Particle Dynamics in a Diblock-Copolymer-Based Dodecagonal Quasicrystal and Its Periodic Approximant by X-Ray Photon Correlation Spectroscopy

Temperature-dependent x-ray photon correlation spectroscopy (XPCS) measurements are reported for a binary diblock-copolymer blend that self-assembles into an aperiodic dodecagonal quasicrystal and a periodic Frank-Kasper $\sigma$ phase approximant. The measured structural relaxation times are Bragg scattering wavevector independent and are 5 times faster in the dodecagonal quasicrystal than the $\sigma$ phase, with minimal temperature dependence. The underlying dynamical relaxations are ascribed to differences in particle motion at the grain boundaries within each of these tetrahedrally close-packed assemblies. These results identify unprecedented particle dynamics measurements of tetrahedrally coordinated micellar block polymers, thus expanding the application of XPCS to ordered soft materials.

Figure 1

(a) Thermal treatment of SB-B to form DDQC and $\sigma$ phases for XPCS studies. The dotted line from $\sim 1500\mathrm{s}$ to $\sim 6000\mathrm{s}$ refers to a 72 h off-beam isothermal annealing step at $150°\mathrm{C}$. (b)–(e) 2D-SAXS detector images and corresponding 1D-SAXS traces for (b),(d) DDQC and (c),(e) $\sigma$. Shaded boxes labeled with ($hkl$) reflections in (d),(e) correspond to the $\mathrm{\Delta }q$ slices in which XPCS analyses were performed.

Figure 2

Acceptance regions on the 2D detector used to generate $g_2(q,\phi ,dt)$ autocorrelation functions employed in this study. (a) Regions of the 2D detector used for XPCS analysis of the SB-B DDQC phase with (b)–(e) resulting autocorrelation functions $g_2(q,dt)$, from inner to outer Debye-Scherrer rings. (f) Regions of the 2D detector examined for XPCS analysis of the SB-B $\sigma$ phase. (g) Azimuthal binning of $\sigma$ Debye-Scherrer rings, with (h) one bin isolated for clarity and (i) its corresponding autocorrelation functions $g_2(q,\phi ,dt)$. Blue curves in (b)–(e), (h) are fits to Eq. (1).

Figure 3

Temperature- and $q$-dependent relaxation times (bottom) and stretching exponents (top) measured by XPCS for DDQC (solid blue points) and $\sigma$ (open red points). 1D SAXS traces of the DDQC (bottom blue curve) and $\sigma$ (top red curve) are provided for reference in the relaxation time panels. The “${150}^{*}°\mathrm{C}$” label corresponds to $\sigma$ data, the DDQC data in this panel was collected at $145°\mathrm{C}$.

Figure 4

2D square and triangle tiling patterns of (a) $\sigma$ and (b) DDQC phases in the (a),(b) plane and projected along the $c$ axis. The three tiles used to construct each phase and the relative positions of particles decorating each tile are also shown. (c) Three types of DDQC grain boundaries associated with (1) grains with parallel periodic $c$ axes but incommensurate stacking of the columns with aperiodic order in the $a\text{−}b$ planes; (2) grains with coincident $c$-axis periodic order and incommensurate rotational symmetry in the $a\text{−}b$ planes; and (3) grains with nonparallel $c$ axes.