Tuning polymer architecture for quasicrystal self-assembly

Abstract

Using computer simulations and theory, we investigate the ultrasoft interactions between dendrimers formed of a central polymer connected by stiff linkers to a corona of flexible polymers, forming `pompoms' at the ends of the linkers. We show that the resulting coarse-grained interaction potential between pairs of dendrimers exhibits tunable lengthscale competition based on properties of the core and corona polymers. We present a simple model for this pair potential, which we confirm using accelerated Monte Carlo methods. We then demonstrate the connection between dendrimer structure and mesoscopic phases by presenting parameter choices that result in stable dodecagonal quasicrystals, and show that the size of the region in the phase diagram where quasicrystals are stable can be controlled by tuning details of the polymer architecture alone. These results pave the way for experimental realization of soft matter quasicrystals by identifying what overall molecular architecture leads to their stability.

Figures (4)

• Figure 1: A sketch of a pair of $N = 18$ arm molecules, each with the yellow core A particle tethered to $N$ blue B particles. These have radii $R_{\rm AA}$ and $R_{\rm BB}$, respectively; c.f. Eq. \ref{['pair-pots']}. The potentials in Eq. (\ref{['tether-pot']}), tethering each A to the $N$ surrounding B particles, are represented as gray springs.
• Figure 2: The effective coarse-grained pair potential $V(r)$ as a function of normalized distance between centers of mass $r/\ell_0$. Each molecule has $N=18$ arms, with parameters $(\beta\varepsilon_{\rm AA},\beta \varepsilon_{\rm AB},\beta\varepsilon_{\rm BB}) = (0,5.4,5)$, $(R_{\rm AB}/\ell_0,R_{\rm BB}/\ell_0) =(0.06,0.045)$. The red points are the MC simulation results, while the blue solid line is the fitting using Eq. (\ref{['eqn:Coarse-grained-PP']}), with parameters $(g_{\rm BB},g_{\rm AB},r_0/\ell_0,w/\ell_0) = (0.5002,0.8787,0.9902,0.2277)$. The black dot-dashed and dashed lines are the individual contributions from the second and third terms in Eq. \ref{['eqn:Coarse-grained-PP']}, respectively. Also displayed are snapshots from the MC simulations at separations $r/\ell_0 = 0$, 0.95, 1.9 and 3.25, highlighting the various interaction stages.
• Figure 3: Examples of three potentials obtained from the coarse-grained pair potential (\ref{['eqn:Coarse-grained-PP']}) for $(w/\ell_0)^2 = \frac{1}{2}$ (I), $\frac{1}{8}$ (II) and $\frac{1}{18}$ (III). We present each potential (a) in real and (b) in Fourier space at parameter values giving QC lengthscales. Panel (c) plots the value of $R_{\rm BB}$ as a function of $R_{\rm AB}$ for which these pair potential families have two equal minima, with panel (d) showing the ratio between the two wavenumbers $k_2/k_1>1$ as a function of $R_{\rm BB}$. The dashed horizontal line marks the DDQC lengthscale of $2\cos (\pi/12)\approx1.93$.
• Figure 4: Phase diagrams in the average density versus $R_\textrm{BB}/R_\textrm{AB}$ plane, for the three coarse-grained potentials I--III in Fig. \ref{['fig:3']}, in panels (a)-(c), respectively. The coexistence regions between neighboring phases are colored red. Logarithms of typical density distributions for each phase in case II appear in panels (d)-(f) with stars in (b) indicating their location in the phase diagram.