Joseph Webber
About Me
I am a research fellow at the Warwick Mathematics Institute at the University of Warwick, working in the group of Professor Tom Montenegro-Johnson on the Leverhulme-funded project Shape Transforming Active Matter. Before this, I was a PhD student in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, supervised by Professor Grae Worster.
I’m an applied mathematician working at the interface between fluid mechanics and soft matter, specifically studying poroelasticity (the behaviour of porous, deformable media). I have a particular interest in hydrogels, soft elastic solids formed of a polymer matrix surrounded by water molecules, and their swelling and drying behaviour. The publications below or my CV show more details on what I’ve been working on, but don’t hesitate to get in touch if you have any questions.
To view resources related to teaching, follow the link.
Publications
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Physical Review E (2024) 109:044602 doi.org/10.1103/PhysRevE.109.044602We 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 formulation...
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Journal of Fluid Mechanics (2023) 960:A38 doi.org/10.1017/jfm.2023.201We consider the multidirectional swelling and drying of hydrogels formed from super-absorbent polymers and water, focusing on the elastic deformation caused by differential swelling. By modelling hydrogels as instantaneously incompressible, linear-elastic materials and considering situations in...
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Journal of Fluid Mechanics (2023) 960:A37 doi.org/10.1017/jfm.2023.200We introduce a new approach for modelling the swelling, drying and elastic behaviour of hydrogels, which leverages the tractability of classical linear-elastic theory whilst incorporating nonlinearities arising from large swelling strains. Relative to a reference state of a fully swollen gel, in which the polymer scaffold may only comprise less than...
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Environmental Fluid Mechanics (2021) 21:1119-1135 doi.org/10.1007/s10652-021-09811-8Motivated by shallow ocean waves propagating over coral reefs, we investigate the drift velocities due to surface wave motion in an efectively inviscid fuid that overlies a saturated porous bed of fnite depth. Previous work in this area either neglects the large-scale flow between layers...
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Philosophical Transactions of the Royal Society A (2020) 378:20190531 doi.org/10.1098/rsta.2019.0531In his famous paper of 1847 (Stokes GG. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441-455), Stokes introduced the drift effect of particles in a fluid that is undergoing wave motion. This effect, now known as Stokes drift, is the result of differences between the Lagrangian and Eulerian velocities...
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Quarterly Journal of Mechanics and Applied Mathematics (2020) 73:1-23 doi.org/10.1093/qjmam/hbz019In a recent article, Ball and Huppert (J. Fluid Mech., 874, 2019) introduced a novel method for ascertaining the characteristic timescale over which the similarity solution to a given time-dependent nonlinear differential equation converges to the actual solution, obtained by numerical integration, starting from given initial conditions. In this article, we apply this method to a range of different partial differential equations...
Talks and posters
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Buckling and swelling instabilities of super-absorbent gels
Abstract
The formation of a wrinkled pattern on the surface of hydrogels as they are brought into contact with water and begin to swell is a familiar phenomenon discussed extensively in the literature, and leads to a wide array of behaviours as the gel continues to swell. The 'reticulated' wrinkle pattern can coarsen, crease, fold or even disappear entirely as more and more water is drawn in to the gel scaffold, but studies usually only focus on the criteria for the exchange of stability. In this talk, I introduce how a linear stability analysis for the onset of stability can be carried out in a poroelastic framework, with different mechanisms for patterning at different times.
At early stages, when only a thin skin on the surface of the gel is swollen, buckling occurs as a purely elastic phenomenon, with horizontal compressive stresses competing against anchoring from the base of the gel to form the pattern. This is then seen to smooth out at the same rate as the growth of the boundary layer, an observation that can be explained in terms of classical plate theory. At late stages, however, I show that wrinkling can only occur with volumetric change, and pattern formation is swelling-dominated. Understanding these two distinct processes allows us to describe observations of patterning made in experiments, including criteria for transient instabilities that disappear entirely as the gel approaches its steady, swollen, state.
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Wrinkling instabilities of swelling hydrogels
Abstract
It has long been known that gels attached to a solid surface and brought into contact with water can form wrinkles, and this has often been explained as resulting from an elastic instability akin to Euler's analysis of buckling beams. However, experiments also show that the characteristic wavelength of wrinkles increases in time and, in some cases, they are seen to smooth out entirely as the hydrogel approaches its steady state. We use our linear-elastic-nonlinear-swelling model to conduct a linear stability analysis of the interface and calculate growth rates of normal modes. We find that, unlike the Euler beam, there are no incompressible instabilities when the gel is tethered to a rigid base and that elastically-driven instabilities are limited by water transport through the gel. If the base state itself is differentially swollen, such as when a gel is first exposed to water, we find a different, osmotically-driven, instability, and the interplay between osmotically-driven flow and elasticity selects a wavelength that scales with the thickness of the swollen region. As the base state approaches its uniform steady state, the characteristic wavelength increases and the effect of this latter mechanism becomes less important, reducing the growth rate and smoothing the wrinkles.
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Linear stability analysis for the formation of wrinkles on confined swelling hydrogels
Abstract
Wrinkling, buckling and creasing instabilities are some of the most familiar phenomena observed in the study of soft materials including hydrogels. They arise when there is mechanical confinement, for example from a fixed base or from hoop stresses in swelling spheres, leading to the preferential formation of wrinkles to relieve shear stresses from the confining strain. These have long been studied as purely elastic instabilities, with a mechanism akin to Biot's classic stability analysis of an elastic half-space under pre-stress. Here, we argue that the swelling process itself is a key part of the mechanism driving these instabilities by carrying out a linear stability analysis of the swelling of a finite layer of gel under horizontal confinement. This stability analysis uses our own linear-elastic-nonlinear-swelling theory for hydrogels that captures the nonlinearities arising from the large isotropic strains when a gel takes on water but allows for an analytically tractable approach through linearising around small deviatoric strains. Under this theory, the physical processes driving the swelling and drying that forms the wrinkles can be easily seen, unlike in fully nonlinear approaches where only a condition for marginal stability can be derived through minimisation of free energy. Furthermore, the growth rate of a given instability is deduced, allowing us to determine the separate influences of wavenumber, layer thickness and material properties on the stability of the water-gel interface. It is observed that the anchoring effect of the fixed base of the gel layer stabilises low-wavenumber (long-wavelength) wrinkles, whilst the growth rate increases unboundedly with the wavenumber, leading to an 'ultraviolet catastrophe' whereby infinitesimally small wavenumbers grow at an infinite rate. We propose solving this by introducing a surface tension at the gel-water interface that serves to stabilise short-wavelength instabilities. The effect of this surface tension is quantified for two different mechanisms; firstly, a surface tension that arises as a bulk elastic discontinuity in stress, and secondly as one arising from a discontinuity in the pore pressure of water between the liquid and gel phases. Quantitative differences between these two mechanisms are discussed, and the evolution of the most unstable wavenumber in time is evaluated and compared to the smoothing and healing of these instabilities seen in experiments, where wrinkles are known to coarsen and, in some cases, disappear entirely as the gel layer imbibes more water. We show that our theory, with the addition of surface tension, can describe all these observations and postulate how further experimentation could determine the true physical origin of the surface tension at the interface.
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A linear-elastic-nonlinear-swelling theory for hydrogels: displacements and differential swelling
Abstract
The shapes of hydrogels as they swell or dry in one spatial dimension, for example when a bead of gel is placed in water, can be determined straightforwardly using polymer conservation. For problems in higher dimensions, we derive an expression for the displacement field of individual gel elements, which allows us to describe the shape of arbitrarily complicated gel geometries given the polymer-fraction field. In a result with parallels in classical linear elasticity, we find that the displacement field satisfies a biharmonic equation forced by gradients in the polymer fraction. As a demonstration, we investigate the drying of slender cylinders of hydrogel by evaporation into the air, with their bases submerged in reservoirs of water. At leading order, the gel locally contracts isotropically as it loses water, with the deviatoric shrinkage arising from differential drying remaining small. Experiments show the formation of a concave top surface and convex bottom surface, phenomena that are explained qualitatively by our model. We also show how our results are equivalent to a mathematical description of the cylinders as stacked disks, with each disk satisfying equations of classical plate theory and coupled dynamically to the disks above and below it.
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Multidirectional gel swelling and drying: a linear-elastic-nonlinear-swelling theory for hydrogels
Abstract
Super-absorbent polymers can form hydrogels with polymer fractions of under 1% by volume when placed in water, with the water molecules being adsorbed by the hydrophilic polymer to form an elastic material. This water is not fixed in place: the polymer scaffold creates a porous structure through which the water can flow to drive swelling and drying processes. Existing studies of the behaviour of these two-phase materials tend to either rely on poroelasticity, coupling the interstitial flow with a constitutive relation to describe the deformation stresses on the gel, or use a microscopic chemical understanding of the water-polymer interactions, minimising the energy density to find both the steady and transient swelling states. In our work, we have derived a constitutive and dynamic model which allows for nonlinearity in the swelling strains but linearises around the macroscopic elastic behaviour of the gel, in effect treating such hydrogels as instantaneously linear-elastic. For one-dimensional problems, such as a gel sphere swelling in water, the swelling state can be described fully by the polymer fraction, which also gives the radial extent of the sphere through a polymer conservation constraint. However, in higher-dimensional problems, the polymer fraction alone cannot describe the gel, since there may be differential swelling in different directions. In such cases, we illustrate how an equation can be derived to describe the displacement field for the gel, which is seen to satisfy a modified biharmonic equation forced by the polymer fraction field, a direct analogue of the biharmonic equation for the displacement field as seen in linear elastostatics. Relying solely on the founding assumption of small deviatoric strains, we can determine the shape of a gel as it is allowed to swell and dry, as well as the transient state as water flows throughout. As an illustration of the utility of this approach, the problem of a cylinder, with its base immersed in water, drying in the air, is considered. In this situation, there is both radial shrinkage and shrinkage along the axis of the cylinder, with the top of the cylinder drying to a greater extent than the base as water is drawn up and evaporated away, with such a problem requiring more than just a polymer fraction for a full description. These modelling assumptions are shown to result in a description of the gel which both agrees with experiments and with a Lagrangian description of the differential drying of the cylinder as a series of stacked elastic plates. Experiments have shown that the cylinder becomes concave on its top surface and convex at its base before reaching a steady state, and both of these phenomena can be described using this displacement formulation, providing further evidence of its ability to solve these more complicated problems.
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