4.3 Results
4.3.1 Homopolymer Conformation in Dilute Solution
Small angle neutron scattering patterns for mono-domain samples of both end-on (CB) and side-on (BB) homopolymers in d5CB at nematic temperatures (Figure 4.2) are anisotropic; the anisotropy disappears above the nematic-to-isotropic transition temperature (Tni).
Figure 4.2: Two dimensional SANS patterns for End-On (left) and Side-On (right) homopolymers
(structures shown in Figure 4.1) in d5CB both the nematic phase (top) and isotropic phase (bottom) using a sample to detector distance of 4m on beamline NG-3 (NIST). The top set of patterns is in the nematic phase 7℃ below the nematic-isotropic phase transition and the bottom set of patterns is in the isotropic phase 5℃
above the transition.
The nematic scattering patterns for the two mesogen attachment geometries are oriented orthogonally to each other, consistent with previously observed ellipsoidal conformations for SGLCPs in small molecule LCs[5, 7, 8]. Taking into account the rotational axis of symmetry about the nematic director, the end-on polymer coil is an oblate spheroid with its long axes perpendicular to the director and the side-on is a prolate spheroid with its long axis parallel to the director.
Sector averages parallel and perpendicular to the director for 1wt% solutions are compared to those for 5wt% solutions of the same polymers (Figure 4.3) by rescaling the intensity to account for the effects of concentration and path length. The q-dependence and relative magnitude of Ipar(q) and Iperp(q) are indistinguishable at high q. In contrast, at low q (<0.04Å-1) the scattering in the direction of the polymer’s short axis (Ipar for CB and Iperp for BB) is significantly greater in the present 1wt% solutions than in the earlier 5wt% solutions. This change is consistent with the hypothesis that the 5wt% solutions are semidilute[22]. The scattering in the opposite direction was similar for the two concentrations, with a deviation visible only at the very lowest q values; this is consistent with the “blobs” in the semidilute solution having an anisotropic shape that is similar to that of the individual chains. The scattering patterns for 1.2wt% and 1.5wt% solutions are similar to the 1wt% data; therefore the 1wt% solutions are regarded as dilute. The scattering pattern of a 3wt% solution is intermediate between that of the 1.5% solution and that of the 5wt%, indicating that the overlap concentration between 1.5 and 3wt%.
Figure 4.3: One dimensional sector averages along the director Ipar (top) and perpendicular to it Iperp
(bottom) from SANS patterns for solutions of End-On and Side-On polymers of 2000 DP at 1wt% and 5wt% in nematic d5CB at T=Tni -7. . The higher concentration data was obtained by Neal Scruggs and Rafael Verduzco[22] and is scaled to compensate for the higher concentration.
The conformational anisotropy is clearly visible in the sector averaged scattering patterns in the nematic phase at Tni – 7℃ (Figure 4.4): the side-on polymers scatter more strongly in the direction perpendicular to the nematic director and the end-on polymers have higher intensity scattering in the parallel direction. The side-on polymers are clearly
more anisotropic and their scattering has a greater difference in functional form between the parallel and perpendicular directions. Where data is available, the isotropic scattering (Tni + 5℃) falls between the two sector averages for the nematic phase and is closer in both form and magnitude to the sector corresponding to the short axis of the polymer (Ipar
for CB and Iperp for BB), consistent with the segment density distribution being more perturbed along the direction of greater stretching.
The aspect ratio of the coil was estimated by visually fitting an ellipse to an iso- intensity contour in the 2D scattering patterns (Table 4.2, 2D). A second estimate of Rlong/Rshort (Table 4.2, right column) was found from 1D sector averages by comparing qpar and qperp values for a given intensity in a region of q where the two curves are approximately parallel (i.e., using two points on an iso-intensity contour). The two methods generate similar aspect ratios (±30%) both for the mildly oblate end-on SGLCP and the strongly prolate side-on SGLCP. The aspect ratios appear to be independent of molecular weight to within the uncertainty inherent to the methodology and accord with the aspect ratios reported for semidilute solutions of these polymers[22].
Table 4.2: Aspect ratios estimated for homopolymers in the nematic phase: (center) by visually fitting an ellipse to isointensity contours in the 2D scattering patterns and (right) by comparing q-values corresponding to equal intensities for the sector averages parallel and perpendicular to the director.
Figure 4.4: SANS intensity for parallel (red) and perpendicular (blue) sector averages for nematic solutions and circular average (green) for isotropic solutions at 1wt% in d5CB of both end-on (left) and side-on (right) SGLCPS having backbone DP=880 (top) and DP=2000 (bottom).
An upturn in intensity can be seen at low q in some of the scattering patterns (e.g.
Ipar of nematic CB880 and BB880 in Figure 4.4). This is not expected for polymer coils in the dilute regime, which should have scattering that is almost independent of q as q goes to zero. This sort of behavior has previously been observed in dilute solutions of PEO in
water[34, 35] and dilute solutions of SGLCP in isotropic solvent[36-38] and is generally attributed to interaction between the polymer coils. In the present systems, the feature was observed in less than half of the samples; when it was present, it would often be associated with the nematic phase (the feature would disappear upon heating to the isotropic phase). For most of the scattering data this effect was only seen at the lowest q- values, making it possible to fit a form factor (below) to the high-q data; however, when this feature was present, it precluded the use of a Guinier fit to estimate the overall size.
Polymer coils are fractal objects with the relationship between mass and radius for a polymer consisting of N segments and having radius R, described by R~Nν where ν is the scaling exponent. The Debye function is frequently used to model the scattering from polymer coils[24] and assumes that ν=0.5 (polymer in theta solvent or in the melt) resulting in I(q)~q-m where m=1/ν=2. A similar scattering model function has been derived for a polymer with excluded volume (more expanded conformation) and scaling ν
=1/m where m is the Porod exponent[39]. This is given by:
𝐼(𝑞) =
1𝜈U1 2𝜈�
𝛾 �
2𝜈1, U� −
𝜈U11 𝜈�𝛾 �
1𝜈, U�
(4.1)Here γ(x,U) is the incomplete gamma function and U is given by
𝑈 =
𝑞2𝑅𝑔2(2𝜈+1)(2𝜈+2)6 (4.2)
This model describes a flexible polymer chain with 1/3 < ν < 1, where the lower limit corresponds to a collapsed globule and the upper limit corresponds to a rigid rod.
This functional form is most reliable for the excluded volume polymer region of ν~0.6.
At intermediate values of q the scattering intensities for solutions of SGLCP in 5CB followed a power law scaling with m<2 (i.e., ν>0.5). For end-on SGLCPs the slopes in
the power law portions of Ipar and Iperp are seen to be similar; in the side-on SGLCPs Iperp
sometimes has a shallower slope than Ipar. Using the SANS analysis software in IGOR Pro developed by Steve Kline[33], equation 4.1 was fit to the sector averaged I(q), setting the background to zero and neglecting the very noisy high q (q>0.15Å-1) region corresponding to incoherent scattering with the other parameters allowed to float freely.
Successful fits were obtained for all isotropic solutions and for the sector average that corresponds to the shorter axis of the ellipsoidal isointensity contours in the nematic phase, i.e., Ipar for CB and Iperp for BB. (resulting Rg and ν values are in Table 4.3).
Table 4.3: Results obtained from fitting an excluded volume polymer scattering function to the scattering shown in Figure 4.4. Blank positions on the left for the nematic phase correspond to sectors for which this form factor was not a good fit. Data for CB880 in the isotropic phase was not available.
It was not possible to obtain values for Rg for the long dimension of the ellipsoids in the nematic phase (Iperp for CB and Ipar for BB) due to the lack of a plateau in the intensity at low q, with the power law scattering region extending out to the lowest available q. This indicates that the long axis of the ellipsoid is too large for the available q-range. In earlier studies, the apparent Rg was seen to be independent of the molecular weight[22] consistent with the hypothesis that the measured size and anisotropy where those of a “blob”[13] in the semidilute solution. Here, the Rg increases with molecular weight and the increase accords with the scaling exponents of ν~0.6 to within the experimental uncertainty. This is consistent with regarding the 1wt% solutions as dilute.
The minor radii of both end-on and side-on polymers in the nematic phase are seen to be slightly smaller than their isotropic radii, consistent with stretching of the polymer along the other axis. The scaling exponents obtained are consistent with behavior intermediate between a Gaussian chain (ν = 0.5) and an excluded volume polymer (ν = 0.6) with the exception of BB880 (ν=0.77 perpendicular to the director). This anomalously high value may be related to the segment density profile in the direction of the very narrow dimension (6.8nm) of the coil, as this phenomenon is not observed for the longer BB2000.