Delayed Buckling and Guided Folding of Inhomogeneous Capsules

Colloidal capsules can sustain an external osmotic pressure; however, for a sufficiently large pressure, they will ultimately buckle. This process can be strongly influenced by structural inhomogeneities in the capsule shells. We explore how the time delay before the onset of buckling decreases as the shells are made more inhomogeneous; this behavior can be quantitatively understood by coupling shell theory with Darcy’s law. In addition, we show that the shell inhomogeneity can dramatically change the folding pathway taken by a capsule after it buckles.

Figure 1

(a) Schematic showing the capsule geometry investigated. (b) Upper: buckling of a capsule; scale bar is $20\mu \mathrm{m}$. Lower: buckling begins at the thinnest part of the shell for capsules with thickness inhomogeneity $\delta /h_0\approx 0.84$; scale bars are $50\mu \mathrm{m}$. (c) Fraction of capsules buckled over time, for three different osmotic pressures $\Pi$. Capsules have mean shell thickness $h_0=1.2\mu \mathrm{m}$, outer radius $R_0=70\mu \mathrm{m}$, and $\delta /h_0=0.20$. Smooth lines show exponential fits. (d) Total fraction of capsules that ultimately buckle over time for varying $\Pi$, for capsules with $h_0$, $R_0$, and $\delta /h_0=1.2\mu \mathrm{m}$, $70\mu \mathrm{m}$, and 0.20 (gray circles), $1.3\mu \mathrm{m}$, $67\mu \mathrm{m}$, and 0.23 (red triangles), and $5.5\mu \mathrm{m}$, $55\mu \mathrm{m}$, and 0.19 (blue squares). Smooth curves are fits to the data using the cumulative distribution function of the normal distribution. Inset shows the mean osmotic pressure of each fit versus $h_0/R_0$; vertical and horizontal error bars show the standard deviation of each fit and estimated variation in $h_0/R_0$, respectively. Straight line shows $(h_0/R_0{)}^2$ scaling. (e) Time delay before the onset of buckling, $\tau$, normalized by $h_0^2$, for varying $\Pi$, for the same capsules as in (d). Closed points show $\Pi >{\Pi }^{*}$ while open points show $\Pi <{\Pi }^{*}$. Vertical error bars show uncertainty arising from estimated variation in $h_0$. Black line shows ${\Pi }^{-1}$ scaling. (f) Time delay $\tau$ decreases with the wait time before a shell is polymerized, $t_w$; capsules have $h_0=1.2\mu \mathrm{m}$ and $R_0=70\mu \mathrm{m}$, and are buckled at $\Pi \approx 0.86\mathrm{MPa}$ $>{\Pi }^{*}$. Black line shows theoretical prediction coupling shell theory and Darcy’s law, as described in the text, with $k\approx 3.5\times {10}^{-24}{\mathrm{m}}^2$.

Figure 2

Folding pathways for different shell inhomogeneities. (a)–(c) Optical microscope images exemplifying buckling at $\Pi \approx 0.86\mathrm{MPa}$ of (a) slightly inhomogeneous capsules polymerized in situ ($t_w\approx 0$), with $\delta /h_0\approx 0.2$, (b)–(c) very inhomogeneous capsules polymerized after a wait time $t_w=1$ day, with $\delta /h_0\approx 0.84$. Very inhomogeneous capsules buckle through the formation of either (b) one single indentation or (c) two indentations. $\Delta t$ is the time elapsed after buckling. Scale bars are $35\mu \mathrm{m}$. (d)–(e) Examples of simulated shells with similar geometries as the capsules shown in (a)–(c), for varying fractional volume reduction $\Delta V/V_0$. Color scale indicates the spatially varying shell thickness.

Figure 3

Colloidal capsules with two or three spherical interior compartments, schematized in left panels, buckle at “weak spots” (arrows). This forms shapes with two or three equally spaced circular indentations after buckling (right panels). Scale bars are $100\mu \mathrm{m}$.