Engineering single-polymer micelle shape using nonuniform spontaneous surface curvature

Conventional micelles, composed of simple amphiphiles, exhibit only a few standard morphologies, each characterized by its mean surface curvature set by the amphiphiles. Here we demonstrate a rational design scheme to construct micelles of more general shape from polymeric amphiphiles. We replace the many amphiphiles of a conventional micelle by a single flexible, linear, block copolymer chain containing two incompatible species arranged in multiple alternating segments. With suitable segment lengths, the chain exhibits a condensed spherical configuration in solution, similar to conventional micelles. Our design scheme posits that further shapes are attained by altering the segment lengths. As a first study of the power of this scheme, we demonstrate the capacity to produce long-lived micelles of horseshoe form using conventional bead-spring simulations in two dimensions. Modest changes in the segment lengths produce smooth changes in the micelle's shape and stability.

Figure 1

Illustration of lock and key mechanism, adapted from [7]. The enzyme, in yellow, is meant to interact specifically with a substrate, shown in green. To avoid unwanted interactions, the enzyme has a specific shape to which only a substrate of complementary shape may bind. More generally, the term “lock and key mechanism” may refer to any interaction controlled by shape.

Figure 2

Two views of a model multiblock copolymer. The multiblock contains two species of monomer beads, shown in red (light gray) and blue (dark gray), connected by bonds shown in black. One view of the multiblock is as a collection of homopolymer blocks. The other view represents the multiblock as a collection of diblocks joined end to end.

Figure 3

Illustration of micelle shape dependence on diblock composition. Diblocks with a very small solvophobic block tend to form highly curved spherical micelles. Diblocks with a more symmetric composition form flat bilayers. The case of cylindrical micelles is intermediate to these two. Figure adapted from [40].

Figure 4

Schematic of proposed mechanism for making micelles of designed shape. A shape-designed micelle, shown in two dimensions for simplicity, has a solvophobic interior (shown in red). At the surface, there are diblocks containing both solvophobic and solvophilic (shown in blue) blocks. The interface between the solvophobic and solvophilic regions is shown in black, and it has a concave dimple. The inset shows how our shape-design mechanism gives rise to the designed shape: regions of the micelle surface where a convex curvature is desired are populated with diblocks having a larger solvophilic block and consequently preferring a convex curvature, while regions to be made concave are populated with diblocks containing larger solvophobic blocks, thereby preferring more concave curvature. Bonds, indicated in black, connect the diblocks end to end forming a multiblock copolymer in order to fix diblocks in their intended positions.

Figure 5

Schematic of multiblock bond architecture. The disks in the top row and the red (light gray) disks from the bottom rows represent solvophobic beads, while blue (dark gray) disks represent solvophilic beads. Black segments represent bonds between beads. The multiblock begins with a core segment composed of solvophobic beads shown in tan (top of schematic). To this segment solvophobic-rich diblocks, outlined in black as in Fig. 6, are successively attached end to end. Last, solvophilic-rich diblocks are attached end to end. The “$•••$” symbols represent further diblocks not shown.

Figure 6

Illustration of how a micelle designed to have a dimple is constructed in our model. The micelle contains both species of beads present in our model: solvophobic, shown in red (light gray) or tan (darker gray beads in the micelle interior), and solvophilic, shown in blue (dark gray beads along the micelle surface). The micelle consists of a long “core segment” of solvophobic beads shown in tan (darker gray beads in the micelle interior), and a collection of diblocks. There are two species of diblocks: a “solvophobic-rich” species with 2 solvophilic beads and 13 solvophobic beads (outlined with black), and a “solvophilic-rich” species with 4 solvophilic beads and 12 solvophobic beads. These diblocks are joined end to end. The dimple is intended to appear in the region occupied by the solvophobic-rich diblocks, as shown (indeed, the above configuration was taken from a simulation).

Figure 7

Initial relaxed configuration of simulated model shape-designed micelle, as described in the text. Color coding is as in Fig. 6. The resulting micelle configuration has the intended topology and surface diblock ordering, but is not biased toward its intended dimpled shape.

Figure 8

Illustration of the difference metrics ${\mathrm{\Delta }}_{\mathrm{ext}}$, defined in Eq. (6) and ${\mathrm{\Delta }}_{\mathrm{int}}$ defined in Eq. (10). In (a), two shapes, $\mathbb{a}$ and $\mathbb{b}$, are shown, together with the displacement vectors $\mathrm{\Delta }{\mathbf{r}}_1, \mathrm{\Delta }{\mathbf{r}}_2$, and $\mathrm{\Delta }{\mathbf{r}}_3$ connecting corresponding vertices of the two shapes. As specified by Eq. (5), both shapes are geometrically centered on the origin. The difference ${\mathrm{\Delta }}_{\mathrm{ext}}$ is given by the sum of square lengths of these vectors: ${\mathrm{\Delta }}_{\mathrm{ext}}(\mathbb{a},\mathbb{b})={\sum }_{i=1}^3\mathrm{\Delta }{\mathbf{r}}_i^2$. In (b) the same two shapes are shown, except $\mathbb{a}$ is rotated about the origin by the angle $\check{\theta }$ that minimizes ${\mathrm{\Delta }}_{\mathrm{ext}}$. Consequently, the displacement vectors $\mathrm{\Delta }{\mathbf{r}}_i$ are clearly smaller here than in (a). The intrinsic difference ${\mathrm{\Delta }}_{\mathrm{int}}$ is defined as this minimum value of ${\mathrm{\Delta }}_{\mathrm{ext}}$.

Figure 9

Illustration of the average $\overline{\mathbb{r}}$ of three shapes ${\mathbb{r}}_1, {\mathbb{r}}_2$, and ${\mathbb{r}}_3$. The original ${\mathbb{r}}_i$ are not shown; instead, the result ${\check{\mathbb{r}}}_i$ of aligning ${\mathbb{r}}_i$ with $\overline{\mathbb{r}}$ is shown. For definiteness, we have fixed the otherwise arbitrary orientation of $\overline{\mathbb{r}}$ by positioning its first junction point on the $x$ axis.

Figure 10

Graphical representation of an average micelle shape and its fluctuations and uncertainty. The central green curve linearly interpolates the average position of each diblock junction point. The region of the surface occupied by solvophobic-rich diblocks is outlined in black. For each junction point, a large blue ellipse and a smaller red ellipse is drawn. A blue ellipse represents the $40\%$ confidence region, corresponding to one standard deviation from the mean, for a junction point assuming a Gaussian distribution with variance given by the shape variance $⅀$ of Eq. (16). In the same way, a red ellipse represents the variance in the mean shape $⅀$ of Eq. (19).

Figure 11

Two different types of malformed micelles. The curves and ellipses have the same meaning as in Fig. 10. In (a), the average micelle shape contains two junction points in the concave region which approach the opposite surface of the micelle. This occurs because the junction points crossed to the other side during the simulations. In (b), the micelle is broken up into multiple disconnected regions outlined by the junction points. Such malformed micelles are not expected to take the designed shape, and so they are excluded from our analysis.

Figure 12

Illustration of the curvature ratio definition. The curvature ratio is defined as the ratio of the average curvature $c_{-}$ in the region occupied by the solvophobic-rich diblocks to the average curvature $c_{+}$ in the region occupied by the solvophilic blocks. The curvature ratio is 1 for a circle, and is negative for a shape with a dimple such as the one shown in the figure, becoming more negative as the dimple grows more pronounced.

Figure 13

(a) Plot of the mode amplitude variance ${\mathrm{\Sigma }}_{\xi \ell }$ vs mode number $\ell$ and (b) the correlation time ${\tau }_{\xi \ell }$ (in units of the sampling interval) from a single simulation run (i.e., single value of $\xi$) for a low-contrast micelle containing 848 core beads. The three modes corresponding to rigid motions are omitted because their amplitude variance is zero. The variances range over three orders of magnitude. (c)–(e) Plots of the $\ell =1, \ell =10$, and $\ell =50$ modes. The average shapes are plotted as well as a deformation of the shape in the direction of the mode ${\mathbb{m}}_{\xi \ell }$. In (c), the size of this deformation is $\sqrt{{\mathrm{\Sigma }}_{\xi 1}}$, which represents one standard deviation of sampled shape distribution in the direction of ${\mathbb{m}}_{\xi \ell }$. In (d) and (e), the size of the deformation is increased to five standard deviations for clarity.

Figure 14

Plot of ${\chi }_{\xi \ell }^2$ [defined in Eq. (28), black circles] vs $\ell$ for the representative simulation run of Fig. 13 of a low-contrast micelle with 848 core beads. As in Fig. 13, the modes are ordered by their amplitude, so that the first mode ($\ell =1$) is the mode with highest amplitude. The horizontal lines bracket the $90\%$ confidence interval for the ${\chi }_{\xi \ell }^2$ statistic, assuming the mean shape distribution is Gaussian. Most of the ${\chi }_{\xi \ell }^2$ fall in this range, with no apparent systematic dependence on $\ell$. For contrast, we present ${\chi }_{\xi \ell }^2$ with correlations ignored [by substituting ${\tau }_{\xi \ell }\to 1$ in Eq. (19), red diamonds]. In this case, the ${\chi }_{\xi \ell }^2$ show a clear dependence on $\ell$, in that ${\chi }_{\xi \ell }^2$ is much larger than unity for small $\ell$.

Figure 15

Plot of two estimates of the uncertainty in the normalized fluctuation $\delta$ for the different micelle compositions of Table 1. Black (red) symbols represent high (low) contrast micelles. Closed symbols represent the first estimate defined using Eqs. (32, 33, 34); open symbols represent the second estimate described below Eq. (34). Since simulations of the high-contrast micelle with 1000 core beads only resulted in one well-formed shape, the second estimate of its normalized fluctuation uncertainty cannot be made. Besides this case, and the case of the high-contrast micelle with 700 core beads, where the two simulation runs giving well-formed micelles had very similar normalized fluctuations, the estimates agree to within a factor of 2.

Figure 16

Plot of ${\chi }_{\ell }^2$ defined in Eq. (29) vs mode number $\ell$. The two horizontal gray lines demarcate the $90\%$ confidence interval.

Figure 17

Plots of normalized fluctuation $\delta$ and curvature ratio $c_{-}/c_{+}$ vs micelle composition. The micelle compositions are those of Table 1. In (a), $\delta$ is plotted against the size of the micelle core. Black (red) points represent high (low) contrast micelles. The data are fit to a model containing only constant and gradient terms: $\delta ={\delta }_0+m_{c}c+m_{r}r$, with $c$ being the number of core beads and $r$ being the solvophobic-rich diblock asymmetry ratio [defined in Eq. (2)]. Best fit parameters are found to be $m_c=(-1.0\pm 0.3)\times {10}^{-4}$ and $m_r=0.37\pm 0.08$. The reduced chi-square for this fit is 1.9. The curvature ratio is treated analogously in (b). The best fit parameters for the curvature ratio are given by $m_c=(1.7\pm 0.1)\times {10}^{-3}$ and $m_r=4.5\pm 0.3$. The reduced chi-square for this fit is 14. After omitting the outlier at a core size of 600 and $n_{\mathrm{int}}:n_{\mathrm{ext}}=27:4$, the reduced chi-square drops to 4.4.

Figure 18

Schematics of alternate bond topologies for single-polymer micelles. The color scheme is the same as that of Fig. 2. In (a), diblocks have been attached as side chains to a solvophobic homopolymer. In (b), the diblocks are instead attached on their solvophilic ends to a solvophilic homopolymer. Last, in (b), the diblocks are represented as a pair of side chains attached to a solvophilic homopolymer chain.

Figure 19

Plots of three unduloids interpolating between a cylinder and a string of spheres. Each surface shown in (a)–(c) has a uniform mean curvature, and furthermore the three curvatures are the same. In fact, these shapes can be continuously deformed into one another with the mean curvature held fixed.

Figure 20

Plots of nonbonded interaction potential. Energies and lengths have been nondimensionalized using $k_{B}T$ and $L_{\mathrm{therm}}$ respectively, as described in the second paragraph of Sec. 2a. In (a), the purely repulsive interaction potential $U_r$ of Eq. (A2) for a bead pair containing a solvophilic bead is plotted. In (b), the interaction potential for two solvophobic beads is plotted. This interaction potential contains an attractive term in addition to the repulsive potential $U_r$ plotted in (a).

Figure 21

Visualization of simulated homopolymer melt. The simulated system contains 30 solvophobic homopolymer chains, each having 35 beads, shown as red disks. The springs connecting adjacent beads are shown as red segments. The system is periodic, its boundary indicated by the black square.

Figure 22

Solid curve: expected functional form Eq. (C1), fitted to the simulation values shown as points. The resulting reduced chi-square is 0.5, indicating that this functional form is consistent with the data. The best-fit value of $d_{\infty }$ is $0.30146\pm 0.00001$.

Figure 23

Two independent fits to determine the compressibility $\beta$. In (a), the density $d$ of the homopolymer system of Fig. 21 with 30 polymer chains and 15 beads per chain is plotted against the applied pressure (data points). The solid line shows a fit of the functional form $d\left(p\right)=d_0\left(1+\beta p\right)$ to the data, which yields a compressibility $\beta$ of $0.242\pm 0.002$, and reduced chi-square of 0.65. In (b), data from the same simulations of Fig. 22 are plotted. The variance of the sampled densities, normalized by the square density $d{\left(n\right)}^2$ found from the fit of Fig. 22, are plotted against the reciprocal of the chain length. Two systems having 45 and 50 beads per chain produced anomalous results, which could not be reproduced after several attempts. These anomalous results are shown as partially transparent data points, and the solid data points at the same chain lengths are results from representative repeats of the simulation. A fit of the expected functional form $\beta d\left(n\right)/\left(N_{c}n\right)$ to the data excluding the outliers was performed, where $N_c$ is the number of chains (namely, 30) in the simulation. The fit gives a best-fit compressibility of $0.2372\pm 0.0007$, and the chi-square of the fit is 1.4.

Figure 24

Plot of the mean-square end-to-end chain distance $\langle r^2\rangle$ as a function of chain length for the same systems of Figs. 22 and 23. The data are fit to the theoretical expectation given in Eq. (C2), resulting in a best-fit value of $b^2$ equal to $6.41\pm 0.05$ and a reduced chi-square of 1.1.

Figure 25

Typical configuration of a strip of homopolymer in (implicit) solvent. The simulated system contains 143 solvophobic homopolymer chains, each having 15 beads, shown as red disks. The springs connecting adjacent beads are shown as red segments. The system is periodic in one direction, and the fixed periodic boundaries are shown as black lines.

Figure 26

Plot of inferred surface tensions vs number of chains for system simulated in the geometry of Fig. 25. A best fit to a constant yields a surface tension of $0.717\pm 0.003$. The reduced chi-square of the fit is 1.1, indicating that the data are consistent with the surface tension being independent of the number of chains in the system, as theoretically expected.