Modeling the formation of double rolls from heterogeneously patterned gels

Both stimuli-responsive gels and growing biological tissue can undergo pronounced morphological transitions from two-dimensional (2D) layers into 3D geometries. We derive an analytical model that allows us to quantitatively predict the features of 2D-to-3D shape changes in polymer gels that encompasses different degrees of swelling within the sample. We analyze a particular configuration that emerges from a flat rectangular gel that is divided into two strips (bistrips), where each strip is swollen to a different extent in solution. The final configuration yields double rolls that display a narrow transition layer between two cylinders of constant radii. To characterize the rolls' shapes, we modify the theory of thin incompatible elastic sheets to account for the Flory-Huggins interaction between the gel and the solvent. This modification allows us to derive analytical expressions for the radii, the amplitudes, and the length of the transition layer within a given roll. Our predictions agree quantitatively with available experimental data. In addition, we carry out numerical simulations that account for the complete nonlinear behavior of the gel and show good agreement between the analytical predictions and the numerical results. Our solution sheds light on a stress focusing pattern that forms at the border between two dissimilar soft materials. Moreover, models that provide quantitative predictions on the final morphology in such heterogeneously swelling hydrogels are useful for understanding growth patterns in biology as well as accurately tailoring the structure of gels for various technological applications.

Figure 1

Schematic overview of the system. (a) Top view of a flat system. A strip of unswollen thickness $t$, width $w$, and length $2L$ is divided into two regimes at $z=L$ (gray line). At the initial, unswollen state, both strips have the same Young's modulus $E$. (b) Side view of a swollen 3D state. The thin orange (light gray) and thick purple (dark gray) curves correspond, respectively, to the strips with the low and high swelling constants ${\mathrm{\Phi }}_1$ and ${\mathrm{\Phi }}_2$. The radius of the roll at $z=L$ is given by $R_L$. Deviations of the 3D shape from this radius are denoted by the radial displacement $u_r\left(z\right)$, where the coordinate $z\in [0,2L]$ represents the arclength of the original, unswollen, state. In the limit of a small thickness, all the deformation is localized around the center point $z=L$. Asymptotically, far away from the transition layer, the radial displacement $u_r\left(z\right)$ approaches the constant amplitudes $A_1$ and $A_2$. (c) and (d) Two results from our gLSM simulations [27] where one of the strips contains polymeric loops that unfold as the temperature is reduced and thereby generate internal stresses in the gel. In the left plot of these figures, the internal stresses are too weak to cause buckling and therefore all the deformation remains in plane. The right plots correspond to a lower temperature, where more loops are opened. This additional increase in excess length increases the internal stresses in the gels and causes buckling. The final configurations are part of a body of revolution, whose cross section is described in (b). The difference between (c) and (d) is in the numbers of prescribed loops. The sample in (d) contains more loops (eight Kuhn segments) than in (c) (four Kuhn segments) and hence the final gradients at the center of (d) are steeper than the gradients in (c). Colors in these plots represent the polymeric volume fraction of the gel elements.

Figure 2

Quantitative comparison between the analytical solution and the numerical optimization of the energy, Eq. (5). In all of these plots we use ${\mathrm{\Phi }}_2=1.1, {\mathrm{\Phi }}_1=1, \nu =1/2, L=1, \varphi ={\mathrm{\Phi }}_2/{\mathrm{\Phi }}_1$, and $e={\mathrm{\Phi }}_1/{\mathrm{\Phi }}_2$ [we choose ${\left(c_0{v}_0\right)}_i=1$ in Eq. (9)]. (a) A log-log plot of the normalized reference radius $R/L$ as a function of the normalized thickness $t/L$. The analytical prediction (27a) is given by the dashed line and numerical results are indicated by orange (light gray) and blue (dark gray) circles, which correspond to nonlinear and FvK strains, respectively. Both sets of strains give almost identical results, which agree nicely with the analytical solution. (b) A log-log plot of the energy (27b) (dashed line) compared to the numerics. (c) The configuration of the sheet for several values of the thickness. The final configuration is a body of revolution that is obtained by rotation of these curves around the horizontal axis. The solid lines are the analytical predictions (3), where the displacements are given by the FvK solution in Sec. 3b. The dashed lines result from numerical optimization of the energy using the nonlinear strains, Eqs. (7).

Figure 3

Quantitative comparison between the approximated analytical solution (solid blue line) and the gLSM simulations (blue circles). In all plots logarithmic scales are used for both axes. In (a) and (b) we plot the radius of the roll $R_L/t$ [see Eq. (30)] as a function of its length $L/t$. The light blue shaded area around the theoretical radius $R_L/t$ corresponds to the minimum and maximum radii of a given shape, i.e., $(R_L+A_1)/t$ and $(R_L+A_2)/t$, where the $A_i$ are given by Eqs. (23). In (a) one strip is embedded with four loops and the entire gel is cooled to $5{}^{\circ }\text{C}$. Using Ref. [27], we convert this setup into our theoretical swelling factors ${\mathrm{\Phi }}_2=1.95$ and ${\mathrm{\Phi }}_1=1.47$, where $\varphi ={\mathrm{\Phi }}_2/{\mathrm{\Phi }}_1$ and $e={\mathrm{\Phi }}_1/{\mathrm{\Phi }}_2$ [see Eq. (9)]. The final configuration at $L=10$ (third gLSM point) is plotted in Fig. 1. In (b) we embed eight loops in one strip and cool the system to $15{}^{\circ }\text{C}$. Using Ref. [27], this gives ${\mathrm{\Phi }}_2=2.14$ and ${\mathrm{\Phi }}_1=1.38$, where $\varphi$ and $e$ are as in (a). The 3D configuration in the case where $L=10$ (first gLSM point from the right) is plotted in Fig. 1. (c) The radius of the roll $R_L/{\mathrm{\Phi }}_{1}t$ as a function of $\varphi ={\mathrm{\Phi }}_2/{\mathrm{\Phi }}_1$. In this plot we keep the length constant, $L=8$, and vary the number of loops within the gel from three to eight (left to right). While at three loops the temperature was lowered to $5{}^{\circ }\text{C}$ in order to obtain a buckled shaped, at the other points (four to eight loops) buckling into a roll is presented at $15{}^{\circ }\text{C}$.

Figure 4

Quantitative comparison between the approximated analytical solution and the experiments in Ref. [20]. In all panels the solid blue line is the analytical prediction and blue circles are points from the experimental data. The data in (a), (b), and (c) are taken respectively from Figs. 2C, 2D, and 2B in Ref. [20]. In all plots logarithmic scales are used for both axes. We emphasize that the experimental radius $R_{K=0}$ is defined slightly differently than the analytical radius $R_L$. For this reason we added a shaded area around the analytical prediction. This area is obtained when $R_L$ is replaced by either the minimum radius $R_L\to r_{min}={\mathrm{\Phi }}_{1}R$ or the maximum radius $R_L\to r_{max}={\mathrm{\Phi }}_{2}R$ of a given 3D shape, where $R$ is given by Eq. (27a). (a) The radius $R_L$ of the roll (30a) as a function of the unswollen thickness $t$, where $2L=200\mu \text{m}$ is the total unswollen length of the gel. (b) The radius $R_L$ (30a) as a function of the unswollen total length $2L$, where the unswollen thickness is $t=11\mu \text{m}$. (c) Length of the transition layer $\mathrm{\Delta }$ as a function of the radius $R_L$. For a given thickness we calculate the parametric line $(R_L\left(t\right),\mathrm{\Delta }\left(t\right))$, where $\mathrm{\Delta }$ is given by Eqs. (31) and (32) and $R_L$ by Eqs. (30). The total length $2L$ is the same as in (a).