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Polymer gels

Zhigang Suo's picture


Rui Huang's picture

Dear Zhigang,

Thank you for posting the note, although this is not the first time I see it. Still, I find the discussions on osmotic pressure at the end quite interesting. This is certaintly a confusing concept in the study of gels. Coming from a background in solid mechanics, the term "osmotic pressure" sounded like a foreign language to me when I first read the literature on gels. A similarly confusing term is "pore pressure" in poroelasticity, which appears to play a similar role as the chemical potential in gels. These different terms are commonly used in different communities (polymer scientists, geo-scientists, mechanician...), a unified language would be very helpful.

As a particular example, I noticed that the osmotic pressure in a gel may not be properly defined if the gel swells under constraint. For a freely swollen gel,  the pressure inside the gel equals the pressure ouside the gel. If we ignore the hydrostatic pressure of the external solvent, the stress is zero everywhere in the gel. As you point out, in this case, the osmotic pressure in the gel is balanced by the tension due to elastic stretch of the polymer network. When the gel is constrained, for example, by a rigid substrate, a nonzero stress is induced in the gel upon swelling. As a result, the pressure inside the gel is no longer equal to the pressure outside the gel. The discontinuity in the pressure across the interface does not violate any boundary conditions in solid mechanics, but it may be non-intuitive for others. This appears to be similar to the phenomenon of osmosis, where the pressure is discontinuous across the semi-permeable membrane. However, it is not clear to me how to define the osmotic pressure for the constrained gel.


Zhigang Suo's picture

Dear Rui,

Many thanks for the comment.  Indeed, I have found these ideas difficult to teach.  As you have just pointed out, some confusion is about terminology, which may be easy to clarify.  The chemical potential of water divided by the volume per water molecule has the dimension of pressure.  This quantity enters the material model of the gel as a hydrostatic pressure (page 6 of the notes).  The same quantity is called the pore pressure in poroelasticity, and the water potential in physiology of plants.

The disscusion of osmosis in gels is perhaps more than terminology.  On page 9 of the notes on pressure, I included a description of osmosis in liquid solution.  This description may be helpful in thinking about osmosis in a gel immersed in a solvent.  The issue of osmosis of the gel came to my attention when I taught the course last time, in 2009.  I added a section in the notes then. In reading the section this time, I felt that the issue might be too distracting, and simply placed the section at the end of the notes, as an appendix.  I have not improved this section this time.  I'll try to imrpove the section next time, possibly in 2013.

I have one more lecture to make up for ES 241 Advanced Elasticity.  I get to teach this course every other year, and this is the third time I'm teaching it.  I have focussed the course on the combination of thermodynamics and finite deformation.  Examples are drawn mostly from elastomers and gels. By now I have updated all the notes used for the course in this semester. 

Kosta Volokh has also taught a course on the mechanics of soft materials.  His notes are also online.

Lianhua Ma's picture

Dear Prof. Huang,

For a gel immersed in aqueous environment (or pure water), I don't think that the pressure inside the gel equals the pressure outside the gel. In such a case, the chemical potential of water in the gel equals that of external solution, whereas the pure pressure does not. That is, the pressure is discontinuous at the solution/gel interface even in the freely swollen gel. In my opinion, the pressure discontinuity across the interface  (both in freely swollen gel and constrained gel) is caused by the non-zero osmotic pressure difference.


Rui Huang's picture

Dear Lianhua,

I disagree. For a freely swollen gel, the pressure must be continuous across the interface, as dictated by the boundary condition along with the hydrostatic stress states both in the water and in the gel. Of course, the chemical potential of water is continuous too.


Lianhua Ma's picture

Dear Rui,
I still persist in my own opinion. Let's consider the special case where the dry gel is fully immersed in pure water and freely swells to an equilibrium state. The pressure and chemical potential of pure water outside the gel both remain zero if ignoring the hydrostatic pressure of the external pure water. The total stress of the gel and corresponding chemical potential of water inside the gel are also zero. Because of the zero chemical potential, the total stress comes from two contributions: elastic stress of the network + osmotic pressure (also called mixing pressure), and the osmotic pressure is balanced by the elastic stress of network. The fluid pressure inside the gel can be represented by the magnitude of osmotic pressure, and it is certainly not zero.

Hope to discuss this further if necessary.

Rui Huang's picture


The pressure as we defined in the continuum theory is zero everywhere in the freely swollen gel. Unless you want to consider the microstructures of the gel by separating fluid from solid, you cannot decompose the pressure to two parts.  In other words, by the continuum theory, every point of the gel is a mixture of fluid and solid, and they are inseparable. By all means, there is no physical base (to my knowledge) to take the osmotic pressure as the fluid pressure in gels. This is probably one of the most confusing parts in the study of gels.


Lianhua Ma's picture

Dear Rui,

Thank you for the explanation.

I noticed the definition of the pressure in gels is where our opinions diverge from each other.

In your opinion, the gel is regarded as a continuum where every point is a mixture of fluid and solid. The pressure inside the gel as you defined denotes the total stress of the gel (including the elastic stress of the pure network).

In my previous view-point, I viewed the gel as a porous media where the total stress is the difference between the elastic tensile stress of pure network and the osmotic pressure: subtracting osmotic pressure from elastic tensile stress of pure network.

My previous understanding of the pressure in the gel was different from you. In my post above, the contribution of the elastic stress of pure network was not incorporated in the pressure as I defined.

In fact, I am still wondering if it is reasonable to look upon the total stress as the pressure inside the gel?

Following your perspective, if a gel is bonded to a rigid substrate or subjected an external load, the stresses in the 3 principal axis directions of the gel may be different from each other, then how to characterize the pressure inside the gel? Px=Sx, Py=Sy, Pz=Sz? Or P=(Sx+Sy+Sz)/3?
If the answer is the former, then the pressure is direction-dependent which could violate our physical cognition that classical pressure is defined as a scalar quantity.If it is the latter, the so-called pressure can be viewed as the hydrostatic stress of the constrained gel. 

Anyway, we can bury our differences by means of the free energy function of the gel without discriminating the osmotic pressure, the total stress and the pressure inside the gel. As for gels, the definition of the terminologies: osmotic pressure, pore pressure and pressure inside the gel, is really confusing part in the study.

Many thanks,

Zhigang Suo's picture

Dear Rui and Lianhua:  It was fascinating to follow your discussion.  The discussion may appear to be confusing, but I believe it goes to the heart of continuum mechanics.  Here are a few thoughts for your consideration.

  1. Let's think of a rubber band.  No solvent yet.  We can pull the band with a force, and measure the length of the band.  The experimental data are the force-length curve.
  2. We divide the force by the cross-sectional area of the undeformed band, and call the quantity the stress.  We divide the length of the deformed band by the length of the undeformed band, and call the quantity the stretch.  Now we can convert the experimentally measured force-length curve to a stress-stretch curve.
  3. The merit for doing this conversion is well appreciated:  the stress-stretch curve allows us to predict the force-length relation of another rubber band, made of the same material, but of a different cross-sectional area and of a different length from the first rubber band. 
  4. For this prediction to work well, we should be clear what we mean by "the same material".  We need to know the molecular picture of the material.  The rubber is a network of covalently crosslinked polymers.  The distance between neighboring crosslinks defines the mesh size of the rubber.  Over the length larger than the mesh size, the network is homogeneous.  Thus, if both rubber bands are made of a similar network and are both larger than the mesh size, the procedure described in 2 should work well.
  5. The knowledge of material constitution is useful in other ways.  We can understand the extraordinary large and elastic deformation of the rubber through the idea of entropy.
  6. Is the rubber a solid or a liquid? The rubber has the attributes of both.  At the size scale larger than the mesh size, the rubber behaves like an elastic solid:  the rubber band is capable of elastic deformation.  At the size scale smaller than the mesh size, the rubber behaves like a liquid:  the molecules can change neighbors.
  7. We now wish to generalize the above idea, and predict the response of a block of the rubber subject to multiaxial forces.  The block is large compared to the mesh size.  The deformation of the black is homogeneous.  We model the rubber by writing the free energy as a function of stretches in three directions.  The form of the function is chosen by guess work of various sophistication, and by fitting to experimental data.  
  8. We can then predict the behavior of a piece of the rubber, of any shape, in response to arbitrary loads.  The piece is large compared to the mesh size of the network.  The deformation in the piece is inhomogeneous.  However, we assume that the variation of the deformation is negligible over the size somewhat larger than the mesh size.  Thus, we think of the piece as many small blocks, each undergoing homogeneous deformation.  We use the free energy function in 7 to represent each block, and use kinematics and Newton's law to describe the interaction of all the blocks.  The procedure is what we teach in finite elasticity.

The theory for gels described in the notes follows the same outline as above.  Now the rubber band is immersed in a solvent and is being pulled.  The loading parameters are the chemical potential of the external solvent, as well as the applied force.  It takes some time for the solvent molecules to migrate into the rubber band.  We will focus on the condition of equilibrium between the rubber, the external solvent, and the applied force.  In a state equilibrium, we record the chemical potential of the solvent, the applied force, the lengths of the rubber band in three directions, and the number of solvent molecules imbibed by the rubber.  We can repeat the experiment with different values of the chemical potential, and different values of the applied forces. 

This set of data will be sufficient for us to predict the behavior of another rubber band, of a different size, immersed in a similar solvent, subjected to an external force.  We can then generalize the procedure for inhomogeneous deformation.

Notice that we do not need speak of osmotic pressure in the above experiment and prediction.  You may wish to define an osmotic pressure, and may wish to invent an experimental method to measure what you have defined.  But the osmotic pressure, however you define it, will not affect the above experiment and prediction.  Then why bother?

Here is a strong reason.  You wish to relate your experimental data to some molecular picture.  To do so you will need to have a molecular model.  For example, on page 8 of the notes, I have included an expression of stress in terms of the stretches and the chemical potential.  This expression is derived from the Flory-Rehner model.  You may call part of the expression elastic stress, another part the osmotic pressure.  Osmotic pressure in a gel is a coinfusing idea because it is associated with specific microscopic models.

You may now have the satisfaction of some molecular understanding.  But however you split the terms and however you name them, the split has no effect when you use the same equation of state to predict the behavior a rubber in a solvent.

  • Macroscopic behavior can be predicted by equations of state in terms of stresses, stretches, the chemical potential and the concentration.  Calling some term in the equations of state the osmotic pressure will have no effect on predicting macroscopic behavior.  
  • Osmosis is a useful notion to think of the molecular process of swelling. 
Rui Huang's picture

Thanks to Zhigang for a thorough discussion above and below.

Lianhua: it appears that you may not be convinced. Allow me to re-emphasize two facts:

(1) Osmotic pressure is NOT pore pressure or fluid pressure in gels.

(2) Pressure in a solid or liquid is well defined as P = (Sx + Sy + Sz)/3 in all cases (constrained or not). Any theory that splits the pressure into different parts would be confusing and unnecssary.


Lianhua Ma's picture

Dear Rui, many thanks for the response.
In our discussion above, it seems that I did not think outside the box of porous media (poroelasticity). Also, you did not jump out of the framework of continuum media.

If we are talking about pure water or pure solids, the pressure is well defined and this is surely not controversial.

The pressure in my previous posts, denotes the pore pressure in the porous media, rather than the hydrostatic stress of the whole media (if you have to consider the hydrogel as a continuum).
From the point of view of the porous media, the pore pressure may not be identical to the hydrostatic stress of the whole structure.

If we treat the hydrogel as a homogenous continuum and discuss the hydrostatic pressure within the continuum framework, I agree with everything you said above.


Lianhua Ma's picture

Dear Prof. Suo,

Thank you for posting the note on the diffusion-deformation coupled theory of soft gels.

I'm a little confused about the experimental measurement of chemical potential and pressure.

For the following description in your note, "Experimentally, when the polymers are crosslinked, one can measure the chemical potential of water and the applied forces, but cannot measure anything like osmotic pressure.", my question is how to measure the chemical potential of solvent . To my knowledge, the solvent chemical potential can be computed by a formula rather than a measuring instrument. Why can't we measure the osmotic pressure? Consider the special case where the gel fully immersed in pure water, the chemical potential of water in gels is equal to zero at equilibrium state, and the osmotic pressure is balanced by the tension of network. In this case, the osmotic pressure can be viewed as a pure pressure of water in gels. Ideally, if we have a micro pressure gage embedded or inserted in the fluid domain of the gel, we can read a number which should be the magnitude of the so-called osmotic pressure.  

In the general porous media theory, we usually use displacement u and pressure p as the fundamental physical quantities( degrees of freedom) for analyzing the deformation-flow couplings of porous material.
In your theoretical framework, the displacement u and chemical potential miu are utilized as the corresponding degrees of freedom. I would like to ask that which one (p or miu) is more basic or critical to the coupled theory?

From the point of view of experimentation, it seems that the pressure p is more suitable for characterizing the fluid flow. The pressure can be measured experimentally by an instrument(pressure gage), whereas the chemical potential miu is usually determined by a formula rather than an existing instrument. Furthermore, the formula of chemical potential is used for aqueous solution at equilibrated state. For example, in the case where the gel is in contact with external solution and reaches an equilibrium state, here the chemical potential in gels equals that of external solution, and we can easily give the magnitude of the external water chemical potential by a formula. However, for the transient diffusion process of solvent in gels, the solvent chemical potential varies with position and time, neither a formula nor an instrument can be used to measure the transient solvent chemical potential. In such a case, it is relatively easy to measure the pure pressure by a pressure gauge.  From this perspective, the chemical potential does not have a direct physical meaning, it seems more like an imaginary physical quantity, whereas the pressure is an actual measurable quantity. Indeed, it is also easier to interpret the fluid pressure than the chemical potential. 

Lastly, I have another question.
What would happen if we have the gel exposed to the environmental air? In our daily life, some gel-like materials such as jelly and Chinese Liangfen commonly exist in air. The theory in this note may be limited in soft gels immersed in an aqueous environment. For the gel set in air and subjected to external tension or compression, how to describe its deformation-diffusion coupling behavior?  

Thanks again for sharing these valuable lectures which gave us a more clear understanding of thermodynamics, chemical potential and osmotic pressure etc.



Zhigang Suo's picture

Dear Lianhua:  Many thanks for these questions.  Here are my thoughts on the issues.

How to measure chemical potential of water in a hydrogel.  As discussed in the notes on chemical potential, one measures the chemical potential of water in a hydrogel by equilibrating the hydrogel with another system, in which the chemical potential of water is known, or can be read from another parameter.

For example, one can immerse the hydrogel in a moist air.  Water molecules can leave or enter the hydrogel, until water molecules in the hydrogel and water molecules in the air equilibrate.  In this state of equilibrium, the chemical potential of water in the hydrogel equals the chemical potential of water in the air.  The chemical potential of water in the air is related to the relative humidity of the air, or partial pressure of water in the air.  The latter can be measured.

Thus, measuring chemical potential is analogous to measuring temperature.  One measures the temperature of a system by equilibrating the system with another system--a thermometer.  The temperature of the thermometer can be read from another parameter, such as the volume of mercury.

Your proposed method to measure the osmotic pressure in a hydrogel.  The mesh size in the hydrogel is typically about a few nanometers.  You will have to be more explicit about the working principle of the "pressure gauge", so we can decide what this device is capable of measuring.

The osmotic pressure is balanced by the tension of the network.  This molecular picture describes how a hydrogel equilibrates with water.  Indeed, in the Flory-Rehner model, the equation of state can be interpreted as

(stress applied on the gel) = (stress due to the stretching of the network) - (osmotic pressure due to the mixing of water and the polymer network, relative to the external solution).  See p.8 of the notes on gels.

This molecular interpretation provides an understanding of how the gel works.  However, neither the stress due to the stretching of the network, nor the osmotic pressure can be measured. 

We write the same equation in the following form:

(stress applied on the gel) = (a function of stretches) - (chemical potential of water)/(volume per water molecule).  See p.8 of the notes on gels.

This equation relates the measurable quantities:  stress, stretch, and chemical potential.

Which one, p or mu, is more basic or critical to the coupled theory?  The pore pressure in poroelasticity is identical to the (chemical potential)/(volume per water molecule).  Thus, p and mu are equivalent.  In Biot (1941), the pore pressure (his sigma) is energy-conjugate to water content (his theta).  Recall the definition of the chemical potential of water in a system:  The change in the free energy of the system associated with adding a water molecule to the system.

Chemical potential of water in a hydrogel can be measured while water migrates in the hydrogel.  Once again, the story is analogous to measuring the time-dependent field of temperature in a solid while heat conduction is going.  When water migrates in a hydrogel, the chemical potential of water in the hydrogel varies from place to place, and from time to time.  So long as the variation is small over the length and over time scales suitable to our measuring device, we can measure the chemical potential as a function of position and time.  As usual in continuum mechanics, the gel is thought of as many small blocks, each block is in a homogeneous state.

There is the practical challenge as to how to equilibrate the measuring device with a small block inside the hydrogel.  The challenge is reminiscent of that measuring temperature field.  Perhaps we can borrow some ideas from thermometry.  Can we invent a noncontact technique to measure the chemical potential, something analogous to measuring temperature by radiation? 

Nonetheless, the time-dependent field of chemical potential of water in the gel is well defined, and can be calculated by solving the initial and boundary value problems, as described in the notes on the migration of matter in elastic solids.

Gel in the air. In addition to water molecules, the air also contains other molecules, such as nitrogen and oxygen.  These molecules have low solubility in water, and should not modify the theory of hydrogel significantly.  Of course, if you are interested in the transport of oxygen in the hydrogel, you should then include oxygen in the theory.  The theory of dilute solution will do.  

Lianhua Ma's picture

Dear Prof. Suo,

Thanks very much for your quick response. I am new in this field and studying the multi-physics couplings of soft materials, especially from your notes and papers posted in iMechanica.It is a pleasure to discuss these questions with you. 

I have read carefully your comments and still have some confusing questions to clarify.

1.The experimental measurement of chemical potential.

As you pointed out above, one usually measure the chemical potential by an equivalent approach in which an equilibrium condition of the gel with another system (aqueous solution) is required. For the transient chemical potential of water in gels, varying with position and time, there is no instrument analogous to a thermometer for measuring the transient chemical potential. Certainly, we can consider inventing a thermometer-like instrument to measure the chemical potential, which is beyond the scope of the discussion. For the kinetics of solvent, from the experimental point of view, if we can measure the transient pore pressure of the gel (by an existing instrument for measuring pressure), maybe it is more reasonable and meaningful to characterize the solvent migration using the pore pressure rather than the chemical potential.   

From my perspective, the chemical potential cannot be directly measured by an experimental instrument at present. I agree with you that the chemical potential behaves like the temperature. However, in reality, it seems there is no instrument like a thermometer (or a pressure gauge) for measuring the chemical potential. For the convenience of experimental investigation, we may have to recur to another measurable physical quantity-the pore pressure to examine the solvent migration without having to invent a new instrument to measure the chemical potential of water in the gel. So the problem becomes that how to measure the pore pressure in the gel.

2.The experimental measurement of osmotic pressure(pore pressure) in a freely swollen gel.

For the PURE elastomer, I believe the mesh size in the network is very small even at the nano scale as you mentioned. To some extent, different from the pure elastomer, the hydrogel is indeed a composite (or porous medium) consisting of polymer network and the pore space (void) filled with water (although we can treat the gel as a fictitious continuum by a superposition assumption of solid and fluid phases at every point within the theoretical framework). Considering the dry state of the gel, the mesh size in the network may be about a few nanometers. When we have the dry gel immersed in pure water, the network may imbibe a very large quantity of water and form a hydrogel where the water content may be dominant in the whole volume of the gel. Consequently, we have reasons to believe that the mesh size in the network or the pore space (void) occupied by water should be much larger than that of pure elastomer(or dry gel). If so, then why can't we measure the pore pressure (osmotic pressure) by a traditional pressure gauge inserted in the pore space filled with water? (I guess the pore space may not be at the nano scale in a swollen gel)

3.The relationship between the chemical potential and the pore pressure inside a porous medium

I agree that the pore pressure p is identical to the (chemical potential)/(volume per water molecule) in traditional porous media (rocks, soil, cements, sponges, biological tissues...) without containing ions and enthalpy of mixing. However, I don't think the relationship, p=(chemical potential)/(volume per water molecule), is satisfied in all instances. To my knowledge, only when the osmotic pressure is zero can the formula be satisfied. For the gels and other porous media containing ions, the relationship, p=(chemical potential)/(volume per water molecule), may not be tenable.


Many thanks

Tony Rockwell's picture

Dear Lianhua,

 I also came to the topic of gels after first learning porous media theory as formulated for soils or fibrous membranes or media. It is natural to try to place Prof. Suo's notes in that context, but there are important differences. In a fibrous medium one can conceive of the network as static, with discrete pores filled with pure fluid. It really is two phases - a liquid and solid phase - and in the absence of flow the hydrostatic pressure is indeed continuous from the pure fluid outside to the pores-space within the medium.

However, attempting to understand a gel of polymer and solvent in terms of the above picture leads only to headaches. The polymer network is not static, but has many parts that flip around exchanging places with solvent molecules. Because of this, we must regard flow through the network as a diffusive process. Also because of this, it is the concept of 'pore pressure' that is fictitious - chemical potential is real! The only way to have a volume of pure fluid in which to insert your nano-pressure gauge is if you have a fluid inclusion (domain of pure fluid) inside your gel which is then in chemical equilibrium with the solvent in the gel, and in mechanical equilibrium with the gel - NOT mechanical equilibrium with the 'pure fluid fraction' of the gel. The point I think is that you have to consider the gel as a single homogenous phase. (For fluid inclusions in gels see Wheeler and Stroock Langmuir 2009).

In summary, in classical porous media theory, the Darcian idea is to represent a mix of phases as fictitious, homogenous phase with material properties that depend in a straightforward manner on the volume fractions of the underlying phases. Gels behave as a single phase, and the material properties depend not just on volume fractions but interactions between polymer and solvent, leading to not at all straightforward behaviors like volume transitions after small perturbations (Doi 2009) - the idea of the gel as a superposition of two independent phases is in this case artificial!

Note: I do not presume to speak for Prof. Suo - any  errors are my own!



Lianhua Ma's picture

Dear Tony,

Thanks for your clarification.

The gel is actually a mixture of stretching polymer and solvent, despite we can treat the gel as a homogenous continuum where every point is a mixture of fluid and solid as Prof. Huang pointed out. It is worthy noted that, just as you mentioned, the superposition of solid and fluid at every point is an artificial assumption, and not in real case. The fictitious superposition treatment of multiphase mixture is also the basic idea of the traditional continuum theory of mixture (porous media theory) (Bowen,1976; Lai,1991, Hueckel,1992; Huyghe,1997).  If we jump over the theoretical framework and just consider the real microstructure of the freely swollen gel in pure water, the gel is a composite consisting of polymer network--solid phase, and open cavities filled with water--fluid phase, NOT a continuum material. I believe that a pore pressure (represented by osmotic pressure) could be found in the domain of fluid phase in the freely swollen gel. Do you agree?  

I think the most importance difference between the gel and the general porous media (mixture) is that the enthalpy of mixing (interaction between polymer and solvent) is included in the free energy of gels, whereas this mixing effect does not exist in general porous media.  

And I am not sure if we can consider the Prof. Suo's theory for soft gels as a special version or an extended version of the classical porous media theory (theory of mixture)? 


Zhigang Suo's picture

In looking over my 2009 version of the notes on poroelasticity, I felt that the notes had too many components, and decided to split the notes into three parts:

I first taught the course in 2007.  I read Biot (1941), but did not know much else in the literature then.  I put together what I felt should be a clean theory that couples finite deformation of a network and absorption of a solvent.

When I posted the notes on iMechanica in 2007, within days Guru pointed me to the work of Gibbs.  I soon learned that Gibbs in his 1878 paper had the exactly same theory--a theory as clean as I had.  He also used deformation gradient and nominal stress.  No reason for me to have the illusion that "great minds think alike".  In this case, I was simply indirectly influenced by Gibbs, through many other authors.

Biot seemed to be unaware of Gibbs's work.  In formulating the theory proroelasticity, was Biot also indirectly influenced by Gibbs?  In Biot (1941), he linked the kinetic part of the theory to Darcy, but did not link the thermodynamic part of the theory to Gibbs. 

Lianhua Ma's picture

In some ways, it was Gibbs that lay the theoretical foundation for poroelasticity.
As we know, Gibbs is a great theoretical physicist and mathematician in the 19th century. It seems a little strange that why Biot (1905-1985) was unaware of Gibbs's groundwork in his 1878's paper.

"In the period between 1935 and 1962 Biot published a number of scientific papers that lay the foundations of the theory of poroelasticity"(, I guess, in that period of Biot, the 1878's paper of Gibbs maybe too old to be traced for reading, hah... Smile

Dear Profs. Zhigang Suo, Rui Huang,

 I have followed your valubale discussion and series of papers using poroelsticity theory for modeling hydrogel swelling. I have a question and appreciate if you can help me. Basically I want to know lis, et say for the constrained 1-d and free swelling of a cube that have been investigated in your papers, how was  the local rate of swelling at a give point? how the mesh size was changing locally upon swelling? what is the difference between local rate of swelling in linear and non-linear poroelasticity theory?

Thank you very much


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