Using indentation to characterize the poroelasticity of gels

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Entry by Jianyu Li, AP225, Fall, 2010

Abstract

The paper reports using indentation to determine the mechanical properties of the gel such as elastic constants and permeability. The indenter is firstly pressed into a gel to a fixed depth. While the solvent in the gel migrates, the force on the indenter relaxes.

Theory

The theory of poroelasticity is used in this paper:

  • The displacement field <math>u_i(x_1,x_2,x_3,t)</math>.
  • The strain <math>\epsilon_ij=(\partial u_i/\partial x_j+ \partial u_j/\partial x_i)/2</math>.
  • The conservation of solvent molecules <math>\partial C/\partial t=-\partial J_k/\partial x_k</math>, C is the concentration of the solvent in the gel, <math>J_k</math> is the flux.
  • The stress is so small that the increase of the volume of the gel equals the volume of the solvent absorbed,<math>\epsilon_kk=\omega(C-C_0)</math>,<math>\omega</math> is the volume per solvent molecule.
  • The stress <math>\sigma_ij=2G[\epsilon_ij+\epsilon_kk\delta_ijv/(1-2v)]-\delta_ij(\mu-\mu_0)/\omega</math>,<math>G</math> is the shear modulus, <math>v</math> is the Poisson's ratio, <math>\mu</math> is the chemical potential of the solvent in the gel.
  • The mechanical equilibrium <math>\partial \sigma_ij/\partial x_j=0</math>.
  • The flux driven by the gradient of the chemical potential <math>J_i=-(k/\eta^2)\partial \mu/\partial x_i</math>, <math>k</math> is the permeability, <math>\eta</math> is the viscosity of the solvent.
  • The diffusion equation <math>\partial C/\partial t=D\nabla^2C</math>, <math>D=[2(1-v)/(1-2v)]Gk/\eta</math>.

An indenter is pressed into the gel at a fixed depth <math>h</math>, the force starts to relax as a function of time, <math>F(t)</math>, as showed in Fig. 1.

  • <math>t=0</math>,<math>C=C_0, \epsilon_kk=0</math>, the gel is like an incompressible elastic solid, <math>F(0)\propto G</math>.
  • <math>t>0</math>, <math>\frac{F(t)-F(\infty)}{F(0)-F(\infty)}=g(\frac{Dt}{a^2})</math>, define the normalized time <math>\tau=Dt/a^2</math>,<math>g(\tau)</math> is dimensionless function specific to the shape of the indenter.
  • <math>t>>0</math>, <math>/mu=/mu_0</math>, the gel is like a compressible elastic solid, <math>F(\infty)\propto G/[2(1-v)]</math>.
FIG. 1. Schematic of indentation on the gel from: Yuhang Hu, Xuanhe Zhao, Joost Vlassak, Zhigang Suo, Using indentation to characterize the poroelasticity of gels. Applied Physics Letters 96, 121904 (2010).

The poroelastic contact problems are solved by the finite element software ABAQUS.The calculation results are showed in Fig.2 and Fig. 3. Fig.2 lists the formulas of <math>F(0)</math> for different indenters. Fig.3 is the plot of <math>g(\tau)</math> for different indenters.

FIG.2. Indenters of several types from:Yuhang Hu, Xuanhe Zhao, Joost Vlassak, Zhigang Suo, Using indentation to characterize the poroelasticity of gels. Applied Physics Letters 96, 121904 (2010).
FIG.3. <math>g(\tau)</math> for different indenters from:Yuhang Hu, Xuanhe Zhao, Joost Vlassak, Zhigang Suo, Using indentation to characterize the poroelasticity of gels. Applied Physics Letters 96, 121904 (2010).

Based on the expressions for <math>F(0)</math> and <math>g(\tau)</math> in Fig.2, <math>G</math> is determined by <math>F(0)</math>, <math>v</math> is calculated from <math>\frac{F(0)}{F(\infty)}</math>. The measured curve <math>F(t)</math> is matched to Eq.(1), determines <math>D</math>.

Experiment

  • Materials: alginate hydrogel with diameter <math>30mm</math> and thickness <math>20mm</math>, water.
  • Indenter: a conical aluminum indenter with half angle <math>\theta=70^o</math>.
  • Load schedule: the indenter is pressed into the gel at a depth of <math>h=1,2,3mm</math> within <math>10s</math>, then held for hours.

Result

  • The relaxation curve depends on the depth of indentation, as illustrated in Fig. 4(a).
  • After the force is normalized by <math>h^2</math>, the curves do not overlap, as Fig. 4(b). It indicates the relaxation mechanism is not viscoelasticity.
  • The force and time are both normalized by <math>h^2</math>, so the curves collapse into one single one which is consistent with poroelasticity, showed in Fig. 4(c).
the relaxation curves from:Yuhang Hu, Xuanhe Zhao, Joost Vlassak, Zhigang Suo, Using indentation to characterize the poroelasticity of gels. Applied Physics Letters 96, 121904 (2010).
  • In calculation, <math>G=27.9kPa, v=0.28, D=3.24*10^-8 m^2/s</math>.

Summary

Based on the theory of poroelasticity, force relaxation curves are obtained in a simple form. Indentation is used to determine the elastic constants and permeability. Application of this method lies in the quantitative characterization of gels and soft tissues.