This paper presents theory and simulation of viscous dissipation in evolving interfaces and membranes under kinematic conditions, known as astigmatic flow, ubiquitous during growth processes in nature. The essential aim is to characterize and explain the underlying connections between curvedness and shape evolution and the rate of entropy production due to viscous bending and torsion rates. The membrane dissipation model used here is known as the Boussinesq-Scriven fluid model. Since the standard approaches in morphological evolution are based on the average, Gaussian and deviatoric curvatures, which comingle shape with curvedness, this paper introduces a novel decoupled approach whereby shape is independent of curvedness. In this curvedness-shape landscape, the entropy production surface under constant homogeneous normal velocity decays with growth but oscillates with shape changes. Saddles and spheres are minima while cylindrical patches are maxima. The astigmatic flow trajectories on the entropy production surface, show that only cylinders and spheres grow under the constant shape. Small deviations from cylindrical shapes evolve towards spheres or saddles depending on the initial condition, where dissipation rates decrease. Taken together the results and analysis provide novel and significant relations between shape evolution and viscous dissipation in deforming viscous membrane and surfaces.
Biological plywoods are multifunctional fibrous composites materials, ubiquitous in nature. The chiral fibrous organization is found in chitin (insects), cellulosics (plants) and collagen I (cornea and bone of mammals) and is a solid analogue of that of cholesteric liquid crystals. The surface and interfaces of plywoods are distinguished by hierarchical topographies and nano-wrinkling. In this paper, we present a theory to model the emergence of these surfaces and interfaces using novel liquid crystal-based shape equations that directly connect material properties with geometric wrinkling. The model applies to liquid crystal precursors of the plywood solid analogues. We focus on wrinkling geometry, wrinkling mechanics, and the mechano-geometry relationships that underlie multifunctionality ubiquitous in biological surfaces. Scaling wrinkling laws that connect mechanical pressures and stresses to folding and bending are formulated and quantified. A synthesis of the connections between mechanics and geometry is achieved using the topology of stress curves and curvature of the wrinkles. Taken together the results show that anchoring is a versatile surface morphing mechanism with a rich surface bending stress field, two ingredients behind many potential multifunctionalities.
In this paper, we generalize the internal viscosity model developed by Professor Williams to semiflexible polymers, biofilaments, and worm‐like micelles, where molecular dissipation is generated by bending. Current models for viscoelasticity with internal viscosity in semiflexible polymers and filaments are based on generalizations of the worm‐like model, but they neglect potential electromechanical couplings such as flexoelectricity. In this paper, inspired by the early work of Professor Williams, we develop a model for worm‐like viscoelastic flexoelectric filaments based on the “line liquid crystal model”. The electroelastic free energy and entropy production are formulated and used to derive the shape equation for these filaments undergoing thermal fluctuations. The resulting time relaxation spectrum is a useful tool to characterize experimentally viscoelastic material parameters. We show that flexoelectricity or polarization‐induced bending softens the filaments. The predicted time relaxation spectrum shows that at longer wavelength modes, the filament behaves like a rigid rod in a viscous solvent, but at shorter wavelengths, it reaches a plateau defined by the bending time scale. The key effect of flexoelectricity is to shift the entire spectrum to higher values, slowing down the response. The model itself is validated using the worm‐like chain and the viscoelasticity of liquid crystals, and the predictions are shown to be in qualitative consistency with the data. Since filament flexoelectricity is associated with 1D sensor‐actuator functionalities, the presented model has many potential novel applications in reduced geometries.
Surface wrinkling is closely linked to a significant number of surface functionalities such as wetting, structural colour, tribology, frictions, biological growth and more. Given its ubiquity in nature’s surfaces and that most material formation processes are driven by self-assembly and self-organization and many are formed by fibrous composites or analogues of liquid crystals, in this work, we extend our previous theory and modeling work on in silico biomimicking nanowrinkling using chiral liquid crystal surface physics by including higher-order anisotropic surface tension nonlinearities. The modeling is based on a compact liquid crystal shape equation containing anisotropic capillary pressures, whose solution predicts a superposition of uniaxial, equibiaxial and biaxial egg carton surfaces with amplitudes dictated by material anchoring energy parameters and by the symmetry of the liquid crystal orientation field. The numerical solutions are validated by analytical solutions. The blending and interaction of egg carton surfaces create surface reliefs whose amplitudes depend on the highest nonlinearity and whose morphology depends on the anchoring coefficient ratio. Targeting specific wrinkling patterns is realized by selecting trajectories on an appropriate parametric space. Finally, given its importance in surface functionalities and applications, the geometric statistics of the patterns up to the fourth order are characterized and connected to the parametric anchoring energy space. We show how to minimize and/or maximize skewness and kurtosis by specific changes in the surface energy anisotropy. Taken together, this paper presents a theory and simulation platform for the design of nano-wrinkled surfaces with targeted surface roughness metrics generated by internal capillary pressures, of interest in the development of biomimetic multifunctional surfaces.
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