# Poroelastic swelling kinetics of thin hydrogel layers: Comparison of theory and experiment

（Entry by Jianyu Li, AP225, Fall, 2010）

## Introduction

This paper reports a method by embedding fluorescent particle into the swollen PNIPAM hydrogels under two conditions: as freestanding layers and as substrate-attached layers. Due to defocusing of the particle during swelling, the authors managed to measure the swelling kinetics for thin layers with 100um thickness. The linear poroelastic theory modified for hydrogels was used to deconvolute the swelling behavior and characterize the material properties of the gels.

## Summary of the Theory

This theory is adapt to analyze migration of small solvent molecules within polymer networks, firstly invented by Biot. The change in the free energy is due to the work input, is quadratic in the strain. The stress is taken to be the differentiate of the free energy, which is expressed by <math>\sigma_{ij}=2G(\epsilon_ij+\frac{\nu}{1-2\nu}\epsilon_{kk}\delta_ij)-\frac{\mu-\mu_0}{\Omega}\delta_ij</math>. The condition of mechanical equilibrium takes the following form where the gradient of the chemical potential is treated as the body force, <math>G(\frac{\partial^2mu_i}{\partial x_k\partial x_k}+\frac{1}{1-2\nu}\frac{\partial^2\mu_k}{\partial x_k\partial x_i})=\frac{\partial \mu}{\partial x_i}</math>. Consider the constraint swelling behavior of a thin gel layer, the following boundary conditions are applied, <math>\epsilon_{xx}=\epsilon_{yy}=0, \sigma_{xx}(z,t)=\sigma_{yy}(z,t), \sigma_z=0</math>. Insert these conditions, the displacement field and the change of the thickness can be obtained. For the free-standing thin layer, after the gel equilibrium, all components of the stress vanish, <math>\sigma_{ij}=0</math>. The thickness changes according to <math>\Delta(\infty)=\frac{(1-2\nu)(\mu-\mu_0)H}{2(1+\nu)G\Omega}</math>.

## Summary of Results

1. The fluorescent particles were assumed to fix in the network, and move in vertical direction during the swelling process. Due the change of height, the apparent sizes of the beads revealed their distance from the focal plane. The representative graphs and swelling curves are illustrated in Figure 3 and Figure 4.

2. According to the poroelastic thoery discussed above, the experimental data was replotted by the form, <math>\frac{\Delta(t)}{H}=\frac{2\Delta(\infty)}{H^2}\sqrt{\frac{Dt}{\pi}}</math>. The diffusion coefficient <math>D=1.5*10^{-11}m^2s^{-1}</math>, as showed in Figure 5.

3. Swelling of the constrained gel produces an equibiaxial in-plane compressive stress proportional to the increase in layer thickness, thus, by measuring the substrate radius of curvature, we can calculate the compressive force.