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A theory of coupled diffusion and large deformation in polymeric gels

Wei Hong's picture

   A large quantity of small molecules may migrate into a network of long polymers, causing the network to swell, forming an aggregate known as a polymeric gel.  This paper formulates a theory of the coupled mass transport and large deformation.
The free energy of the gel results from two molecular processes:  stretching the network, and mixing the network with the small molecules.  Both the small molecules and the long polymers are taken to be incompressible, a constraint that we enforce by using a Lagrange multiplier, which coincides with the osmosis pressure or the swelling stress.  The gel can undergo large deformation of two modes.  The first mode results from the fast process of local rearrangement of molecules, allowing the gel to change shape but not volume.  The second mode results from the slow process of long-range migration of the small molecules, allowing the gel to change both shape and volume.  We assume that the local rearrangement is instantaneous, and model the long-range migration by assuming that the small molecules diffuse inside the gel.  The theory is illustrated with a layer of a gel constrained in its plane and subject to a weight in the normal direction.  We also predict the scaling behavior of a gel under a conical indenter.

PDF icon Kinetics 2007 10 22 correct.pdf267.9 KB


Xiaodong Li's picture

Thanks Wei for posting your paper. This paper is of great interest to me, in particular, the indentation creep part.  I used to do nanoindentations on polymers. At that time, I did not find a good theory to explain my results. I think now I can use your theory to heighten my understanding. Your theory is very timely. I believe that your Eq. 7 can be used to predict or fit a lot of experiment curves. This is also applicable for tissues like collagen. Aging behavior of tissues under stress is actually a time-dependent behavior and your theory may be extended to that point. I hope to use your theory in my future experimental work.  

Wei Hong's picture

Thank you for your interest on our paper, Xiaodong!

We are also eager to see experiments that can shine light onto our theory.

Our theory is more like a framework or platform, even more input, for example the material law of free energy and kinetic relation, is left open  and awaiting experimental results.


Konstantin Volokh's picture

Interesting paper! I have questions concerning the use of thermodynamics. Is not the gel an open system where mass is supplied from outside? Does the thermodynamics make sense for open systems? Is the gel the same thermodynamic system when extra particles are supplied? What is the meaning of entropy for an open system? I am asking the questions because I failed to answer them constructing a somewhat similar theory for a description of tissue growth.

Zhigang Suo's picture

Dear Kosta:  Thank you for your interest!  The school starts today here, and John Hutchinson has set me up to teach a large unbdergarduate course on complex analysis.  Need to do some preparation.  I'll be brief here. 

I tend to avoid open system.  You can do this by including the source of molecules as part of the system.  See Fig. 2 in the paper.  All that the pump does is to do work to the molecules.  The combined system conserves the number of molecules.  Here is a little write-up on chemical potential .

The subject of gels is new to my group.  We are not sure we got everything right in this paper, although we have tried hard.  The literature also scatters in several disciplines, I'm sure we have not read all the relevant papers.  Even among the papers we did read, we are not sure we understand them all.  At the end, we have tried our best to be fair to other authors, and to ourselves.  (As a practical matter, we just cannot spend all our lives decyphering old texts.)  If you see that we miss anything, please do let us know.  The paper has just been submitted to a journal.  We will have opportunities to revise. 

I'm interested in the paper on tissue growth, and hope to read it when things start to settle.

Konstantin Volokh's picture

Dear Zhigang,

With your approach there are no open systems: everything can be considered closed. Well, thermodynamics is a philosophical (theological?) subject and it makes no sense to argue about it. In any case, complex analysis is more important than thermodynamics...



ErwanVerron's picture

Dear Suo,

I had a look on this very interesting paper and I have two comments. I am not totally sure of what I will write in the following: some ideas might be good, but we should make further investigations. It is just a feeling !!

1. I am not an expert in thermodynamics, but I think that your model can be derived in another way. If we consider that your continuum is a melt of an elastomer and some particles, the free energy will be written as W (F,c) (it is what you wrote just above Eq. (9)). Then writing the change in entropy will lead to the constitutive equations:
s = dW/dF and mu = dW/dc
Moreover, the dissipation will be given in terms of the entropy flux that is related to the flux of c (something like the fourth term in Eq. (11)).
So in order to consider the evolution of the variable c, the simplest ways is to postulate that its flux J is related to its affinity which might be something like dmu/dX. Considering a linear evolution (a sort of Fick law), it may lead to Eq. (15).

2. My second comment concerns the definition of the free energy. You wrote it as the sum of two functions: Ws(F) + Wm(c) (Eq. (20)). The coupling between deformation and c is then obtained through Pi (the osmotic pressure).
Considering Eq. (16) in Eq. (22), it might be possible to write Wm as an explicit function of c and F and then to perform directly the calculation of the stress without the Lagrange multiplier. In this case, I think the coupling between the variables would be easier to understand for the reader.

So, I will try to write in details the equations in order to check if what I wrote just above is not just mistakes ...



Wei Hong's picture

Dear Erwan,

 Thank you for your interest.  Let me try and see if I've got your point:

1) Yes the one you described is another approach that will lead to same result if equivalent assumptions are made.  Before our paper, there are alreay many theoretical works, each of them contains considerably amout of assumptions, many of which named "xxx laws".  I am not sure about others, but to us, these theorys are hard to follow and even harder to make extension.  The purpose of our paper is to construct a framework rather than to postulate some specific material law, so we tried to make clear in the framework that some "laws" can be changed (eg. W(F, C) and the kinetic relation) while some can not (e.g. conservation of mass and momentum).  Within the framework, even the constitutive equations you wrote can be changed, for example, mu might not equal to dW/dC, especially for the case when there is chemical reaction, etc.

2) You are right, in some simple cases, the Pi can be canceled through different equations, and we will have a more explicit coupling between F and C.  We are just trying to be more general, as it is not always like that.


Hua Li's picture

Hi Zhigang & Wei,

Thank you very much for your email to let me know your latest output and also thank you to mention our work in the paper.

My comment here is on the contribution of ionic concentration difference between the interior gel and exterior solution into the free energy. In the previous discussion with Wei, he also agreeed this contribution should be considered, and he said it has been considered but I failed to find out there. Similarly, in this paper, it may appear in Eq (20) - Ws(F) + Wm(c). I mean it may be the sum of three functions, one more W represents the the contribution of ionic concentration difference into the free energy.

In addition, I am slightly confused about your view on the generalized Darcy's law for moving porous media, described in pages 14~15. Are you sure of that?


Zhigang Suo's picture

  • We have limited this paper to nonionic gels.  Perhaps we should add a statement in the paper to make this limitation explicit.  A vesrion that includes the effect of ions has been placed in iMechanica:  Polyelectrolyte gels.  We are working on a more extended version.
  • Section 4, Kinetics, is perhaps the most novel (or risky?) part of the paper, we believe.  We discussed our concerns with Darcy's law and some other existing formulations.  We then stipulate diffusion as the mechanism for mass transport.  Please do let us know which apsepct of this section troubles you.
  • Regardless the detailed transport mechanism, the platform set up in Section 2 is more generally valid.  This section constructs a plateform that is consistent with thermodynamics.  The approach is rather standard.
Hua Li's picture

Hi Zhigang,

To my understanding, the osmotic pressure results from the concentration difference, wich is not limited to ionic gels. It is also available for neutral/nonionic gels. In your system furthermore,  the osmotic pressure is sole driving source to balance the elastic stress. This is why I think the contribution of concentration difference should be included.

In terms of  Darcy's law, let me take time to clarify myself first. I will post my clear questions late on if I can.



Wei Hong's picture

Hi Hua,

Sorry that I didn't make it clear in the previous discussion.  Let me try to clarify it.

By saying that the contribution of concentration difference is considered, I mean the concentration of solvent (water) itself, not ions.  Here it is eaxatly the same.  Wm(C) is the energy of mixing, where C is again the concentration of the solvent, so that dW/dC is the driving force due to the concentration difference (not only between the gel and exterior, but also inside a gel, when there is a concentration gradient).

In the current paper, the osmotic pressure never solely depends on the elastic stress.  As can be seen in Eqs. (23) and (24), it serves as the linkage that couples the stress and chemical potential.  In return, the osmotic pressure PI itself has contributions from both.

It can be seen in a very special case in Eq.(38), which the whole system reduces to a PDE of lambda_3.  Among the 3 terms in the parenthesis in front of dlambda_3/dX_3, the first term comes from the entropy of mixing, the second from the enthalpy of mixing, and only the last from the elasticity of the polymer network.  (A reminder: under our incompressible assumption, lambda1*lambda2*lambda3 = 1+vC, just don't let the equations that do not explicitly have C in it give you the impression that it does not depend on C)

 Thanks again for you interest, and we are eager to see the questions related to Darcy's law or diffusion.


Hua Li's picture

Hi Wei, 

you said that "so that dW/dC is the driving force due to the concentration difference (not only between the gel and exterior, but also inside a gel, when there is a concentration gradient)".

- Do you mean your computational domain also covers the exterior solution surrounding the gel? To my understanding, your domain covers the gel only.

 In fact, you and I are arguing 2 different topics with different original source terms.

my source term = the ionic concentration difference between the interior gel and exterior solution -> the free energy -> osmotic pressure -> ... ...

your source term = energy of mixing Wm(C) -> dW/dC is the driving force due to the concentration difference -> ... ...

I said that "In your system furthermore,  the osmotic pressure is sole driving source to balance the elastic stress", based on your equ (19), as you also said (see the bottom of page 12) " the elastic stress must balance the osmotic pressure".

Have a nice weekend,


Zhigang Suo's picture


Wei is back to China for a visit.  Let me try to respond to you.

  1. The domain of the problem can be either the gel or the gel plus the external solvent.  Mostly, the external solvent is in a state of equilibrium, so that nothing is really interesting in the external solvent.  If this is the case, the domain of compution will be the gel.  The external solvent only provides boundary conditions to the gel, as discussed in Section 5.
  2. I agree with your clarification of osmotic presseure.  See also my comments on osmosis in gels.

Thank you for your interest in our work.  Your papers have been important to us.



  I found this paper to be very interesting and a nice approach on a tough problem.  I actually agree with your statement regarding Darcy's law, in the sense that in our work we found the application of it to gels to be problematic from a number of levels.  In addition to the extent to which it is physically questionable for gels, it is sometimes used to derive a "collective diffusion" equation, which as employed in many cases is not properly invariant.  

  Finally, I just want to point out that we took a similar approach in our work on chemically-induced swelling of gels.  Thank you for citing the JMPS paper, but in fact the theory is explained much more fully in our CMAME paper.  We also work with the chemical potential as a primary field there, and I think if you just consider a single phase of the gel many of the ideas are common.  In the general theory we allow for the mobility to be a function of the left Cauchy-Green tensor and the chemical potential, and then specialize later on.    I think the primary differences concern the motivation for some of the constitutive assumptions, and incompressibility.  We did not take the gels to be incompressible, and my sense is that there is quite a range of opinion on that.  

I don't understand some of the basic thermodynamics in the original paper.  Could you point me to a source that explains the approach in a concise manner (without too many technicalities)?  More precisely, I'm looking for a source where thermoelasticity is derived in the same manner (and with similar notation) as in Wei's paper.

Also, John, could you send me a copy of your CMAME paper?


Zhigang Suo's picture

Dear Biswajit:  The approach we followed is that of nonequilibrium thermodynamics.  In the paper we cited two classics:  Prigogine (1967) and Coleman and Noll (1963).  Both are very readable.

Wei Hong's picture

Hi John,

Thank you for your interest and insightful comments!

We have studied your CMAME paper.  Indeed the approaches are very similar and the framework share many identical equations.

We both have stress equilibrium, ofcourse; Eq. (2.45) in your paper is equivalent to our (13), (14) and (15).

However, we use different free energy functions your (3.3) vs. our (20-22).   The treatment of incompressibility is a minor issue while the major difference is in the treatment of the contribution from mixing (between water and polymer network.)  You model this contribution as a "configurational force" while we just combine the entholpy of mixing into the free energy and get this driving force directly as part of the constitution law.  While practically the two different approaches might yeild similar results.

Another major difference is on the mobility.  In your specified mobility form (3.6), you treated it as a constant isotropic tensor,  while we took it as anisotropic and depend of deformation, even under the simplest assumption (section 4).


Zhigang Suo's picture

Dear John: 

I've just read your 2005 CMAME paper.  This is a very interesting and rich paper, and I have a feeling that I'll have to return to it for more readings in coming years.  Here I'll note a few thoughts coming to me as I read your paper, and make comparisons with our paper.

  1. Your paper uses the approach of nonequilibrium thermodynamics, and uses the nominal quantities.  As you have pointed out in your comments above, your theory and our theory are identical.  They look almost exactly same if we focus on the governing equations:  yours (2.24) and ours (13)-(15).
  2. I don't think that you or we want to claim this basic formulation is really novel.  Gibbs (1878) and Biot (1941, 1973) were way ahead of us, not to mention so many other authors who have also followed Gibbs and Biot.
  3. Your paper went one step further, and treated the moving boundary between two phases of a gel.  This is indeed the focus of your paper and is very novel.  I hope to return to this aspect of your paper in future.  For now, I'll focus on the case that there is no phase transition.
  4. In our paper, to avoid the known controversy over what we mean by stress in poroelasticity, we elect to define the stress via the weak statement.  We have had a detailed discussion on this point in a paper on deformable dielectrics.  A similar discussion appears in a recent thread of iMechanica discussion.
  5. We show that (13) is a consequence of local equilibrium assumption.  That is, (13) is not always correct; it is correct when we neglect the dissipation associated with viscous rearrangement.
  6. The nonequilibrium thermodynamic theory leaves open the free-energy function and mobility.  As you have pointed out, many options exist as to how to specify them.  In your paper, you wrote down several expressions (3.3) and (3.6).  In our paper, we adopted the free energy function according to an existing molecular model (20)-(22).  We formulated a kinetic law on the basis of molecular diffusion.  The resulting mobility (30) has two interesting aspects:  the mobility is an anisotropic tensor, and the mobility vanishes when the gel is dry.  Both aspects have simple interpretations, and should reappear in any model, I believe.
  7. As we all know, it is important to connect continuum theories to molecular models.  It is particularly so for complex materials, as the number of molecular variables is large.  For a gel, an experimentalist can vary many parameters, such as cross link density.  She also has intuition as how such a variation will affect behavior of a gel at a macroscopic scale.  It would be nice if a continuum theory can recover her intuition.
  8. In Section 3 of our paper, we enforced the condition of molecular incompressibility.  This leads to a discussion of osmosis pressure.  Because a large number of physical chemistry papers on gels have organized their experiments and molecular models in terms of the osmotic pressure, the idea has also become part of intuition in the field.  It is nice that the continuum theory recovers this intuition.

Thank you again for letting us know your paper.  We'll have much good time together talking about this fascinating material in coming years.  I hope to learn more from you. 


Thanks for the kind comments.  

I completely agree with your second statement and did not mean to suggest otherwise.  The basic formulation is not novel in either case.  

I also very much appreciate the sixth comment and the important choices that were made in your work.  We used relatively simple constitutive models in our work as a starting point, in part because the focus was on the phase transition as you've indicated.   

While a phase transition has not been explicitly modeled in your work, certainly one could consider the "dry" and "wet" polymer as phases, no?  Along these lines, it would be interesting to see if your model is capable of predicting the formation of a relatively sharp front separating the two.  We would expect that front to have a finite thickness in your model, whereas in ours it is assumed sharp.  Both cases actually present considerable challenges for numerical implementation.   

I look forward to discussing the models and materials more in the future with you and your group as well.  There is much to do and learn.   

Wei and Zhigang,

Very nice paper. Sorry for my late to get back to you.

I want to follow up with Hua's comment on osmotic pressure. By definition, osmotic pressure is due to the concentration difference and causes diffusion. When the concentration difference disappears, osmotic pressure drops to zero. This seems to be different from what you said that "osmosis pressure is also called the swelling stress". The swelling stress is due to penetrant occuping space therefore pushing the macromolecluar network to swell. During swelling, as penetrants go into the network, the osmotic pressure will drop and swelling stress will increase.  Did I miss something here? I know this is not a critical point in the theory developed in your paper, but clarifying this concept might be helpful.

Again, very nice paper.  


Wei Hong's picture

Hi Jerry,

Thank you for your kindly compliments.

We really did not use the idea of osmotic pressure in our paper.  We
introduced a lagrange multiplier just to enforce the incompressibility,
and it just happens to share the same expression as the osmotic
pressure.  I am not so sure about the definition of "swelling stress"
(or if there is any:)).  We say it is a swelling stress, merely because
it is the driving force that causes the gel to swell.  Could you please
let us know if you can find the definition of a swelling stress? 
Thanks again for your clarification.


Zhigang Suo's picture

  1. When a semi-permeable membrane seprates a pure solvent and a liquid solution, the osmotic pressure is a well defined quantity.  I find on the Internet a nice drawing of osmosis.
  2. For a gel in contact with a pure solvent, no semi-permeable membrane is needed.  Solvent molecures can enter and leave the gel freely, but the long polymers are cross linked into a network and cannot leave the gel.  The long polymers are the solute, and the small molecules are the solvent.  The concentration of the long polymers in the gel will never drop to zero, and there is no long polymers in the external solvent.  This imbalance in the concentration of the long polymers set up the osmotic pressure.  This osmotic pressure will not drop to zero.
  3. When the gel and the external solvent reaches equilibrium, the osmotic pressure is balanced by the elatsic stress of the network.
  4. Osmotic pressure and swelling presure is used interchangiably; see Treloar's book, p. 131.  Perhaps the most respected book on ion exchange is the one by Friedrich Helfferich (1962).  As far as I can tell, he uses the concept of swelling pressure in the same way as osmotic pressure.  Thus, when the phenomenon occurs to a liquid solution, we use the phrase osmosis pressure.  When the same phenomenon occurs to a gel, we use the phrase swelling pressure.
  5. I believe that in general, osmotic pressure or swelling pressure of a gel is not a well defined concept.  Their status is rather like the Maxwell stress:  widely used but is on a shaky foundation.  We have not really developed this theme in the paper, though.  However, from (13) and (14) in our paper, you can already tell you can make all the predictions without ever need to mention the word osmosis.  In general, osmosis is a superfluous notion for gels.  But the concept does come up naturally under the condition of molecular incompressibility.

Thanks, Zhigang. Your explaination clarified my confusion here.


Aaron Goh's picture

Dear Wei and Zhigang, thank you for initiating a very lively thread.  I am just beginning to learn about flow in deformable porous media.  Could you possibly suggest experiments, or point to references which contain experimental details, to test your theory?


Zhigang Suo's picture

  1. Our paper is about polymeric gels.  Gels are usually not porous.  Small molecules and long polymers always fill the space.  Even before the polymer network imbibes small molecules, the polymer network itself fills the space, with no pores.  Of course, there might be free volumes at the molecular scale, like any condensed matter.  Consequently, the image of fluid flowing in small channels might be misleading.
  2. There is a large body of experimental literature on gels, ranging from material characterization, to intriguing phenomena, and to device operation.  Here is an experimental paper to get you started:  Kim et al., Surprising shrinkage of expanding gels under an external load, Nature Materials 5, 48-51 (2006).
  3. Our theory has several parts.  The thermodynamic part has followed the Flory-Huggins theory, which has been tested for over 50 years.  It's not perfect, but it captures the main trend.  A summary of this experimental literature is given by Horkay and McKenna, which was pointed out by Jerry Qi in the Theme of July.
  4. The other part of our theory is kinetics.  Kinetics is less well treated in the literature.  We are in the process to compare our theory with the existing experimental data.  Stay tuned.  Jury is still out.
Yanfei Gao's picture

Zhigang and Wei:

Thanks a lot for your paper. It seems to me that the
constitutive behavior of your material has a creeping component and an
evolution equation of concentration (or some measure of order parameter). For
indentation on such kind of material, here is my view, in complementary to Xiaodon'g note.

(1) Indentation on standard creeping solid
In this case, the
creep behavior is solely governed by stress. If elasticity is further
neglected, one can use Hill's cumulative superposition method to derive a
similarity analysis. The indentation response, being a dead weight or
strain-rate controlled, can be easily derived. See Allan Bower's paper in Royal Society Proceeding in 1992.

(2) Indentation on creeping solid
with structural evolution
If the creeping behavior also depends on a local
evolution of internal state variable, indentation behavior is not fully understood in
literature. Dr. George Pharr and I recently have done an interesting material,
amorphous selenium, which might be described by Spaepen's free volume model.
Indentation will either cause Newtonian flow or shear band. However, our amorphous selenium indentation behavior only deals with
free volume evolution in a local form. If you have a diffusion equation for
the internal order parameter, I have never seen any work (I might be ignorant here).

Yanfei Gao,

Rui Huang's picture

Hi Wei,

I am reading your paper, "A theory of coupled diffusion and large deformation in polymeric gels", which I like very much. I am trying to do some simple calculations myself, which leads to a minor question here. For the uniaxial creep problem, I reached an equation similar to Eq. (32) in your paper, but different for the second last term on the right hand side. Instead of 1, I have 1/lambda3. Solving this nonlinear equation for s = 0 with Matlab,  I could not get the same stretch, lambda = 3.125. With your equation, I got 1.294, and with my equation I got 3.390. I wonder if I have missed something somewhere. I would appreciate it if you can check your equation and solution to let me know.



Wei Hong's picture

Hi Rui,

Thank you for reading our paper so carefully and pointing out the errors.

Actually these were just typos we had on the first version of our manuscript.  We have corrected them on the later versions.

Besides the two places you have identified, there are more:
1) the chi number we used should be 0.2 instead of 0.1
2) the lambda value sould be 3.215 instead of 3.125

I am uploading the new version.  Please take a look at this version and sorry for the misleading typos.
You can also check on the final version on JMPS website at



Zhigang Suo's picture

Dear Rui:  Thank you very much for your interest, and for going through the calculation.  We have since made a number of applications, which have been posted at

Working through these specific problems, we are trying to learn about applications of hydrogels, and to connect the theory to experimental observations.  The experimental literature on hydrogels is huge, and will take many theoreticians many years to sort out.


Rui Huang's picture

Dear Wei and Zhigang,

Thank you for your responses and for pointing to your recent works on hydrogels. Indeed, I have been following your works (quietly so far).  I have several experiments in mind and try to develop models, with little success so far. One particular experiment involves patterned hydrogel lines constrained by a substrate. The swelling and deformation in this case is highly inhomogeneous and anisotropic. What caught my attention at the beginning is that these lines buckle into wavy structures. The question here is how to relate the buckling phenomenon to the material and geometry properties of the hydrogel lines. 


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