Session L02: Focus Session: Fluids Next: Flow in Soft Porous Media

Capillary entry pressure of a hydrogel packing

The capillary entry pressure of a porous medium is the applied pressure at which a non-wetting fluid will first invade the pore space by displacing the wetting fluid from the largest pore throats. For a rigid porous medium, the entry pressure is a characteristic of the two fluids, the solid material, and the pore structure. For a soft porous medium, however, the applied pressure will also compress the medium, thereby changing the pore structure and thus the entry pressure itself. This coupling complicates the basic concept of entry pressure as a material property. Here, we use experiments and modelling to study the capillary entry pressure of a model soft porous medium: a packing of hydrogel beads (diameter ~1 mm), in which the individual grains are themselves soft porous media on a much smaller scale (pore size ~10 nm). We show that the measurement of entry pressure provides a sensitive probe of the complex mechanics of these materials. We highlight in particular the strong interactions between entry pressure, viscoelastic creep, and stress-strain hysteresis.   

Flow-induced compaction of soft porous media and the interaction with gravitational and mechanical loading

Flow-induced compaction of deformable porous media is characteristically non-uniform due to gradients in the fluid pressure. The constitutive laws for effective pressure and permeability encode the rheology of the solid matrix; these laws separate porous media into two 'types' based on the limit of large applied pressure difference. This classification of types is found to be intrinsically linked to the well-known poroelastic diffusivity. Industrial and geographical applications motivate the consideration of porous media that naturally slump due to gravitational stresses, the significance of which is characterised by a non-dimensional gravity term that captures the relative importance of gravitational stress and elastic stresses. Further, if a medium is mechanically compressed between two plates, as is the case in various industrial processes, then it takes up an external load which must be relieved before any bulk flow-induced compaction can occur. In particular, in this 'pre-strained' state, the flow can compact some regions and decompress others, such that it is possible for the overall depth to remain fixed which is not possible in the non-pre-strained regime. The interplay between these distinct types of compaction is explored, and consideration is given to how understanding of behaviour can inform industrial choices.   

A theory of hydrogel mechanics that couples swelling and external flow

Two aspects of hydrogel mechanics have been studied separately in the past. The first is the swelling and deswelling of gels in a quiescent solvent bath triggered by an environmental stimulus such as a change in temperature or pH, and the second is the solvent flow around and into a gel domain, driven by an external pressure gradient or moving boundary. The former neglects convection due to external flow, whereas the latter neglects solvent diffusion driven by a gradient in chemical potential. Motivated by engineering and biomedical applications where both aspects coexist and potentially interact with each other, this work presents a poroelasticity model that integrates these two aspects into a single framework, and demonstrates how the coupling between the two gives rise to novel physics in relatively simple one-dimensional and two-dimensional flows   

A reciprocal theorem for poroelastic materials

In studying the transport of particles and inclusions in multi-phase systems we are often interested in integrated quantities such as the net force and the net velocity of the inclusions. In the reciprocal theorem, the known solution to the first and typically easier boundary value problem is used to compute the integrated quantities, such as the net force, in the second problem without the need to solve that problem. Here, we derive a reciprocal theorem for biphasic materials which are composed of a linear compressible viscoelastic solid phase, permeated by a viscous fluid. As an example, we analytically calculate the time-dependent net force on a rigid sphere in response to point-forces applied to the elastic network and the Newtonian fluid phases of the biphasic material. We find that when the point-force is applied to the fluid phase, the net force on the sphere evolves over timescales that are independent of the distance between the point-force and the sphere; in comparison, when the point-force is applied to the elastic phase the timescale for force development increases quadratically with the distance, in line with the scaling of poroelastic relaxation time.  Finally, we discuss some applications of these solutions, including their utility in developing pseudo-analytical solutions, analogous to Stokes flow microhydronamics, to a mixture of spherical particles in a poroelastic material.   

Freeze fracturing of hydrogels

At moderate temperatures below the freezing temperature of water, the pore water inside a hydrogel cannot freeze because the surface energy of an ice–water phase boundary mitigates against the nucleation of ice crystals inside the nanoscale pores.  This is a well-known consequence of the Gibbs-Thomson effect.  However, if ice forms adjacent to a hydrogel or within a large pore then it can continue to grow by drawing water out of the hydrogel by cryosuction.  Equilibrium between hydrogel and adjacent ice occurs at a liquidus temperature related to the osmotic pressure of the gel and therefore to its polymer concentration.  We first study the fundamental interplay between temperature gradients and osmotic pressure gradients leading to fluid flow through the gel towards a solidification front in one dimension when a saturated gel is brought into contact with an isothermal cold boundary.  We then study the growth of an ice wedge in two dimensions when a slab of hydrogel is subject to a temperature gradient imposed along the slab.  Ice growth proceeds by drawing water predominantly transverse to the slab but the resulting longitudinal gradient of osmotic pressure draws water additionally along the slab.  Ice growth also compresses the hydrogel, inducing a deviatoric elastic strain field that is characteristic of either mode I or mode II crack opening, depending on the conditions imposed on the displacement field at the boundaries of the slab.  These studies are relevant to the cryopreservation of biological tissue, for example.   

XOXO, Gossip Gel: oscillating chemical reactions facilitate communication between responsive hydrogels

Responsive hydrogels promise a new world of soft and smart devices that interact with environmental stimuli, such as light-levels, ambient pH, and temperature. In pH-responsive gels, a chemical signal can result in a dramatic shape change which, in turn, squeezes solute out of the gel. This can locally change chemical concentrations, mediating long-distance signalling between isolated hydrogels.

In this talk, I will discuss how coupling an oscillating chemical reaction to a chemo-responsive hydrogel leads to regular repeated shape changes, with the resultant release of solute feeding back into the reaction rate, allowing two spatially-separated hydrogels to “communicate” via synchronisation of their separate chemical reactions. This chemical signalling leading to a physical change allows us to view active gels as proxies for simple life and giving insights into ways in which colonies of microorganisms can signal to each other.   

Rapid, nonlinear diffusiophoretic swelling of chemically responsive hydrogels

Chemicaly responsive hydrogels can store and release chemical signals, such as polyacrylic acid (PAA) gels storing copper or calcium divalent ions and freeing them upon adding acid. While the PAA hydrogel interface is fully permeable to the released ions and acid, interactions between the free ions and the gel polymer network lead to a diffusiophoretic gel actuation at a rate faster than the characteristic poroelastic deformation rate. We have recently shown this effect theoretically and experimentally by focusing on linear deformations [1]. However, in light of potential applications such as hydrogel-based soft robotics and drug delivery, comprehending large diffusiophoretic deformations in hydrogels is imperative for increased strain rates and power output. We present a continuum nonlinear poroelastic theory to model large diffusiophoretic gel swelling, induced by high acid concentrations or by steady stimulus flow at arbitrary rates. The theory incorporates the interplay between nonlinear poroelasticity, variable hydrogel permeability, and the diffusiophoretic swelling along with the transport dynamics of chemical agents within the gel. With tunable diffusiophoretic interaction strength between the free ions and the gel backbone, we theoretically show that a wide array of chemically induced nonlinear hydrogel deformations are attained.   

Interface conditions for a polyelectrolyte gel in a viscous salt solution in the thin-EDL limit

Polyelectrolyte gels are electro-active soft materials that are often surrounded by a viscous salt solution. The formation of an electrical double layer (EDL) at the interface between the gel and the solution can give rise to fluid-structure interactions driven by Coulomb forces. However, resolving the EDL is computationally difficult because the EDL thickness is often orders of magnitude smaller than the typical gel size. In this talk, we show how matched asymptotic expansions can be used to derive interface conditions that capture the impact of a thin EDL on the gel and surrounding fluid. A key result is the derivation of a Helmholtz--Smoluchowski slip condition that describes how shear stresses in the EDL lead to a bulk flow in the surrounding salt solution. The talk will conclude with some model predictions of how the EDL can impact the gel behaviour in surprising ways.   

Diffusiophoretic transport of colloids in 2D porous media

Understanding, predicting and controlling the transport of colloids in porous media is important in many applications, from contaminant remediation in subsurface flows to drug delivery to cancer cells. Chemical gradients and flows are often the common feature of these environments. Chemical gradients lead to diffusiophoretic migration of colloids. Yet, how the interplay between chemical gradients and flows in porous media governs the transport of colloids is not known. In this work, we combine numerical simulations and theoretical modeling with experimental observations to show how chemical gradients at the scale of pores (micron-scale) influences the macroscopic transport and dispersion of colloids.   

Transport and lymphatic uptake of biotherapeutics after subcutaneous injection

The subcutaneous injection has emerged as a common approach for self-administration of biotherapeutics due to patient comfort and cost-effectiveness. Here we aim to find drug distribution in the tissue and lymphatic uptake after subcutaneous injection. A high-fidelity computational model of poroelastic tissue is developed to find the biomechanical response of the tissue to injection. The effects of layered tissue properties with primary layers, including epidermis, dermis, subcutaneous, and muscle layers, on tissue biomechanical response and drug transport are discussed. The role of secondary tissue elements like the deep fascia layer and the network of septa fibers inside the SQ tissue is investigated. The process of plume formation, interstitial pressure, and drug transport is explored. Finally, the computational model is validated against experimental studies available in the literature.   

Entrainment by biogenic bubbles enables long-range microbial dispersal in yield-stress environments

Microbial communities typically inhabit spatially-constrained 3D environments, such as soils and sediments, foods, and gels and tissues in the body. While some microbes can disperse in their surroundings using motility, many are non-motile and can only grow and proliferate locally. Here, we show how even these non-motile microbes can break free of their local microenvironments and disperse over long ranges by riding bubbles they produce through metabolism. We study non-motile yeast growing in transparent 3D granular hydrogel matrices. We show that through fermentation, yeast produce carbon dioxide (CO₂), which initially dissolves in the medium. As fermentation progresses, the medium gradually reaches supersaturation, resulting in the nucleation of CO₂ bubbles. These biogenic bubbles then grow, deform the surrounding matrix, and ultimately rise, entraining yeast cells in their wake over large vertical distances. The motion of these bubbles leaves a lasting imprint in the matrix, acting as a nucleation site for subsequent bubbles. The sequential entrainment by the train of rising bubbles ultimately culminates in the formation of a conduit within the matrix, encapsulating the colony and giving rise to a distinct columnar morphology. Simulations of bubble dynamics in a viscoplastic medium underscore the critical role of sequential entrainment in shaping the columnar structure. Our study provides a quantitative insight into the entrainment process driven by biogenic bubbles and demonstrates its connection to the microbial dispersal. It highlights the pivotal role of biogenesis in the proliferation and transport of living matter within complex environments, which mirrors many biogeological processes in nature.   

On the Study of the Role of Arachnoid Trabeculae in Traumatic Brain Injury Under Translational Impact

Traumatic brain injury (TBI) is a significant health issue resulting from incidents like blasts, falls, and car accidents, often involving translational impacts. The brain is cushioned within the skull by cerebrospinal fluid (CSF). Additionally, the brain's delicate structure is supported by the arachnoid trabeculae (AT), a thin, porous layer connecting the arachnoid mater and pia mater. To investigate the protective role of trabeculae within the brain, we developed an experimental setup that mimics this structure using a thin, porous layer of polyurethane (PU) foam. This foam layer, approximately 3 mm thick, was placed on the surface of a hydrogel brain model within a head surrogate. The head surrogate, filled with cerebrospinal fluid (CSF) and encased in a transparent plastic skull, allowed for detailed observation of brain deformation and pressure dynamics during impact. Two pressure sensors were positioned within the surrogate to measure CSF pressure at the top and bottom regions of the brain. Results indicated that the trabeculae-mimicking PU foam significantly reduced peak pressure at the top region compared to tests without the foam layer, while the bottom pressure remained unchanged. This reduction suggests that trabeculae protect the brain from cavitation effects during impacts. By demonstrating the pressure-mitigating effects of trabeculae, our study provides new insights into brain injury mechanisms and potential strategies for enhancing brain protection in TBI scenarios.