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# Stretching and polarizing a dielectric gel immersed in a solvent

This paper studies a gel formed by a network of cross-linked polymers and a species of mobile molecules. The gel is taken to be a dielectric, in which both the polymers and the mobile molecules are nonionic. We formulate a theory of the gel in contact with a solvent made of the mobile molecules, and subject to electromechanical loads. A free-energy function is constructed for an ideal dielectric gel, including contributions from stretching the network, mixing the polymers and the small molecules, and polarizing the gel. We show that the free-energy function is non-convex, leading to instabilities. We also show that mechanical constraint markedly affects the behavior of the gel.

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## Comments

## Stretching - Swell only?

Xuanhe, really appreciate Prof Zhigang's email to let me know the latest output from you. Also thank you very much to mention our work in your paper. It is a really interesting work, and also difficult for me to understand well. In order to get an insight into the paper, several questions come from my site as: (1) You use the word "Stretching" in the title, could I understand that your model is limited to the gel swelling only? I mean how about the gel DIswelling? If yes, may I know which terms in your model cause the limitation? (2) Equation (6) provides the condition of molecular incompressibility. Where is it required in your following formulation? (3) The free energy of the gel in your model considers the contributions of stretching, mixing and polarizing. How about the contribution coming from the small molecular concentration, especially when you mentioned the equation (6)? Thank you in advance for your input.

## Swelling and Deswelling

Thanks for your interest, Dr. Li Hua.

Let me try to resolve your concerns:

1) I am not sure whether you mean "deswelling" or not. If so, we are only considering the equilibrium states, so we don't really care whether it is swelling or deswelling. In other words, there is no arrow of the process as we do not care about kinetics in this paper, the gel does not know the history.

Also, if I understand your concern right, I think you might misunderstood out assumption. We use the "dry state" as the reference state of the network, in which no solvent is present. So in that sense, comparing to the reference state, the gel could only "swell".

2) Eq. (6) has been used in the following calculation so that C is no longer an independent variable in the free energy function W.

3) The contribution from C is just the energy of mixing (14). (mixing between the solvent molecules and the network.)

Ofcouse, we replace C by a combination of lambdas in later calculation as of the incompressibility (6).

Hope this helps,

Wei

## Hi, Wei, thank you for your

Hi, Wei, thank you for your response, but i still want you to clarify that "3) The contribution from C is just the energy of mixing (14)." In your system, is there difference of concentrations between the interior gel and exterior solution?

## Yes, there is difference

Hi Hua,

There is difference. Inside the gel, the solvent is mixed with network (We define concentration as number of molecules per unit volume, not just the concentration of ions with respect to liquid only.) , while ourside, it is either pure solvent or the mixture of solvent with something else.

Wei

## contribution of this difference into free energy?

Hi Wei,

Sounds great. Do you think if this difference also makes its own contributions into the free energy of your gel system?

## Yes, it has been considered

Hi Hua,

Yes, it can be understood as such, that this difference drives the swelling of the gel.

(The difference will drive solvent molecules diffuse into the gel. To be more exact, it should be the difference in chemical potential, the partial derivative of the free energy with respect to the concentration.)

If there were no elastic force of the network, the gel will keep swelling forever, as the concentration is always lower than the exterior.

However, as for the elastic force, there will be a equilibrium state, in which the driving force from the difference in concentraion balances the elastic force.

Wei

## Dr Li, thank you for your

Dr Li, thank you for your comments and questions on our work. I think Wei's answers will resolve your concerns. One additional comments on "swelling" and "deswelling". The formulation allows us to take an arbitary configuration of the gel as the reference state. Whether the gel will swell or deswell depends on the concentration of solvent molecules in the gel at equilibrium state compared with the reference state.

## conjugacy

Dear Zhigang,

I looked again through your JMPS paper and the DorfmanOgden paper in Acta Mechanica and found the following mismatch. You conclude that the true electric and electric displacement fields are not work-conjugate while Dorfmann and Ogden state their conjugacy - Eq.(25)_2. Am I missing something?

-Kosta

## Re: conjugacy

I don't have Dorfman-Ogden paper with me, but it is very easy to see that, for a deformable dielectric, the true electric displacement and the true electric field is not work-conjugate. See three very short equations (3.7)-(3.9) in our JMPS paper.

## Yes! Your conclusion seems

Yes! Your conclusion seems reasonable. I am curious, however, where is the flaw in DorfmanOgden (if any)? I'll email you their PDF.

## Notes on two formulations of the theory of dielectrics

Dear Kosta: Thank you for the PDF of the Dorfmann-Ogden paper (Acta Mechanica 174, 167-183 (2005)). Our JMPS paper is now online (PDF of pre-print or JMPS website). Here are some notes.

I have just forwarded this thread to Luis Dorfmann. Maybe he can shed some light into this.

## Re: Notes

Concerning your remark #3:

In the beginning of section 4 they introduce free energy in the form ksi(F,E). If I calculate ksi_dot, I have: ksi_dot=dksi/dF*F_dot+dksi/dE*E_dot. If, however, we define dksi/dE=-P (Eq.(25)_2) then E and P are work conjugate. Am I missing the point?

## Re: Re: Notes

As you can verify by a direct calculation, P and E are not work-conjugate. Thus, ksi in (25) should not be really the free energy. Rather, Omega defined in (34) is a free energy.

To avoid confusion, I stick to the language of thermodyanmics, and define free energy using work.

## Well, in this case, it seems

Well, in this case, it seems that ksi(F,E) should not exist at all...