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# Inhomogeneous swelling of a gel in equilibrium with a solvent and mechanical load

A network of polymers can imbibe a large quantity of a solvent and swell, resulting in a gel. The swelling process can be markedly influenced by a mechanical load and geometric constraint. When the network, solvent, and mechanical load equilibrate, the gel usually swells by a field of inhomogeneous and anisotropic deformation. We show that this field in the swollen gel is equivalent to that in a hyperelastic solid. We implement this theory in the finite-element package, ABAQUS, and analyze examples of swelling-induced deformation, contact, and bifurcation. Because commercial software like ABAQUS is widely available, this work may provide a powerful tool to study complex phenomena in gels.

The source code of the UHYPER program is attached below.

Attachment | Size |
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gel.for_.txt | 2.32 KB |

gel_in_equilibrium_2008_05_06 submit.pdf | 711.69 KB |

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## Comments

## a concern

Hi, Zhigang, thank you very much for your info. Also Hello, Wei, Great congratulation to you for your new position in Iowa State University. Hope to keep in close touch in future.

This paper is of great interest to me, especially on the simulation of 2-D complex gel with commercial software ABAQUS. May I have a concern to be clarified, that is, how to understand that the chemical potential of the solvent molecules, mu=kT(p/p0) and is also defined as Eq.(2), say mu=dW(F,C)/dC? Thank you very much fro your time.

## About the chemical potentials

Dear Hua,

Thank you very much for the good words. We will definitely keep in touch.

The chemical potential is defined as the work needed (or increase in the free energy) when adding one extra atom (or particle).

By this definition, there could be chemical potential of the solvent molecules in the vapor and that in the gel: mu_vap = dW_vap/dC=kT(p/p0), mu_gel = dW(F, C)/dC=.... In general, the two are not equal, and the chemical potential can be a field variable. However, we are looking at equilibrium state here, so they must be equal and homogeneous, mu_vap=mu_gel. We didn't put on the subscripts on the two mu's, but they mean different things and they are equal only in equilibrium.

Hope this resolves your concern. Thanks again for your interest!

## Question about the chemical potential

Dear Wei Hong,

Thank you for sharing your source code.

I'm trying to use it for a swelling deformation problem. I have a question about the chemical potential. Since the initial free swelling is an input parameter, shoud we specify the initial chemical potential accoroding to the free swelling equation? If so, since the chemical potential is mimicked by a temperature-like variable in ABAQUS as stated in your paper, How do we specify the increment of the chemical potential or temperature?

Thanks,

Min Kyoo Kang

## setting chemical potential

Dear Min Kyoo,

Thank you for your interest in our work.

Yes, you should specify the initial chemical potential according to the given initial free swelling ratio (3rd material parameter)

The chemical potential is specified using pre defined fields (temperture) in abaqus input.

Let me know if you have further questions.

Wei

## additional questions

Thank you for your prompt response.

I have additional questions. In defining the temperature in the predifined fields, we usually need to define thermal expansion coefficient in the property module, otherwise we don't see any deformation. I wonder in your simulation whether you also need to specify thermal expansion coefficient, and if that's the case, how does the thermal expansion coefficient relate with the chemical potential?

Your chemical potentials are in the range of -0.05 to 0. Does this means we have to use the same values for the temperture input in the predefiend field which actually means contraction not swelling in the point of temperature?

I'm bothering you with many questions and I really appreciate your help.

Thanks,

Min Kyoo

## Re: additional questions

We don't need to specify the thermal expansion coefficient. We use the "T" just as a general field parameter, not as temperature, so it has nothing to do with thermal expansion.

Due to the definition of the chemical potential, it is always negative. It can be negative infinity to 0. (-0.05 is just an arbitrary number we picked.) As we start from -0.05 and end in 0, so it is still swelling instead of contracting. Just don't read it as temperature.

Please feel free to let me know if you have further concerns.

Wei

## good job.

Dear Wei Hong,Your investigation on mechanics of gel is very good job.

you had implemented the theory in the finite element package, ABAQUS. the simulation of large deformation of similar hyperelastic material requires the satisfaction of quasi-static mechanical behavior. so , the FEM simulation in your paper did not reflect the diffusion process of solvent in gel. In other words, you assumed that the chemical potential is constant in one simulation, In fact, the chemical potential is variable in different position of gel. the theory in your paper may be only suitable for final equilibrium state of diffusion.

another question: the key of your paper is the free energy funtion W . For other soft materials, If we have no corresponding free energy funtion put forward by predecessor, How can we investigate the mechanical behavior of soft materials? can you give me any advice? thanks for your paper! Hope to keep in touch with you.

THANKS

L.H. MA

## Limitation of the current work

Dear Lianhua,

Thank you for your interest in our work!

As you mentioned, the current implementation does have its limitations: it is only suitable for the final equilibrium state of diffusion. Although the chemical potential is not required to be constant in the simulation, it is a predefined field. In other words, we can not solve for the chemical potential, it must be given. Therefore a steady-state calculation of a complex domain might not be possible either.

No theory could ever predict a general free-energy function, although some theoretical abstraction might give insights to some specific material behavior, Florry-Huggins, for example. The right way to investigate the mechanical behavior of a material would always be experimental. Instead of doing one experiment on one material, one should do a series of experiment on a same material, using different loading conditions, different sample shape/sizes. Instead of starting from nowhere, I think it is always better to start from a theoretical model, and see the deviation. If there is no deviation, good, we extract the material parameter; if, most likely, there is deviation, we either modify the theory to say why, or just use the test result numerically, if a result is important. Also instead of testing the static/equilibrium behavior together with the kinetic properties, I suggest to do separate tests for a same material. These are just my general thoughts. Let's keep on the discussion if you have further interest.

Thanks,

Wei

## Dear Wei, Thanks for

Dear Wei,

Thanks for your explanation for the free-energy function. As you said,we must do some experiments on soft materials which have no corresponding free energy funtion. the acquirement of the free-energy function for a new soft material may be very difficult, and the experimental test is very importrant. For a new soft material without free-energy function,experimental tests may be the only way to describe its mechanical behavior!

THANKS

Lianhua

## Questions about your paper

Dear Dr. Hong.

I am trying to find an idea for my proposal presentation for my Ph.D and I have found interest in yor work about gels. So, I read two papers that you wrote about it.

I have two simple questions related with the paper: W. Hong, Z. Liu, and Z. Suo, Inhomonegeous swelling of a gel in equilibrium with a solvent and mechanical load.

International Journal of Solids and Structures46,3282-3289 (2009).1) Why do you relate I = F_ik F_ik with the first invariant of the deformation gradient (eq.(17)) ? Could you give me some reference? Now I am reading the Gerahard Holzapfel's book about continuum mechanics, and I have not found any reference to it.2) Is there a lambda_0^3 term missing in the second log, in eq.(23) ? I have done the mathematic and I just found a discrepancy in the numerator of the second log.I am a Ansys user and I want to reproduce your excellents results, but using Ansys.

Thank you,

Mario J. Juha

Civil and environmetal Engineering DepartmentUniversity of South Floridahttp://www.eng.usf.edu/~mjuha/

## Re: Questions about your paper

Dear Mario,

Thank you for your interest in our work.

I don't quite understand your first question though. I = F_iK F_iK is an invariant of the deformation gradient F. It is not the first invariant of F. In fact, it is the first invariant of the right Cauchy-Green deformation tensor. You can find reference on these quantities on almost any textbook on finite-deformation continuum mechanics, or even on wikipedia.

You are right that on the final published paper, the lambda_0^3 is missing in eq (23). The version here on imechanica was correct. Thank you for pointing this out, I appologize for our carelessness on proofreading the final print.

Wei

## Molecular incompressibility and the free-energy function

Dear Dr. Hong. I have been busy studying your papers and I have more questions than answer (to me that is good). In: W.Hong, X. Zhao, J. Zhou, Z. Suo.

A theory of coupled diffusion and large deformation in polymeric gels.Journal of the mechanics and physics56, 1779 - 1793 (2008), section 3, you specify the molecular incompressibility as:1 + vC = det(F)

I am familiar with the incompressibility constraint of det(F) = 1, but not with the above equation. Could you give more insight about it? Probably it is an standard constraint in gel theory, but I am not familiar with it.

Other question that I have is related with C(X), distribution of solvent molecules in the gel. Do I need to understand this notation as C(X) = C(phi^-1(x))? , where phi(x)^-1 is the inverse of the motion, lower x is the spatial coordinate and capital X is the reference coordinate. In general terms, what I have understood is that C is an inhomogeneous field, is it correct?

cordially,

Mario J. Juha

www.eng.usf.edu/~mjuha/

## Re: Molecular incompressibility

Dear Mario,

It is actually similar to the constraint you are familiar with, the only difference being the volume of solvent. When solvent migrate into a polymer network, the network expands. Here we assume the total volume to be a constant.

Your understanding of C(X) is correct.

Wei

## Future work...

Dear Dr. Hong.

I am really interested in this work. I have been studied it for weeks and I would like to extend your results. In section 8.3, last paragraph, you said something about instabilities that you can not solve currently with ABAQUS. I would like to know if I can collaborate with you and or your students.

thank you,

Mario

## Re: Future work

Dear Mario,

Sure you can! Please let me know if you need more information than those on the paper.

Wei

## Material model implementation

Dear Dr. Hong.

I have implemented your material model in a non-linear finite element program that I wrote. I am using a Continuum Mechanics Total Lagrangian formulation to deal with nonlinearities, and my measure for stress and strain are Second Piola-Kirchhoff stress tensor and Green-Lagrange strain tensor, respectively. In order to test the program, I would like to reproduce your results. I have followed what you nicely explained in the paper, but I feel I need an extra help.

If it is appropiate, could you give me extra information about the finite element model?, that is, what exactly boundary conditions you used and how many substeps?.

To use your model and doing the following:

1) Specify the initial chemical potential and then calculate lambda_0 using Eq.(19). This value of lambda_0 is what I am going to use in Eq.(23)

2) Increment linearly the current chemical potential using the following relation:

current_mu = mu_0 + delta * ( current_mu - mu_0 ), where delta is the increment corresponding to the load step.

current_mu is what I am going to use in Eq.(23)

3) Solve nonlinear equations using Newton-Raphson.

Thanks in advance,

Mario

## I already found the answer.

I already found the answer. Thanks I was measuring lambda respect to the free-swelling state and not from the dry network state. Now I have same results. But if you could send the info, I would appreciate it.

Thanks,

Mario

## Dear Dr. Hong. I am a

Dear Dr. Hong.

I am a postgraduated student,my major reasearch direction is about finite element implementation of swelling of polymeric hydrogels. I have some questions about modeling and subroutine.

1)In your paper ,it says that the chemical potential is mimicked by a temperature-like variable, and I am confused that how to defined this variable in ABAQUS, in other words ,is the chemical potential a loading parameter when I apply the load to model like that I apply the force to the beam. But I can not find a parameter of temperature.

2)There is a note at the end of the six part of this paper that is ABAQUS uses I1=J^-2/3I, rather than I, what does this mean?

I am sorry with my poor English and Thank you very much for your time.

Dan Dan