Hexagonal Order in Crystalline and Columnar Phases of Hard Rods

We report a study of colloidal suspensions of highly monodisperse semiflexible chiral rodlike viruses, denoted $fd$, in the range of high concentrations. Small angle x-ray scattering experiments reveal the existence of two hexagonal phases: the first one is crystalline and the second one is hexatic columnar, as shown by its short-range positional order. The suspension of rodlike viruses is the first experimental system showing the whole phase sequence with increasing particle concentration theoretically predicted for systems of hard rods, ranging from the chiral nematic via the smectic to columnar and crystalline phases.

Figure 1

Schematic representation of the crystal (Cr), columnar (Col), and smectic (Sm) phases of rodlike particles.

Figure 2

(a) X-ray scattering pattern of the columnar monodomain exhibiting a sixfold symmetry and (b) its averaged radial profile showing the (100), (110), and (200) Bragg reflections [21] as the typical signature of a hexagonal packing [10]. Inset in (b): Porod representation of the scattered intensity for a better visualization of the different Bragg peaks. (c) From the sixfold azimuthal intensity modulation (open circles) is extracted the orientational order parameter, $C_6$, by a fit (red solid line) according to Eq. (1). (d) Optical texture of the columnar phase observed by polarizing microscopy. The scale bar indicates $30\mu \mathrm{m}$.

Figure 3

(a) Coexistence of the columnar (left) and smectic (right) phases observed by DIC microscopy (digitally enhanced picture). The scale bar indicates $10\mu \mathrm{m}$. (b) Log-log scale representation of the virus concentration (or the corresponding volume fraction $\varphi$) dependence of the (100) Bragg peak position (red triangles, $I=50\mathrm{mM}$ [7]). The red solid line is a data fit $q_{100}\propto {\varphi }^{1/2}$, characterizing a swelling law of 2D order. Blue circles show the corresponding correlation length $\xi$ and the blue dashed line indicates the instrumental resolution [21]. The phase coexistence ranges are shown by dashed regions.

Figure 4

Average radial intensity profile of the synchrotron SAXS patterns in the (a) smectic, columnar, and (c) crystalline phases [23]. (b),(d) Fits (solid lines) according to Eq. (2) of the (100) Bragg reflections (open symbols) which are not resolution limited. The dashed line shows the subtracted linear background.