Self-organization of tip-functionalized elongated colloidal particles

Weakly attractive interactions between the tips of rodlike colloidal particles affect their liquid-crystal phase behavior due to a subtle interplay between enthalpy and entropy. Here we employ molecular dynamics simulations on semiflexible, repulsive bead-spring chains where one of the two end beads attract each other. We calculate the phase diagram as a function of both the volume fraction of the chains and the strength of the attractive potential. We identify a large number of phases that include isotropic, nematic, smectic-$A$, smectic-$B$, and crystalline states. For tip attraction energies lower than the thermal energy, our results are qualitatively consistent with experimental findings: We find that an increase of the attraction strength shifts the nematic to smectic-$A$ phase transition to lower volume fractions, with only minor effect on the stability of the other phases. For sufficiently strong tip attraction, the nematic phase disappears completely, in addition leading to the destabilization of the isotropic phase. In order to better understand the underlying physics of these phenomena, we also investigate the clustering of the particles at their attractive tips and the effective molecular field experienced by the particles in the smectic-$A$ phase. Based on these results, we argue that the clustering of the tips only affects the phase stability if lamellar structures (“micelles”) are formed. We find that an increase of the attraction strength increases the degree of order in the layered phases. Interestingly, we also find evidence for the existence of an antiferroelectric smectic-$A$ phase transition induced by the interaction between the tips. A simple Maier-Saupe-McMillan model confirms our findings.

Figure 1

Schematic representation of the initial configurations and snapshots of resulting configurations after 20 000 000 time steps of simulations starting from each of them at approximately constant volume fraction $\varphi =$ 0.52 (at a fixed pressure 1.7 $\epsilon /D^3$) for the attraction strength of 0.6 $k_{B}T$. In the initial configuration in (a), all particles have the same orientations, in (b), particles alternate orientation within the same layer, and in (c), all particles have the same orientation within the layer, but the orientation alternate from consecutive layers.

Figure 2

Calculated phase diagram of repulsive, rodlike particles that have a single attractive tip as a function of the attraction strength between the end tips (in units of thermal energy $k_{B}T$) and the volume fraction $\varphi$. The particles have a base aspect ratio $L_0/D=10.77$ and flexibility of $L/L_P=0.3$ (see the main text.) The following phases are identified: isotropic (orange circle), nematic (yellow square), smectic-$A$ (green up-triangle), smectic-$A_2$ (dark green down-triangle), smectic-B (blue diamond), smectic-$B_2$ (dark blue diamond), crystalline (purple pentagon), and ${\mathrm{crystalline}}_2$ (dark purple pentagon). Snapshots of the data points highlighted by red circles in the phase diagram are presented in Fig. 3.

Figure 3

Snapshots of the simulations at approximately constant volume fraction $\varphi$ in the (a) isotropic phase at $\varphi \simeq 0.30$, (b) nematic phase at $\varphi \simeq 0.38$, (c) smectic-$A$ and smectic-$A_2$ phases at $\varphi \simeq 0.51$, and (d) smectic-$B$ (left) and crystal phases (right) at $\varphi \simeq 0.56$ obtained. From left to right, a pair of snapshots for each value of the volume fraction is given for attraction strengths of 0.4 and 1.6 in units of $k_{B}T$. The corresponding data points are highlighted by red circles in the phase diagram in Fig. 2.

Figure 4

(a) Average aggregate size for various attraction strengths between the end groups as a function of the volume fraction of particles. Attraction strengths 0.2, 0.6, 0.8, 1.2, 1.6, and 2.0 in units of $k_{B}T$ from right to left. The inset is an enlarged view of the graph for lower volume fractions for attractive strength equally spaced between 0 and $1k_{B}T$. (b) Fraction of monomers, dimers, trimers, or larger aggregates, which are connected via the attractive ends for attraction strength 0.2 in units of $k_{B}T$. The particles have aspect ratio $L_0/D=10.77$ and a flexibility of $L_0/L_P=0.3$. The following phases are identified: isotropic (orange circle), nematic (yellow square), smectic-$A$ (green up-triangle), smectic-$A_2$ (dark green down-triangle), smectic-$B_2$ (dark blue diamond), and ${\mathrm{crystalline}}_2$ (dark purple pentagon).

Figure 5

Average aggregation number for the attraction strengths between 0 and 2 in units of $k_{B}T$ divided by pressure and presented as a function of the volume fraction $\varphi$ in the isotropic (orange circle) and nematic (yellow square) phases. The particles have a base aspect ratio $L_0/D=$ 10.77 and flexibility of $L/L_P=$ 0.3.

Figure 6

Scaled interlayer distance $\lambda /(L+D)$ of repulsive, rodlike particles with a single attractive tip as a function of the volume fraction $\varphi$ for attraction strengths between the end groups ranging from 0 to 2 in units of $k_{B}T$ from top to bottom in the smectic-$A$ (green up-triangle), smectic-$A_2$ (dark green down-triangle), smectic-$B$ (blue diamond), smectic-$B_2$ (dark blue diamond), crystalline (purple pentagon), and ${\mathrm{crystalline}}_2$ (dark purple pentagon) phases. Particles have aspect ratio $L_0/D=10.77$ and a flexibility of $L_0/L_P=0.3$. The inset is an illustration representing the interlayer distance $\lambda$.

Figure 7

Average interlayer distance relative to the average rod length ${\lambda }_{\mathrm{odd}}/(L+D)$ for odd and ${\lambda }_{\mathrm{even}}/(L+D)$ for even interlayer distances of repulsive, rodlike particles that have an attractive tip in the smectic-$A$ (green up-triangle), smectic-$A_2$ (dark green down-triangle), smectic-$B_2$ (dark blue diamond), and ${\mathrm{crystalline}}_2$ (dark purple pentagon) phases for attraction strengths of (a) 0.2, (b) 0.6, and (c) 1.0 in units of $k_{B}T$. Particles have aspect ratio $L_0/D=10.77$ and flexibility of $L/L_P=0.3$.

Figure 8

Antiferroelectric order parameter $\delta$ of repulsive, rodlike particles that have a single attractive tip as a function of volume fraction $\varphi$ for various values of the attraction strength between the attractive tips. In the inset we compare the antiferroelectric $\delta$ and smectic $\sigma$ order parameters for the attraction strength of 0.7 in units of $k_{B}T$. Particles have aspect ratio $L_0/D=10.77$ and flexibility of $L_0/L_P=0.3$. The following phases are identified: nematic (yellow square), smectic-$A$ (green up-triangle), and smectic-$A_2$ (dark green down-triangle) phases.

Figure 9

Smectic order parameter $\sigma$ of repulsive, rodlike particles that have a single attractive tip as a function of $\mathrm{\Delta }\varphi /{\varphi }_{N-\mathrm{Sm}}$, where $\mathrm{\Delta }\varphi$ is the difference between the volume fraction and the volume fraction at the nematic–to–smectic-$A$ (for $\epsilon =$ 0) or smectic-$A_2$ phase (for $\epsilon =$ 0.7 $k_{B}T$) transition, $\mathrm{\Delta }\varphi =\varphi -{\varphi }_{N-\mathrm{Sm}}$, for attraction strengths of 0 and 0.7 $k_{B}T$. Particles have aspect ratio $L_0/D=$ 10.77 and flexibility of $L_0/L_P=$ 0.3. The following phases are identified: nematic (yellow square), smectic-$A$ (green up-triangle), smectic-$A_2$ (dark green down-triangle), smectic-$B$ (blue diamond), and smectic-$B_2$ (dark blue diamond) phases.

Figure 10

Smectic ordering potential of repulsive, rodlike particles with a single attractive tip. Particles have an aspect ratio of $L_0/D=10.77$ and a flexibility of $L_0/L_P=0.3$. (a) Height of the smectic ordering potential of repulsive as a function of the attraction strength between tips in the smectic-$A$ (green up-triangle) or smectic-$A_2$ (dark green down-triangle) phases at a volume fraction of $\varphi \cong 0.52$. In the inset, there is the same height of the smectic ordering potential of repulsive as a function of the volume fraction for attraction strengths 0, 0.4, 0.8, and 1.2 in units $k_{B}T$. (b) Width of the smectic ordering potential of repulsive as a function of the attraction strength between tips in units of $k_{B}T$. In the inset, the smectic ordering potential $\mathrm{\Delta }U\left(z\right)$ is presented as a function of the position along the director, normalized by the particle length $z/L$. The increase of the attraction strength drives the stabilization of the smectic-$A$ and smectic-$A_2$ phases.

Figure 11

Theoretical approach to describe the antiferroelectric phase transition for perfectly parallel, hard rods that have a single attractive end. Phases identified are as follows: nematic (yellow square), smectic-$A$ (green up-triangle), and smectic-$A_2$ (dark green down-triangle). (a) Smectic and antiferroelectric order parameters numerically calculated for the attraction strength $\tilde{\epsilon }=6$ and the adjustable parameter $\gamma =5$, chosen to adjust the volume fraction interval for convenient viewing. (b) Phase diagram obtained from classification based on the order parameters for the same value of the adjustable parameter $\gamma =5$. We find that the nematic phase is destabilized in favor of the smectic-$A_2$ phase and that the smectic-$A$ phase is suppressed at high-enough attraction strength $\tilde{\epsilon }$.

Figure 12

Energy term $\beta \mathrm{\Delta }{U}_{\mathrm{Sm}{A}_2}\left(z\right)$ or smectic ordering potential as a function of the position normalized by the layer thickness numerically calculated for the attraction strength $\epsilon =6$ and the adjustable parameter $\gamma =5$ in the nematic (at volume fraction $\varphi =0.3$), smectic-$A$ (at $\varphi =0.5$), and smectic-$A_2$ (at $\varphi =0.7$) phases. In the inset, the same energy term $\beta \mathrm{\Delta }{U}_{\mathrm{Sm}{A}_2}\left(z\right)$ calculated for the volume fraction $\varphi =0.5$ and the adjustable parameter $\gamma =5$ in the smectic-$A$ (at attraction strengths $\tilde{\epsilon }=0$ and 6), and smectic-$A_2$ (at $\tilde{\epsilon }=12$ and 18) phases.

Figure 13

Height of the effective smectic ordering potential is given by $\sigma \varphi (\gamma +\tilde{\epsilon }{\delta }^2)$ as a function of the volume fraction for the attraction strengths $\tilde{\epsilon }$ from 0 to 24 at the interval of 3. The following phases are identified: nematic (yellow square), smectic-$A$ (green up-triangle), and smectic-$A_2$ (dark green down-triangle) phases. The linear dependence of the amplitude of the smectic ordering potential with the volume fraction is also what we find in our simulations. See the inset in Fig. 10.

Figure 14

Height of the effective smectic ordering potential is given by $\sigma \varphi (\gamma +\tilde{\epsilon }{\delta }^2)$ as a function of the attraction strength at the volume fraction 0.5. The following phases are identified: smectic-A (green up-triangle) and smectic-$A_2$ (dark green down-triangle) phases. The linear dependence of the amplitude of the smectic ordering potential with the attraction strength is also what we find in our simulations. See Fig. 10.