-packed layers. Using this model two stacking sequences of the double hexagonal close-packed type with12layers were found to match the experimental data. Both of them are in good agreement with the results of the phase retrieval which shows that the suggested method is a feasible new route for the analysis of finite-size objects.
Our results are of a significant importance for the further progress and developments of CXDI methods with an aim to resolve the three-dimensional structure of nanocrystals with atomic resolution. They are remarkably related to the phase problem in crystallography. In essence, this is a successful experimental realization of the idea suggested by Sayre [65] (for details see Section 2 in Chapter 2), which before that was never applied for X-rays.
Chapter 6
Coherent X-ray diffraction studies of colloidal crystals upon heating
6.1 Heating and annealing treatment of colloidal crystalline structures
An important aspect of possible applications of colloidal systems in photonics and nanolithog-raphy is their behavior under heating treatment [164, 165, 166, 167, 168, 169]. On one hand, it has been shown that occasional appearance of defects in a three-dimensional colloidal crys-tal leads to significant degradation of its optical characteristics [170]. On the other hand, the photonic band gap properties can be deliberately modified by sintering and annealing at ele-vated temperatures [171,172,173]. Furthermore, the glass transition temperature for polymers is known to be greatly influenced by free surfaces [174, 175, 176]. After transferring a free-standing polystyrene film to a substrate, and thus reducing the free-surface-to-volume ratio by a factor of two, an increase of50K of the glass transition temperature has been observed [176]. In a range of temperatures below the glass transition temperature of a polymer, the crystal retains a long-range order and undergoes a blue shift of the optical attenuation bands. The tempera-ture region beyond the long range-ordered phase, when the crystal starts to deteriorate, is not well studied, although it has important technological aspects regarding the tolerable temperature range of a photonic device.
Structural studies of spin-coated polystyrene (PS) colloidal thin films during annealing us-ing a combination of grazus-ing incidence small-angle X-ray scatterus-ing and optical ellipsometry have been reported by Herzog et al. [168]. It was observed that colloidal particles flatten during annealing, and it was suggested [166] that a coalescence process takes place. Similar observa-tions were made in the work of Chen et al. [167] where the structural evolution of latex films was studied as a function of annealing time using SAXS. Two main effects due to dry sinter-ing were identified: particle deformation and aggregation of particles due to interdiffusion of polymer chains.
Presently, it is not well studied what happens with a 3D assembly of colloidal particles in
a crystal upon heating. We can generally assume that the glass transition temperature of the polymer should directly influence the melting temperature of the colloidal crystal. The physical processes taking place during heating and annealing of the colloidal crystal involve particle shape change, thermal shrinkage, interpolymer diffusion and particle melting [171, 172,166].
In the present work, we aim at a quantitative investigation of lattice distortions and particle shape changes in colloidal crystals upon heating treatment. For this purpose, we perform a detailed analysis of Bragg peaks in diffraction patterns of PS colloidal crystals measured in situ during incremental heating in a range of temperatures below and above the glass transition temperature of bulk polystyrene (373K).