View source for Poroelasticity of a covalently crosslinked alginate hydrogel under compression

''Written by Kevin Tian, AP 225, Fall 2011''

--[[User:Ktian|Ktian]] 20:03, 11 September 2011 (UTC)

'''Title:''' Poroelasticity of a covalently crosslinked alginate hydrogel under compression

'''Authors:''' Sengqiang Cai, Yuhang Hu, Xuanhe Zhao, and Zhigang Suo

'''Journal:''' Journal of Applied Physics, 108 113514 (2010)

'''Keywords:''' [[poroelasticity]], [[gels]], [[indentation]], [[compression]], [[relaxation]]

==Paper Summary==
This paper by the Suo group studies poroelasticity in an alginate hydrogel system by performing two experiments on the gel; one involving compression, the other involving an indentation method.  The results of the experiments were analyzed by using the theory of linear poroelasticity.  By comparing the theory and the experimental results, the authors were able to determine various elastic properties of the gel.  

It was found that the results obtained from both compression and the indentation method agreed well with theory, thus supporting the validity of the two methods.  An interesting side effect of the compression was that during relaxation the gels were found to fracture.  This is believed to be due to inhomogeneity of the water in the relaxing gel, which can be observed via theory.  However the exact mechanics of the fracture observed has yet to be determined.

==Theory==

'''Background'''

Gels have been the subject of study in recent years because of the wide variety of materials that consist of gels, including biological tissues, foods and drug delivery mediums.  As such the investigation of the mechanical behavior of these gels is important in determining how they may be applied.
Gels are known to have a time dependent mechanical behavior, due to the dynamic nature of the gel itself; namely the migration of water in and out of the polymer network.
By using the linear theory of poroelasticity one can determine the expected behavior of the gels and relate the relaxation data to the several material properties.

'''Biot's Theory of Poroelasticity''' 

We note that there are various symmetries to be noted for a disk under compression by parallel plates.  In addition various aspects of the situation allow us to write the governing equations for the poroelastic situation.  These considerations are as follows:
*The disk is taken to deform under the conditions of generalized plane strain
*Axial strain, <math>\epsilon_z</math>, is homogenous but varies with time
*Deformation of the disk can be taken to be axisymmetric
*The number of solvent molecules is conserved, namely: <math>{\partial C \over {\partial t}} + {\partial {r J} \over {r \partial {r}}} = 0 </math>
*The gel is always in mechanical equilibrium, and thus radial and hoop stresses must satisfy: <math>{\partial {\sigma_r} \over {\partial r}} + {{\sigma_r - \sigma_\theta} \over r} = 0 </math>
*Shear stresses and strains vanish
*The gel is not in diffusive equilibrium
*Since gel stresses are typically small, we assume polymer and solvent molecules are incompressible.  Thus the increase in gel volume is due only to the volume of the absorbed solvent (water in this case) or that: <math>\epsilon_r + \epsilon_{\theta} + \epsilon_{z} = \Omega (C - C_0)</math>

*Assume that the gel is isotropic and that gel stresses are linear in strains.  From this assumption we can write equations of state (relating stresses to strains).

Combining all the above allows us to finally write the governing equations for the concentration, displacement and chemical potential fields (<math>C(r,t), u(r,t),  \mu (r,t)</math>):

*<math> {{\partial (r u)} \over {r \partial r}} + \epsilon_z (t) = \Omega (C - C_0)</math>

*<math>{{2(1-\nu)} \over {(1-2\nu)}} {\partial \over \partial r} {\left ({\partial{(r u)} \over {r \partial r}} \right)} = {\partial {\mu} \over {G \Omega \partial{r}}}</math>

*<math>{{\partial C} \over {\partial t} } = {{D \partial} \over {r \partial r}} {({r \partial C} \over {\partial r})}</math>

Diffusivity is defined by the expression:
<math>{D} = {{2(1-\nu)Gk} \over {(1-2\nu)\eta}}</math>

Boundary conditions are defined by noting that
*Chemical potential of the solvent in the gel is equal to the external solvent at all times: <math>\mu (a,t) = 0\, \!</math>
*Radial stresses on the gel’s edge vanish at all times <math>\sigma_r (a,t) = 0\,\!</math>

'''Relaxation Curves'''

Solving the set of differential equations we can then obtain a closed form for the various stresses in the gel.  Integrating the axial stress throughout the area of the gel yields the following form for the compression force <math>F(t)</math>:

<math>{ F(t) \over {G \epsilon \pi a^2} } = 2(1 + \nu) - \sum_{n=1}^{\infin} {B_n \left (-4J_1 (\lambda_n) + {{2(1-\nu} \over 1-2\nu} \lambda_n J_0 (\lambda_n)  \right) exp\left(-{\lambda_{n}^2 {{D t} \over a^2}} \right) } </math>

'''Short and Long time limits'''

''Compression Case''

*We can consider two cases with regards to the deformation: a short-time limit (<math>t=0</math>) and a long-time limit (<math>t=\infin</math>).  

*We are quick to see that in the short-time limit:  <math> {F(0)} = 3G \epsilon \pi a^2 \,\!</math>,

*and in the long-time limit:  <math> {{F(\infin) \over F(0)}} = {2(1+\nu) \over 3} </math>

''Indenter Case''

*In a very similar fashion to the compression case, we can do the same for the indenter geometry as well.  We know for the indenter, with a half included angle <math>\theta</math >, is pressed into and held at a fixed depth ‘h’.  The radius of contact, ‘a’, is given by:  <math> a = {2 \over \pi} h tan(\theta) </math>

*In the short-time limit the gel behaves much like an incompressible elastic solid, due to the short time scales.  This prevents the solvent from having sufficient time to migrate, and yields expression: <math> {F(0) \over 4 a h} = G </math>

*In the long-time limit the gel has arrived at a new state of equilibrium, since the solvent (water) has had sufficient time to migrate.  This yields an indenter force of <math> F(\infin) = {2 G a h \over (1-\nu)} </math>

==Experimental Design==
[[Image:kt-3-fig1.jpg|thumb|200px|Figure 1. Illustration of the experimental design.  (a) depicts the case of compressing the gel between two parallel plates.  (b) depicts the indentation with a conical indenter.]]
The experiment first submerges a disc of the alginate hydrogel in water and allows it to fully swell for 24-48 hours.  Three disks of radii [3,4,5]mm are punched out of the gel and deformed in one of two ways:
 1. The gel is suddenly compressed between two parallel plates with some known force.
     *The plates are made of stainless steel, which approach the gel surface at a rate of 2<math>\mu m/s </math>.
     *When the measured force increases the thickness of the plate is observed.
     *Total vertical compressive strain is 20%
     *Initial contact time is 10s, and relaxation time is between 3-8h.
 2. The gel is suddenly indented with a conical indenter with a known force.
     *The indenter is made of aluminium with a half-included angle of 70&deg; the same approach rate of 2<math>\mu m/s </math> is used.
     *Initial contact time is 10s, and relaxation time is between 3-16h.
Afterwards the displacement is then kept constant and the force on the plate is measured using an AR rheometer.  During this time the gel undergoes a relaxation, as water migrates out of the gel, causing the gel to move towards a new state of equilibrium.  

In order to prevent variation due to gel preparation, the same batch of alginate hydrogel is used in both experiments.

==Results==
Using the above derived theory one can compute various elastic quantities from the relaxation curve obtained from experiment.  Observing fig [2a] we can see the raw relaxation curve data, and in fig [2b] the normalized relaxation curves (where conversion of force measured to a nominal stress has been performed).  

[[Image:kt-3-fig2.jpg|thumb|200px|Figure 2. Plots of the force on the parallel planes.  The first plot is the unmodified relaxation curves obtained.  The second normalizes the force to the area of the disc, and the time by disk radius squared.  The matching theoretical relaxation curve is also included in the second plot]]

More specifically one can see that computation of Poisson’s ratio, shear modulus and the diffusivity are possible using the data and the known functional form of <math>F(t)</math>.

'''Shear Modulus''': 	In the short time limit we observe that:
        For the compression case (parallel plate): <math> {F(0) \over \pi a^2} = 3G \epsilon  </math>
                  ...and for the indentation case: <math> {F(0) \over \pi a h} = 4 G </math>

'''Poisson’s Ratio''':	In the long time limit we observe that:
        For the compression case (parallel plate): <math> {{F(\infin) \over F(0)}} = {2(1+\nu) \over 3} </math>
                  ...and for the indentation case: <math> F(\infin) = {{2 G a h} \over (1-\nu)} </math>


'''Diffusivity''':		We perform a comparison of the theoretical and experimental relaxation curves for various diffusivities was made for various values of D until a match occurs.  This is true for both indenter and parallel plate methods.

==Discussion & Conclusions==
Comparing the material properties measured by the two separate methods, there was a 5.2% difference in shear modulus, 4.6% difference in Poisson’s ratio, and a 6% difference in diffusivity.  This is considered excellent agreement and supports the validity of the two methods.

Both methods have advantages and disadvantages.  With the compression test, it is absolutely necessary to have parallel top and bottom contact surfaces, otherwise this disrupts the test from functioning correctly.  Indentation does not have this requirement, instead requiring the detection of the starting contact point (which can be easily done for spherical and conical indenters).  However indentation additionally needs a significantly thicker sample, a minimum of 10 times the indentation depth.  

Depending on the situation, either requirement can be difficult to satisfy in practice.  However the validity of both methods in determining elastic parameters of the material has been supported by this analysis.  Thus the importance of this development is the ability to both understand and predict the behavior of the gel with established knowledge, as well as a new technique for measuring several elastic/material properties of gels.