Revisiting the wrinkling of elastic bilayers I: linear analysis

Wrinkling instabilities of swelling hydrogels

We investigate the formation of wrinkling instabilities at the interface between layers of hydrogel and water, which arise to relieve horizontal compressive stresses caused by either differential swelling or confinement. Modelling the gel using a linear-elastic-nonlinear-swelling approach, we determine both a criterion for marginal stability and the growth rates of normal modes. Furthermore, our formalism allows us to understand the influence of differential swelling on the stability of hydrogels brought into contact with water, and we find three distinct phases of the instability. Initially, when only a thin skin layer of gel has swollen, buckles grow rapidly and the gel deforms as an incompressible material. A balance between normal elastic stress and pore pressure selects a wavelength for these buckles that increases with the square root of time. At late times, when the gel approaches a uniformly swollen state, buckles can only grow by differential swelling on much slower timescales determined by solvent transport. At intermediate times, growth is driven by the same fluid transport process as at late times but gradients in fluid pressure in the gel as it swells destabilize the interface, driving faster growth of wrinkles. We also explain why some instabilities can be transient, "healing" as time progresses, while others must remain for all time.

A one-step, tunable method of selective reactive sputter deposition as a wrinkling approach for silver/polydimethylsiloxane for electrically conductive pliable surfaces

The wrinkle period and morphology of a metal thin film on an elastic substrate is typically controlled by modifying the substrate before carrying out additional metal deposition steps. Herein, we show that a simultaneously selective and reactive sputtering plasma that modifies the surface of a polydimethylsiloxane (PDMS) substrate while not reacting with the metal during the deposition process decreases the wrinkle wavelength and induces additional wrinkling components and features such as ripples or folds. The selective reaction of the nitrogen plasma with PDMS functionalizes the siloxane surface into silicon oxynitride. This hardens the immediate surface of PDMS, with a quadratic increase in the Young's modulus as a function of the sputtering flow ratio. The increase in the critical strain mismatch and the corresponding presence of folds in the nitrogen-modified wrinkled silver film form a suitable plasmonic platform for surface-enhanced Raman spectroscopy (SERS), yielding an enhancement factor of 4.8 × 10⁵ for detecting lipids. This enhancement is linked to the emergence of electromagnetic hotspots from surface plasmon polariton coupling between the folds/wrinkles, which in turn enables the detection of low concentrations of organics using SERS. Furthermore, when strained, the nitrogen-modified wrinkles enhance electrical conductivity by a factor of 12 compared with unmodified films. Finally, the optical properties of the substrate can be tuned by altering the N₂ content. The simple addition of nonreactive nitrogen to silver sputtering enables simultaneous PDMS hardening and growth of the silver film and together provide a new avenue for tuning wrinkling parameters and enhancing the electrical conductivity of pliable surfaces.

In vivo visualization of butterfly scale cell morphogenesis in Vanessa cardui

Many organisms exhibit functional micro- and nanoscale materials with structural definition and performance that challenge synthetic fabrication techniques, yet we know little about the processes that enable their formation. Using butterfly scales as a model system for functional biomaterials, we establish a timeline of scale formation and quantify relevant structural parameters for developing painted lady butterflies. We overcome challenges of previous efforts by imaging structure formation directly in living organisms, which allows us to continuously observe the evolving wing tissue and the fine details of individual scale cells. Visualization of scale structure formation in live butterflies forms the basis for modeling the underlying biomechanical processes and opens avenues for their translation into advanced fabrication strategies.

During metamorphosis, the wings of a butterfly sprout hundreds of thousands of scales with intricate microstructures and nano-structures that determine the wings’ optical appearance, wetting characteristics, thermodynamic properties, and aerodynamic behavior. Although the functional characteristics of scales are well known and prove desirable in various applications, the dynamic processes and temporal coordination required to sculpt the scales’ many structural features remain poorly understood. Current knowledge of scale growth is primarily gained from ex vivo studies of fixed scale cells at discrete time points; to fully understand scale formation, it is critical to characterize the time-dependent morphological changes throughout their development. Here, we report the continuous, in vivo, label-free imaging of growing scale cells of Vanessa cardui using speckle-correlation reflection phase microscopy. By capturing time-resolved volumetric tissue data together with nanoscale surface height information, we establish a morphological timeline of wing scale formation and gain quantitative insights into the underlying processes involved in scale cell patterning and growth. We identify early differences in the patterning of cover and ground scales on the young wing and quantify geometrical parameters of growing scale features, which suggest that surface growth is critical to structure formation. Our quantitative, time-resolved in vivo imaging of butterfly scale development provides the foundation for decoding the processes and biomechanical principles involved in the formation of functional structures in biological materials.

Computational models of cortical folding: A review of common approaches

The process of gyrification, by which the brain develops the intricate pattern of gyral hills and sulcal valleys, is the result of interactions between biological and mechanical processes during brain development. Researchers have developed a vast array of computational models in order to investigate cortical folding. This review aims to summarize these studies, focusing on five essential elements of the brain that affect development and gyrification and how they are represented in computational models: (i) the constraints of skull, meninges, and cerebrospinal fluid; (ii) heterogeneity of cortical layers and regions; (iii) anisotropic behavior of subcortical fiber tracts; (iv) material properties of brain tissue; and (v) the complex geometry of the brain. Finally, we highlight areas of need for future simulations of brain development.

Folding drives cortical thickness variations

The cortical thickness is a characteristic biomarker for a wide variety of neurological disorders. While the structural organization of the cerebral cortex is tightly regulated and evolutionarily preserved, its thickness varies widely between 1.5 and 4.5 mm across the healthy adult human brain. It remains unclear whether these thickness variations are a cause or consequence of cortical development. Recent studies suggest that cortical thickness variations are primarily a result of genetic effects. Previous studies showed that a simple homogeneous bilayered system with a growing layer on an elastic substrate undergoes a unique symmetry breaking into a spatially heterogeneous system with discrete gyri and sulci. Here, we expand on that work to explore the evolution of cortical thickness variations over time to support our finding that cortical pattern formation and thickness variations can be explained - at least in part - by the physical forces that emerge during cortical folding. Strikingly, as growth progresses, the developing gyri universally thicken and the sulci thin, even in the complete absence of regional information. Using magnetic resonance images, we demonstrate that these naturally emerging thickness variations agree with the cortical folding pattern in n = 9 healthy adult human brains, in n = 564 healthy human brains ages 7-64, and in n = 73 infant brains scanned at birth, and at ages one and two. Additionally, we show that cortical organoids develop similar patterns throughout their growth. Our results suggest that genetic, geometric, and physical events during brain development are closely interrelated. Understanding regional and temporal variations in cortical thickness can provide insight into the evolution and causative factors of neurological disorders, inform the diagnosis of neurological conditions, and assess the efficacy of treatment options.

Phase separation in swelling and deswelling hydrogels with a free boundary

We present a full kinetic model of a hydrogel that undergoes phase separation during swelling and deswelling. The model accounts for the interfacial energy of coexisting phases, finite strain of the polymer network, and solvent transport across free boundaries. For the geometry of an initially dry layer bonded to a rigid substrate, the model predicts that forcing solvent into the gel at a fixed rate can induce a volume phase transition, which gives rise to coexisting phases with different degrees of swelling, in systems where this cannot occur in the free-swelling case. While a nonzero shear modulus assists in the propagation of the transition front separating these phases in the driven-swelling case, increasing it beyond a critical threshold suppresses its formation. Quenching a swollen hydrogel induces spinodal decomposition, which produces several highly localized, highly swollen phases which coarsen and are then ejected from free boundary. The wealth of dynamic scenarios of this system is discussed using phase-plane analysis and numerical solutions in a one-dimensional setting.

Mechanics of human brain organoids

Organoids are prototypes of human organs derived from cultured human stem cells. They provide a reliable and accurate experimental model to study the physical mechanisms underlying the early developmental stages of human organs and, in particular, the early morphogenesis of the cortex. Here we propose a mathematical model to elucidate the role played by two mechanisms which have been experimentally proven to be crucial in shaping human brain organoids: the contraction of the inner core of the organoid and the microstructural remodeling of its outer cortex. Our results show that both mechanisms are crucial for the final shape of the organoid and that perturbing those mechanisms can lead to pathological morphologies which are reminiscent of those associated with lissencephaly (smooth brain).

Growth and remodelling of living tissues: perspectives, challenges and opportunities

One of the most remarkable differences between classical engineering materials and living matter is the ability of the latter to grow and remodel in response to diverse stimuli. The mechanical behaviour of living matter is governed not only by an elastic or viscoelastic response to loading on short time scales up to several minutes, but also by often crucial growth and remodelling responses on time scales from hours to months. Phenomena of growth and remodelling play important roles, for example during morphogenesis in early life as well as in homeostasis and pathogenesis in adult tissues, which often adapt to changes in their chemo-mechanical environment as a result of ageing, diseases, injury or surgical intervention. Mechano-regulated growth and remodelling are observed in various soft tissues, ranging from tendons and arteries to the eye and brain, but also in bone, lower organisms and plants. Understanding and predicting growth and remodelling of living systems is one of the most important challenges in biomechanics and mechanobiology. This article reviews the current state of growth and remodelling as it applies primarily to soft tissues, and provides a perspective on critical challenges and future directions.