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Drying-induced bifurcation in a hydrogel-actuated nanostructure

Wei Hong's picture

Hydrogels have enormous potential for making adaptive structures in response to diverse stimuli.  In a structure demonstrated recently, for example, nanoscale rods of silicon were embedded vertically in a swollen hydrogel, and the rods tilted by a large angle in response to a drying environment (Sidorenko, et al., Science 315, 487, 2007).  Here we describe a model to show that this behavior corresponds to a bifurcation at a critical humidity, analogous to a phase transition of the second kind.

The structure adapts to the drying environment in two ways.  Above the critical humidity, the rods stand vertical, enabling the hydrogel to develop tension and retain water.  Below the critical humidity, the rods tilt, enabling the hydrogel to reduce thickness and release water.  We further show that the critical humidity can be tuned.


Xiaodong Li's picture

Wei, this is a paper of great interest to me. The model makes sense and provides guidelines for nanoactuator/sensor design. This may be a humidity sensor or other sensors/actuators. My other thought is that if we use magnetic rods, we may get multifunctionalities. The magnetic rods tilt will change magnetic fields/signals. I think that the concept may be extended to biosensing. From the experimental point of view, I think it will be great to grow nanorods embedded in hydrogel.

Wei Hong's picture

Hi Xiaodong, Thank you for your interest in our paper.  Your thought about the magnetic rods is indeed very interesting, both experimentally and theoretically.  Also, as we mentioned in the paper, similar devices are easy to made to be sensitive to other stimuli, such as temperature, ph, light, etc.


  I haven't yet had the chance to read your paper but it certainly sounds interesting.  I look forward to reading it soon. 

  At the moment I just thought I would mention that magnetic particles have been embedded in thermally-responsive hydrogels for the purpose of faster heating.  One of the issues with using temperature to induce volume change is that it can be a slow process.  The idea is that a magnetic field gives rise to local heating in the particles, inducing more rapid heating throughout the gel, and thus collapse.  Here's a recent article that employed the technology in drug delivery. 

   Happy New Year to all at iMechanica!

Wei Hong's picture

Dear Prof. Dolbow,

Thank you for your interest in our paper, and thanks for bringing the interesting paper to our attention.

While the magnetic particles can induce rapid heating in a gel, it might also affect the swelling behavior of the gel, as for the interactions between particles, which is an interesting physical phenomenon to study by itself.

On the other hand, if we use an array of aligned magnetic rods in a gel, we would have a composite structure that can be triggered directly by a external magnetic field.  Even more interestingly, would one have a ferromagnetic gel that can converts magnetic signal to macroscopic deformation or vice versa?  It might have special applications due to its softness.

Happy New Year!

Zhigang Suo's picture

Dear Xiaodong:  Thank you so much for all the suggestions for our work.  I really appreciate your generosity.  We are making baby steps in learning about gels, and are not always sure that we are going in the right direction.  Feedback from experimentalists is of particular value to us.

Although gels have always been part of our life, as tissues and as food, several major pieces have fallen into place in recent time and have made a compelling case for gels as an engineering materials for diverse applications.  These pieces are:

Materials.  Many gels, natural or synthetic, now exist and are known to swell greatly in response to diverse stimuli.

Fabrication.  Micro- and nano-fabrication methods can integrate gels with other materials in hybrid structures with small features.  Multiple materials enrich functionality.  Integration reduces cost and increases reliability.  Small features shorten response time.

Theories.  Theories on the basis of statistical mechanics and continuum mechanics have become sophisticated enough to deal with many phenomena in gels, as well as their integrations with other materials.

A unique opportunity is now open to engineers:  to create new desires and new designs by using gels.  Your remarks clearly point to this direction. 

Hua Li's picture

Hi, Zhigang, Sorry for the delay. The paper you mentioned is really interesting as it has opened another window for me. However, I still have several questions to be clarified. (1) As it is assumed that the hydrogel is bonded to the rods of silicon (see page 3), how to understand the hydrogel developing a state of isotropic tension near the rods along the rod direction? (2) How to understand that " the process is reversible" (see the line 6 from bottom of Page 2) from the formulations? (3) do you think it is necessary to make some assumptions to use the continuum theory for the present nanoscale domain?

Wei Hong's picture

Hi Dr. Li,

Let me try to answer your questions:

(1) The gel is assumed to be bonded to both the rods and the substrate.  Constrained in all directions, the gel is just like in a cage, and it simply can not deform (except for tilting which involves additional thredshold as described in the paper).  When dried, the gel has an natural tendency to reduce in volume.  If without the constraints, the gel would shrink freely.  However, it is now constrained in a cage.  The only thing it can do before tilting is building up an internal stress against shrinking.  As both the initial condition and the constained are assumed to be isotropic, the tensile stress within can only be isotropic.

(2) The process is reversible is that it can shrink when dried or swell back when hydrated.  Not as the "reversible process" in thermodynamics terminology, such a process does dissipates energy.  Zhigang, shall we consider another word here?

(3) I think the assumption necessary is that the size of the nanostructure is still larger than the microstructure of the material itself.  In this case, for example, the assumption should be that the nanorods are still way bigger than the polymer molecules. 

Jinxiong Zhou's picture

Dear Wei and Xuanhe,

Congratulations for your new publications. Very excellent works!  I just give the paper a glance and have a very basic question. I understand that the embeded rods and the substrate provides different constraints. The swelling hydrogel can deform through the sapcing of two adjacent rods but can not penetrate through the substrate. So the vertical state may not be homogeneous state. Do I misunderstand the physical picture?

BTW, Happy new year!


Wei Hong's picture

Dear Jinxiong, Thank you for your interest in the paper! The homogeneous state is our assumption, and it corresponds to a limiting case. Both the rods and the substrate are rigid, so that the layer of gel very close to the rods or substrate can not deform. If the rods are very close to each other, most part of the gel is constrained by the rods and the substrate, and can not deform freely, except for the gel near the top surface. In the case when the rods are very long, the inhomogeneous part near the surface can be neglected. This is the picture we have in mind. Wei

Zhigang Suo's picture

I'd like add to a discussion on rate processes between Taher Saif and Wei Hong.

  1. Viscoelasticity vs. poroelasticity.  This issue has been discussed by Michelle Oyen and Jerry Qi, among others.  Both viscoelasticity and poroelasticity appear as time effects, such as in creep test and relaxation test.  However, viscoelasticity and poroelasticity correspond to different microscopic processes.  The difference can be readily understood in a polymer.  Viscoelasticity is due to the delay in rotating individual molecular joints and sliding polymer chains.  Poroelasticity is due to delay in migration of mobile molecules.  Thus, viscoelasticity is a short-range effect, and poroelasticity is a long-range effect.  When the size of a sample is large compared to the size of individual molecules, the viscoelastic effect is independent of the size of the sample, but the poroelastic effect will happen over a longer time scale for a larger sample.  Clearly, one can design macroscopic experiments to differentiate the two effects.  Of course, in a given material, both effects can occur:  the viscoelastic effect will dominate over short time and in large samples, while the poroelastic effect will dominate over long time and in small samples.  Learning examples of this difference will be illuminating.
  2. Cases in which only equilibrium poroelasticity is important.  Rate effects are hard to study.  Thus, it is significant to focus on equilibrium states whenever detailed rate effects are unimportant.  In the nanostructure studied in this paper under discussion , the time scale can be estimated by the diffusion time, L^2/D, where L is the length of the rods and D is an effective diffusivity.  Either this time scale is good enough for an application or not.  A more detailed analysis on rate effects will not add much in practice.  In this particular case, because the small scale, the rate of switching seemed to be limited by transport in the environment, rather than transport in the gel, as shown by experimental observations.  Also, equilibrium state itself can be very interesting, as illustrated by this example.  As another example, when a gel reaches a state of equilibrium, the distribution of the mobile species can be inhomogeneous.
  3. Cases in which rate effects are important.  A primary example is drug delivery systems, where the amount of release as a function of time is the name of the game.  As another example, when a structure has many equilibrium states, which one is reached will depend on the path of evolution.  The path is determined by kinetics.
  4. The state of the art of modeling rate processes in gels.  My group has recently reviewed the literature on modeling rate effects in gels.  If you have read the paper or have had some experience with the subject, you will probably know that much work need be done.  On the other hand, successful drug delivery systems have been designed with existing understanding, aided by trial and error.  To make further progress in modeling, and to see how better modeling might aid in design and discovery, one point of entry is to learn from the successful practices in which kinetic effects are significant.  

 The paper you mention as a review of rate effects in gels is interesting.  To add to this discussion for those that are interested in the transport of low molecular weight  species in polymeric systems, I would like to point out the following series of papers some old, some new, which also include some interesting references that are not included in Dr. Suo's review.  The first paper deals mainly with modeling, the second with both some modeling and then solution issues, the third solely with solution issues and the fourth with both modeling and solution issues.  The modeling also includes the cases where the polymers can undergo a rubber-glass phase transformation and one of the papers looks at the case where additionally chemical reactions can take place (semi-conductor lithography).


One interesting point to note that we do not assume the additivity of volumes that is typical in the literature and which is embedded in the Flory-Huggins theory of mixing.  Instead we rely upon an extension of the Hildebrandt theory of mixing.  This allows for independent deformation and concentration fields in the formulations -- something that is not possible if one invokes the Flory-Huggins theory of mixing. 


  1. Govindjee, S.
    & Simo, J.C., ``Coupled Stress--Diffusion: Case II," J. Mech. Phys. Solids,
    41, 863-887 (1993)
  2. Vijalapura, P.K. and Govindjee
    S., ``Numerical simulation of coupled-stress case II diffusion in one dimension,"
    Journal of Polymer Science Part B: Polymer Physics, v41, 2091--2108 (2003).

  3. Vijalapura, P.K., Strain, J.A., Govindjee, S.,
    "Fractional step methods of index-1 differential-algebraic equations,"
    Journal of Computational Physics, v203, 305-320 (2005).
  4. P.K. Vijalapura and Govindjee,
    S, ``An adaptive hybrid time-stepping scheme for highly nonlinear strongly
    coupled problems," International Journal for Numerical Methods in Engineering,
    v64, 819-848 (2005).






Prof. Dr. Sanjay Govindjee
University of California, Berkeley

Zhigang Suo's picture

Dear Sanjay:  Thank you very much for pointing out these papers.  My
students and I will follow them up.  Incidentally, why did you allow
compressibility of molecules?  Is it for numerical implementation or
for some physical reasons?  We thought that since deformation of gels
are so much larger than volumetic change of molecules, the latter is
negligible.  Best, Zhigang 

Dear Zhignag, In our first paper (Govindjee & Simo), I used a hard sphere model for the diluent.  But this can be problematic numerically and for some systems it can be physically important.  It really depends quite a bit on the boundary conditions.  That is why in the latter papers, it is included in the formulation -- it allows for more general problems to be treated and it also helps with the numerics.  What also matters is the ratio of volumetric compressibility of the diluent to the volumetic compressibility of the polymer.  One other thing to observe about these papers is that the mixing energy is phrased in terms of two independent variables, J = det[F] and c.  Without this independence of fields, the volumetic deformation is simply a slave to the diffusion field, which in many cases is simply not true -- just think of rapidly expanding a cube of polymer in a solvent bath, the volume changes quickly but the solvent will take its own time to diffuse in to fill the chemical potential void.  Of course W_mix(c) is just fine for solely equilibrium situations with free swelling mechanical boundary conditions.  -sanjay


Prof. Dr. Sanjay Govindjee
University of California, Berkeley

Wei Hong's picture

Dear Zhigang,

 I think the major difference between our formulation and Sanjay's is on the treatment of volume: we assume molecular incompressibility and allow no free volume in the polymer, but Sanjay allow such free volume to exist.  This is very important to the study of case II diffusion, as the rubber-glass transition is determined by the free-volume concentration. (with Dolittle's model I suppose).  I believe this is also the major difference between Hildebrand's free-energy of mixing and the Flory-Huggins model.

Physically, the free volume in a polymer must exist.  The question is just whether it can be neglected or not.  We are looking at gels, with up to 90% of the total volume taken by solvent molecules.  Compared to that amount of volume, the free volume may be neglected, as we did in the last paper.  However, in cases Sanjay studied, the concentration of solvent molecules is so low that the free volume is essential in the physical processes.

This is just my superficial understanding after reading the papers briefly.  Please correct me if I am wrong, Sanjay.


Zhigang Suo's picture

Dear Sanjay and Wei:

Thank you so much for the clarification.  Here are a few thoughts running through my mind as I read your comments.

  1. I have no problem with including molecular compressibility for the ease of computation.
  2. Sanjay mentioned that if the outside volume swells substantially, how can the inner part respond without volume expansion.  Well, the inner part can do several things to adapt without great expansion.  It can develop a high stress, of multiple of NkT, the shear modulus of the dry network, as shown in a recent calculation of a gel containing a hard core.  Or the inner part can damage, by forming macroscopic voids (cavitation, rather than molecular level free volume).
  3. In the case that the inner part develops a stress of multiple of NkT, the magnitude of the stress is really just of order some MPa.  When we talk about molecular compressibility, we are talking about a modulus, say on the order 10 GPa.  This stress will only cause molecular compressibility of less than 10^3.  This strain is much, much smaller than the volumetric expansion due to uptake of solvent, as Wei has just pointed out.
  4. Will this small molecular compressibility be ever important in any physical situation?  Wei mentioned the increase of free volume, and its possible effect on diffusion.  To evaluate its importance, we will have to look at numbers.  For example, how much will diffusion in a liquid or a polymer solution be increased under a stress of MPa?  Will this increase significantly impact stress or damage in a gel?

I'm interested in your take on these issues. 

Wei and Zhigang,

1. Free volume was indeed a needed concept in what we were looking at for precisely the reason mentioned --  Doolittle's theory.   [Just a word of warning, do not think of free volume as actually volume!  It is energetically (easily) accessible space.]  Hildebrand's theory is also like the Flory-Huggins theory, an equilibirum entropy of swelling model.  In fact, the common version of F-H is the same as the Hildebrand expression at the end of the day.  Hildebrand's deriviation is however a bit easier to manipulate if you do not want to assume equilibirum swelling -- which is why we used it.  The same can be done with F-H but the algebra is a bit more involved.

2. In the highly swollen gel (depending upon the process being analyzed), ignoring free volume is perhaps justifiable.  It all depends upon the actual parameters.  One needs to check the numbers -- especially any relaxation effects in the polymer.

3. The example mentioned on fast expansion was just a thought experiment.  Of course other things can happen.  But I do note that you essentially use the same example on page 10/11 of your theory paper. Smile

4. Regarding the relative compressibilities, I have always had in mind essentially solid materials with little void space (a few percent at most) which is quite different than what I think your are aiming at.  And then the possibility of deformations that could be volume changing.

5. Stress can have an enormous impact on diffusion depending upon the system.  Your kitchen sponge is a classic example (ignoring of course that this is really a porous media problem).  In semi-conductor lithography, it leads to highly undesirable corner rounding, for example.  In the newer papers mentioned above there are some calculations that hint at the issue of stress effects on diffusion in polymer systems.

6. The issue of incompressiblity for problem solutions is an interesting one.  As you have noted, with it you can effect some nice hand analysis.  However, numerically Lagrange multipliers can be a bit messy --  certainly not undoable but messy nonetheless. 

7. One last random comment.  F-H involves the equilibirum free swelling assumption as well as the additivity of volumes assumption.  This last assumption always needs to be checked for the system under consideration.  It is easy to add two nominally incompressible liquids of volumes V1 and V2, for example, and end up with a final volume less than V1+V2.   The devil is in the details so they should be checked before using F-H.  In a complex situation with constraints etc., the only way I know how to check both  main assumptions in F-H is to use a more general theory and then check the magnitude of the extra terms.

8. btw, another nice reference on such models (albeit linear) is the paper of Carbonell and Sarti (1990).  This paper, like mine and yours, essentially provides the same model at a high level.  The only differences of substance are in the final selection of the free energy function.




Prof. Dr. Sanjay Govindjee
University of California, Berkeley

Zhigang Suo's picture

Dear Sanjay:  Thank you so much for relating your experience.  They are indeed very helpful.  Wei is studying your papers, and will explain to me in some detail.  As you can probably tell from these conversations, he is a very special researcher.  I think we all agree on the following:

  1. If a network imbibes a large quantity of solvent molecules (often greater than 90% by volume for a hydrogel), the volume change is mainly the sum of the volumes of the individual molecules.  Molecular compressibility due to stress is very small by comparison.  Volumetric change due to molecular interaction (your point 7) is also small by comparison.  This is the picture for hydrogels, a situation that my group is focusing on.
  2. However, if the uptake is not that large, then all the above effects can become important.  From your comments, this situation seems to be the one that you are focusing on.  I'll learn more when I read your papers.
  3. The stress may (or may not) affect diffusivity.  It depends on the level of stress and material.  I had experience with this issue mostly in the area of electromigration in aluminum and copper.  There, the main effect of stress is through its gradient as a driving force to motivate diffusion.  The effect of stress on diffusivity itself is a small effect even when the stress is as high as several hundred MPa.  I suspect that the effect of stress on diffusivity in a hydrogel is also small, just because the level of stress is not very high compared to molecular compressibility.  But again, I totally agree with you that one should check numbers for a given material.

My group has no prior experience with gels.  We are learning from the literature, and by talking to people.  Thank you very much for pointing out new papers.  The continuum theory seemed to be first formulated by Gibbs (187x).  His theory is for large deformation, using deformation gradient, nominal stress and chemical potential.  He characterized material law using a free-energy function, but he did not give any specific form of this function.  His theory is limited to equilibrium.  Biot (1941) is commonly credited to add kinetics into the theory.  Of course, following Biot, a huge literature now exists on poroelasticity.

It appears that the continuum theory is basically set by Gibbs and Biot.  As you pointed out, the devil is in the details of specifying the free-energy function and kinetic laws.  The details are extraordinarily rich, and are essential in connecting the theory to experiments, and in discovering new phenomena.  Also, experiments in recent decades have shown that hydrogels are responsive to diverse stimuli, further enriching the details.  The field is wide open.

Ahh, a point of mis-understanding on my part...I agree stress will likely have little to no effect on the molecular mobility M, as in j = -M grad[mu], concentration yes, but not stress.  Stress affects the chemical potential (gradient) is what I was trying to say.

I will have to look up the paper by Gibbs as I have not seen it.  Biot of course should also gets a lot of credit; somehow I did not see/appreciate those papers when I first saw them because (if I recall correctly) I was blinded by my drive toward finite deformation polymeric systems and the first paper of Biot I had encountered was only linear.

One other side note on Gibbs since I think you are also interested in Statistical Mechanics.  I taught a course on Statistical Mechanics in Elasticity last year.  I found Gibbs book Elementary Principles of Statistical Mechanics (1902 reprinted 1981) quite good for teaching purposes.



Prof. Dr. Sanjay Govindjee
University of California, Berkeley

Just to add more completeness to the referencing, here are two interesting citations of Biot's which deal with finite deformation.

M. Biot, Theory of Finite Deformations of Pourous Solids, Indiana Univ. Math. J. 21 No. 7 (1972), 597–620

M. A. Biot, Variational Lagrangian-thermodynamics of nonisothermal finite strain mechanics of porous solids and thermomolecular diffusion, International Journal of Solids and StructuresVolume 13, Issue 6, , 1977, Pages 579-597.



Prof. Dr. Sanjay Govindjee
University of California, Berkeley

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