4.4 Discussion
4.4.4 Development of Toy Models to Test Plausibility of Interpretations for Scattering Patterns
4.4.4 Development of Toy Models to Test Plausibility of Interpretations for
𝐼(𝒒)~𝑑Ω𝑑𝜃 =∑𝑁𝑖=1∑𝑁𝑗=1𝑏𝑖𝑏𝑗〈cos �𝒒 ∙ �𝒓𝑖− 𝒓𝒋��〉 (4.7)
This form will always produce a real result for the intensity and is easily solved using elementary functions given a set of position vectors and their corresponding SLDs. The primary limitation is the number of scatterers used in the model system as the inherent computational scaling is O(N2). Two approaches were used to generate sets of ri that describe the hierarchically anisotropic micelles that are formed by SGLCP-b-PS self- assembly while keeping N to a value compatible with calculations on current personal computers (N<106).
The first toy model approach was to generate the shapes of the core and corona of the micelle on a lattice. The lattice consisted of a 2D grid of M points on each side spaced Δx apart with each point having an associated vector ri and SLD bi. Calculations were done in units of the lattice spacing with each vector ri having the lattice coordinates as its components. The SLD was chosen to represent the density of scatterers in that region of the lattice. The structure consisted of two nested ellipses with the inner having its long axis oriented orthogonal to that of the outer with the dimensions of each specified by Rpar
and Rperp , parallel and perpendicular to the nematic director. The scattering was calculated along the parallel and perpendicular directions using Equation 4.7.
The scattering predicted for one such structure is shown in Figure 4.11 with a cartoon of the structure inset. For this particular model the following parameters were used: Δx = 1nm, M=150, Rpar,core = 22nm, Rperp,core = 11nm, Rpar,corona = 46nm, Rperp,corona = 64nm, bcore = 8, and bcorona = 1. The size of the core was estimated from the data for CB900-PS1150 with the corona consisting of SGLCP coils surrounding the core. The
SLD values were chosen to be consistent with the estimated 80% polymer in the core and 10% polymer in the corona. The calculation of the scattering intensity took approximately 30 minutes to calculate on an Intel i7 computer.
Figure 4.11: Scattering pattern from ellipsoid model. The dashed red line is the scattering parallel to the nematic director while the blue solid line is the scattering perpendicular to the director. The inset on the left side of the graph is a cartoon of the structure used to generate the scattering.
The second approach incorporated much more molecular level detail into the toy model representation. The structure was built up of scattering points representing the polymer segments with each polystyrene segment having an SLD of bi = 1 and each SGLCP segment having an SLD of bi=3 to represent the larger size of the side-group.
The overall structure was constructed by placing individual polymer coils representing the SGLCP around a central region representing the polystyrene core. A schematic of the structure is inset in (Figure 4.12). Each SGLCP coil was generated using a 3D, lattice- free random walk with the segment length (0.5nm) and the number of segments (900) corresponding to CB900-PS1150. Excluded volume was included by rejecting configurations in which segments crossed. The anisotropy and orientation of an SGLCP
was included by making steps parallel to the director more probable to occur and adjusting the probability until the aspect ratio of the resulting walk was approximately equal to that of an end-on SGLCP homopolymer (~1.6). For the core, another random walk was started at the end of the SGLCP coil with higher density achieved by ignoring the excluded volume and reducing the length of the segments until the density was similar to the 80% polymer that was estimated for the core.
Figure 4.12: Scattering pattern from random walk model. The dashed red line is the scattering parallel to the nematic director while the blue solid line is the scattering perpendicular to the director. The inset on the left side of the graph is a cartoon of the structure used to generate the scattering.
The predicted scattering from one such structure is shown in Figure 4.12. This structure was based on CB900-PS1150 and consisted of 23 polymer coils, with the SGLCP corona coils having their centers positioned around the central core coils, resulting in an anisotropic micelle structure. In order to take into account the ensemble averaging of a real system several replicates of the structure were generated. Each replicate had independently generated random walks for each of the polymer coils, with the starting points of the walk fixed in order to maintain the micelle structure. Scattering
was calculated for each of the replicates using Equation 4.7 and then all of the scattering patterns were averaged. The scattering calculation were much slower for this model due to the greater number of scatterers used, and the same computer used for the first model took several days to calculate the scattering pattern shown in Figure 4.12.
Both approaches qualitatively reproduce some of the features seen in the scattering curves for CB900-PS1150 (Figure 4.9). Both the ellipse (Figure 4.11) and the random walk (Figure 4.12) models show low-q plateaus in the scattering both parallel and perpendicular to the director. For both models Ipar is higher at the lowest accessible q and plateaus earlier, corresponding to an overall micelle structure with the long axis oriented perpendicular to the director. Both show an intermediate plateau in Iperp, in accord with the experimental scattering data. For the ellipse model (Figure 4.11), the scattering at higher q values is dominated by the beat patterns, resulting from the discrete lattice structure and sharp boundaries between the inner and outer scattering features. For the random walk model (Figure 4.12) the scattering shows a smooth power law dependence for q>0.08Å-1 with the slope consistent with a polymer in good solvent.
Although many of the features of experimental scattering are reproduced by the toy models, there are some discrepancies. In the scattering for both models, the two intensities Ipar and Iperp cross only once, having a low-q region with Ipar>Iperp and a high-q region with Ipar<Iperp. In the experimental data (Figure 4.9) the curves cross twice with Ipar>Iperp at both high-q and low-q and an intermediate region 0.01Å-1 <q<0.02Å-1 in which Ipar<Iperp. The locations and intensities of the experimentally determined scattering are also not quantitatively reproduced by either of the toy models.
Although neither toy model quantitatively reproduced the scattering data, they both capture some of the unique features of the scattering by SGLCP-b-PS block copolymers in aligned 5CB. We are optimistic that further refinement of the toy model and further sampling of the structure space will enable more quantitative agreement between the model and the experimental data. Work on the refinement of the toy models is ongoing and should provide greater insight into the size and conformation of hierarchically structured anisotropic micelles.