Swelling and de-swelling of gels under external elastic deformation Robyn H. Pritchard, Eugene M. Terentjev * Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK article info Article history: Received 28 September 2013 Received in revised form 4 November 2013 Accepted 4 November 2013 Available online 13 November 2013 Keywords: Gel swelling Rubber elasticity Poisson ratio abstract We show that the equilibrium Poisson ratio of electrically neutral gels depends on their shear modulus. When immersed in a good solvent, gels increase their volume on imposed external deformation, but stiffer gels swell less and exhibit a larger Poisson ratio, closer to 0.5, while the gels with a higher solvent content (and correspondingly lower shear modulus) approach a Poisson ratio of 0.25. We monitor the full process of stress and shape relaxation after an instantaneous deformation by using the technique of digital image correlation (DIC), and show that the amount of stress relaxation in uniaxially strained gels is proportional to the shear modulus of the free swollen state and a change in effective strain. Experi- ments were conducted on polyacrylamide (PAAm) gels in a custom built setup to give the Poisson ratio to high accuracy and time resolution, as well as verification of homogeneous deformation in equilibrium. In addition to water, hydrophilic gels were stretched in three poor solvents: silicone oil, mineral oil, and in air. All three exhibited water loss on imposed deformation and a resulting increase in stress, with mineral oil presenting the smallest change due to its lower permeability to water. Mineral oil and silicone oil are of particular interest as they are often used in mechanical testing to prevent solvent loss. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction It is well known that polymer gels immersed in a good solvent will swell, often to a great many times their dry volume. Variations in experimental conditions, such as temperature, pH, and solvent can modify the extent of swelling, and the response can vary markedly with gel composition [1]. This tunability, as well as the potential biocompatibility of hydrogels, has led to extensive research into areas as diverse as drug delivery, tissue engineering, and soft actuation; the recent developments in these fields are summarised in Refs. [2e4]. Furthermore, implications in gel research also extend to biological materials, including connective tissue: fascia, ligaments, tendons, etc. which can be considered as fibre reinforced gels [5]. In many applications or natural situations, gels operate under mechanical stress. Therefore, in addition to other environmental conditions, the effects on volume change (i.e. a forced change of solvent content) induced by stress or externally imposed defor- mation should be considered. Typically, it is thought that stretching will induce the intake of solvent whereas compression will expel the solvent e both driving a change in volume as well as reduction in stress [6]. The material property governing the relative volume change is the Poisson ratio, defined here via a ratio of Henky strain components [7]: n ¼ lnðl t Þ ln  l k  ; (1) where the Henky strain is ln(l i ) ¼ ln(1 þ ε i ), with l i ¼ 1 þ ε i repre- senting the stretch ratio (the deformed length over the initial length) and ε i the engineering strain in the ith direction, respectively. The Poisson ratio in this form is known as the ‘true Poisson ratio’, which, for a homogeneous material of constant volume, remains at 0.5 throughout deformation. This differs from the Poisson ratio often expressed via infinitesimal strain theory (i.e. n ¼ε t =ε k ), which will deviate from 0.5 as strain increases. Of course, both expressions agree within the small deformation limit, where lnðlÞzε. As gels are semi-open systems that permit the exchange of sol- vent, the Poisson ratio becomes a time dependent quantity. For instantaneous deformations, n ¼ 0.5 as there is no time for solvent exchange (and we assume that, as in any soft material, the shear modulus of the gel is many orders of magnitude less than the compression modulus). Following deformation, solvent diffuses into the gel (or out of it, depending on the outside conditions) and the Poisson ratio will relax along with stress to some equilibrium value. With slower deformation rates the Poisson ratio is less than 0.5, and there would be a characteristic slow deformation rate where the gel would deform at the equilibrium Poisson ratio [8]. The timescale of * Corresponding author. E-mail address: emt1000@cam.ac.uk (E.M. Terentjev). Contents lists available at ScienceDirect Polymer journal homepage: www.elsevier.com/locate/polymer 0032-3861/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.polymer.2013.11.006 Polymer 54 (2013) 6954e6960