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Journal Club Theme of July 2007: Mechanics of Hydrogels

Before we start this issue of J-club, I would like to recommend Prof. Langer's lecture for his MRS Von Hippel Award in the 2005 MRS Fall Meeting (Langer, 2006). His lecture not only delineated the history of the new exciting field of drug delivery and controlled release, but also told us many interesting stories happened in his career development. With Prof. Langer's pioneer work, many new materials are developed for designing new drug delivery and controlled drug release systems.

Environmentally responsive (ER) hydrogels are one of such materials. ER hydrogels can experience an abrupt volume change and hence hold or release a large amount of solvent under the change of environmental conditions, such as temperatures, pH values, and electric signals. Two recent review papers (Qiu and Park, 2001; and Bromberg and Ron, 1998) provide excellent overviews of the state-of-the-art and challenges of ER hydrogels for drug delivery and controlled release applications. The mechanism of large volume change of ER hydrogels is due to the competition between polymer-polymer macromolecule interaction and polymer-solution macromolecule interaction; this competition can be controlled through environmental stimuli. This can be illustrated through negatively thermoresponsive hydrogels (NTRH), which shrink as the temperature increases. At temperatures below LCST (Lower Critical Solution Temperature), hydrogen bonding between hydrophilic segments and water molecules are dominant, rendering enhanced water solubility and swollen state of hydrogels. As the temperature increases, hydrophobic interactions among hydrophobic segments of the macromolecular chains become strengthened, resulting in inter-macromolecular association through hydrophobic interactions and shrinking of hydrogels. Among thermally ER hydrogels, PNIPAAm (poly(N-isopropylacrylamide)) is the most extensively studied.

In many applications of ER hydrogels, it is very important that the volume change of hydrogels as a function of environmental stimuli and time can be precisely controlled, which requires high fidelity constitutive models that can be combined with numerical methods, such as finite element methods. The above-mentioned mechanism for swelling/shrinking described the fundamental reason for equilibrium swelling/shrinking. A review paper by Prof. McKenna (McKenna) describes the modeling consideration for equilibrium swelling/shrinking behaviors. Developing constitutive models of swelling/shrinking process (a diffusion process) suitable for numerical methods, however, is a major challenge. As one can imagine, during sorption the polymer network chains begin to stretch by the increase in mass uptake. Diffusion in polymers is therefore a complicated rate controlling process between the mechanical relaxation effects associated with the viscoelastic behavior of polymers and the relative diffusion time scale associated with the mutual diffusion process. In many cases, diffusion in polymers does not follow the simple Fick's law. Briefly, when the characteristic diffusion time is much longer than the polymer relaxation time, the diffusion process is Fickian in nature. Alternately, when the characteristic diffusion time is much shorter than the polymer relaxation time, the diffusion process is associated with a so-called Case-II behavior. Lastly, when the two characteristic times are comparable, the diffusion process in the glassy polymer is anomalous behavior. Case II behavior is characterized by the existence of a sharp boundary (or front) between the swollen and dry polymers and the mass uptake has a linear relationship with time. Other abnormal non-Fickian behaviors include intermediate anomalous behaviors, two-stage behaviors, super case II behaviors, overshoot behaviors, etc.

Although swelling behaviors of polymers have been studied both experimentally and theoretically for a long time, major challenges still exist. Emerging new applications of ER hydrogels also demand better models to be developed. First, as pointed out in the paper by De Kee et al, the attempts to develop a unified model that can consider both Fickian and non-Fickian behavior were not quite successful in the past. Is it necessary to develop such a model? Is it possible to develop such a model with the assistance of other technique, such as numerical method? Second, many theoretical models have been developed in the past to address different aspects of diffusion in polymers. In contrast, efforts using numerical investigations, such as finite element methods, appear to be surprisingly limited, at least as reflected from the number papers in the literature. Third, studies on coupling between external stress and strain with polymer diffusion are also limited. In this issue of J-club, we will start our discussion on above mentioned questions or comments. The discussions are not limited to these; many other interesting problems also exist related to ER hydrogel applications and diffusion. For example, damage of ER hydrogels is a very important factor in designing hydrogel-based devices. For another example, due to similarity between hydrogel structures and biological tissues and cells, problems of diffusion in polymers may also exist in such materials. To start the discussion, I listed three papers: 

1. De Kee, D., Liu, Q. and Hinestroza, J. 2005. Viscoelastic (non-fickian) diffusion, Canadian Journal of Chemical Engineering. 83(6): 913-929

2. Olsen, M.G., Bauer, J.M., Beebe, D.J. 2002, Particle imaging technique for measuring the deformation rate of hydrogel microstructures, Applied Physics Letters, 76(22): 3310-3312

 3. Ji, H., Mourad, H.,  Fried, E.,  and Dolbow, J., 2006, Kinetics of thermally induced swelling of hydrogels, International Journal of Solids and Structures 43(7): 1878-1907.

The first paper provides a good review of diffusion in polymers and general considerations in developing models. For those who are not familiar with Case-II diffusion, the second paper provides an experimental visualization of the existence of a sharp front. The third paper is written by one of the frequent participants of J-club, Prof. Dolbow. This paper appears to be one of the first papers to address directly the motion of sharp front in hydrogel swelling/shrinking.

Comments

Thanks very much, Jerry, for mentioning our work on this topic! 

Stimulus-responsive hydrogels (SRHs, as we call them) are materials that most mechanicians should find interesting.  In some ways, they're the latest and greatest class of "smart materials".  On a more fundamental level, I suspect many of us find problems where stress is coupled to diffusion of interest. 

Our work on this topic began with this paper published in JMPS.  The details of the theory are actually provided in a subsequent work that appeared in CMAME, which may prove to be a more useful reference for the jclub.   

We're not the first to develop models that explicitly treat the interface between swelled and collapsed phases.  Tomari and Doi have a much earlier paper along these lines that appeared back in 1994 (see our JMPS paper for the reference).  I think what's fair to state is that we're the first to employ concepts from modern continuum and computational mechanics to address the problem.   In particular the notion of configurational force balance and its numerical treatment in this context.  

Of course many of us owe a debt of gratitude to the late Toyoichi Tanaka on this subject.  I've had more than a few people relate to me how interesting his lectures on "smart gels" were.  Apparently he used to begin the lecture by placing a small spherical gel (collapsed) next to him on the lectern.  By the time he was finished, it would have swelled to the size of a tennis ball.  Background on his career at MIT and work on this subject is provided here.

I look forward to more discussion of these fascinating materials here on jclub.  

Thanks, John, for mentioning many other people's work. Due to the space limitation, I didn't include those people's names and works.  It's a shame that Tanaka passed away.

Your work is particularly interest because it employs the finite element, which shed the light for the possiblity of using FE to solve gel swelling/shrinking problem in a more complicated geometry setup. I personally think this shoud be the direction to go.

There is another group of people (K.N. Christodoulou, E.C. Achilleos, I.G. Kevrekidis) who have done very good work toward finite element modeling. But their work seems discontinued after 2003.

Henry Tan's picture

It seems that the very interesting behaviour of hydrogels could be applied to explosion science.

Henry, I haven't seen any works using hydrogel for explosion. I think the main problem may come from the fact that hydrogels typically have low modulus and relatively slow response. For example, the response time for hydrogels are typically in the range of seconds to minutes, depending on the method for heating.

MichelleLOyen's picture

Henry, semi-related to your question...  There is a lot of interest in gel materials in the context of ballistics experiments for simulation of soft biological tissues .  See recent work from Mark VanLandingham et al.

Rui Huang's picture

Hydrogel seems to be a very interesting material to work with. I know nothing about it and thus would ask an elementery question: 

In addition to the kinetics (diffusion, swelling/shrinking, etc.), what do we know about the mechanical properties of hydrogels (e.g., elastic, viscoelastic, etc.)? For example, if we want to simulate a hydrogel film under nanoindentation, what type(s) of the constitutive model should we use in general? 

Thanks. 

RH

The modulus of hydrogels is in the range of hundreds of KPa, which is very low. This is due to the small crosslinking density, which allows the large volume change but at the same time reduces the modulus of elasticity. In the past, people have tried different methods to increase the modulus of elasticity to the range of hundreds of MPa but were not very successful. Therefore, most of hydrogel applications are in the area where the mechanical propeties are not critical and small modulus is desired. For example, hydrogels have found applications in microfluidics, lens, cell encapsulation, drug delivery, etc.

Regarding constitutive models, McKenna's paper provide a good overview for constitutive model on equilibrium swelling behavior. Basically, one can modify a hyperelasticity model by considering the additional marcomolecular chain stretch due to swelling. In addition, one need to consider the constraints due to crosslinking sites when the two crosslinking sites get cloers to each other. This can be modelded using models such as Flory-Erman (Flory, P.J., and Erman, B., 1982. Theory of Elasticity of Polymer Networks. 3. Macromolecules, 15(3): 800-806. )

It is diffcult to consider the non-equilibrium swelling behavior in a general 3D deformation setting. Because of the complicated kintics of swelling, to my best knowledge, constitutive models that can be used with FEA to simulate complicated deformation cases, such as indentation, do not exist. This remains a major challenge.

 Jerry

Hua Li's picture

Jerry, I just know this interesting  Journal Club by Zhigang's email sent to Lu Chun. Zhigang will stay in IHPC of Singapore from 25 July to 5 Aug. I look forward to an appointment with him to share our work on modeling of smart hydrogels responding to pH, externally applied electric voltage and environmental temperature, respectively.

 To my understanding, the modulus of hydrogels is governed not only by crosslinking density, also by others, especially for environmentally responsive (ER) hydrogels. obviously, the modulus of pH-sensitive hydrogel also varies with solution pH. Therefore, development of constitutive models of hydrogel becomes a challenge. A large variety of hydrogels, from very soft state like mucus to hard state like solid, makes it more chahallenging.

Rui Huang's picture

Is it possible that at the nanometer scale (<1000 nm) all the kinetic processes are soon finished and thus what observed in experiments is mainly equilibrium state (suppose that the loading conditions are time-independent)? If so, would an elastic constitutive model (most likely nonlinear due to large deformation) be sufficient to describe or predict the equilibrium state? Thanks. 

RH

Zhigang Suo's picture

Rui:  Here are a few points relevant to your comments.

  1. Indeed, hydogels have been used in small devices so that time delay is very small.  One example is hydrogel used to control flows in microfluidics (Beebe 2000).  In such a case, once you know the time scale to attain equilibrium, the diffusion process is irrelevant.  All you need is an equilibrium theory.
  2. An equilibrium theory of swelling has long been available.  For example, you can find the field equations in my lecture notes, and a free energy function from chapter 7 in the book by Treloar.
  3. When the feature size approaches the Debye length, electrical effects can be significant.
Hua Li's picture

Zhigang, this is a good topic with wide-range applications. Howerer, one of disadvanges of the smart hydrogels  is their rather slow response to surrounding stimuli. Maybe an approach is to increase the surface of hydorgel componemt  in MEMS devices by optimal design of geometry. Unfortunately, this often weakens the mechanical strength of the hydorgel. I am really interested in any comments on this concern. 

Zhigang Suo's picture

A continuum theory of hydrogels needs to describe migration of molecules in a solid.  As such, the same theory applies to soils and bones, where the theory is known as poroelasticity.  The same theory also applies to solutes diffusing in crystals. 

Last semester, I included a section on the subject in my second-year graduate course on Advanced Elasticity, and posted on iMechanica the lecture notes titled Poroelasticity, or Diffusion in Elastic Solids. I took an established approach in formulating a continuum theory.  The kinematics is described by deformation gradient and concentration.  The field equations are set up to be consistent with thermodynamics. 

The notes assume some knowledge of finite deformation, thermodynamics and diffusion.  The notes are rather terse, written mainly as a reminder for myself and the students attending the class.  I suppose the notes can also be adapted by other instructors in their classes, but am not sure that they are helpful to beginning students. 

I learned quite a few things from comments on the lecture notes.  In particular, Mogadalai Gururajan pointed out that absorption of fluid in a solid was treated by Gibbs in his classic paper published in 1870s.  Gibbs treated the problem in the setting of finite deformation, using deformation gradient and nominal stress.  Of course, he also used chemical potential.  He only formulated an equilibrium theory, and did not treat diffusion.  The theory of poroelasticity is commonly attributed to Biot's 1941 paper.  Evidently Biot was unaware of Gibbs's work.

Comments from other people have led to quite a few instructional materials and research papers.

Applications of the theory to soils, bones and crystals have mostly invoked small, elastic deformation.  For hydrogels, several effects are prominent:

  • Finite deformation is obviously important.
  • The migration species may contain charged species. (Electric effects have already been studied extensively for bones.)
  • The gel may undergo viscoelastic or even plastic deformation, as discussed in Jerry's original post.
  • The density of cross links can be readily modified by things like pH and enzymes.
  • Deformation affects the coefficient of diffusion.

I share the enthusiasm of Jerry Qi and John Dolbow for two reasons: 

  1. The mechanics of hydrogels is so rich that mechanicians of any background (theoretical, experimental, computational, etc.) can make contributions. 
  2. Hydrogels have such a large array of existing and emerging applications that mechanicians can interact with people in several other fields.

To me, a missing link between 1 and 2 is a set of questions that are both important in applications and challenging to mechanics.  I'd like to hear more from Jerry and John, as well as others, on this link.

Wei Hong's picture

Zhigang,

 The lecture notes is helpfull in the sense that it gives out a very simple picture, and it lumps all unknowns to a misterious variable, the chemical potential.  For traditional porose material, it might work alright, but it is really different inside a hydrogel.  One important thing is the electric field, created by the charged particles (ions) in the network and in the aqueous solution.  Not only are they influenced by the exernal electric field, they themselves will also modify the electric field.  Another difficulty is when the charged particles move in the solution, they create an electric current in the gel, the external batteries do work through the process.  Also one needs to consider the creation and anilation of the particles by electrolysis.

In general, the "chemical potential" should be a function of the concentration field (not cocentration of a single point, as for the electric field created by other particles), the position (in respect to external electric field), and the deformation state.

If you look into these, the picture is really not as clear as the one we have for dielectrics.

 

Zhigang Suo's picture

Dear Wei:  This is my last day at IFMA.  Tomorrow will be the French National Holyday.  Denian, Michael, and I will spend the weekend in Lyon over the weekend, and I'll be back to Cambridge on Tuesday. 

In response to your concerns over electrical effects, I've written some notes on polyelectrolyte gels in between visits to Puy-de-Dome and Michelin Museum   The notes are very rough, and likely contain errors, but I think the basic ideas are sound.  Please look at them and we should discuss next week.

MichelleLOyen's picture

Great points, although I'd note that so-called "triphasic " formulations for poroelastic media with coupled mechano-electrical effects have been considered far more in cartilaginous tissues (articular cartilage and spinal intervertebral discs) than in bones.  I think this is another case where a large literature already exists in the biomechanics field that may be simply unknown to more traditional mechanicians; the level of complexity found in current cartilage modeling is quite remarkable.  Large deformations, coupled viscoelastic-poroelastic effects, and anisotropy have all been the subject of recent study.

Update: a more recent paper (accessible as a pdf) also containing "biphasic" vs "triphasic" formulations in cartilage tissues is here

Zhigang Suo's picture

Mishelle:

I'm rereading the thread initated by you on viscoelastic contact, and find many discusions of viscoelastcity and poroelasticity there are very relevant to the Theme of this month.  Thank you.

Hua Li's picture

Zhigang, thank you for your email sent to Lu Chun to let me know this interesting  Journal Club. I know you will stay in IHPC of Singapore from 25 July to 5 Aug. I look forward to an appointment with you to share our work on modeling of smart hydrogels responding to pH, externally applied electric voltage and environmental temperature, respectively.

 To my understanding, there is a large variety of hydrogels, from very soft state like mucus to hard state like solid. Thus it is difficult to develop the constitutive models of hydrogel by a generalized formulation, especially for environmentally responsive (ER) hydrogels. Elastic, vicoelastic or plastic behaviors are observied in various hydorgels. 

Zhigang Suo's picture

Dear Hua:  Great to hear from you!  Although we have not met, I have been studying your JAP paper on stimuli-responsive hydrogels.  Your paper has put many modeling issues together.  I look forward to meeting you in person next week.

Hua Li's picture

Zhigang, many thanks go to you to take a look on our work. So far we have developed 3 models for simulation of hydrogels responding to solution pH, externally applied electric voltage and surrounding temoerature, respectively. All are based on Nernst-Planck-Poisson system. They are really preliminary work and there are a lot of rooms to improve. I most sincerely appreciate any comments and discussions on them. Further, I also look forward to researchers who are interested in work with us for opening one more window. Yes, I also look forward to meeting you soon.  

MichelleLOyen's picture

Our recent preliminary work--on implementing poroelastic contact mechanics models from the literature for analyzing experimental indentation data--has just appeared in proceedings format (link to the paper on the MRS website --free download for members).

Zhigang, Thanks for the nice discussion on poroelasticity and diffusion. I absolutely agree with you that mechanics of hydrogels is a very rich field that mechanicians of any background can contribute. In addition, since swelling/shrinking of hydrogels is governed by diffusion and mass transportation, it is also a field that might be familiar to many mechanicians.  

For hydrogels, in addition to the important effects to mechanics of hydrogels you mentioned, I would like to stress one additional challenge: as penetrants enter the polymer network, they also serve as platicizers which effectively change Tg and hence the viscoelastic behavior of the material. If swelling/shrinking happen around the Tg, one would imagine a dramatic mechanical property change. In many cases, the swollen portion is in the rubbery state but the shrunk portion is the glassy state. Such property change has profound effects to the swelling/shrinking kinetics. 

Regarding the “missing link”, in addition to the lack of an effective constitutive model that can be combined with FEA to assist product design, I would like add a few cents:

 

  1. Fracture of hydrogels. Fracture is an important issue since most hydrogels are fragile. In some cases, the presence of a moving sharp front can cause hydrogels to break. In other cases, external mechanical constraints can break hydrogels.
  2. Repeatability of swelling/shrinking behaviors or fatigue. In many hydrogel applications, it is expected that hydrogels can go through cyclic swelling and shrinking without damage. It is therefore important to understand the fatigue behavior of hydrogels under cyclic swelling/shrinking.
  3. Precise determination of mechanical properties at small scale. Major issues in determining mechanical properties of hydrogels are 1) low modulus and 2) the requirement to immerse hydrogels in solutions. The next issue of J-club is about indentation on soft matters. It will be interesting to look at the discussions in that issue.

Jerry,

This is a great point and I'm glad you brought it up.  We've been looking into the failure characteristics of hydrogels for awhile here at Duke, and the results are really interesting. We see many phenomena that are not observed in other "soft" polymers.  I hope to find some time to write this up soon.  

SRHs also exhibit interesting tribological behavior, with apparent frictional coefficients changing by an order of magnitude with phase state.  This is another aspect that is often neglected when hydrogels are discussed for small-scale devices: friction can be a show-stopper.  Our work recently appeared in Langmuir .    

Thanks, John. Very nice work on the tribological behaviors of hydrogels. I am also eager to see your work on the failure characteristics of hydrogels.

Jerry

Aaron Goh's picture

Since you mentioned fracture, Jerry, I was wondering, that when a gel sets, is there any residual stress in the gel, such that when the gel is broken into two, the sum of the two parts, in terms of volume, is greater/smaller than the original, and as a result, the water is absorbed/released.  I was wondering whether it is common to have mechanically responsive gels.

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