Using indentation to characterize the poroelasticity of gels

When an indenter is pressed into a gel to a ﬁxed depth, the solvent in the gel migrates, and the force on the indenter relaxes. Within the theory of poroelasticity, the force relaxation curves for indenters of several types are obtained in a simple form, enabling indentation to be used with ease as a method for determining the elastic constants and permeability of the gel. The method is demonstrated with a conical indenter on an alginate hydrogel.


Using indentation to characterize the poroelasticity of gels
Yuhang Hu, Xuanhe Zhao, Joost J. Vlassak, and Zhigang Suo a͒ When an indenter is pressed into a gel to a fixed depth, the solvent in the gel migrates, and the force on the indenter relaxes.Within the theory of poroelasticity, the force relaxation curves for indenters of several types are obtained in a simple form, enabling indentation to be used with ease as a method for determining the elastic constants and permeability of the gel.The method is demonstrated with a conical indenter on an alginate hydrogel.© 2010 American Institute of Physics.͓doi:10.1063/1.3370354͔][7][8][9][10][11][12][13][14] A challenge has been to relate the response of indentation to the properties of the material.
Consider a gel that aggregates a network of crosslinked polymers and a species of mobile solvent molecules.The deformation of the gel is time-dependent, resulting from following two concurrent molecular processes: the conformational change of the network, and the migration of the solvent molecules.At a macroscopic scale, the two processes result in viscoelastic and poroelastic deformation.So long as the mesh size of the network is much smaller than the contact size of the indentation, the viscoelastic relaxation time is independent of the contact size.By contrast, the poroelastic relaxation time is quadratic in the contact size.Thus, the two types of deformation can be differentiated by indentation.Furthermore, when deformation is poroelastic, the contact can be made small to shorten the time needed for the experiment.
This paper focuses on the indentation of gels undergoing poroelastic deformation ͑Fig.1͒.Oyen and co-workers [11][12][13] have developed a method in which a sphere is pressed into a gel with a force ramping at a constant rate.To extract poroelastic parameters of the gel from the experimental data, the method requires a large number of precalculated master curves and a computational algorithm.By contrast, Hui et al. 8 have proposed a method in which a cylinder ͑in a plane-strain configuration͒ is pressed into a gel to a fixed depth.While this configuration is difficult to use in practice for characterizing materials, Hui and co-workers obtained a theoretical force relaxation curve in a remarkably simple form.The same form is also applicable to a cylindrical punch. 9n this paper, we focus on spherical and conical indenters, commonly used configurations to characterize materials.We show that when these indenters are pressed into a gel to a fixed depth, the theoretical force relaxation curves also take a simple form.Consequently, the force relaxation curves can be used with ease to characterize the poroelasticity of gels.We solve for the theoretical force relaxation curves, and demonstrate the method experimentally with a conical indenter on an alginate hydrogel.
6][17][18][19] The initial gel is taken to be in a homogenous state, subject to no mechanical load, with C 0 being the number of solvent molecules per unit volume of the gel, and 0 being the chemical potential of the solvent in the gel.When the gel deforms, the displacement is a timedependent field, u i ͑x 1 , x 2 , x 3 , t͒, and the strain is ij = ͑‫ץ‬u i / ‫ץ‬x j + ‫ץ‬u j / ‫ץ‬x i ͒ / 2. The conservation of solvent molecules requires that ‫ץ‬C / ‫ץ‬t =−‫ץ‬J k / ‫ץ‬x k , where C is the concentration of the solvent in the gel, and J k the flux.
The stress in the gel is typically so small that the volume of each molecule is commonly taken to be constant.Consequently, the increase of the volume of the gel equals the volume of the solvent absorbed, namely, kk = ⍀͑C − C 0 ͒, where ⍀ is the volume per solvent molecule.When the chemical potential of the solvent in the gel changes from 0 to , the stress in the gel is given by ij =2G͓ ij where G is the shear modulus, Poisson's ratio.
The gel is in mechanical equilibrium, so that the field of stress satisfies ‫ץ‬ ij / ‫ץ‬x j = 0.The gel, however, is not in diffusive equilibrium.The gradient of the chemical potential drives the flux of the solvent according to Darcy's law,  As illustrated in Fig. 1, a gel is submerged in a solvent of chemical potential 0 .After an indenter is pressed into the gel to a fixed depth h, the force on the indenter relaxes as a function of time, F͑t͒.At the instance of indentation, the solvent has no time to migrate, C = C 0 and kk = 0, so that the gel behaves like an incompressible elastic solid, and the instantaneous force is the same as the force on an indenter pressed into an incompressible elastic solid, F͑0͒ ϰ G.After a long time, the solvent in the gel equilibrates with the external solvent, = 0 , so that the gel behaves like a compressible elastic solid, and the force in equilibrium is the same as the force on an indenter pressed into a compressible elastic solid, F͑ϱ͒ ϰ G / ͓2͑1−͔͒.The two limits are related as F͑0͒ / F͑ϱ͒ =2͑1−͒.Figure 2 lists the formulas of F͑0͒ for indenters of several types. 20,21or the gel to equilibrate, the solvent in the gel needs to migrate over a distance comparable to the size of the contact, a.At time t, the solvent migrates over the length scale ͱ Dt.

Write the function F͑t͒ in the form
where = Dt / a 2 is the normalized time, and g͑͒ is a dimensionless function specific to the type of the indenter.The ratio on the left-hand side measures how far the gel is away from the state of equilibrium.The poroelastic solution for a plane-strain cylindrical indenter exhibits two remarkable features. 8First, if the depth of the indentation, h, is held fixed while the force relaxes, the size of the contact, a, remains fixed over time, and the function a͑h͒ is independent of Poisson's ratio.Second, g depends on all parameters in the problem through only.Both features are shared by the solution for a cylindrical punch. 9We next confirm that the same features are present for spherical and conical indenters.We solve the poroelastic contact problems for indenters of several types by using the finite element software ABAQUS.In the calculation, the size of the contact is 20 times smaller than the size of the gel, and more than 16 elements are under the contact area in all calculations.Numerical results are insensitive to further increase in the size of the gel or refinement of the mesh.The initial conditions are set by the solution of an indenter pressed into an incompressible elastic solid.The boundaries of the gel outside the contact is traction-free and is in diffusive equilibrium with the external solvent, = 0 .The indenter is taken to be rigid, frictionless, and impermeable.The calculated functions g͑͒ for indenters of various types are plotted in Fig. 3, and the approximate formulas are included in Fig. 2. We have verified the two features mentioned above by varying Poisson's ratio, the half angle ͑for cones͒, and the ratio R / h ͑for spheres͒.Our calculation reproduces the function for cylindrical punch given in Ref. 9 but does not reproduce the function for the plane-strain cylindrical indenter given in Ref. 8.
Equation ͑1͒ and relation F͑0͒ / F͑ϱ͒ =2͑1−͒, along with the expressions for F͑0͒ and g͑͒ in Fig. 2, lead to a simple method to extract the three poroelastic constants, G , , D, from an experimentally measured relaxation curve F͑t͒.The measured value F͑0͒ determines G.The measured ratio F͑0͒ / F͑ϱ͒ determines .The measured curve F͑t͒, when matched to Eq. ͑1͒, determines D.
We prepared a covalently crosslinked alginate hydrogel following an existing protocol. 22Each sample had a diameter of 30 mm and a thickness of 20 mm, and was submerged in water during indentation.We made a conical indenter of aluminum, half angle = 70°.The indenter was pressed into the gel to a depth of h = 1, 2, and 3 mm.For each depth, the force was recorded as a function of time.To minimize the effect of the initial loading rate, the time used to press the indenter into the gel ͑10 s͒ is set to be much shorter than the relaxation time ͑hours͒.
Figure 4͑a͒ shows that the measured relaxation curve strongly depends on the depth of indentation.For a conical indenter, the depth of indentation is the only independent length scale. 10When the force is normalized by h 2 , but the School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA ͑Received 11 December 2009; accepted 1 March 2010; published online 23 March 2010͒ where k is the permeability and the viscosity of the solvent.The volume per molecule ⍀ and the viscosity are well known for commonly used solvents.The three other parameters, G , , k, are to be determined.A combination of the above equations leads to the familiar diffusion equation ‫ץ‬C / ‫ץ‬t = Dٌ 2 C, with the diffusivity being D = ͓2͑1−͒ / ͑1 −2͔͒Gk / .In poroelasticity, the diffusion equation usually a͒ Electronic mail: suo@seas.harvard.edu.

FIG. 1 .
FIG. 1.͑Color online͒ A gel is submerged in a solvent.After an indenter is pressed into the gel at a fixed depth, the solvent in the gel migrates, and the force on the indenter relaxes.