Experimental determination of equations of state for ideal elastomeric gels Jianyu Li, Yuhang Hu, Joost J. Vlassak * and Zhigang Suo * Received 24th February 2012, Accepted 8th May 2012 DOI: 10.1039/c2sm25437a A polymer network can absorb a solvent and swell, forming an elastomeric gel. The model of ideal elastomeric gels is based on two assumptions. First, the volume of a gel is the sum of the volume of the dry network and that of the solvent. Second, the free energy of the gel is the sum of the free energy due to stretching the network and that due to mixing the polymer and the solvent. These assumptions lead to a set of equations of state, which can be tested experimentally without invoking any specific models of statistical mechanics. Here we test the model of ideal elastomeric gels by conducting experiments with polyacrylamide hydrogels, and by extracting from the literature four sets of data on polyacrylamide hydrogels and polyacrylamide–water solutions. For an ideal elastomeric gel, the effect of mixing the polymer and the solvent is represented by the osmotic pressure as a function of the swelling ratio. We show that this function obtained by several distinct experimental methods is consistent. Specifically, the function obtained from a gel under different states of applied stress is the same, the function obtained from a free-swelling gel is the same as that obtained from the constrained- swelling gel, the function is independent of the crosslink density, and the function obtained from the gels is similar to that obtained from the solutions. We further show that the Flory–Huggins model of mixing with a constant Flory–Huggins parameter does not fit the experimental data well, but does capture the trend of the data over four orders of magnitude in the osmotic pressure. 1. Introduction A polymer network can absorb a solvent and swell, forming an elastomeric gel (Fig. 1). The amount of swelling depends on the molecular interaction between the polymer and the solvent, and changes greatly in response to environmental stimuli such as temperature, 1 pH 2,3 and salinity. 4 Such stimuli-responsive gels are being developed as vehicles for drug delivery, 5 sensors and actuators in micro-devices, 6,7 and packers in oilfields. 8 The gels in devices are typically constrained by hard materials. The devices operate by exploiting the chemomechanical interaction of the gels: how the mechanical constraint affects swelling and how stimuli generate mechanical forces. The chemomechanical interaction of gels has been described by many theories. 9,10 The classic theory of Flory and Rehner 11 combines two models of statistical mechanics: the Gaussian- chain model describes the elasticity of the network and the Flory–Huggins model describes the mixing of the polymer and the solvent. 12,13 While these models of statistical mechanics relate macroscopic behavior of gels to molecular processes of the network and the solvent, the relation is inexact: the models are often modified in various ways to fit experimental data. 14,15 For applications of gels in devices, it is desirable to develop experimental methods that characterize the chemomechanical behavior of gels without being constrained by models of statis- tical mechanics. In this connection, a commonly held notion is useful: stresses applied on a gel are balanced by the elasticity of the network and the osmosis of the solution. Cai and Suo 16 showed that this notion can be developed from two basic assumptions made in the Flory–Rehner theory. First, the volume of a gel is assumed to be the sum of the volume of the dry network and that of the solvent absorbed. Second, the free energy of the gel is assumed to be the sum of two parts: the free energy associated with stretching the network and the free energy associated with mixing the polymer and the solvent. The first assumption is known as molecular incompressibility, 17 and the Fig. 1 A polymer network absorbs a solvent and swells, forming an elastomeric gel. School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA. E-mail: vlassak@seas.harvard.edu; suo@ seas.harvard.edu This journal is ª The Royal Society of Chemistry 2012 Soft Matter, 2012, 8, 8121–8128 | 8121 Dynamic Article Links C < Soft Matter Cite this: Soft Matter, 2012, 8, 8121 www.rsc.org/softmatter PAPER