The swelling and shrinking of spherical thermo-responsive hydrogels

Abstract

Thermo-responsive hydrogels are a promising material for creating controllable actuators for use in micro-scale devices, since they expand and contract significantly (absorbing or expelling fluid) in response to relatively small temperature changes. Understanding such systems can be difficult because of the spatially and temporally varying properties of the gel, and the complex relationships between the fluid dynamics, elastic deformation of the gel and chemical interaction between the polymer and fluid. We address this using a poro-elastic model, considering the dynamics of a thermo-responsive spherical hydrogel after a sudden change in the temperature that should result in substantial swelling or shrinking. We focus on two model examples, with equilibrium parameters extracted from data in the literature. We find a range of qualitatively different behaviours when swelling and shrinking, including cases where swelling and shrinking happen smoothly from the edge, and other situations that result in the formation of an inwards-travelling spherical front that separates a core and shell with markedly different degrees of swelling. We then characterise when each of these scenarios is expected to occur. An approximate analytical form for the front dynamics is developed, with two levels of constant porosity, that well-approximates the numerical solutions. This system can be evolved forward in time, and is simpler to solve than the full numerics, allowing for more efficient predictions to be made, such as when deciding dosing strategies for drug-laden hydrogels.

A New Class of Single‐Material, Non‐Reciprocal Microactuators

“…We expect this rate to depend on the geometry of the hydrogel structure, and may increase as the maximum distance to the nearest free surface decreases or as the fluid viscosity decreases. [ 41,44,46 ]…”

“…This is well documented by the "skin" formation effect observed with rapidly collapsing pNIPAM gels, where the surface gel layer shrinks to form a barrier, inhibiting the water escape from the gel center. [43,44] Hence, simple changes of the gel geometry, by means of its surface-to-volume ratio, would affect the out-of-equilibrium collapse and swelling rates. Therefore, the porous struts display faster collapsing and re-swelling, as the pores reduce the distance water molecules must travel from within the gel to the external environment and vice versa.…”

A linear-elastic-nonlinear-swelling theory for hydrogels. Part 1. Modelling of super-absorbent gels

“…It is this problem that was treated linearly by Tanaka & Fillmore (1979) and in fully nonlinear models by Tomari & Doi (1995), Bertrand et al. (2016) and Butler & Montenegro-Johnson (2022). On the assumption that the swelling bead is at all times spherically symmetric, and therefore all displacements are in the radial direction, both polymer and water velocities can be assumed to be purely radial.…”

“…Perhaps the most common quasi-one-dimensional swelling problem considered in the literature is that of a swelling spherical bead of gel, initially held at some uniform polymer fraction φ * > φ 0 , before being placed into bulk water and allowed to expand. It is this problem that was treated linearly by Tanaka & Fillmore (1979) and in fully nonlinear models by Tomari & Doi (1995), Bertrand et al (2016) and Butler & Montenegro-Johnson (2022). On the assumption that the swelling bead is at all times spherically symmetric, and therefore all displacements are in the radial direction, both polymer and water velocities can be assumed to be purely radial.…”

“…(2016), Chester & Anand (2011), Hong et al. (2008) and Butler & Montenegro-Johnson (2022) invoke fully nonlinear descriptions of gels. Many such approaches are based on the work of Flory & Rehner (1943 a , b ), who derive a free-energy density function that is separated into contributions from mixing of water and polymer components and stretching of polymer chains .…”