Interaction of chiral rafts in self-assembled colloidal membranes

Colloidal membranes are monolayer assemblies of rodlike particles that capture the long-wavelength properties of lipid bilayer membranes on the colloidal scale. Recent experiments on colloidal membranes formed by chiral rodlike viruses showed that introducing a second species of virus with different length and opposite chirality leads to the formation of rafts—micron-sized domains of one virus species floating in a background of the other viruses [Sharma et al., Nature (London) 513, 77 (2014)]. In this article we study the interaction of such rafts using liquid crystal elasticity theory. By numerically minimizing the director elastic free energy, we predict the tilt angle profile for both a single raft and two rafts in a background membrane, and the interaction between two rafts as a function of their separation. We find that the chiral penetration depth in the background membrane sets the scale for the range of the interaction. We compare our results with the experimental data and find good agreement for the strength and range of the interaction. Unlike the experiments, however, we do not observe a complete collapse of the data when rescaled by the tilt angle at the raft edge.

Figure 1

Illustration of mesh refinement for a single raft of radius 4 centered at (10,10). A much finer grid was utilized in obtaining our results.

Figure 2

Tilt angle ${\theta }_0$ (stars) at the edge of a single raft as a function of the raft radius $R$ (measured in units of $\lambda$) with $q=0.71$ in a background membrane with chirality equal in magnitude and opposite in sign. The solid circles are the corresponding tilt angle values from Ref. [16] for circular membranes surrounded by depletant.

Figure 3

Tilt angle $\theta$ (solid line) in a single raft surrounded by a background membrane of equal and opposite chirality ($q=0.71$) as a function of the $x$ coordinate (units of $\lambda$) measured from the center of the raft. The values of $\theta$ are compared with the results (dashed curve) found in Ref. [16] for circular membranes surrounded by depletant. The centers of the raft and the circular membrane are both located at the origin. The vertical dashed-dotted line denotes the edge of the raft and circular membrane.

Figure 4

Dependence of ${\theta }_0$, the tilt angle at the raft edge, on mesh refinement $h$ with $q=0.71$ for four different values of raft radius (measured in units of $\lambda$). The horizontal axis is the finest mesh size level (i.e., near the raft edge), measured in units of $\lambda$.

Figure 5

Tilt angle $\theta$ profile for a system of two rafts with radius $R=5$ and $q_{\text{b}}=-q_{\text{r}}=0.71$, for four values of the edge-to-edge separation $D$ of the rafts (which is given by their center-to-center separation minus the raft diameter). The value of $\theta$ is indicated by the gray scale bar to the right of each figure. Here we choose ${\lambda }_{\text{r}}={\lambda }_{\text{b}}=\lambda$ and measure lengths in units of $\lambda$. The figure shows a portion of the total system, which is a square of side $L=60$. The maximum tilt angle ${\theta }_0$ occurs at the edge of the rafts. (a) $D=0$; (b) $D=1$; (c) $D=2$; (d) $D=4$.

Figure 6

Raft interaction potential $\varphi$ (units of $k_{\mathrm{B}}T$) as a function of edge-to-edge separation. (a) and (b): ${\lambda }_{\text{r}}=0.5,R=3{\lambda }_{\text{r}}$; in (a) all lengths are in units of ${\lambda }_{\text{r}}$; in (b) all lengths are in units of ${\lambda }_{\text{b}}$; (c) and (d): ${\lambda }_{\text{b}}=0.5,R=3{\lambda }_{\text{b}}$; in (c) all lengths are in units of ${\lambda }_{\text{r}}$; in (d) all lengths are in units of ${\lambda }_{\text{b}}$. Note the collapse of the curves in (b) and (d) where the unit of length is ${\lambda }_{\text{b}}$.

Figure 7

Dependence of raft interaction $\varphi$ (units of $k_{\mathrm{B}}T$) on $q_{\text{b}},q_{\text{r}},C_{\text{r}}$, and $R$. All lengths are measured in units of ${\lambda }_{\text{b}}$ and $C_{\text{b}}=4$. The exponential expressions in the legend are fits to the data. (a) $q_{\text{r}}=-0.355,{\lambda }_{\text{r}}=1.333,R=2.231$, (b) $q_{\text{b}}=2,{\lambda }_{\text{r}}=1.333,R=2.231$, (c) $q_{\text{r}}=-1.42,q_{\text{b}}=1.42,R=2.231$, and (d) $q_{\text{r}}=-1.42,q_{\text{b}}=1.42,{\lambda }_{\text{r}}=1.333$.

Figure 8

Raft interaction potential $\varphi$ (units of $k_{\text{B}}T$) of two rafts of radius $R=2.231$ (units of ${\lambda }_{\text{b}}$) for five different linear chirality profiles where the chirality varies linearly from $q_{\text{r}}=1.42$ to $q_{\text{b}}=-1.42$ over a distance $2L_{\text{tran}}$ centered on the edge of the raft. Here $C_{\text{r}}=2.25$ and $C_{\text{b}}=4.0$.

Figure 9

Tilt angle $\theta$ in a single raft of radius $R=2.231$ (in units of ${\lambda }_{\text{b}}$) as a function of the radial coordinate (units of ${\lambda }_{\text{b}}$) measured from the center of the raft for five different chirality profiles at the raft edge. The chirality profile is assumed to be linear with a width $2L_{\text{tran}}$ centered at $x=R$. In each case $q_{\text{r}}=1.42,q_{\text{b}}=-1.42,C_{\text{r}}=2.25$, and $C_{\text{b}}=4.0$.

Figure 10

Comparison of theoretical results for the raft interaction potential $\varphi$ (symbols) with experimental data (lines) [20]. Here, $q_{\text{r}}=-1.42,q_{\text{b}}=2,{\lambda }_{\text{r}}=1.33{\lambda }_{\text{b}}$, chosen by eye to provide the best fit. Theoretical units were converted to physical units using ${\lambda }_{\text{b}}=0.65\mu \text{m}\text{and}K=125k_{\mathrm{B}}T$.