Modelling discrete differential swelling in a bi-strip rolled gel

We derive an analytical model that allows us to quantitatively predict the features of 2D-to-3D shape changes in polymer gels that encompass different degrees of swelling within the material and thus, can model different regions of growth within the sample. Such gels can be realized, for example, by introducing variations in the cross-link density within the network or polymerizing the chains to be relatively longer in one area of the sample than another. Focusing on a bi-strip gel that swells into a ``bi-roll'', we determine the radii and amplitudes within a given roll, and the length of the transition layer between the two rolls. The predictions from our model agree quantitatively with available experimental data. In addition, we carry out numerical simulations that account for the complete non-linear behavior of the gel, and show good agreement between the analytical predictions and the numerical results. Models that provide quantitative predictions on the final morphology in such heterogeneously swelling hydrogels are useful not only for understanding growth patterns in biology, but also for establishing how to accurately tailor the structure of gels to meet the requirements of various technological applications.