A theory of coupled diffusion and large deformation in polymeric gels

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.

• 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.

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?

Thanks, 

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. 

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.

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.

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

• Drying-induced bifurcation in a hydrogel-actuated nanostructure  
• Inhomogenous and anisotropic equilibrium state of a swollen hydrogel containing a hard core

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.

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. 

RH