4.4 Discussion
4.4.3 SGLCP-b-PS Structural Length Scales
The scattering patterns for the block copolymers in the nematic phase show a hierarchical structure (Error! Reference source not found.) that does not conform to standard form factors for a core-shell micelle with spherical, cylindrical, or ellipsoid cores. This is due to the fact that these models assume a uniform corona thickness in all directions and cannot account for a corona that is anisotropic but with an orientation perpendicular to that of the core. In the absence of an applicable model, it is common to use model-independent scattering functions that allow the approximate length scales of scattering features to be determined without any further information about the underlying structure. We apply this approach to the sector averaged scattering patterns of nematic solutions for sectors parallel and perpendicular to the director.
The scattering corresponding to the shorter axis of the corresponding homopolymer SGLCP (parallel to the director for end-on polymers and perpendicular to the director for side-on polymers) only contains a single feature in the accessible range of q; we extract a length scale associated with this feature using a Guinier function at low q and approximate the homopolymer-like scattering at high q using Porod function (I(q) ~ q-n) [23]. The resulting values of this “Guinier radius” increase with PS block length: in
the end-on type SGLCP-b-PS this apparent Guinier radius (parallel to the director) increases from approximately 20nm for PS400 and PS550 to approximately 30nm for PS800 and PS1150, and in the side-on type SGLCP-b-PS (apparent Guinier radius perpendicular to the director), from approximately 15nm to 35nm (Table 4.6). These apparent radii agree with the half the diameter observed in unstained TEM (Table 4.5).
Therefore, we interpret these as these apparent Gunier radii as those of the larger dimension of the PS core.
The orthogonal sector averages appear to have two features, but data at even lower values of q (outside the accessible range in these experiments) would be needed to extract an overall size. It is, nevertheless, possible to determine the length scale of the intermediate q (0.01Å-1 ≤ q ≤ 0.03Å-1) scattering feature. Because of the superposition of real-space scattering features in this region of q it is not possible to adequately describe the scattering by a single model. In such situations it is a reasonable first approximation to linearly superimpose the scattering intensity of multiple form factors, and this approach has been used for a variety of systems with hierarchical structures[42, 43] and is similar in concept to the Beaucage fitting approach[44]. Thus two models were combined, a Guinier-Porod model for the structure and a Lorentzian correlation length model to account for scattering contribution of the intra-chain correlation. The following function was fit to the more complicated scattering:
𝐼(𝑞) = 𝐷𝑒
−13𝑞2𝑅𝑔2+ 𝐵 for 𝑞 ≤ 𝑞
1(4.4)
𝐼(𝑞) = 𝐴
𝑞
𝑛+ 𝐶
1 + (𝑞𝜉)
𝑚+ 𝐵 for 𝑞 > 𝑞
1(4.5)
Here the parameters A, C, and D are the scaling factors of the Porod, Lorentzian and Guinier functions, respectively, and B is the incoherent background contribution which was set to a very low value, as most of the background scattering was accounted for by the subtraction of solvent scattering from the data. The Guinier term (equation 4.4) reproduces the plateau part of the hump (0.01Å-1 ≤ q ≤ 0.03Å-1) and is matched at q=q1 to the Porod and Lorentzian scattering (equation 4.5). In this case the Porod term has an exponent n~4 corresponding to surface fractal scattering, most probably from the core- corona interface. The Lorentzian term describes the scattering for a polymer chain at high q where the ξ is the correlation length and the power m=1/ν is the scaling exponent of the chain, in this case corresponding to a polymer in good solvent. The values of n and m were found to be very similar for all molecular weights of both end-on and side-on polymers and were fixed to the physically reasonable values of n=4 and m=2 in order to reduce the number of floating parameters. The resulting apparent radii again increase with PS block length: in the end-on type SGLCP-b-PS this apparent Guinier radius (perpendicular to the director) increases from approximately 10nm for PS400 and PS550 to approximately 14nm for PS800 and PS1150, and in the side-on type SGLCP-b-PS (apparent Guinier radius perpendicular to the director), from approximately 10nm to 16nm (Table 4.6). These values are small enough that they could be spanned by the PS blocks and we believe they correspond to the thinner dimension of the PS core.
Table 4.6: Rg from SANS for SGLCP-b-PS diblocks in nematic LC.
Linear fits were obtained for the sector averaged intensity parallel to the director in end-on polymers (Error! Reference source not found., left side, open red circles) in the region (0.005Å-1<q<0.008Å-1) which also satisfies the condition that qRg<1.2 needed for the Guinier approximation. For the side-on polymers, the sector perpendicular to the director (Error! Reference source not found., right side, filled blue squares) showed a good plateau for BB1250-PS400 and BB1050-PS550 but had fewer points available for BB1100-PS800 and BB900-PS1150, although a good linear fit was nevertheless obtained. The combined model described earlier provided a good fit to the perpendicular sector of the end-on polymers(Error! Reference source not found., left side, filled blue squares) and the parallel sector of the side-on polymers(Error! Reference source not found., right side, open red circles) for q > 0.01Å-1. The pair of dimensions obtained from the Guinier fits for all of the block copolymers is consistent with our interpretation of the data as showing an anisotropic micelle core.
4.4.4 Development of Toy Models to Test Plausibility of Interpretations for