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42421-AC10
Deformation and Disorder in Hydrogel Photonic Crystals
The award focuses on understanding the use of hydrogels in deformable photonic crystal materials. The effects of nonlinear deformation and disorder are of particular interest, and both mechanical behavior and the electromagnetic behavior are to be considered computationally.
In the second period of the award duration, both mechanical and optical behavior of the highly deformed hydrogel structure have been explored. The objectives of this part of the work are to understand, in a predictive sense, the deformed shape of the highly irregular inverse FCC hydrogel photonic crystal structure upon swelling, including nonuniform localized deformation observed experimentally; and to predict the effects of the deformed geometry and the strain-affected dielectric properties on the reflection/transmission spectra of the photonic crystal structure.
Progress on this task has spanned three areas: (i) study of the origins of the localized deformation in the swollen inverse FCC hydrogel photonic crystal structure, as observed experimentally, (ii) investigation of the effects of the strain on the effective refractive index tensor, and (iii) investigation of the reflection spectrum shift as a function of swelling, in comparison with the experimentally observed trends.
• A coupled fluid-diffusion/poroelastic finite element analysis model has been developed, whereby the hydrogel material is modeled as a porous material with an initial volume fraction of material pores. The matrix material is modeled as a linear elastic material with Young's modulus decreasing with the change in void ratio (which is defined as the ratio of material pore volume to that of the matrix material). Fluid flow is modeled using Darcy's law which is valid at low fluid velocities. The swelling model is based on specifying a volumetric strain as a function of the saturation (defined as the fraction of material pores filled with fluid). The material sorption curve is defined whereby the saturation increases as the elemental fluid pressure increases. Using this formulation, we observed that the initially circular voids in the photonic crystal structure take on a triangular shape under swelling, as observed experimentally. The orientation of the buckling of the thin ligaments is correlated over short distances, but changes direction over larger length scales due to proximity to grain boundaries or other defects in the photonic crystal. These observations are consistent with recent experimental reports of transition from ABCABC packing structure to ABCA'B'C' packing structure, or an FCC to L12 transition.
• The refractive index of the hydrogel material changes due to the effect of the swelling on the molecular structure of the material. This is modeled using a statistical thermodynamics-based photoelastic formulation whereby the stress optic relation is defined in which the stress is given implicitly as a function of the strain. The stress optic coefficient depends on the average refractive index of the hydrogel material in the initial state which remains constant. The evolving indices then maintain their average about the initial mean value. Effective anisotropic refractive indices are calculated using a rule of mixtures based on the volume fraction of material and fluid. We find here that the refractive index increases in the <111> direction (from 1.52 to 1.57) and decreases uniformly in the other two coordinate direction (from 1.52 to 1.47).
• A finite element based electromagnetics simulation is carried out in which the hydrogel structure is meshed with a high density of linear vector elements. A scattering boundary condition is applied at the top of the unit cell to simulate a plane wave incident in normal direction, and periodic boundary conditions are applied on the six lateral surfaces along with a perfect magnetic boundary condition. A scattering condition is applied at the bottom surface at which the transmittance is calculated. Power is calculated by integrating the Poynting vector over the surface area, and the reflectance (ratio of reflected power to incident power) of the unit cell is obtained by assuming that there is no absorption of the field by the hydrogel material. The optical reflectance is calculated for a range of wavelengths of the incident plane wave. These results indicate that if one considers the anisotropy of the refractive index and the effect of material swelling into the voids, the computed diffraction wavelength falls within the experimentally observed ranges for large swelling factors.
In conclusion, we have made significant progress in understanding the mechanics and electromagnetics of the hydrogel inverse FCC structure subjected to large swelling deformations. We have focused on accounting for sources of both geometrical and material nonlinearity, which dominate the observed experimental deformation patterns. Concluding the analysis of these mechanics and electomagnetics issues, and summarizing our results in a series of journal articles to be submitted in the coming year will be the focus of the remaining work on the project.
