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# A Theory of Ionic Polymer Conductor Network Composite

Ionic polymer conductor network composite (IPCNC) is a mixed conductor consisting of a network of loaded ionomer and another network of metallic particles. It is known that the microstructure of the composite, especially that of the electrodes, plays a dominating role in the performance of an IPCNC. However the microstructures of IPCNC have seldom been addressed in theoretical models. This letter formulates a continuum field theory for IPCNC by considering a supercapacitor-like microstructure with a large distributed interface area. The theory is then applied to the study of the equilibrium deformation and electrochemistry in a thin-sheet IPCNC actuator.

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## Comments

## Nice work!

Hi Xiao，

Thanks for posting the interesting paper on a new theoretical model of the fascinating material-IPCNC. I have a question about the solution of governing equations in your model.

With the help of the proposed free energy function W(e，c+，c-，q)， the coupled constitutive relations have been explicitly obtained(i.e. eqs. (7) (8)), and the mechanical equilibrium equation (4) can be further derived as a function of u ,c+,c-. To my understanding, the quantities u, c+, c-,and q are unknown variables , however only one mechanical equilibrium equation (4) and two conservation equations of species are given, these conditions seem insufficient to solve all of the unknown variables.

As explained in your paper, Eqs. (4) and (5) were solved by the COMSOL multiphysics software. How to treat eq.(5) in FEM? Can you simultaneously solve the four physical quantities？

I would be grateful if you could assist me with this question.

Very nice work, thanks for sharing!

Regards,

Lianhua

## Hi, Lianhua, Thank you

Hi, Lianhua,

Thank you for your interest in our work. I am more than glad to answer your questions.

We assume the system to be electrically neutral everywhere, Eqn (3) serves as another constraint and c+, c- and q are not independent variables. Since this IPCNC is a capacitor, equal amount of electronic charge is stored on the two electrodes, which is another hidden condition. Nevertheless, it is automatically satisfied by enforcing neutrality.

Eqn (5) is not treated explicitly in our codes, in stead, we simpliy apply Eqn (6) as the weak form, together with the three constraints: conservation of ions (2 Eqs) and neutrality, then e, c+, c- are solves simultaneously.

Let me know if you have further concerns.

Regards.

Xiao

## Thanks

Hi, Xiao, Much more

clearly. Thanks for the clarification. Combining one

weak form (eq (4)) and three constraint conditions can give the four

quantities.

From the

paper, it seems that the equilibrium equation eq(4) is applied on the whole homogenized

model. Due to the different microstructures of the tri-layer IPCNC, the material property parameters and

equations of state may differ from each other.

Based on

the developed tri-layer model, my understanding is that the weak form of eq (4)

is applied only on the domains of the two composite electrodes (solid-like

material) of the IPCNC. Because

the ions are distributed in the whole model, the conservation of ions (2 Eqs)

and neutrality conditions should be accordingly applicable for the whole

domain. So my another question is how to deal with the middle domain ( ionic

conductive only) of your tri-layer

model? What is the governing equation applicable to the middle layer

(liquid-like material) ?

Thanks,

Lianhua

## Hi, Lianhua, This is

Hi, Lianhua,

This is a very insightful question. A short answer is that since there are no contribution from double layer energy in the middle layer, we simply drop this term and now the system is just a function of c+, c- and e.

The physical meaning of doing this is that although we assume the middle layer obeys the same governing equation as that of the composite layer, as there are not actual electrode connected to material particles in the middle layer, one has to enforce the electronic charge on the electrode to be zero. The electrode in middle layer is just a conceptually define electric potential and as a measure of the electric work done by transporting ions.

Please let me know if you have more questions. Regards.

Xiao

## Hi, Xiao, Thanks a lot for

Hi, Xiao, Thanks a lot for your explanation.

In your work, the model was numerically solved by FEM. Maybe it can also be analytically solved by the clear equilibrium eq and boundary conditions because of the regular shape and boundary of the IPCNC.

Cheers

Lianhua

## Hi, Lianhua, Yes, if the

Hi, Lianhua,

Yes, if the deformation is blocked, then Eqn (7) and (8) are decoupled, we could actually solve for the concentration of ions, and calculate the bending moment. However, the difficulties are: 1. Non-linearity is essential. If one linearize the equation, there is no actuation. In this case, I am not able to get any close form solution yet. 2. We have several different domains (e.g. tri-layers), for each domain there is one set of such equations, and the solutions are connected between domains. Therefore it is a practicle problem of how to deal with this large amount of coupled equations.

Regards.

Xiao