Phase Transition to Bundles of Flexible Supramolecular Polymers

We report Monte Carlo simulations of the self-assembly of supramolecular polymers based on a model of patchy particles. We find a first-order phase transition, characterized by hysteresis and nucleation, toward a solid bundle of polymers, of length much greater than the average gas phase length. We argue that the bundling transition is the supramolecular equivalent of the sublimation transition, which results from a weak chain-chain interaction. We provide a qualitative equation of state that gives physical insight beyond the specific values of the parameters used in our simulations.

Figure 1

(a) Geometry of the patches. ${\theta }_i$ is the angle between the direction ${\overrightarrow{p}}_i$ of patch $i$ and the interparticle vector ${\overrightarrow{r}}_{ij}={\overrightarrow{r}}_j-{\overrightarrow{r}}_i$. ${\theta }_j$ is the angle between the direction ${\overrightarrow{p}}_j$ of patch $j$ and ${\overrightarrow{r}}_{ji}=-{\overrightarrow{r}}_{ij}$. ${\theta }_{ij}$ is the angle between ${\overrightarrow{p}}_j$ and $-{\overrightarrow{p}}_i$. (b) Patch potential for three values of ${\theta }_j$ for $w=0.4$. The strongest attraction occurs for ${\theta }_i={\theta }_j={\theta }_{ij}=0$ and decreases rapidly for increasing angles. Particles are depicted at half their size.

Figure 2

Temperature dependence of the aggregation fraction $\eta$ [Eq. (2)] for several values of the flexibility $w$ for $M=1330$, $\rho =1.13\times {10}^{-3}{\sigma }^{-3}$. $\eta =1/2$ defines the polymerization temperature $T^{*}$, indicated by an asterisk on the curves and on the horizontal axis. At $w=0.4$ a transition from a gas of chains to a solid bundle at $k_{B}T=0.057ϵ$ is indicated by the dotted line. Symbols denote simulation results, and solid lines are the WTPT. The dashed line is a fit of the IFA model to the curve of $w=0.3$, with $v=1.4\times {10}^{-4}{\sigma }^3$ and $G=0.91ϵ$.

Figure 3

Top: Representative configuration for $w=0.4$ at two very close temperatures. $M$, $\rho$ as in Fig. 2. (a) $k_{B}T=0.057ϵ$, where a solid bundle has nucleated. (b) $k_{B}T=0.058ϵ$, where the system is still a polymer gas. (c) Average chain length $\overline{N}(T)$ for $w=0.4$. The solid line shows the jump of $\overline{N}$ for $k_{B}T\approx 0.057ϵ$ on cooling. The dashed line shows the hysteresis while raising $T$ from an equilibrated system initially at $k_{B}T=0.057ϵ$. Inset: Same curve for several values of $w$. No hysteresis loop is shown. Notice that the bundling temperature increases with $w$.

Figure 4

Simulated gas-solid coexistence line (circles) and polymerization line (squares) for $w=0.4$, together with the theoretical monomer sublimation line ${\rho }_{\mathrm{sub}}^m$ [Eq. (7), dotted line], polymer sublimation line ${\rho }_{\mathrm{sub}}^p$ [Eq. (8), solid line], and polymerization line ${\rho }^{*}$ [Eq. (9), dashed line] based on the WTPT description of a polymer gas in equilibrium with an Einstein crystal, with $k=1.2\times {10}^3ϵ{\sigma }^{-2}$ and ${ϵ}_{\mathrm{sol}}=1.05$. In the inset we show the same, but on a linear density scale.