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 |
|---|---|
| gel.for_.txt | 2.32 KB |
| gel_in_equilibrium_2008_05_06 submit.pdf | 711.69 KB |
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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