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Journal Club Theme of April 2015: Mechanical Characterization of Soft Hydrated Materials

Yuhang Hu's picture

Mechanical characterization of soft hydrated materials

By Yuhang Hu, University of Illinois at Urbana Champaign

 

Importance

Soft materials are ubiquitous in nature, from biopolymers, cells, and tissues to organs. Soft materials in recent years also become important engineering materials. Various stimuli-responsive hydrogels have been widely explored in microfluidics, tissue scaffold, drug delivery, actuators and sensors etc. Some swellable elastomers have been used in oil industry as packers. Fundamental studies of the deformation mechanisms of soft materials and optimal designs in different applications urgently need robust methods for material characterization. 

 

Practical challenges

(1) Soft materials are soft. The typical modulus of hydrogels are in the order of 1~102 kPa, which is more than 6 orders of magnitude lower than metals, and 3 orders of magnitude lower than plastics. Most commercial equipment is designed and built for testing hard materials. It is very difficult to be directly used on soft materials. In recent years, with the increasing interests in studying soft materials, many companies have started to introduce low load transducers into their testing systems, but it still remains challenging to get reliable and repeatable data from low load measurements. 

(2) Soft materials are often saggy. Soft of them are also slippery and/or brittle. Traditional methods such as tension, bending, and torsion are very difficult to apply. Sample casting and clumping are big issues. 

(3) Most biological materials are inhomogeneous and/or anisotropic. Local testing technique is required.

(4) Soft materials often need to be kept hydrated during test, which requires special experimental setup. 

 

Theoretical challenges

(1) The polymeric gels usually can absorb large amount of solvent and have huge volume change. Developing constitutive model to describe the coupled deformation and diffusion behavior of gels itself is a challenging task, not to mention using these advanced models to develop mechanical testing method.

(2) Gels and biological tissues are composed of both polymeric components and solvent molecules. The deformation of these materials can involve two molecular processes: the conformational change of the polymer network, and the migration of the solvent molecules.  The two processes result in the macroscopic behavior of viscoelasticity and poroelasticity.  In viscoelasticity, solvent molecules move relative to the polymer network over a short range. In poroelasticity, solvent molecules move relative to the polymer network over a long range. This behavior raised the issue of differentiating the two time scales in experimental measurements.

 Figure 1

Current status

(1) General Overview

The mechanical properties of gels have been studied through various testing methods.  Widely used techniques for measuring the modulus and/or dynamic modulus of gels include tensile testing [1-3], compression testing [4,5], beam bending [6,7], shear testing [8], cavitation rheology [9], micro-bead rheology [10,11], and nano- or macro-indentations [12-23]. 

The studies on the Poisson’s ratios of gels are controversial.  When a load is suddenly applied to a gel, solvent molecules do not have enough time to migrate into or out of the polymer network right away.  Consequently, the volume of the gel is conserved and the instantaneous value of Poisson’s ratio is very close to 0.5[1-3]. Given enough time, however, the solvent molecules diffuse into or out of the network until a state of equilibrium is reached.  The volume of the gel changes and Poisson’s ratio decreases to a value below 0.5.  Based on the method of poroelastic relaxation indentation (PRI), the equilibrium Poisson’s ratio of the gel can be attained [17-23]. 

Besides the mechanical properties of gels, i.e. shear modulus and Poisson’s ratio, the transport properties are also important. Transport in gels is characterized by the diffusion rate of small molecules through the network. The most widely used methods for measuring diffusivity of gels include free swelling and constrained swelling [24-27]. These methods measure the average diffusivity of a solvent in a gel as it evolves from its dry state to swollen state. It has been observed experimentally that the diffusivity of gels changes with solvent concentration [28]. The swelling methods, however, cannot measure the diffusivity of a gel in a particular swollen state.  The PRI method, in contrast, measures the diffusivity of a gel at a particular solvent concentration of the gel after it reaches chemical equilibrium with the environment [20,21]. Another way to measure the diffusivity of a gel at a particular concentration is through membrane transport [29-31], which is limited to stiff gels because of practical issues.

The advantages and disadvantages as well as the applicability of each of the above-mentioned methods are summarized in Table 1. 

 

 Table 1

(2) Indentation of soft hydrated materials

Among the various mechanical testing methods, indentation is recognized as the most practical technique for soft materials. It requires minimum specimen preparation, can probe local properties, and can be used at various length scales. It is also relatively easy to keep the sample in hydrated environment. The difficulty for using indentation lies in how to relate the experimentally measured data to the materials properties. Indentation is a mixed boundary value problem. It is very difficult to solve, especially when the time-dependent behavior of the material is considered. 

(i) Separation of Viscoelasticity and Poroelasticity 

As discussed above, the time-dependent behavior of gels could be viscoelastic or poroelastic corresponding to the reconfiguration of polymer network and migration of solvent respectively. In the context of indentation, it means that the viscoelastic relaxation time is independent of the contact size, while the poroelastic relaxation time is quadratic in the contact size. Therefore, the two time scales can be differentiated by a proper size of indentation. Indeed, this concept has been proved by us and other researchers. [18-20,32] 

(ii) Poroelastic relaxation indentation (PRI) method

To focus on poroelastic behavior, a method of relaxation indentation was proposed [16-21]. As shown in Fig. 2, an indenter is pressed into a swollen gel to a certain depth and is held for a period of time. Meanwhile, the force on the indenter is measured as a function of time. Both the indenter and the gel are submerged in solvent during testing. At the instance of indentation, the solvent has no time to migrate, 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. Therefore, using classical elastic solution and setting the Poisson’s ratio equals 0.5, we can directly calculate the shear modulus of the gel, G, from the instantaneous force F(0). After a long time, the solvent in the gel equilibrates with the external solvent, 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. The two limits are related as F(0)/F(∞) = 2(1-ν). From the equilibrium force, we can obtain the Poisson’s ratio ν. In the transient state, for the gel to equilibrate, the solvent in the gel needs to migrate over a distance comparable to the size of the contact, a. A good thing about relaxation test in contrast to creep text is the contact radius is kept constant during relaxation, because the classical elastic solution shows the contact radius is independent of material properties. As a result, the relaxation follows a simple scaling, Dt/a2. In one of our previous work, we showed that a single master curve could be derived, [F(t)-F(∞)]/[F(0)-F(∞)]=g(Dt/a2), and it only depends on the shape of the indenter. Fitting the experimental data to the master curve, we can obtain the diffusivity D. Figure 3 lists the solution for several shapes of indenters. So far, the PRI method has been used to measure the mechanical and transport properties of various soft gels and cell.[17-23,33]

Figure 2

figure 3_1

(iii) Poroelastic property vs. thermodynamic properties

To this point, you may want to argue the validity of using poroelasticity in the linear region. The rational is the following. Although gels can absorb a large amount of solvent and generate large deformation, indentation is carried out at a particular swollen state of the gel. Since we usually use shallow indentation in testing soft materials, the deformation generated is small. It is valid to use linear poroelasticity. 

For those who are interested in the large deformation and nonlinear behavior of gels, we have derived the relation of G and  to the thermodynamic parameters NkBT and Χ in the Flory-Huggins theory through linear perturbation at a particular swollen state of the gel. Details can be found in reference [20].

 

Future direction

Although indentation has been used to characterize diverse materials [34-36], the growing interest in developing indentation as a method to characterize soft and hydrated materials such as gels, cells, and tissues remains an immature stage. (1) A standardized procedure is waiting to be developed. (2) As discussed in the previous section, the time dependent behavior of soft materials is vsico-poroelastic in nature. While viscoelasticity is characterized by a material intrinsic time , the poroelastic behavior is characterized by the rate of solvent migration D. The two parameters define a material intrinsic length (D)1/2. Similar as the Ashby chart for hard material, a map of the characteristic length and time for soft hydrated materials is waiting to be constructed. It will provide an important guideline for material and device design. More detailed discussion regarding this point can be found in our paper on viscoelasticity and poroelasticity of gels [37].

 

References

 

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14.J. D. Kaufman, G. J. Miller, E. F. Morgan, and C. M. Klapperich, Time-dependent mechanical characterization of poly(2-hydroxyethyl methacrylate) hydrogels using nanoindentation and unconfined compression. J. Mater. Res. 23, 1472-1481 (2008).
15.M. Galli, K. S. C. Comley, T. A. V. Shean, and M. L. Oyen, Viscoelastic and proelastic mechanical characterization of hydrated gels, J. Mater. Res. 24, 973-979 (2008).
16.C. Y. Hui, Y.Y. Lin, F.C. Chuang, K.R. Shull, and W.C. Ling, A contact mechanics method for characterizing the elastic properties and permeability of gels. J. Polymer Sci B: Polymer Phys.43, 359-370 (2006).
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20.Y. Hu, X. Chen, G. M. Whitesides, J. J. Vlassak, and Z. Suo, Indentation of polydimethylsiloxane submerged in organic solvents. J. Mater. Res. 26, 785-795 (2011).
21.Y. Hu, J.-O. You, D. T. Auguste, Z. Suo, and J. J. Vlassak, Indentation: a simple, nondestructive method for characterizing the mechanical and transport properties of pH-sensitive hydrogels. J. Mater. Res. 27, 152-160 (2012).
22.Y. Hu, E. P. Chan, J. J. Vlassak, and Z. Suo. Poroelastic relaxation indentation of thin layers of gels. J. Appl. Phys.110, 086103 (2011).
23.E. P. Chan, Y. Hu, P. M. Johnson, Z. Suo, C. M. Stafford. Spherical indentation testing of poroelastic relaxations in thin hydrogel layers. Soft Matter8, 1492-1498 (2012).
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26.S. K. De, N. R. Aluru, B. Johnson, W. C. Crone, D. J. Beebe, and J. Moore, Equilibrium swelling and kinetics of pH-responsive hydrogels: models, experiments, and simulations, J. Microelectromech. Syst. 11, 544-555 (2002).
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32.Q. Wang, A.C. Mohan, M.L. Oyen, and X. Zhao. Separating viscoelasticity and poroelasticity of gels with different length and time scales. Acta Mechanica Sinica30, 20-27 (2014).
33.E. Moeendarbary, L. Valon, M. Fritzsche, A. R. Harris, D. A. Moulding, A. J. Thrasher, E. Stride, L. Mahadevan, and G. T. Charras. The cytoplasm of living cells behaves as a poroelastic material. Nature Materials 12, 253-261 (2013).
34.W. C. Oliver and G. M. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments J. Mater. Res.7, 1564-1583 (1992).
35.W. D. Nix, Elastic and plastic properties of thin films on substrates: Nanoindentation techniques, Mater. Sci. Engng. A234, 37-44 (1997).
36.Y. T. Cheng, and C. M. Cheng, Scaling, dimensional analysis, and indentation measurements, Mater. Sci. Engng. R44, 91-149 (2004).
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Comments

Cai Shengqiang's picture

Hi Yuhang, 

Many thanks for initatiting the nice review of mechanical characterization of soft materials. Mechanical characterization of soft materials is extremely important considering the intensive developement of soft materials in different applications. Compared to hard materials, mechanical properties or behaviors of soft materials have been much less investigated. For instance, it is known that fracture toughness of a material depends on the fracture mode mixdity. However,  when I tried to find similar information for soft materials such as rubber, I cannot found any relevant literature talking about it. (I will appreciate it if some one can point out one or more papers talking about the toughness of rubber as a function of mode mixidity). As another example, we have recently tried to measure the fatigue properties of soft materials, namely, crack propagation rate as a function of energy release rate, which turns out to be extremely challgenging. Therefore, I conclude that there are many research opportunities and open quesitons in characterizing mechanics of soft materials. 

shengqiang

Yuhang Hu's picture

Good comment. Exactly, there are a lot of room for development in this area. I also feel there is a need to promote this topic. It was less emphasized in the past in the mechanics of soft materials community.

Li Han's picture

Hi Yuhang, nice topic and review! I am particularly interested in your work on indentation study of gels. The coupling of viscoelasticity and poroelasticity are among the key factors complicating the analysis, and your approach has made it quite tractable. For any new material systems, however, it still will take some trial and error to find out the right time and size scale to work with experimentally. Any comment on this regard?

Han 

Yuhang Hu's picture

Thank you Han for your commonts. Good point. At this stage, it is very difficult to give a unified recipe. But if you think about it, it is no more complicated than conventional material. For instance, plane strain fracture toughness vs. plane stress toughness. First, people have to try a lot of materials with different sizes. After gathering enough data, people come up with a standard of sample size for measuring plane strain fracture toughness. The same analogy here. The difficulty here is just we have not gathered enough information to  make any recommendation yet. I hope as more and more people join in this study, in a few years we will be able to draw such a conclution that if you do indentation in, hypothetically 100um length scale, you will get pure poroelastic effect or viscoelastic effect. It is actually what I meant by "a standard needs to be set" in my original blog in the "future direction" part.

If you are considering really really new material which could have unexpected behavior, then trial and error is unavoidable. But it will be the same for getting other propeties of the material as well.

linst06's picture

Hi Yuhang,

Thank you for so nice and inspiring summary. Since indentation is local measurement, is it possible to utilize this techinique to achieve some full field measurement. Something like DIC, in addition to the local paramters, distributions in the full field can be obtained as well. However different from DIC (digital image correlation) just for strain measurement, indentation-based full field measurement definetly can provide more complete information. I am thinking about some problems such as what's going on around the crack tip, if there is any topological evolution when a PAAm-alginate gel is loaded and unloaded.  The information from DIC is quite limited. Some other ways such as dye fluorescene can be tried as well. However, indentation can measure a number of critical local parameters such as modulus, diffusivity, etc, which can be really helpful to the further understanding the intrinsic mechanics. But I can imagine, it can be sort of challenging for full field measurement. Thanks for comments.

Shaoting

Yuhang Hu's picture

Shaoting, thank you for pointing it out. Measuring local property is a good advantage of indentation technique. People have already used AFM based indentation to map the stiffness of gels and cells. It is certainly applicable. The practical issue to consider is perhaps the time-scale of a perticular phenomenon of interest vs. the measurement time. The measurement time is mostly constrained by equipment. Also because indentation is measured point by point, it is intrinsically a time-consuming process if you want to map something. I guess combining indentation technique with some other techniques might offer a way out of this constraint, and might be an important direction to think about for future study.

venapa's picture

Dear Yuhang Hu,

thank you for your nice review on the mechanical characterization of soft hydrated tissues.

I just would like to contribute to this issue by bringing to your knowledge additional papers of ours, recently published on this matter.

In particular, on nanoindentation of cartilage:

1. Taffetani, M., R. Gottardi, D. Gastaldi, R. Raiteri, and P. Vena. 2014. Poroelastic response of articular cartilage by nanoindentation creep tests at different characteristic lengths. Medical Engineering and Physics 36 (7): 850-8. 

2. Taffetani, M., M. Griebel, D. Gastaldi, S. M. Klisch, and P. Vena. 2014. Poroviscoelastic finite element model including continuous fiber distribution for the simulation of nanoindentation tests on articular cartilage. Journal of the Mechanical Behavior of Biomedical Materials 32 : 17-30. 

 

On dynamic nanoindentation of cartilage:

3. M.Taffetani, R. Raiteri, R. Gottardi, D. Gastaldi and P. Vena, A quantitative interpretation of the resposne of articular cartilage to atomic force microscopy-based dynamic nanoindentation tests, Journal of Biomechanical Engineering (accepted manuscripts) (2015); doi: 10.1115/1.4030175

 

 

4. Taffetani, M., E. Bertarelli, R. Gottardi, R. Raiteri, and P. Vena. 2012. Modelling of the frequency response to dynamic nanoindentation of soft hydrated anisotropic materials: Application to articular cartilage. CMES - Computer Modeling in Engineering and Sciences 87 (5): 433-60.

 

If needed additional details can be provided on the above papers.

Thank you very much.

Pasquale Vena

Politecnico di Milano

Dept. of Chemistry Materials and Chemical Engineering

Laboratory of Biological Structure Mechanics (LaBS)

www.labsmech.polimi.it

Yuhang Hu's picture

Dear Pasquale, thank you for the reference. It is very interesting. Since you mentioned dynamic nanoindentation, I actually have a question to consult you. For materials' time-dependent behavior, creep, relaxation or dynamic testing actually give the same information about the material. Is there a pratical reason to do dynamic test in the experimental perspective?

venapa's picture

Well, dynamic loading allows you to assess material properties at different strain rates. In some applications (like biological tissues) this can be an important piece of information.

From the experimental perspective: when using AFM, it is perhapes easier to perform dynamic tests rather then creep or relaxation tests; while, using typical nanoindentation equipments, creep tests are the most frequently performed  tests.

In the mentioned papers you find both time domain and frequency domain experiments. The nice thing of nanoindentation is that you may easily change the characteristic length of the experiment (by changing tip radius or penetrarion depth). This should allow you to discriminate between poro-elasticity and visco-elasticity.

Regards

Pasquale V.

D.Rittel's picture

Since the subject of dynamics of soft matter was briefly touched upon, I would refer to work done by G. Subhash and his colleagues, as well as our own work on the response of gelating at high strain rates.

  D.   Richler and D. Rittel, (2014), “On the testing of the dynamic mechanical properties of soft gelatins, Experimental Mechanics, 54, 805-815.

 

 

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As to the reason "why dynamics"?, one could reply that many situations (e.g. traumatic) involve dynamic loading of soft matter (e.g. flesh). There are no proven models to describe this case, yet pathologists need such information for simulations.

Finally....poroselasticity....viscoelasticity? The time scales involved in dynamic loading conditions are quite short, to an extent on could argue that those times are shorter than the characteristinc times scales of diffusion. If so, the problem get greatly simplified.

With best regards :=) 

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